Add Batch d03a9d01-50b7-4ef9-b049-075484685bbb data

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@@ -2455,3 +2455,67 @@ saved_model/**/* filter=lfs diff=lfs merge=lfs -text
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2025/Protein Language Model Fitness is a Matter of Preference/7bad12b8-258c-41ff-afb9-26a1c6c24092_origin.pdf

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2025/Protein Language Model Fitness is a Matter of Preference/full.md

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+# PROTEIN LANGUAGE MODEL FITNESS IS A MATTER OF PREFERENCE
+
+Cade Gordon, Amy X. Lu & Pieter Abbeel
+
+University of California, Berkeley
+
+textttcadegord@berkeley.edu
+
+# ABSTRACT
+
+Leveraging billions of years of evolution, scientists have trained protein language models (pLMs) to understand the sequence and structure space of proteins aiding in the design of more functional proteins. Although they have shown ability to improve efficiency in engineering, it remains unclear if such models capture true biological patterns or artifacts of the training data. We aim to predict the circumstances in which pLMs can successfully perform zero-shot fitness estimation. Our work studies trends observed over hundreds of deep mutational scans across multiple different fitness objectives. We find that the likelihood, or abstractly, implicit preference of a certain protein sequence imbued during pretraining is predictive of fitness prediction capabilities. Both over-preferred and under-preferred wild type sequences harm performance. Using influence functions to causally understand how individual data points increase protein likelihoods, we find that there exists a power law tail due to sequence homology. Lastly, under-performance on low likelihood wild type proteins can be remedied by unsupervised finetuning. These findings that pLM zero-shot fitness estimation can be predicted by the likelihood of the engineered sequence can motivate and improve pLMs' deployment in protein maturation campaigns.
+
+# 1 INTRODUCTION
+
+Protein Language Models (pLMs) have been thought to encapsulate millions of years of evolutionary information through unsupervised pretraining on protein databases. Works have shown that their likelihoods can infer evolutionary trajectories, improve design campaigns, and predict zero-shot mutational effects (Hie et al., 2022; Biswas et al., 2021; Meier et al., 2021). However, more recent works have begun enumerating cases where pLMs likelihoods are influenced by training data compositions that are not direct consequences of natural evolution (Weinstein et al., 2022; Ding & Steinhardt, 2024).
+
+To better understand how training data selection in biological pretraining leads to biases in the learned patterns of pLMs, we propose to study performance through the lens of preference. At the individual comparison level, the Bradley-Terry model of preference (Bradley & Terry, 1952) has been shown to be a flexible tool, grounding both modern alignment techniques such as Direct Preference Optimization (Rafailov et al., 2024) and ranking techniques such as ELO. As Ding & Steinhardt (2024) used sequence likelihood to construct species-level ELO scores, we can generalize likelihood to be a measure of sequence preference. By framing our analysis through the lens of preference, we aim to uncover how pLMs implicitly "prefer" certain protein sequences over others. Crucially, we posit that this implicit preference is not just a property of the model, but stems from the preferences encoded in the training data itself. We hypothesize that this multi-layered preference structure, quantified by sequence likelihood, can explain variations in pLM performance across different proteins and tasks.
+
+We start by showing that variations in downstream protein engineering performance can be explained by the likelihood of the starting sequence. To do so, we use deep mutational scan (DMS) datasets to see if underlying log likelihood of starting sequences can be predictive of zero-shot fitness prediction capabilities, and find that mutation effect prediction on lower likelihood starting (i.e. wild type) sequences have worse performance, and that high likelihoods can become harmful after
+
+![](images/85dd4dffabba0cc61f24adb196a151842598dcc8954414880c1e6a16f8412265.jpg)  
+a
+
+![](images/1b663dd2c7d3765d90f4b2ede3dc77d13a0d222b6ccf27c571c2723af7b1b296.jpg)  
+2. Mutate
+
+![](images/6405cee053ca9650773cb0fff82731aab68ffdca1e909ede76d5677ac79b8dc2.jpg)  
+3. Assay
+
+![](images/2330232869abde4afb391cefb08e7f2270e1da99d6199671009ace8707e3ad98.jpg)  
+4. Score
+
+![](images/9de8828491531b3e3f4b7b26ae63043260d16f8d8f77cdf3e84a1ace6f6caffe.jpg)  
+b  
+Figure 1: Understanding DMS and a theory of pLM evolution capabilities. (a) A deep mutational scan follows by choosing a protein then selecting residues to mutate. Mutations are performed, expressed, then assayed for function (e.g. binding, fluorescence, or stability) to determine a fitness score. (b) We refine the efficient evolution hypothesis of Hie et al. (2024) suggesting that ability of pLMs for economic protein maturation is dependent on more than nature's plausible mutations. Mutation effect prediction success is reliant on the underlying likelihood or preference towards the non-mutated protein. Ultimately, model performance can be seen as a result of the model's implicit preference and human preference during the data curation pipeline.
+
+![](images/9d62e957db7a43af9453e049be596ef3409731ce3602079c9da7bcb50a273260.jpg)  
+i
+
+![](images/3ff4bdf8339c12b1deb9a48d6d67c2f28af1b2fc31d65ef0a8ceacd6a30de242.jpg)  
+ii
+
+![](images/1dfdd42d48fa4270d4df7c1df2bcd2b2d8c1e2b8dc47a2da4c7d01f0205b0c30.jpg)  
+iii
+
+a certain threshold. From this finding, we generalize the failure pathology from species-level (Ding & Steinhardt, 2024) to individual sequence probabilities. The ability to calculate pseudo log likelihood on datasets of evolutionary-scale magnitude is enabled by a new derivation of pseudo log likelihood calculation reducing the number of inference passes from  $\mathcal{O}(L)$  to  $\mathcal{O}(1)$  without any post-training.
+
+A question then emerges: what is causing these sequence likelihoods? We utilize influence functions to understand what proteins from the underlying training dataset increase the likelihood of certain wild type proteins. Our studies reveal that the distribution of influential data points follow a power law distribution, and that highly influential sequences can be quickly found using a search tool such as mmseqs2 (Steinegger & Söding, 2017).
+
+Combining these results, we motivate a past method known as evo-tuning (Alley et al., 2019) to protein designers or test-time training (Sun et al., 2019) to the machine learning community. We leverage our results relating to likelihood to suggest that evo-tuning on low likelihood wild types improves performance and evo-tuning on high likelihood wild types harms performance. This finding matches the intuition garnered from the zero-shot plots and helps to explain counterintuitive dynamics seen in Hsu et al. (2022) that unsupervised finetuning can sometimes worsen performance. Together, our findings improve real-world applicability of increasingly powerful pLMs, also providing theoretical and granular analysis to how homology to training data sequences affects performance.
+
+# 2 RELATED WORKS
+
+Protein Language Models Modern pLMs come in two forms: masked or autoregressive models. Of the former, transformer based language modeling efforts like ESM-1B, ESM-1V, and ESM-2 became dominant with BERT-like pretraining objectives (Vaswani et al., 2017; Rives et al., 2021; Meier et al., 2021; Lin et al., 2023). In contrast, ProGen (Alley et al., 2019; Madani et al., 2023; Nijkamp et al., 2023) and other works embraced the GPT-like pretraining style and opted for autoregressive training scheme.
+
+As a result of these training tasks, pLMs have afforded many capabilities surrounding evolution through their likelihoods. Biswas et al. (2021) use pLMs to do property engineering with a limited
+
+number of labeled sequences. Meier et al. (2021) showed that pLMs are capable of zero-shot mutation prediction by comparing the log odds ratios between sequences of interest. Extending this, Hie et al. (2022) proved pLMs can predict the evolutionary trajectory of a range of different proteins and selective pressures. Not only can pLMs predict evolution, Hie et al. (2024) demonstrated that an ensemble of pLMs can improve the affinity of monoclonal antibodies without any information about the antigen.
+
+Limitations Protein Language Models of Training Data Though data research is becoming more and more prevalent within the language modeling, its treatment within the domain of pLMs has not yet been as extensive thus far. Fannjiang & Listgarten (2024) provides an introduction to the relationship between data and model performance in both protein specific and general purpose modeling regimes. On the matter of training as a whole, Weinstein et al. (2022) argued that the density of pLM training data alone doesn't specify the fitness functions of interest, rather due to misspecification pLMs learn to model the stationary distribution enabling their success in fitness tasks. Recently, Ding & Steinhardt (2024) showed how species bias within protein databases has lead to biases in the underlying pretraining sets for most pLMs. It further demonstrated that pLMs fail on design tasks catered towards low-likelihood species and would often revert to over-represented homologs. Lastly, Hermann et al. (2024) uncovered dataset overlaps between commonly used pLM benchmarks and their pretraining datasets. When these overlapping points were removed from the benchmarks, scores decreased.
+
+Relating Training Data and Model Outputs One of the first attempts to relate a probabilistic model's outputs to its input data comes from influence functions (IFs) in robust statistics (Hampel, 1974). Koh & Liang (2017) ported the classic technique to deep learning. A few years later, Grosse et al. (2023) improved computations of IFs enabling evaluation for models of up to 52 billion parameters.
+
+Other modern schools of thought surrounding data have begun understanding downstream model performance as a function of pretraining data for robustness and pruning. Fang et al. (2022) finds that the data distribution is the cause of the large gains in effective robustness for CLIP. Others used data pruning to outperform classical neural scaling laws (Sorscher et al., 2022). With the growing importance being placed on data, tasks are now developing to understand which data points are most important through the DataComp challenges that have been put forward for multimodal and traditional language models (Gadre et al., 2024; Li et al., 2024). Further evidence can be seen with complex data distributions and post-training regimes in works such as Llama 3.1 (Dubey et al., 2024).
+
+# 3 PRELIMINARIES
+
+# 3.1 ZERO-SHOT FITNESS PREDICTION
+
+One of the most common methods for determining how protein fitness changes with mutations is called a deep mutational scan (DMS). As seen in 1a, deep mutational scans start with a protein of interest, then perform combinatorial set of mutations to select residues in the protein. Those are characterized in the wet lab then compared against the unmutated or wild type starter protein.
+
+We choose to focus on masked language models in this study due to their ease of use in point wise mutations and community adoption. Following ESM-1V we utilize, the difference in model likelihoods to compute zero-shot fitness predictions. We denote the wild type sequence as  $x$  and the set of mutable residues as  $T$ . From here we can calculate a predicted gain in fitness  $f$  of a mutated sequence  $x'$  over  $x$  using parameters  $\theta$  as the log odds ratio of the mutated sequence against the wild type:
+
+$$
+f \left(x ^ {\prime}, x\right) = \sum_ {t \in T} \log P \left(y _ {t} = x _ {t} ^ {\prime} \mid x _ {\backslash t}, \boldsymbol {\theta}\right) - \log P \left(y _ {t} = x _ {t} \mid x _ {\backslash t}, \boldsymbol {\theta}\right). \tag {1}
+$$
+
+Each evaluation of  $f$  can be compared against real world assay values, thus Spearman correlation can be used as a measurement of agreement (Spearman, 1904) making the task equivalent to ranking. This method is how the predictions in ProteingGym are formulated for the ESM suite of models.
+
+Although this is a powerful method of zero-shot precition, it has two major limitations. First is the linear additivity of fitness. As a result it can't model epistatic fitness interactions. Second is the need
+
+for mutants to be identical in length to the wild type sequence. Since we're evaluating the log odds ratios at certain locations to derive fitness, each residue must have an existing counterpart on both proteins to enable evaluation.
+
+# 3.2 INFLUENCE FUNCTIONS
+
+We find ourselves utilizing a training dataset  $\mathcal{D} = \{z_i\}_{i=1}^N$  of  $N$  samples and individual sequences  $z_i$ s. Models are then fit to minimize the empirical risk of a loss function  $\mathcal{L}$  to derive an optimal set of parameters:
+
+$$
+\boldsymbol {\theta} ^ {\star} = \arg \min  _ {\boldsymbol {\theta} ^ {\star}} \mathcal {J} (\mathcal {D}, \boldsymbol {\theta}) = \arg \min  _ {\boldsymbol {\theta} ^ {\star}} \frac {1}{N} \sum_ {i = 1} ^ {N} \mathcal {L} \left(z _ {i}, \boldsymbol {\theta}\right). \tag {2}
+$$
+
+In particular, we are interested in understanding the effect of a single point  $m$ . We choose to weight this data point with some parameter  $\epsilon$  arriving at:
+
+$$
+\boldsymbol {\theta} ^ {\star} (\epsilon) = \arg \min  _ {\boldsymbol {\theta} ^ {\star}} \frac {1}{N} \sum_ {i = 1} ^ {N} \mathcal {L} \left(z _ {i}, \boldsymbol {\theta}\right) + \epsilon \mathcal {L} \left(z _ {m}, \boldsymbol {\theta}\right). \tag {3}
+$$
+
+Setting  $\epsilon = -1$  can be thought of as asking the counterfactual: what if  $z_{m}$  wasn't in the training set? Using a first-order Taylor series of Equation 3, the Implicit Function Theorem, and a few other assumptions we can derive the influence of  $z_{m}$  on  $\theta^{\star}$  in Equation 4. Furthermore, Equation 5 shows that we can calculate the influence of a sequence on a functional evaluation of  $f$  via the chain rule.
+
+$$
+\mathcal {I} _ {\boldsymbol {\theta} ^ {\star}} \left(z _ {m}\right) = \frac {d \boldsymbol {\theta} ^ {\star}}{d \epsilon} \Big | _ {\epsilon = 0} = - \mathbf {H} ^ {- 1} \nabla_ {\boldsymbol {\theta}} \mathcal {L} \left(z _ {m}, \boldsymbol {\theta} ^ {\star}\right) \tag {4}
+$$
+
+$$
+\mathcal {I} _ {f} \left(z _ {m}\right) = \nabla_ {\boldsymbol {\theta}} f \left(\boldsymbol {\theta} ^ {\star}\right) ^ {\top} \mathcal {I} _ {\boldsymbol {\theta} ^ {\star}} \left(z _ {m}\right) = - \nabla_ {\boldsymbol {\theta}} f \left(\boldsymbol {\theta} ^ {\star}\right) ^ {\top} \mathbf {H} ^ {- 1} \nabla_ {\boldsymbol {\theta}} \mathcal {L} \left(z _ {m}, \boldsymbol {\theta} ^ {\star}\right) \tag {5}
+$$
+
+$\mathbf{H}^{-1}$  represents the inverse hessian calculated over  $\mathcal{D}$ . As one can imagine, calculating this directly becomes prohibitively memory intensive. Grosse et al. (2023) makes it computationally feasible by assuming layer-wise independence and using of EK-FAC for hessian calculation (George et al., 2018). We use the implementation of Bae (2024) in our analysis.
+
+Although giving insight to a very complex relationship, influence functions are not without their limitations. Basu et al. (2020) show the fragility of the method to training alterations like the number of layers, layer width, and the inclusion of weight decay. Extending upon these findings, Bae et al. (2022) showed how a mixture of complications in assumptions and practical training dynamics differ from the idealized construction in influence functions. The work argued that influence functions better capture the proximal Bregman response function, that is what is the effect of removing a data point while also attempting to maintain current predictions?
+
+# 3.3 PSEUDO LOG LIKELIHOODS
+
+Unlike autoregressive language models, masked language models don't have a natural way to immediately compute the joint likelihood of a sequence. As a result, Wang & Cho (2019) proposed to mask every index of a sequence one-at-a-time then average to derive a PLL (Wang & Cho, 2019):  $\mathrm{PLL}(x) = \frac{1}{L}\sum_{i=1}^{L}\log P(y_i = x_i|x_{\backslash i},\pmb{\theta})$ .
+
+This formulation suffers from the need to run  $\mathcal{O}(L)$  forward passes to compute a perplexity or log likelihood. In response to this, the community only considers autoregressive pLMs when computing fitness values for proteins containing insertions or deletions.
+
+# 4 UTILIZING PLM LIKELIHOODS TO PREDICT ZERO-SHOT SUCCESS
+
+# 4.1 EFFICIENT PSEUDO LOG LIKELIHOOD CALCULATION
+
+For non-autoregressive models, papers (Lin et al., 2023) often use a measure of pseudo log likelihood (PLL) by evaluating  $\mathrm{PLL}(x) = \frac{1}{L}\sum_{i=1}^{L}\log P(y_i = x_i|x\backslash i,\pmb{\theta})$ . The resultant equation requires  $\mathcal{O}(L)$  forward passes to calculate, where  $L$  is the length of the protein. Because of the high computational burden, autoregressive language models are sometimes preferred over masked language models when likelihood-based mutation effect prediction is the desired downstream use. To overcome this, Kantroo et al. (2024) perform post training on ESM-2 to predict the distribution of tokens of as if the underlying token of interest had been masked.
+
+We argue that PLL can be approximated for any masked language model in a single forward pass using Algorithm 2 as a result of Remark 4.1. To achieve this, we define the term "mask-consistent", where a token's masked probability is equal to its probability under the implicit assumption that the token might have been scrambled by the training procedure. Using mask consistency, we produce an approximation of the probability of a masked token when the token is not masked during inference in Remark 4.1. As a result, we can calculate PLL for any model without requiring bespoke finetuning or exhaustive resources.
+
+Remark 4.1 Under a mask-consistent masked language model with a training scheme that sets training tokens to a random token with probability  $\alpha$  and keeps them unchanged with probability  $\beta$ :
+
+$$
+P \left(y _ {i} = x _ {i} \mid x _ {\backslash i}, \boldsymbol {\theta}\right) = \frac {\alpha + \beta}{\alpha} P \left(y _ {i} = x _ {i} \mid x, \boldsymbol {\theta}\right) - \frac {\beta}{\alpha}. \tag {6}
+$$
+
+# 4.2 PROOF OF REMARK 4.1
+
+Recalling the training of BERT (Devlin et al., 2018),  $15\%$
+
+of tokens in a sequence are chosen for training. Of those tokens  $80\%$  are turned into [MASK],  $10\%$  are substituted with random tokens, and the last  $10\%$  are left unchanged. We denote the likelihood of substitution with  $\alpha$  and the probability of being unchanged with  $\beta$ .
+
+Letting  $\phi |x\sim$  Bernoulli  $\left(\frac{\alpha}{\alpha + \beta}\right)$  represent the event of a token being a substituted token given that it's not a [MASK] token. Using the law of total probability, we can now expand the likelihood of a token being identical to its input token.
+
+$$
+\begin{array}{l} P \left(y _ {i} = x _ {i} \mid x\right) = P \left(y _ {i} = x _ {i}, \phi = 0 \mid x\right) + P \left(y _ {i} = x _ {i}, \phi = 1 \mid x\right) (7) \\ = P \left(y _ {i} = x _ {i} \mid \phi = 0, x\right) P (\phi = 0 | x) + P \left(y _ {i} = x _ {i} \mid \phi = 1, x\right) P (\phi = 1 | x) (8) \\ = (1) \left(\frac {\beta}{\alpha + \beta}\right) + P \left(y _ {i} = x _ {i} \mid \phi = 1, x\right) \left(\frac {\alpha}{\alpha + \beta}\right) (9) \\ \end{array}
+$$
+
+$P(y_{i} = x_{i}|\phi = 0,x)$  becomes 1 as the token was unchanged. Now the insight comes at the evaluation of  $P(y_{i} = x_{i}|\phi = 1,x)$ . This is the probability of the  $i$ th token given that the input variable was uninformative. Put in another way, it's the probability if the token was masked. We call the language model "mask-consistent" if  $P(y_{i} = x_{i}|\phi = 1,x) = P(y_{i} = x_{i}|x_{\backslash i})$ . Substituting this identity with some algebraic manipulation completes the proof.
+
+$$
+P \left(y _ {i} = x _ {i} \mid x\right) = \frac {\beta}{\alpha + \beta} + \frac {\alpha}{\alpha + \beta} P \left(y _ {i} = x _ {i} \mid x _ {\backslash i}\right) \tag {10}
+$$
+
+$$
+\frac {\alpha + \beta}{\alpha} P \left(y _ {i} = x _ {i} \mid x\right) - \frac {\beta}{\alpha} = P \left(y _ {i} = x _ {i} \mid x _ {\backslash i}\right) \quad \blacksquare \tag {11}
+$$
+
+Kingdom  
+Figure 2: Single Inference PLL is Consistent with PLL. ESM-2 650M Single Inference PLL plotted against traditional PLL on a diverse set of proteins.  
+![](images/21876c5d6e0ca2b71baae20e07c9a409fc03431e815aa4e16b6a43f073822630.jpg)  
+- Animalia - Euryarchaeota  
+- Plantae - Viridiplantae  
+Fungi Chromista
+
+![](images/58e625e2ac9ab9a57f4d19f06d76ae745c1d3e73fccd200675cbec374f2c1bf3.jpg)  
+a
+
+![](images/f9e4907052172c793b9017e78577f54476946861f1cc276d539eff744177817b.jpg)  
+b
+
+![](images/3104dd3d105e8e07bbb930f93b146d87cf268c3d38b9598bf7345435fd16f68a.jpg)
+
+![](images/7e37c3462db22b5cab27fab81e522b6c1f1fd40d91c77ff32d478316c157132a.jpg)
+
+![](images/6878bca965e29059f21c74ca46314a4269760e887f2bbb9f344bde7339267c6a.jpg)  
+C
+
+![](images/af44c8238ba79f7dd8ecfdd2ff460eaa60abf0d76e30ccb8937e5d733856ada6.jpg)
+
+![](images/dd867831d8fcf19d4b837829b874e8fd890ba1afe3d7b97b62cfb15eb3d9ebc7.jpg)
+
+![](images/1b039f07d57d38726e4c45c351e92dee69f0bd6cef48b6d34fd30b380c5e99e4.jpg)
+
+Figure 3: pLM likelihoods are predictive of mutation effect capabilities. (a Upper) Model parameter counts versus average Spearman correlation on DMS datasets across models. (a Lower) Spearman correlation of PLL and performance at each model size. ESM-2 (b) and ProGen2 (c) of varying scales comparing PLLs DMS wild types against zero-shot performance. Each plots title lists the Spearman  $(\rho)$  and Pearson  $(r)$  correlation of the underlying data. Lastly, a second order polynomial regression is fit to the data in all cases resulting in a concave down parabola.  
+![](images/7245b3ca7da52dbb2bd87d64a46fd9d41251baddac44125d744b6e6424ef4c27.jpg)  
+- Organismal Fitness - Activity - Stability - Binding - Expression - Line of Best Fit
+
+![](images/a221dd98098d06ef1ccf0929430c4a4834b2085136eff56d20cd9d0d8e6b473f.jpg)
+
+![](images/89f3b171429c91137a299261bc5e7174db757c287cd75ba8d4ab4cadb529452d.jpg)
+
+![](images/a23b314f730fb1174c8a0b0e5823a20560e58d5abd084031b287bdd8ea676e9b.jpg)
+
+# 4.3 ESM-2 AND PROGEN-2 LIKELIHOODS PREDICT DMS CORRELATION
+
+Our goal is to see if the apparent capacity for pLMs to be used for zero-shot fitness is primarily driven by data or a true understanding of the fitness landscape. To do so, we evaluate pLMs on various deep mutational scan (DMS) tasks, and assess if task performance is correlated with likelihood. Specifically, we take the wild type proteins in 217 DMS studies from ProteinGym (Notin et al., 2023) and calculating PLLs for each of them. In Figure 3 we plot the relationship between PLL and DMS Spearman  $\rho$  for the most utilized masked and autoregressive pLMs ESM-2 and ProGen-2.
+
+Generally, ProteinGym task averages improve as parameters increase (Figure 3a upper), with ESM-2 performance degrading past 650 million parameters. But, an inverse scaling law emerges for the Spearman correlation between likelihood and DMS correlation. Figure 3a lower suggests that in small models, a lot of performance can be explained by the likelihood of the wild type sequence, which then degrades as parameters increase across both ESM-2 and ProGen-2.
+
+Further looking at the data in Figures 3b & c illuminates why the correlation between likelihoods and DMS performance decreases. Instead of of likelihoods being less explanatory of zero-shot fitness prediction, higher parameter count models exhibit performance degradation at high likelihoods. This effect is magnified when looking at sequences with probabilities near 1 (or 0 on the log scale) for models such as ESM-2 15 billion and ProGen-2 XL. One interpretation of this phenomenon is that the increased learning capacity of larger models has caused them to overfit on certain regions of sequence space, suggesting that the optimal choice of pretrained model is dependent on the downstream task and dataset.
+
+In all, the results shown in Figure 3 corroborate a theory exemplified in Figure 1b. Low likelihood or under preferred wild type sequences struggle to predict beneficial and harmful mutations. As the likelihood increases so does performance, but after a certain threshold too much preference harms predictive capability.
+
+# 5 UNDERSTANDING PLM LIKELIHOODS USING INFLUENCE FUNCTIONS
+
+Motivated by the predictive phenomenon with likelihoods, we aim to study a more rigorous causal relationship between data and downstream likelihoods using influence functions. We determine a structure of the data involved in these likelihoods and thus preference. In Section 6, we leverage these findings to inspire a method of post-training.
+
+# 5.1 PROTEIN LIKELIHOOD INFLUENCE HAS A POWER LAW TAIL
+
+Influence functions (Hampel, 1974; Koh & Liang, 2017) measure the impact of a training sequence on function of interest. We quantified per-datum influence values in the train dataset to find most influential points on sequence likelihood to investigate our examined phenomenon. To approximate ESM-2's training distribution, we randomly sample 10,000 proteins from UniRef50 and trim sequences to be of length at most 1,024. We utilize the Kronfluence library (Bae, 2024) to calculate influences conditioning the inverse Hessian on the random samples. Figure 4 depicts
+
+influence of these 10,000 points as well as points retrieved by searching UniRef50 using mmseqs2 on the likelihood four common proteins of interest: Green Fluorescent Protein (GFP), Cytochrome Complex (CytC), KaiB, and Programmed Cell Death Protein 1 (PD1).
+
+![](images/c14dbfd1dc76aab58306166a216ff72781b3f57453bd359533cf1d2d3edbad95.jpg)
+
+![](images/fb4fb8700b80ea9caa840b20be51e259f0cf318965147d0368d12fba48a73f3c.jpg)  
+Figure 4: pLM influence tails exhibits a power law relationship. ESM-2 650M influences plotted against the complementary cumulative distribution.
+
+Our results suggest two main insights. First, we reproduce the power law tail observed for traditional LMs in Grosse et al. (2023), suggesting similar data dynamics between pLMs and LMs. From a biological perspective, Qin & Colwell (2018) finds a power law tail in the covariance of phylogenetic protein systems, which might lend a way to understand this result. Second, for each of the four proteins, mmseqs2 found some of the most influential proteins when compared to an unfiltered set. This means that search might serve as an efficient way to deduce which training samples can be used to improve performance.
+
+# 5.2 INFLUENCE DIMINISHES WITH EDIT DISTANCE
+
+![](images/209a1aea7d20ca166b5f88000ad41c4d36d26af1afe05fd8e3632d7013f8c008.jpg)  
+Figure 5: Influence on likelihood decreases as hamming distance increases. The diminishing influence suggests that homologous sequences have a the greatest causal link to preference.
+
+As protein search yields many of the influential proteins, a natural thought would be that influence is related to the amount of homology between a training data point and the sequence whose likelihood is in question. To investigate this phenomenon, we selected DMS studies by the maximal number of edits from the wild type sequence taking the top 10 from ProteingGym. From each DMS study, a random 1,000 samples are used as a synthetic set of training examples to examine what would happen were a protein with some number of mutants from the wild type within the train set.
+
+We present a representative example in Figure 5 using ESM-2 650M on Sinai et al. (2021)'s DMS
+
+![](images/3e008ad18f06badd1654e83d8e8d3fcf45fa438ea072aa112b2257c3642ccac9.jpg)  
+a  
+Finetune Epoch 1
+
+![](images/7ec72db1c52b796134ad917ec3b5d6badf7a48adb6132eb6662021869c6369c8.jpg)  
+Finetune Epoch 2
+
+![](images/f12f756332e46fd40adcfa4f2ddbbd3412b7a3a0cd1726f116eca0ae25fd1fbf.jpg)  
+Finetune Epoch 3  
+b
+
+![](images/7d16b69c8f9c14cd6389a8e11cf8d1dd36251e96a2cfa8b5a1a52cc72f3d1165.jpg)  
+Figure 6: Wild type likelihoods are predictive of finetuning success. (a) The change in Spearman correlation on 217 DMS studies over 5 epochs. Each plots upper right corner denotes the slope of a linear model. (b) The cumulative gain in performance for studies below likelihood  $\epsilon$ .
+
+![](images/528c1bad7cec95566ea367f5ad1d0e0840942be35ce2e7c1e21ebe742bffd0cc.jpg)  
+Finetune Epoch 5
+
+![](images/374de856486bfd5fcf4e0c661c2a2af7409242cdd7de16a8cae4b3cc63fd2df8.jpg)
+
+<table><tr><td rowspan="2">Model Name</td><td colspan="5">Selection Criteria</td><td rowspan="2">Mean</td><td rowspan="2">Weighted Mean</td></tr><tr><td>Activity</td><td>Binding</td><td>Expression</td><td>Organismal Fitness</td><td>Stability</td></tr><tr><td>ESM-2 650M</td><td>0.441</td><td>0.327</td><td>0.415</td><td>0.390</td><td>0.523</td><td>0.439</td><td>0.419</td></tr><tr><td>+ Finetune ε = -1.4</td><td>0.4610.020</td><td>0.3560.029</td><td>0.4370.022</td><td>0.4250.035</td><td>0.5340.011</td><td>0.4620.023</td><td>0.4430.024</td></tr><tr><td>+ Finetune ε = 0</td><td>0.3910.050</td><td>0.2710.056</td><td>0.3930.022</td><td>0.3850.005</td><td>0.4440.079</td><td>0.3980.041</td><td>0.3770.042</td></tr><tr><td>ProGen2 XL</td><td>0.404</td><td>0.291</td><td>0.418</td><td>0.389</td><td>0.445</td><td>0.406</td><td>0.389</td></tr><tr><td>EVE</td><td>0.464</td><td>0.354</td><td>0.404</td><td>0.449</td><td>0.487</td><td>0.454</td><td>0.432</td></tr><tr><td>MSA Transformer</td><td>0.459</td><td>0.32</td><td>0.435</td><td>0.416</td><td>0.476</td><td>0.438</td><td>0.421</td></tr><tr><td>TrancaptEVE L</td><td>0.492</td><td>0.359</td><td>0.457</td><td>0.466</td><td>0.5</td><td>0.474</td><td>0.455</td></tr></table>
+
+Table 1: Comparing finetuned ESM-2 650M to state-of-the-art pLMs. ESM-2 650M with and without finetuning at various values of  $\epsilon$  compared to other fitness prediction models. Results for all non ESM-2 models are reported from ProteinGym. In line with earlier findings that high likelihoods can harm performance, we find that applying a threshold to which sequences we finetune on improves performance.
+
+data chosen on the basis of having the second highest edit distance and a fairly uniform edit distance distribution. The influence of a mutant protein against its wild type decreases as a function of its edit distance to the wild type. Further evidence can be seen in Appendix Figures 8 and 9. Across five different ESM-2 model scales (8M, 35M, 150M, 650M, and 3B) and all 10 datasets, we find that as edit distance increases influence decreases. Variations in the clarity of this relationship might be explained by the number of locations mutated on the protein of interest and underlying distribution of edit distances.
+
+# 6 EVO-TUNING PLMS TO IMPROVE FITNESS PREDICTION
+
+Motivated by the finding that low probability sequences underperform, and that mmseqs2 serves as a simple heuristic for finding likelihood-influencing data points, we propose a simple remedy: a pLM can be finetuned to increase the likelihood of the wild type sample for a protein engineering task, but sequences with sufficiently high likelihoods should remain unchanged.
+
+We perform unsupervised finetuning on homologous sequences (sometimes referred to as evo-tuning (Alley et al., 2019; Biswas et al., 2021)) of ESM-2 650M on a set of sequences derived by searching
+
+UniRef100 for the wild type sequence on each of the 217 DMS studies in ProteinGym separately. Post training utilizes AdamW (Loshchilov et al., 2017) with a learning rate of  $1\mathrm{e} - 6$  for 5 epochs on the 1,000 most similar proteins to wild type as determined by E-value of mmseqs2 search with a maximum cut off of 1. Finetuning starts at a batch size of 32 is progressively halved in the occurrence of an out of memory exception. Each run consumes a single 80GB A100 GPU. As our findings above further indicate that too much likelihood harms performance, we only consider finetuning models where the log likelihood falls below a threshold  $\epsilon$ .
+
+Examining the change in Spearman correlation on the DMS test bed from finetuning with respect to initial sequence log likelihood in Figure 6, performance improvement is anti-correlated with starting probability. Low likelihood sequences benefit from training, while high likelihood sequences get harmed, an effect that gets amplified as more training occurs. This effect further corroborates the model shown in Figure 1b. The nontrivial performance harm of too much likelihood might be explained through Weinstein et al. (2022)'s findings that modeling the sample data distribution density isn't what leads to fitness prediction. In our case, further finetuning on high likelihood regions in sequence space might cause memorization of phylogenetic artifacts instead of fitness signals.
+
+If the model is naively finetuned before evaluation on every DMS, then performance degrades as seen in Table 1 at  $\epsilon = 0$ , in accordance with Meier et al. (2021). Figure 6b investigates how only performing finetuning on samples that fall below some likelihood threshold  $\epsilon$  leads to a gain in total correlation across ProteinGym.  $\epsilon$  is evaluated at 11 equally spaced log likelihoods from  $-2$  to 0. The best performance is observed at  $\epsilon = -1.4$ . After accounting for this procedure, ESM-2's scores jump and outperform models that utilize evolutionary information through MSAs like EVE (single) and MSA Transformer (single) (Frazer et al., 2021; Rao et al., 2021), while becoming competitive with the state-of-the-art hybrid MSA and pLM model TranceptionEVE (Notin et al., 2022).
+
+# 7 DISCUSSION
+
+In this work, we proposed studying pLMs through the framework of preference. Our findings demonstrate that sequence likelihood, which can be thought of as implicit preference, can predict pLM performance on a diverse set of fitness prediction tasks. In thinking about pLM capabilities through preference, we illuminate different layers of preference suggesting that the learned pLM behavior is reflective of the user level bias in curation of training data, and may not always reflect evolutionary patterns as one might assume.
+
+We show that both low and high likelihood sequences suffer in performance, suggesting that over or under preferring data is harmful for fitness prediction capabilities. By utilizing influence functions, we're able to link the observed likelihoods to training data in a causal manner suggesting that homologous protein training data is most responsible for the driving the latent preferences of these models. Combining these two findings we arrive at a nuanced way to improve off the shelf models: unsupervised finetuning on regions of low-likelihood space.
+
+Crucially, our work demonstrates that the ability of a pLM to predict fitness is indeed a matter of preference - not just of the model, but of the human choices and resource limitations that shape data curation. This perspective helps us understand when these models will succeed or fail in applied use giving us ways to improve their success.
+
+Looking forward, our findings motivate the need to think more intentionally about pretraining and evaluation. Naive usage of sequence databases and scaling will magnify the biases training data leading to miscalibrated preferences. Researchers should curate the next generation of datasets and supervision schemes to adjust for this. Lastly, understanding the interplay between data and performance suggests that researchers should ensure proper test task representation during pretraining when benchmarking as suggested by Dominguez-Olmedo et al. (2024). Algorithmic differences might be overshadowed by human preferences at the data level confounding whether a model better captures the biology of proteome.
+
+In conclusion, by thinking about pLM behavior in terms of preference, we've provided a new lens through which to understand and improve these powerful tools. As the field of protein engineering continues to evolve, studying pLMs through their likelihoods and training distributions offers a promising path to diagnose and train more effective protein language models.
+
+# ACKNOWLEDGEMENTS
+
+We're grateful for our conversations with Cem Anil surrounding influence functions, and Frances Ding surrounding bias in pLMs. We also thank Seyone Chithrananda and Ludwig Schmidt for their feedback on this work. Our experiments are enabled by the resources donated by NVIDIA. Cade Gordon is sponsored by the Siebel Foundation. Amy X. Lu acknowledges support from NSERC PGS D. Pieter Abbeel holds concurrent appointments as a Professor at UC Berkeley and as an Amazon Scholar. This paper describes work performed at UC Berkeley and is not associated with Amazon.
+
+# REFERENCES
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+
+# APPENDIX A FURTHER EXPERIMENTAL DETAILS
+
+# A.1 PROTEIN SEARCH
+
+We utilize the July 24, 2024 UniProt releases of both UniRef50 and UniRef100 (Suzek et al., 2015). The search performed over UniRef50 used the default sensitivity and a maximum E-value of 1 to be inline with Lin et al. (2023). Similarly, searches on UniRef100 utilized a maximum E-value of 1, but also a sensitivity of 7.5.
+
+# A.2 INFLUENCE FUNCTIONS
+
+Using the Kronfluence (Bae, 2024) we calculate influence functions using a method identical to that introduced in Grosse et al. (2023). As such the computation has three major components that can be altered: the training loss function  $\mathcal{L}$ , the measurable value  $f$ , and the data used to condition the Inverse Hessian  $\mathbf{H}^{-1}$ .
+
+For both the loss function and measurable function we want something that mirrors masked language model training and PLL calculation. To that end, we choose the function  $\frac{1}{L}\sum_{i = 1}^{L}\log P(y_i = x_i|x,\theta)$ . This function is both a rough estimate of the training objective, while removing the need for a stochastic loss estimation of a certain point and is similar to PLL. Although, we do introduce another version of PLL that requires only a single inference, the clipping would result in certain tokens not contributing to the loss. As a result, this form both avoids stochasticity and multiple inference passes to arrive at both our training loss and observed function PLL.
+
+As the Inverse Hessian will approximate some of the curvature of our data, we want to choose conditions that match our study of interest. As a result, we used two different setups to perform this training data conditioning. In Section 5.1, we use 10,000 randomly sampled points from UniRef50 as can be thought of as an estimate of ESM-2's training data. Then in Section 5.2, we condition on each individual study's downstream points as those will help capture the local curvature of the likelihoods around the wild type protein of interest.
+
+# APPENDIX B SINGLE-INFERENCE PSEUDO LOG LIKELIHOOD
+
+In this section, we expand on the implications of Remark 4.1. First we use it to propose Algorithm 2 that leverages this result to reduce the number of inferences from  $\mathcal{O}(L)$  to  $\mathcal{O}(1)$ . Lastly, to ensure that the assumptions weren't too lenient, we evaluate our likelihoods on 7,545 sequences of varying species and length showing tight agreement between our Single-Inference PLL and traditional PLL.
+
+Though Kantroo et al. (2024) also provide a method for enabling PLL calculation in a single pass, it requires training a separate neural network ensemble to estimate the masked quantities. Hence, whenever one wants to calculate PLL in one pass for any new model this adapter ensemble must be trained before PLL calculation. Our method bypasses this need, letting PLL be evaluated out of the box in a single inference pass.
+
+# B.1 ALGORITHM
+
+Algorithm 1 provides detail on the traditional method of PLL calculation. Each residue is masked one at a time, then the log probabilities of being the inputted sequence at the masked location are averaged into a single scalar.
+
+Algorithm 1 Traditional Pseudo Log Likelihood Calculation  
+Require:  $x\in \mathcal{V}^L$  1:  $z\gets 0$  Initialize PLL 2: for  $i\in \{1,\dots ,L\}$  do 3:  $z\gets z + \frac{1}{L}\log P(y_i = x_i|x_{\backslash i})$  Mask and infer 4: end for 5: return  $z$
+
+To overcome the for loop, and thus  $\mathcal{O}(L)$  forward passes, Algorithm 2 relies on Remark 4.1 to derive probability values equivalent to the masked probabilities. As each probability of interest is captured in a single forward pass, we can side step the for loop now only requiring  $\mathcal{O}(1)$  or a single forward pass to calculate PLL. One limitation becomes apparent in calculating our closed form probability of interest, it's plausible that  $\frac{\alpha + \beta}{\alpha} P(y_i = x_i|x) < \frac{\beta}{\alpha}$  leading to a negative probability. Since probability must be a non-negative measure, we therefore clip values that are too low to some  $\epsilon$ .
+
+# Algorithm 2 Single-Inference Pseudo Log Likelihood
+
+Require:  $x\in \mathcal{V}^L$ $\epsilon \in \mathbb{R}^{+}$
+
+1:  $p \gets P(y = x|x)$  
+2:  $p' \gets \max \left( \frac{\alpha + \beta}{\alpha} p - \frac{\beta}{\alpha}, \epsilon \right)$  
+3: return  $\frac{1}{L}\sum_{i=1}^{L}\log p'(y_i = x_i)$
+
+$\triangleright$  Perform inference once,  $p \in [0,1]^{L \times \mathcal{V}}$
+
+$\triangleright$  Use Thm. 4.1 and  $\epsilon$  for negative probabilities
+
+> Vector sum and in place operation
+
+To ground  $\alpha$  and  $\beta$  for the reader, BERT plus the ESM family use 0.1 for both  $\alpha$  and  $\beta$ . For each token included in loss calculation, the models substitute and hold each token with  $10\%$  chance given. We utilize this value for all calculations of Single-Inference PLL within this study.
+
+# B.2 EMPIRICAL VALIDATION OF SINGLE-INFERENCE PSEUDO LOG LIKELIHOOD
+
+To assess the correctness of the Single-Inference PLL calculation, we seek to measure its correspondence with classic PLL calculation. We calculate PLL using both methods using 7, 545 of various lengths, species, and families from Ding & Steinhardt (2024). In Figure B.2 we can see the calibration of the two quantities against one another.
+
+The two methods have strong correlation statistics. Calculating the correlation between PLL and Single-Inference PLL yields Spearman  $\rho = 0.923$  and Pearson  $r = 0.930$ . The  $P$ -values for both underflow the range of Python's floating point precision. Spearman correlation was seen to slightly improve as  $\epsilon$  increased while Pearson correlation would lessen. As our interests lie in rank order statistics, we chose to utilize an  $\epsilon = 10^{-3}$  to have more rank agreement during the studies of the main text.
+
+# APPENDIX C EXTENDED INFLUENCE FUNCTION PLOTS
+
+We plot the relationship between influence and edit distance for the ESM-2 family at scales between 8 million and 3 billion. Each plot depicts a single model examining the effect of adding the mutants of a DMS study into the underlying training set. We selected the 10 studies with the highest maximal edit distance from their wild type sequence.
+
+![](images/fd4199219657eb384dbcf655d802f8d895eadd9f26efafee3905d9712432e8de.jpg)  
+Figure 7: Single-Inference PLL is calibrated with PLL on a diverse set of proteins. We plot the mean and inner  $95\%$  quantile for 11 evenly spaced bins between -4.5 and 0. Each data point is colored by the number of samples used to derive the statistics for the bin. Although low likelihood sequences seem inconsistent, there are few of these points in the dataset making it hard to draw a confident conclusion on their calibration. On the other hand, for the bins with greater sequence counts, Single Inference PLL aligns well with with traditional PLL.
+
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+Figure 8: Influence versus edit distance for the top 5 DMS studies with the most mutations from wild type. Each row represents a unique study an each column an ESM-2 Model 8M, 35M, 150M, 650M, and 3B (left to right).
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+![](images/2d2c1bad47246b95444cb1fbac9a822b6c49c4274d2da2f030eb2bb17d8ae290.jpg)
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+![](images/9642520604de15b7defee49405161f253d59db2806fc6f142cab9cede371809e.jpg)
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+![](images/cc3f0f176c4dce3a4b8126ff7a39b9dec7624b3ac1180791fbebe6e92c6d15a8.jpg)  
+Q6WV13 9MAXI Somermeyer 2022
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+![](images/cd811529c0a96c5885261e7e0afc87463944ce4652fb2bd2648d3de4f845fd30.jpg)
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+![](images/db3a11fc77e1a652177dece3a6f9bfa0193050b7bab8cde82b4d40e0b262cfca.jpg)
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+![](images/9ae6770a07f28e2b0a1dd8a46f3443c29f76b6ebad7aefb20b860137b54c33f5.jpg)  
+Figure 9: Influence versus edit distance for the top 6-10 DMS studies with the most mutations from wild type. Each row represents a unique study on each column an ESM-2 Model 8M, 35M, 150M, 650M, and 3B (left to right).
+
+![](images/d54983b34ebea0fa89b2975aa140b583bb40c606c2f673c9dfd5bffe108e025d.jpg)
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+![](images/33c818f3d1b23e57c76dfe6009e477306ccb85dcfde3a625b70dbaa84d23b9c7.jpg)  
+F7YBW8 MESOW Aakre 2015
+
+![](images/745f5a60dbd89eb9f8f3d148a9b2e03cf36970883be53610728fca908b65e5f5.jpg)
+
+![](images/19e9b158977e642dea06d7fc1b402f0c86617ba9d47ca8d30ca3f5ea35af7bca.jpg)

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2025/ProtoSnap_ Prototype Alignment For Cuneiform Signs/full.md

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+# PROTOSNAP:  
+PROTOTYPE ALIGNMENT FOR CUNEIFORM SIGNS
+
+Rachel Mikulinsky $^{1*}$  Morris Alper $^{1*}$  Shai Gordin $^{2}$  Enrique Jimenez $^{3}$  Yoram Cohen $^{1}$  Hadar Averbuch-Elor $^{1,4}$
+
+$^{1}$ Tel Aviv University  $^{2}$ Ariel University  $^{3}$ LMU  $^{4}$ Cornell University
+
+# ABSTRACT
+
+The cuneiform writing system served as the medium for transmitting knowledge in the ancient Near East for a period of over three thousand years. Cuneiform signs have a complex internal structure which is the subject of expert paleographic analysis, as variations in sign shapes bear witness to historical developments and transmission of writing and culture over time. However, prior automated techniques mostly treat sign types as categorical and do not explicitly model their highly varied internal configurations. In this work, we present an unsupervised approach for recovering the fine-grained internal configuration of cuneiform signs by leveraging powerful generative models and the appearance and structure of prototype font images as priors. Our approach, ProtoSnap, enforces structural consistency on matches found with deep image features to estimate the diverse configurations of cuneiform characters, snapping a skeleton-based template to photographed cuneiform signs. We provide a new benchmark of expert annotations and evaluate our method on this task. Our evaluation shows that our approach succeeds in aligning prototype skeletons to a wide variety of cuneiform signs. Moreover, we show that conditioning on structures produced by our method allows for generating synthetic data with correct structural configurations, significantly boosting the performance of cuneiform sign recognition beyond existing techniques, in particular over rare signs. Our code, data, and trained models are available at the project page: https://tau-vailab.github.io/ProtoSnap/
+
+# 1 INTRODUCTION
+
+The earliest forms of decipherable scripts date back to the late  $4^{\text{th}}$  millennium BCE, with the invention of the cuneiform writing system in ancient Mesopotamia, which came to be used for a number of historically significant ancient languages such as Sumerian and Akkadian (Radner and Robson, 2011; Streck, 2021). Cuneiform signs have complex internal structures which varied significantly across the eras, cultures, and geographic regions among which cuneiform writing was used. The study of these variations is part of a field called paleography, which is crucial for understanding the historical context of attested writing (Biggs, 1973; Homburg, 2021). However, while computational methods show promise for aiding experts in analyzing cuneiform texts (Bogacz and Mara, 2022), they are challenged by the vast variety of complex sign variants and their visual nature: Represented as wedge-shaped imprints in clay tablets which have often sustained physical damage, cuneiform appears as shadows on a non-uniform clay surface which may even be difficult for human experts to identify under non-optimal lighting conditions (Taylor, 2015).
+
+Prior work has focused on digitization of cuneiform tablets at a coarse resolution, localizing and classifying signs from photographs of whole tablets (Dencker et al., 2020; Stötzner et al., 2023a). However, these methods treat sign types as categorical while neglecting sign-internal configurations of strokes in each character, which provides crucial information for identifying rare signs and distinguishing between sign variants. In this work, we aim to recover the fine-grained internal configuration of real cuneiform signs, given coarse-grained categorical information as input. In
+
+![](images/bbea5597b6d5d5694aee43e2f58bd9bf830373adc3e2a68f6661d8efe6bf9036.jpg)  
+Real Cuneiform Tablet+
+
+![](images/b63b7ddbf38c2e6b7e25e44f38197b5e430bc66a8e6a048e6afbf4f91c61d379.jpg)  
+Cuneiform sign Prototype (font+skelton)
+
+![](images/1de60c46e5fcaef9d214af887098a15ec321024de3f9ff693c9f732c68757bd4.jpg)  
+Our output  
+Figure 1: ProtoSnap applied to a full tablet by cropping each sign using existing bounding boxes (such as those depicted in unique colors), and matching prototypes of the signs (illustrated in the center). Our technique "snaps" the skeletons of the prototypes to the target images depicting real cuneiform signs. These aligned results can be used to produce an automatic digital hand copy (right). We also show that our approach can be used to boost performance of cuneiform sign recognition.
+
+![](images/d3ae9426dbc9fc390c655c0627902f51136a6a207e8f076a14cd5b9f9965650f.jpg)  
+Application: hand copy generation
+
+particular, our method is provided with a prototype image and its associated skeleton indicating the canonical structure of a sign, and aligns this structure to a target image depicting a corresponding real cuneiform sign. As illustrated in Figure 1, our technique is analogous to the laborious hand copies produced by expert Assyriologists; when applied to a tablet with existing character-level annotations, this outputs the outlines of signs in the style of the original document. Furthermore, we show that these aligned skeletons may be used to boost optical character recognition (OCR) performance, by training a generative model with structural conditioning as detailed below.
+
+To this aim, we present ProtoSnap, an unsupervised approach leveraging deep diffusion features to snap a skeleton-based prototype to a target cuneiform sign, revealing its structure without requiring any labelled examples of real photographed signs. By using a fine-tuned generative model as a prior on the appearance of cuneiform images, and enforcing global and local consistency, we are able to localize the constituent strokes in real cuneiform images. We make use of the key insight that pairwise similarities between regions of the prototype and target images encode information about both coarse global alignment and fine-grained local deviations of each stroke from its canonical pose. Our technique distills this information with a multi-stage process performing global alignment followed by local refinement of stroke positions.
+
+We provide a new benchmark of expert-annotated photographed signs for evaluation, and show that our system succeeds at identifying their internal structures, significantly outperforming generic correspondence matching techniques. We also show the downstream utility of ProtoSnap for automatic digitization of cuneiform texts, by using aligned prototype skeletons as a condition for a generative model to produce structurally-correct synthetic data to train cuneiform OCR. Our results show that this achieves state-of-the-art results on cuneiform sign recognition, particularly enhancing performance on rare signs where naive synthetic data generation struggles to produce instances of the correct sign.
+
+Stated explicitly, our key contribution are:
+
+- ProtoSnap, an unsupervised prototype alignment method capturing the structure of photographed cuneiform signs.  
+- A novel benchmark with expert annotations, and results showing that our method outperforms generic correspondence methods at this task.  
+- State-of-the-art OCR results for cuneiform when using synthetic data produced using our method's alignments for conditional generation.
+
+# 2 RELATED WORKS
+
+Machine learning (ML) for cuneiform. Ancient texts provide a window into our history, but their decipherment and interpretation require painstaking work and expert knowledge of esoteric languages, complex writing systems, and historical context which serve as a barrier to their translation and analysis on scale. Due to the high societal value of these tasks and the scarcity of expert knowledge and time, machine learning promises to provide an invaluable aid for understanding the ancient world. In the context of a number of works on ML applied to diverse ancient inscriptions (Hassner et al., 2013a; Assael et al., 2019; Yin et al., 2019; Huang et al., 2019; Luo et al., 2021; Hayon et al., 2024), the cuneiform script poses particular challenges. These include the nature of the physical writing media (indentations in textured and often damaged clay under various lighting conditions), and the diverse nature of cuneiform signs which changed over thousands of years of use in vast geographical regions.
+
+Various approaches have been applied to modeling cuneiform signs for the purpose of downstream tasks such as optical character recognition. Some works apply recognition directly to image data (Bogacz et al., 2017; Dencker et al., 2020; Stötzner et al., 2023a), while others have treated cuneiform as structured graphs to recognize signs based on their internal configurations (Kriege et al., 2018; Chen et al., 2024). A handful of works have explicitly modeled cuneiform signs as compositions of wedges, though these mainly focus on segmentation from 3D meshes or localizing strokes on the bounding box level (Bogacz and Mara, 2022; Stötzner et al., 2023b; Hamplova et al., 2024). By contrast, our approach provides a pixel-aligned skeleton indicating the relative positions, sizes, and orientations of the strokes, and unlike prior works we operate exclusively on 2D photographs of cuneiform signs and without any strong supervision.
+
+Skeleton-based template alignment. Our approach to inferring the configuration of cuneiform images as compositions of strokes by aligning a skeleton-based template bears partial similarity to various existing methods that typically operate over generic natural images.
+
+Our method resembles template matching in that we use a template image (our font prototype) and search for relevant regions in the target image. While earlier approaches used naive comparisons of image intensities or low-level features when sliding the template across the target image (Ben-Arie and Rao, 1993; Cole et al., 2004; Kim et al., 2011), this is not robust to complex relations between the template and target. To handle these challenges, more recent works have adopted deep image features along with modeling complex non-rigid deformations (Oron et al., 2017; Talmi et al., 2017; Cheng et al., 2019; Gao and Spratling, 2022). Our method similarly searches for matches to our template using deep features and allowing for deformations, although we differ from conventional template matching in explicitly using the skeletonized graph structure of the template and matching each of its constituent strokes separately.
+
+We also note similarity to pose estimation methods, as we infer the structure of a sign by localizing keypoints. Pose estimation methods align a graph of keypoints to an image, most commonly applied to a single category such as humans (Fang et al., 2022; Zheng et al., 2023), animals (Li and Lee, 2021; Yang et al., 2022), or vehicles (Reddy et al., 2018; López et al., 2019), where the same fixed graph applies to all instances. However, in our setting the number and connectivity of keypoints depends on the sign type under consideration. In this respect our method resembles category-agnostic pose estimation methods (Xu et al., 2022; Hirschhorn and Avidan, 2023), though our input also includes a template image rather than only using an abstract graph.
+
+In the context of images depicting text, a number of works address the problem of text spotting, which searches for matches to a given visual text representation in images (He et al., 2018; Huang et al., 2022; Ye et al., 2023). This may include alignment or dewarping of detected text, but does not typically explicitly leverage the internal shape of symbols as in our method. We also note several works performing transcript alignment using dense correspondence methods, which align visual text but do not explicitly handle character-internal structure (Hassner et al., 2013b; 2016).
+
+# 3 THE CUNEIFORM WRITING SYSTEM
+
+Cuneiform, one of the earliest known writing systems, was a logo-syllabic script indicating both units of sound and meaning with signs. Unlike the Latin alphabet which uses less than thirty basic
+
+![](images/af20eda4f3e25e637b5f21f503517705633707db95f503bd7c7a8be36bb3f96e.jpg)  
+Figure 2: Method Overview. Given a prototype image with annotated skeleton and a target image of a real cuneiform sign, ProtoSnap first extracts best-buddy correspondences from deep diffusion features (extracted with our fine-tuned SD-  $\mathbb{F}$  model), globally aligning the target image to the skeleton of the prototype. Our method then "snaps" the individual strokes into place with a local refinement stage by optimizing a per-stroke transform.
+
+![](images/f885c2c495c5bbda845fc6306df7db4eb73b637fd92e9c93f6a2f3169ae0325d.jpg)
+
+![](images/19b088a3c27d329a2d4acde9b842b5d27d92584abdd2074acd50ec05fc87a690.jpg)
+
+![](images/f1f8d9743ce520f31dec8f9990c7b55d394697902da273223c127223a5c429eb.jpg)
+
+letters to indicate sounds, cuneiform signs numbered upwards of one thousand unique types, which varied dramatically in their realizations across eras, languages, and geographical regions (Walker, 1987). See, for instance, the two right-most examples in Figure 5, both variants of the same sign AN from different eras. Scholars have attempted to collate lists of known signs and their variants (Labat and Malbran-Labat, 1988; Borger, 2003), and canonical representations of the most common variants for different time periods have been encoded in digital fonts.
+
+Cuneiform was written on clay tablets by scribes using a stylus to create wedge-shaped impressions, also known as strokes, which combine to form signs. Scribes used styli with triangular edges to create impressions on the moist clay in three possible directions: horizontal, vertical, or oblique (Cammarosano, 2014; Cammarosano et al., 2014). Various encoding schemes proposed to encode these strokes digitally (Bogacz and Mara, 2022); we adopt the four-keypoint scheme, treating all strokes as being composed of a triangular head indicated with three keypoints and a fourth keypoint indicating the stroke's tail; see Figure 2 (second to left) for an example. We refer to the graph of these keypoints and the edges connecting them as a sign's skeleton. Our method's prototype input consists of both a rasterized font image along with its skeleton encoding the configuration of its strokes; we collect these skeletons via manual annotation as described in the appendix.
+
+# 4 METHOD
+
+Given a prototype consisting of a font image annotated with a skeleton composed of strokes and a target image of a real cuneiform sign, our system aligns the prototype skeleton to match the sign structure in the target image. An overview of this process is shown in Figure 2.
+
+Our system consists of three steps: First, we calculate a semantically-adapted 4D similarity volume which encodes the pairwise feature similarities of each pair of regions in the two images. This similarity volume uses diffusion features to encapsulate the complex geometry and semantics of cuneiform images. We then calculate a global alignment between the prototype font image and the target image, using best-buddies sparse correspondences, extracted from the similarity volume, as a robust signal to fit this alignment. Finally, we perform local refinement to optimize the relative positions of each stroke and their internal configurations, when deviation from the canonical configuration is necessary. We describe each step in turn below, with further implementation details provided in the appendix.
+
+# 4.1 SEMANTICALLY-ADAPTED 4D SIMILARITY VOLUME
+
+Our system is based on the guiding assumption that local similarities between regions in the two images can be used to compute a structurally-consistent matching between the prototype sign structure and the target cuneiform sign. As a backbone for computing meaningful similarity scores, we use diffusion features (DIFT Tang et al. (2023)), which leverage the strong geometric and semantic understanding of a generative text-to-image model to represent image features for discriminative tasks. These features are calculated as intermediate activations the model's denoising component, applied to the input image with added random noise. However, standard generative models
+
+![](images/c9f47bec38f7b6ff742ac0ae326ebc842a9bf7500fdd457ccf07f2d4a584e800.jpg)  
+Figure 3: DIFT-Based Best-Buddies Correspondences. Noised images are passed through our fine-tuned denoising diffusion model SD- to extract deep Diffusion Features (DIFT), used to calculate the 4D similarity volume  $S$ . For each region  $(i,j)$  in the target image, we examine the 2D slice  $S[i,j,\cdot ,\cdot ]$  and determine the indices  $(k,\ell)$  which maximize its value. Symmetrically, for each region  $(k,\ell)$  in the prototype we find the corresponding region in the target by maximizing the 2D slice  $S[\cdot ,\cdot ,k,\ell ]$ . If these two regions correspond to each other, they are identified as best buddies.
+
+are typically pretrained on natural images from the Internet, with cuneiform scans being out-of-distribution. We thus fine-tune the generative model Stable Diffusion (Rombach et al., 2022) on cuneiform image scans, which we indicate as SD-  $\mathbb{F}$ , using the cuneiform sign name as its accompanying text prompt. We use this as our vision backbone for calculating DIFT features.
+
+We apply DIFT with SD- to our prototype and target images to obtain feature vector maps  $F^{(p)} = (\mathbf{f}_{i,j}^{(p)})_{i,j}$  and  $F^{(t)} = (\mathbf{f}_{i,j}^{(t)})_{i,j}$  respectively of unit-normalized feature vectors. Each feature map is a  $C\times H\times W$  tensor, where  $C$  is the number of feature dimensions and  $H$  and  $W$  are the spatial dimension of the feature map (lower-resolution than the original image, as each feature vector has a larger receptive field in the image). Using these features, we calculate the four-dimensional similarity volume  $S = (\mathbf{f}_{i,j}^{(p)}\cdot \mathbf{f}_{k,\ell}^{(t)})_{i,j,k,\ell}$ . This  $H\times W\times H\times W$  tensor, visualized in Figure 3, contains the pairwise cosine similarities between features encoding patches of the prototype and target images.
+
+# 4.2 GLOBAL ALIGNMENT FROM BEST-BUDDIES CORRESPONDENCES
+
+While  $S$  provides a strong similarity measure, it does not encode geometric constraints on the overall matching between the two images. For instance, multiple regions in one image may all have high similarity scores with a single region in the other image, but mapping them all to the same target region will result in a degenerate solution. To robustly identify a sparse set of best-matching pairs of regions, we follow prior work (Oron et al., 2017; Drory et al., 2020) by identifying best buddies, defined as pairs of patches in the two images which are mutual nearest-neighbors according to their similarities scores in  $S$ . Formally, this is defined as pairs of coordinates  $(i,j)$  and  $(k,\ell)$  such that  $(i,j) = \arg \max_{i,j} S_{i,j,k,\ell}$  and  $(k,\ell) = \arg \max_{k,\ell} S_{i,j,k,\ell}$ . See Figure 3 for an illustration.
+
+Using these sparse correspondences, we fit an affine transformation with least squares estimation, defining a global alignment  $G$  of the prototype to the target image. This transformation allows for basic deformations while preserving the overall structure of the prototype. We learn the parameters
+
+$$
+G = \left[ \begin{array}{c c c} g _ {1 1} & g _ {1 2} & g _ {1 3} \\ g _ {2 1} & g _ {2 2} & g _ {2 3} \\ 0 & 0 & 1 \end{array} \right]
+$$
+
+permitting scaling, rotation, and shear  $(g_{11}, g_{12}, g_{21}, g_{22})$  as well as translation  $(g_{13}, g_{23})$ . This is applied in projective space  $\mathbb{P}^2$ , i.e. mapping a point  $\mathbf{v} = [x, y, 1]^T \in \mathbb{P}^2$  to the point  $\mathbf{v}' = G\mathbf{v} = [x', y', 1]^T$ .
+
+![](images/26b38b24f5af40355d503f37bef8c91e357370c05179d37a0f6e54edeb02b963.jpg)  
+Figure 4: Local Refinement via Skeleton-Based Optimization. To adjust the positioning of individual strokes in a sign, our global alignment is followed by a local refinement stage which learns transformations for each stroke. The loss function encourages positioning on salient regions  $(\mathcal{L}_{sal})$  while semantically matching the corresponding regions in the prototype image, as measured by feature similarity  $(\mathcal{L}_{sim})$ . For each stroke (exemplified by the stroke in red above), these objectives are calculated along points sampled from the skeleton (red dots above). The loss also includes a regularization term  $(\mathcal{L}_{reg})$  preventing excessive deviation from the global transformation.
+
+To handle outliers in a robust manner, we perform the fitting with RANSAC. For further robustness, we incorporate a prior on inlier points being spread over the majority of the area of prototype and target images, by performing this procedure multiple times (including the stochastic calculation of DIFT features and correspondences from Section 4.1) and selecting the result with the best spread of inlier points across the relevant regions in the images, following Hassner et al. (2014) and as further described in the appendix.
+
+# 4.3 LOCAL REFINEMENT VIA SKELETON-BASED OPTIMIZATION
+
+Our global alignment procedure can roughly align the prototype to the target image. However, as the target image was written by hand, each stroke's location may deviate from the canonical relative position given by the prototype font image. Therefore, we introduce a local refinement stage, illustrated in Figure 4, which allows each stroke to move from the canonical prototype structure and "snap" into place, while avoiding excessive deviations from the global structure representing the sign's identity. We model each stroke's deviation from the global alignment as a projective transformation, allowing for a higher degree of deformation than the affine global transformation. The local transformation of stroke  $i$  is parameterized by the matrix
+
+$$
+P ^ {(i)} = I + \left[ \begin{array}{c c c} p _ {1 1} ^ {(i)} & p _ {1 2} ^ {(i)} & p _ {1 3} ^ {(i)} \\ p _ {2 1} ^ {(i)} & p _ {2 2} ^ {(i)} & p _ {2 3} ^ {(i)} \\ p _ {3 1} ^ {(i)} & p _ {3 2} ^ {(i)} & 0 \end{array} \right]
+$$
+
+where  $I$  is the  $3 \times 3$  identity matrix. These are applied on top of the global transformation, resulting in per-stroke transformation of the form  $P^{(i)}G$ . As a projective transformation, this maps a point  $\mathbf{v} = [x,y,1]^T \in \mathbb{P}^2$  to  $P^{(i)}G\mathbf{v} = [x',y',z']^T \sim [x'/z',y'/z',1]^T$ . The  $p_{jk}^{(i)}$  are all initialized to zero (i.e. each local transformation is initialized as the identity).
+
+We optimize the parameters  $p_{jk}^{(i)}$  via gradient descent with the loss function
+
+$$
+\mathcal {L} = \lambda_ {s i m} \mathcal {L} _ {s i m} + \lambda_ {s a l} \mathcal {L} _ {s a l} + \lambda_ {r e g} \mathcal {L} _ {r e g}
+$$
+
+where  $\lambda_{sim},\lambda_{sal},\lambda_{reg}$  are constant weights. We proceed to define each loss term.
+
+**Featural Similarity Loss  $\mathcal{L}_{sim}$ .** To encourage semantically-correct positioning of strokes, we define a loss to maximize feature similarities between matching points on the prototype and target images under the local transformation. Using the similarity volume  $S$  from 4.1, we sample points along the lines connecting skeleton keypoints in the prototype image, calculate their corresponding points under the current global and local transformations, and evaluate their similarity via  $S$  with a temperature-weighted softmax applied to each slice of  $S$  over the prototype image. This uses differentiable grid sampling to interpolate values of  $S$ , as  $S$  has a lower spatial resolution in comparison to the images.
+
+Saliency Map Loss  $\mathcal{L}_{sal}$ . To encourage the strokes to cover salient regions (i.e. areas which appear to contain writing), we calculate a saliency map over the target image and use it to define a
+
+![](images/ac7eca7482c42d5d00d099af04a7bf7c8f0acd989ea58f639e69b25482df750c.jpg)  
+Figure 5: Qualitative alignment results, aligning the prototypes (first row) to target cuneiform images (second row). We demonstrate the results after performing global alignment (third row), and the final result after local refinement (fourth row). As illustrated above, the global alignment stage provides a coarse placement of the prototype template, while the refinement stage allows each stroke to slightly diverge from the original prototype, resulting in more accurate alignments.
+
+loss. The saliency map is calculated using the 4D similarity tensor  $S$  from Section 4.1; for each region in the target image, we calculate the difference between mean similarities to foreground (black) and background (white) regions in the prototype font image, and post-process by applying adaptive histogram equalization, scaling and setting low values to zero. This yields an approximate segmentation map of the cuneiform sign visible in the scanned image. The loss  $\mathcal{L}_{sal}$  is calculated as the mean value of the map sampled at points along the transformed skeleton, selecting points and using temperature-scaled differentiable grid sampling as in Section 4.1.
+
+Regularization Loss  $\mathcal{L}_{reg}$ . To avoid invalid solutions that over-optimize the previous objectives, we add a regularization term that penalizes excessive deformations and solutions that stray from the boundaries of the image. This is defined as  $\mathcal{L}_{reg} = \mathcal{L}_{L1} + \mathcal{L}_{oob}$ . L1 regularization loss is given by
+
+$$
+\mathcal {L} _ {L 1} = \frac {1}{N} \sum_ {i, j, k} \left| p _ {j k} ^ {(i)} \right| = \frac {1}{N} \sum_ {i} \| P ^ {(i)} - I \| _ {L 1}
+$$
+
+where  $N$  is the number of strokes and  $I$  is the  $3 \times 3$  identity matrix. This penalizes local transformations which greatly deviate from the identity.
+
+The out-of-bounds loss  $\mathcal{L}_{oob}$  is defined as zero if all transformed keypoints are within the image boundaries, otherwise as the maximum absolute difference between each transformed coordinate and the image bounds. This models the soft constraint that all keypoints must lie within the image, handling edge cases where the global transformation pushes part of a stroke outside the image bounds.
+
+# 5 EXPERIMENTS
+
+We conduct various experiments, evaluating our method both directly on our benchmark of expert-annotated real photographed signs (Section 5.1) and by testing it on a downstream OCR benchmark (Section 5.2). We also present qualitative results (Figure 5 and appendix), which further highlight the complexity and unique challenges of the task and setting addressed in our work. Finally, we discuss limitations of our approach (Section 5.3). Further implementation details and results are provided in the appendix.
+
+<table><tr><td>Method</td><td>F1@20</td><td>F1@30</td><td>F1@40</td></tr><tr><td>SIFT (Lowe, 1999) + RANSAC</td><td>2.56%</td><td>3.78%</td><td>5.01%</td></tr><tr><td>DINOv2 (Oquab et al., 2024)</td><td>12.33%</td><td>21.88%</td><td>31.04%</td></tr><tr><td>DINOv2 + RANSAC</td><td>16.12%</td><td>28.19%</td><td>37.93%</td></tr><tr><td>DIFT (Tang et al., 2023)</td><td>16.14%</td><td>25.80%</td><td>33.79%</td></tr><tr><td>DIFT + RANSAC</td><td>13.13%</td><td>21.96%</td><td>30.15%</td></tr><tr><td>Ours (w/o refinement)</td><td>21.31%</td><td>37.08%</td><td>50.13%</td></tr><tr><td>Ours (full)</td><td>27.14%</td><td>42.09%</td><td>52.43%</td></tr></table>
+
+Table 1: Alignment evaluation, measuring keypoint localization at various distance thresholds. We compare against several generic correspondence matching baselines, including a geometry-based method (SIFT) and deep feature-based methods (DINOv2, DIFT). As illustrated above, our method significantly outperforms these baselines. Furthermore, our local refinement stage provides a performance boost beyond learning simply a global transform.
+
+# 5.1 ALIGNMENT EVALUATION
+
+To evaluate the quality of alignment, we curate a test set of ground-truth (GT) alignments from manual annotations by expert archeologists. Provided with photographs of cuneiform signs from the eBL dataset Cobanoglu et al. (2024), the experts annotated the position of strokes by indicating groups of four keypoints. Images were marked for exclusion from the test set if they were poor quality or show sign variants differing from the corresponding prototype font images. In total, our test set contains 272 images with expert annotations, covering 25 different sign types, which were not seen in training of SD-Ⅱ. A breakdown of this test set and more details on our annotation procedure are provided in the appendix.
+
+To quantify performance on this benchmark, we compare the predicted and GT positions of stroke keypoints; given a fixed distance threshold, we consider a predicted keypoint as a true positive prediction if it is within the given threshold of a GT keypoint. We report F1 scores for several distance thresholds (along with precision and recall in the appendix).
+
+In Table 1 we compare our solution to generic correspondence matching techniques. We compare against a traditional geometry-based baseline that computes SIFT (Lowe, 1999) features and aligns the images using RANSAC. We also compare against two deep feature-based techniques (DINOv2 (Oquab et al., 2024), DIFT Tang et al. (2023)), applied by assigning each keypoint to the corresponding point of maximum similarity in the target image.
+
+We see that ProtoSnap significantly outperforms the baseline methods at localization, as our solution explicitly considers the complex structure of the sign. We also report performance of our approach without local refinement (i.e performing the first step of global alignment alone); we see that the final step of local optimization indeed achieves better alignment as reflected in our metrics. This may also be seen visually in Figure 5, illustrating the overall global alignment and more precise results yielded by local refinement. In the appendix, we show an ablation study of the different components of our system (such as using fine-tuned SD-  $\overline{\mathbb{N}}$  and each of our loss terms). These results provide further motivation for the design of our full system. We also provide additional qualitative results.
+
+To further evaluate the practical usability and effectiveness of ProtoSnap, we conducted a survey of 12 assyriologists, finding that users are approximately twice as likely to prefer scans with our aligned skeleton overlaid to an overlay without our alignment applied. Further details regarding this user study can be found in the appendix.
+
+# 5.2 LEARNING OCR FROM ALIGNED DATA
+
+We show the downstream benefit of our approach by using ProtoSnap alignments to produce structurally-diverse synthetic training data for an OCR system. As cuneiform signs are highly diverse, with number and arrangement of strokes varying significantly between eras, regions, and individuals, a model trained to produce synthetic data conditioned on sign type alone may struggle to depict the correct configurations of signs, particularly rare signs with few to no attestations
+
+<table><tr><td rowspan="2">Training Data</td><td colspan="2">Accuracy</td><td colspan="2">Balanced Accuracy</td></tr><tr><td>All</td><td>Rare</td><td>All</td><td>Rare</td></tr><tr><td>CSDD* (Dencker et al., 2020)</td><td>58.43%</td><td>25.84%</td><td>34.57%</td><td>16.45%</td></tr><tr><td>+SD-¶ data</td><td>61.56%</td><td>38.56%</td><td>39.02%</td><td>31.13%</td></tr><tr><td>+CN-¶ data</td><td>64.14%</td><td>53.17%</td><td>43.57%</td><td>39.98%</td></tr></table>
+
+Table 2: Sign classification performance when training on previously collected real data (CSDD), and with added data generated using our fine-tuned diffusion model (SD-¶) or ControlNet trained using alignments from ProtoSnap (CN-¶). Our solution demonstrates improved OCR results, with structural augmentations showing increased performance relative to direct unconditional image generation, especially on rare signs (signs that have less than 50 occurrences in the real train set).  $\star$  denotes our reproduced model, as further described in the appendix.
+
+in the existing data. Furthermore, conditioning on sign type does not allow to specify the exact variant of the sign, resulting in generation of the most prevalent variant in the data. As we show, conditioning on a skeleton-based structures provides control over the generated fine-grained structures yielding synthetic data which better captures the real configurations. To demonstrate the benefit of our approach for the downstream task of OCR, we align prototype skeletons to a set of cuneiform images from the eBL dataset (Cobanoglu et al., 2024), and fine-tune ControlNet (Zhang et al., 2023) to generate new cuneiform images using such skeletons as a condition, instead of a text prompt. This model, denoted as CN-  $\mathbb{W}$ , allows us to produce new cuneiform sign images with any input structure, not limited to a list of predefined categories or specific structural configurations. We then use this model to generate synthetic training data, as shown in Figure 6. Further details about the model training are provided in the appendix.
+
+We examine the benefit of this generated data for learning cuneiform sign classification. Dencker et al. (2020) report sign classification performance on the CSDD dataset when training a Resnet18 (He et al., 2016) model with supervision from the CSDD training set alone. We compare this to augmenting the training set with the CN-  $\mathbb{T}^{\mathbb{F}}$  -generated synthetic data described above. As an additional baseline, we also compare with using SD-  $\mathbb{T}^{\mathbb{F}}$  (see Section 4.1) generations for augmenting sign types which uses categorical sign names as a textual condition rather than structural conditions. We report classification accuracy following Dencker et al. (2020); as this dataset is highly imbalanced, we also report balanced accuracy and performance on rare sign types (signs with less than 50 occurrences in the real training set). Results are summarized in Table 2. As seen there, augmenting the CSDD dataset with synthetic data significantly improves classification performance, with structurally-conditioned data from CN-  $\mathbb{T}^{\mathbb{F}}$  providing a significant boost over synthetic data from SD-  $\mathbb{T}^{\mathbb{F}}$ . These results reflect that structurally-controlled generation with CN-  $\mathbb{T}^{\mathbb{F}}$  guarantees generation of the correct sign variant, while SD-  $\mathbb{T}^{\mathbb{F}}$  struggles to produce correct configurations from the sign category alone, as seen in Figure 6.
+
+# 5.3 LIMITATIONS
+
+We note various limitations of our work (visualized in Figure 7). Our alignment procedure requires a canonical sign image and will still fail if the scan displays a structurally different variant of the sign in question. Additionally, our method may fail under extreme deformations, or on low-quality tablets or scans which cannot be feasibly interpreted. Future work might investigate how to calculate a confidence measure to detect such failure cases.
+
+# 6 CONCLUSION
+
+We have proposed the ProtoSnap method for estimating the internal structure of cuneiform signs without any direct supervision, by harnessing pairwise comparisons of deep image features in regions of photographed cuneiform images and skeleton-based prototypes. We have curated expert annotations to provide a new benchmark for this task, and have shown that our method significantly outperforms generic correspondence-based techniques. Beyond its direct application for paleographic analysis, we have also shown our method's utility for the downstream task of OCR, by using aligned skeletons for conditional synthetic data generation.
+
+![](images/2417d68db0fbf1fab78512d98ba0a4556ce8c11bc43853b260fcb8de7166b857.jpg)  
+Figure 6: We demonstrate the benefit of producing structurally-controlled synthetic data (denoted as CN-  $\overline{\mathbb{T}}$  above), in comparison to text-conditional generation (denoted as SD-  $\overline{\mathbb{T}}$ ), using two different scenarios: (1) Text-conditional generation cannot succeed in generating signs unseen during training, while our method can correctly adhere to the structural conditioning provided as input, even for rare or unseen signs. (2) A text-conditional model often generates the sign variation most prevalent during training (e.g., the variants on the right sides above), while our method can generate different variants, yielding a more diverse synthetic set for training downstream models.
+
+![](images/9f59ec8142ebf4d0af40661e24512419771f02ab6cb1551a4bcd4e0bb26107ec.jpg)  
+Figure 7: Limitations of our method, illustrating examples with significant deformations from the prototype skeleton (left) and structurally different sign variants (middle) and corrupt sign image (right). We visualize correctly-aligned strokes in green, and misaligned strokes in red.
+
+We foresee a range of applications and possible extensions of our work. Our method represents a step towards paleographic analysis on scale, tracking allographs (sign variants) for each sign as produced by scribes between time periods, cities, archives, and personal handwriting styles. It could also be used to automate the production of hand copies, currently produced manually by experts to illustrate tablet contents in their original context. While we have focused on the case of single sign scans, we foresee future work extending these results to lines of text, as our method could be incorporated into a pipeline including text detection and localization of individual signs applied either to images or directly to 3D scans of inscriptions. Finally, our per-stroke optimization process is designed for cuneiform signs composed of wedge shapes parametrized by four keypoints, future work might extend this to other ancient scripts with different geometric configurations – for example, by using more flexible primitives such as Bezier splines to model curved lines found in ancient Chinese oracle bone inscriptions. We hope that our work will spur future research on using the internal structure of glyphs in complex scripts such as cuneiform to advance downstream tasks.
+
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+
+# APPENDIX
+
+# A EXPERIMENTAL DETAILS
+
+Below we provide additional experimental details. Our code is also provided (zipped in the supplementary material). For all experiments described below, a single A5000 GPU was used. Running the method on a single image takes about 1 minute.
+
+# A.1 IMAGE AND FONT INFORMATION
+
+For all of our tests, we use RGB images with resolution  $512 \times 512$ , resizing images as needed.
+
+Font images are rendered from the Santakku and SantakkuM fonts, designed by Sylvie Vanséveren and available on the Hethitologie Portal Mainz. For a uniform appearance, the white margins of the image are cropped, than 10 pixels of white margins are added to each side, and finally the image is resized to  $512 \times 512$  resolution.
+
+# A.2 MODEL DETAILS
+
+We use the CompVis/stable-diffusion-v1-4 checkpoint as our base Stable Diffusion model. We fine-tune this on eBL classification train dataset (Cobanoglu et al., 2024), for 50K iterations with batch size of 4, learning rate of  $10^{-5}$  and Adam optimizer. For textual prompts, we use a unique code for each sign type indicated in the eBL dataset. This fine-tuned Stable Diffusion model, SD-  $\mathbb{N}$  , is used for our DIFT feature calculations as well as in the synthetic data generation for our OCR application. For tests using ControlNet, we fine-tune the base illyasviel/sd-controlnet-openpose checkpoint, on 932 samples with their paired alignment created by ProtoSnap. We trained it for 20K iteration, with batch size of 4, learning rate of  $10^{-5}$  and Adam optimizer. This fine-tuned ControlNet, CN-  $\mathbb{N}$  , uses another fine-tuned Stable Diffusion model, trained with the same parameters as SD-  $\mathbb{N}$  , but without a prior from a textual prompt, using the same prompt for all signs: "cuneiform single ancient icon".
+
+For DIFT feature calculations, we average an ensemble of results on four random noises, sampled at timestamp  $t = 261$ , following Tang et al. (2023). We concatenate features from the second and third upscaling U-Net layers, bilinearly interpolated to the same spatial resolution, yielding a set of  $64 \times 64$  feature vectors of dimension 1920.
+
+# A.3 GLOBAL ALIGNMENT DETAILS
+
+To fit our global alignment, we apply RANSAC with 2000 iterations. At each iteration, 5 correspondences are used to fit a least-squares affine transformation, with a distance threshold of 50 pixels used to identify outliers. The transformation with the greatest number of inliers is returned.
+
+As a high-quality set of correspondences should explain the relevant regions in both of the images, we incorporate a prior on inlier points being spread across the images, following Hassner et al. (2014). In particular, we perform the above procedure 8 times, and assign each result a score using the convex hulls of the inlier points in the prototype and scanned cuneiform sign images. For the prototype image, we calculate the proportion  $p_{proto}$  of the prototype font foreground contained within the convex hull of inlier points. For the scanned cuneiform sign image, we calculate the proportion  $p_{scan}$  of the total area of the image covered by the convex hull of inlier points. Finally, we select the result with maximum score  $p_{proto} \cdot p_{scan}$ , to encourage the global transformation to be based on matches between regions covering most of the scan and prototype font.
+
+# A.4 LOCAL REFINEMENT DETAILS
+
+Our total loss uses coefficients  $\lambda_{sim} = 1.0$ ,  $\lambda_{sal} = 3 \times 10^{-4}$ , and  $\lambda_{reg} = 10^{-4}$ .
+
+To calculate the saliency map used for the saliency loss, we compute differences in mean similarities as described in the main paper, apply CLAHE histogram equalization with clip limit 10.0 and tile grid size (2, 2), set values below the mean to zero, and scale values to the range [0, 1]. This yields a scalar field of resolution  $64 \times 64$ .
+
+For both semantic similarity and saliency losses, at each iteration values are sampled at both each of the keypoints in the skeleton, and at 8 randomly-sampled points along each line segment in the skeleton connecting keypoints. Loss values are averaged over all of these points. The sampled points are sampled uniformly from the lines in the prototype images, and then transformed using the current global and local transformations to obtain corresponding points in the target scanned cuneiform sign image. Loss values are computed with differentiable grid sampling, using bilinear interpolation over scalar fields (the similarity values in respective slices of  $S$ , and saliency map values) each of which is passed through a softmax with fixed temperature parameter 100.
+
+To perform optimization, we apply gradient descent for 100 iterations with learning rate 0.01 and Adam optimizer, updating the the parameters of the local transformations of all strokes.
+
+# A.5 DATASET DETAILS
+
+Both the training and the test datasets are taken from the eBL classification dataset Cobanoglu et al. (2024) which consists of scanned imaged of cuneiform signs. The dataset was originally split to train and test by source tablet. The train set included 34,868 sign images, representing 362 sign types and was used to train SD-¶. The test set was used to select the 272 images for our test set (as described in 5.1), focusing on signs for which we have an available prototype, and the samples in the set match the prototype structure variant. The final test set consists of 272 samples from 25 different signs, not seen in training, varying from 2 strokes per sign to 8.
+
+The full dataset comprises around  $40\%$  from the Neo-Babylonian period (1000-600 BC), around  $20\%$  from the Neo-Assyrian (1000-609 BCE), and less than  $10\%$  from the following periods: Ur III (2100-2002 BCE), Old Babylonian (2002-1595 BCE), Old Assyrian (1950-1850 BCE), Middle Babylonian (1500-1000 BCE), Late Babylonian (600 BC-100 AD), Persian (539-331 BCE), Hellenistic (331-141 BCE), Parthian (141 BCE-100 CE). The dataset represents the Akkadian and Sumerian languages used at those eras.
+
+# A.6 USER SURVEY DETAILS
+
+For the users survey, we have asked experiences assyriologists to review 35 cuneiform signs. For each signs, we presented them with 3 options - a plain sign, a sign with an overlaid prototype without alignment, and a sign with overlaid prototyped aligned using ProtoSnap. For each sign we asked the user to select the image in which they can most easily identify the sign. 12 assyriologists had answered the survey. In all but three signs, the users preferred an overlaid option to the plain one, and out of those, the users preferred the aligned option  $65\%$  of the time (in 21 signs).
+
+![](images/35684fc6153fac84d8ac9d0e904bc84c61e126d5373fefe7a59b9b0eec1c6d35.jpg)  
+Figure 8: Examples for data generated using our fine-tuned ControlNet model CN-  $\mathbb{N}$  , where the control is an image of a prototype sign (with two different added transformations).
+
+# A.7 OCR EXPERIMENT DETAILS
+
+For the OCR experiment we have generated 50 samples per each sign in the test set (180 signs in total), using our fine-tuned SD-  $\mathbb{F}$  described above. In addition, we used CN-  $\mathbb{F}$  to generate 50 samples for each sign we have available prototype (124 signs in total). To better mimic human handwriting, we augmented the skeletons by applying small random transformation on the entire skeleton and on each stroke individually, creating a diverse set of controls for each sign. Figure 8 shows examples of such generated data. Both generations were done using 50 inference steps and classifier-free guidance scale 7.5.
+
+For the experiment we used 6205 samples from the CSDD data (consisting of data from all URLs which were not broken in the dataset), 3299 samples in the train set and 2906 in the test. Training ResNet18 on this dataset alone was done for 10 epochs, using batch size of 64, learning rate of  $10^{-3}$  and Adam optimizer. Training with generated data (both from SD-  $\overline{\mathbb{N}}$  and CN-  $\overline{\mathbb{N}}$ ) was done for 8 epochs, and then fine-tuned 10 more on real data only, using batch size of 64, learning rate of  $5^{-4}$  and Adam.
+
+Note that our reproduced baseline on CSDD achieves slightly higher accuracy than reported by Dencker et al. (2020), but the test dataset includes URLs which are no longer operational which may contribute to this slight discrepancy in results.
+
+# B ADDITIONAL RESULTS
+
+Figure 9 shows additional examples, illustrating ProtoSnap applied to various samples of the same sign. Figure 10 shows ProtoSnap results on our manually annotated test set, compared to the baseline of applying DIFT directly (assigning each keypoint to the region of maximal feature similarity).
+
+Figure 11 shows ProtoSnap results on a new, previously unseen dataset, JOCCH (Rusakov et al., 2020), which contains signs from the Hittite language, as opposed to the Akkadian and Sumerian from which the training and test set are composed. Those results show that our method is robust and can be generalizable to other usages of the cuneiform writing system.
+
+![](images/ce218fc2bf2798b3ff47eec6127d19606890d2011789993dad97dee27adb1bb2.jpg)  
+Figure 9: Examples of ProtoSnap applied on photographed cuneiform signs of varying structure, illumination conditions and degrees of intactness.
+
+![](images/81ff457ea4c145aa4b2370fc58eeceda6c5f9c0b58d7a4588c1a62d390542ec4.jpg)  
+Figure 10: Results of ProtoSnap on our manually annotated test set, with DIFT and PoseAnything (Hirschhorn and Avidan, 2023) shown for comparison. We can see that our method produces alignments which are much closer to expert annotations and is generally less sensitive to outliers.
+
+![](images/9538789804de763ace80bc59a88c8c83690ffd99e2fdc360639a4f8b9748baa2.jpg)  
+Figure 11: ProtoSnap applied on images from a different dataset and language (Hittite), showing that the method is robust and generalizable to various usages of cuneiform writing system. The 3 images on the left show signs from types (names) unseen in the training data, further emphasizing the generalizability of the method.
+
+<table><tr><td rowspan="2">Method</td><td colspan="3">threshold = 20</td><td colspan="3">threshold = 40</td></tr><tr><td>Precision</td><td>Recall</td><td>F1</td><td>Precision</td><td>Recall</td><td>F1</td></tr><tr><td>SIFT (Lowe, 1999) + RANSAC</td><td>2.59%</td><td>2.53%</td><td>2.56%</td><td>5.49%</td><td>4.61%</td><td>4.91%</td></tr><tr><td>DINOv2 (Oquab et al., 2024)</td><td>12.42%</td><td>12.25%</td><td>12.33%</td><td>32.09%</td><td>30.05%</td><td>31.04%</td></tr><tr><td>DINOv2 + RANSAC</td><td>16.22%</td><td>16.01%</td><td>16.12%</td><td>38.47%</td><td>37.42%</td><td>37.93%</td></tr><tr><td>DIFT (Tang et al., 2023)</td><td>16.18%</td><td>16.10%</td><td>16.14%</td><td>34.32%</td><td>33.27%</td><td>33.79%</td></tr><tr><td>DIFT + RANSAC</td><td>13.15%</td><td>13.11%</td><td>13.13%</td><td>30.43%</td><td>29.88%</td><td>30.15%</td></tr><tr><td>Ours (w/o refinement)</td><td>21.38%</td><td>21.23%</td><td>21.31%</td><td>50.55%</td><td>49.73%</td><td>50.13%</td></tr><tr><td>Ours (full)</td><td>27.17%</td><td>27.10%</td><td>27.14%</td><td>52.76%</td><td>52.10%</td><td>52.43%</td></tr></table>
+
+Table 3: Precision and recall metrics for the alignment evaluation, on top of F1 metric presented in the main paper, at two distance thresholds.
+
+# B.1 ADDITIONAL METRICS
+
+Table 3 shows precision and recall metrics for the alignment evaluation, on top of F1 metric presented in the main paper. Table 4 show alignment evaluation breakdown per signs, and also provides the number of samples in the test set per sign.
+
+# B.2 ABLATION STUDY
+
+We demonstrate the effect of key parts of our system by ablating them and evaluating performance on our test set. In particular, we ablate the following:
+
+- Use of our fine-tuned SD-  $\overline{\mathbb{Y}}^{\dagger}$  (rather than base Stable Diffusion)  
+- Use of best-buddies correspondences for computing the global transformation (rather than using all correspondences between prototype image regions and the best-matching regions in the target image according to DIFT)  
+Each of the three loss terms in our full loss function
+
+As seen in Table 5, most of these ablations have a significant negative effect on quantitative performance. Removing  $\mathcal{L}_{sim}$  slightly improves metrics, but we find this is reflects a qualitative trade-off.
+
+<table><tr><td>Sign Name</td><td># Samples</td><td># Strokes</td><td>F1@20</td><td>F1@30</td><td>F1@40</td></tr><tr><td>ME</td><td>20</td><td>2</td><td>23.13%</td><td>33.75%</td><td>46.88%</td></tr><tr><td>A</td><td>19</td><td>3</td><td>26.75%</td><td>40.79%</td><td>52.41%</td></tr><tr><td>IGI</td><td>18</td><td>3</td><td>12.50%</td><td>25.23%</td><td>37.50%</td></tr><tr><td>AN</td><td>19</td><td>3</td><td>30.04%</td><td>46.05%</td><td>61.39%</td></tr><tr><td>UD</td><td>18</td><td>3</td><td>16.20%</td><td>29.17%</td><td>44.21%</td></tr><tr><td>GISH</td><td>19</td><td>3</td><td>28.29%</td><td>42.11%</td><td>54.17%</td></tr><tr><td>MA</td><td>16</td><td>4</td><td>26.95%</td><td>44.73%</td><td>57.81%</td></tr><tr><td>EN</td><td>5</td><td>5</td><td>20.00%</td><td>36.00%</td><td>47.49%</td></tr><tr><td>IR</td><td>17</td><td>5</td><td>22.94%</td><td>36.17%</td><td>42.04%</td></tr><tr><td>IB</td><td>15</td><td>5</td><td>35.17%</td><td>55.17%</td><td>63.33%</td></tr><tr><td>HA</td><td>3</td><td>5</td><td>23.33%</td><td>35.81%</td><td>48.33%</td></tr><tr><td>UR</td><td>16</td><td>5</td><td>32.66%</td><td>49.06%</td><td>57.50%</td></tr><tr><td>RI</td><td>13</td><td>5</td><td>29.81%</td><td>44.03%</td><td>57.11%</td></tr><tr><td>RU</td><td>15</td><td>5</td><td>24.67%</td><td>37.50%</td><td>45.83%</td></tr><tr><td>DIM2</td><td>2</td><td>5</td><td>25.00%</td><td>35.00%</td><td>42.50%</td></tr><tr><td>DI</td><td>6</td><td>5</td><td>24.17%</td><td>43.71%</td><td>51.25%</td></tr><tr><td>U2</td><td>9</td><td>5</td><td>21.67%</td><td>41.39%</td><td>53.88%</td></tr><tr><td>DIB</td><td>3</td><td>6</td><td>23.61%</td><td>34.01%</td><td>46.52%</td></tr><tr><td>SA</td><td>3</td><td>6</td><td>36.79%</td><td>45.13%</td><td>52.00%</td></tr><tr><td>GI</td><td>1</td><td>7</td><td>28.57%</td><td>46.43%</td><td>53.57%</td></tr><tr><td>DA</td><td>1</td><td>7</td><td>28.57%</td><td>42.86%</td><td>50.00%</td></tr><tr><td>KA</td><td>12</td><td>7</td><td>26.64%</td><td>44.94%</td><td>55.79%</td></tr><tr><td>ZI</td><td>10</td><td>7</td><td>31.43%</td><td>42.50%</td><td>50.35%</td></tr><tr><td>A2</td><td>4</td><td>8</td><td>29.69%</td><td>44.92%</td><td>55.47%</td></tr><tr><td>ZE2</td><td>8</td><td>8</td><td>39.45%</td><td>54.49%</td><td>60.35%</td></tr></table>
+
+We foresee an expansion of our test set or development of additional test metrics for this task to better capture this performance.
+
+Table 4: Performance breakdown by sign type. We report alignment performance of our model over the different annotated signs. We also provide the number of strokes in each sign (#Strokes) and number of samples of the sign in the test set (#Samples).  
+
+<table><tr><td></td><td>F1@20</td><td>F1@30</td><td>F1@40</td></tr><tr><td>ProtoSnap (ours)</td><td>27.14%</td><td>42.09%</td><td>52.43%</td></tr><tr><td>-SD-¶</td><td>19.01%</td><td>31.89%</td><td>41.25%</td></tr><tr><td>-best buddies</td><td>26.76%</td><td>41.76%</td><td>52.76%</td></tr><tr><td>-Lsim</td><td>27.37%</td><td>42.61%</td><td>53.19%</td></tr><tr><td>-Lsal</td><td>21.13%</td><td>37.48%</td><td>50.08%</td></tr><tr><td>-Lreg</td><td>26.72%</td><td>39.66%</td><td>47.41%</td></tr></table>
+
+Table 5: Ablation study results, demonstrating differences in performance when removing key parts of our system. Most ablations negatively impact quantitative results, further explained in Section B.2.
+
+# C ANNOTATION DETAILS
+
+Our expert annotations were performed by Assyriologists who participated in this research.
+
+Below, we provide further details on our annotations collected via crowdsourcing, used to annotate keypoints in prototype font images and in scanned cuneiform signs. We then connected the keypoints manually ourselves, creating the prototype skeleton.
+
+# C.1 IRB APPROVAL, PARTICIPANT SOURCING, AND COMPENSATION
+
+Our annotation tasks, approved by our institution's IRB, were conducted on the Amazon Mechanical Turk (MTurk) crowdsourcing platform. We published our tasks for MTurk workers with at least 1000 completed HITs (MTurk tasks) and a HIT approval rate of at least  $95\%$ . Workers were compensated $0.25 for each font annotation, corresponding to the duration of this task.
+
+# C.2 ANNOTATION TASK INSTRUCTIONS
+
+In this task, you will indicate keypoints on ancient character ("cuneiform") to indicate the location of each stroke. Please indicate each stroke with four keypoints as shown here:
+
+![](images/7f5dbf2067ab5828cf0ed6c664705861935fdb586ae43c30f077d3d18c07f1ae.jpg)
+
+If there are multiple strokes, please indicate each stroke in a separate color using four keypoints per stroke, as in these examples:
+
+![](images/fc8eb5524621228007ee46dac7b753e6ed949754aedc7d3091d247679f75f38e.jpg)
+
+Make sure the four keypoints are in the locations as shown above - three indicating the corners of the stroke's triangular head, and one indicating the end of its tail.
+
+Use the point tool to place points on the requested target(s) of interest: Four each stroke in the character, place four keypoints of the same color, using your mouse to click on each keypoint. Use the four keypoints described in the instructions: for each stroke, three points indicating the corners of the stroke's triangular head, and one indicating the end of its tail. Make sure to indicate every stroke seen in the glyph.

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+    {
+        "type": "text",
+        "text": "PROTOTYPE ANTITHESIS FOR BIOLOGICAL FEW-SHOT CLASS-INCREMENTAL LEARNING",
+        "text_level": 1,
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "Binghao Liu $^{1,2*}$ , Han Yang $^{1,2}$ , Fang Wan $^{3*}$ , Fei Gu $^{1,2}$",
+        "bbox": [
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+    {
+        "type": "list",
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+        "list_items": [
+            "$^{1}$ DAMO Academy, Alibaba Group, Hangzhou, 310023, China",
+            "$^{2}$ Hupan Lab, Hangzhou, 310023, China",
+            "<sup>3</sup>University of Chinese Academy of Sciences, Beijing, 101408, China"
+        ],
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+    {
+        "type": "text",
+        "text": "ABSTRACT",
+        "text_level": 1,
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+            265,
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+            280
+        ],
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+    },
+    {
+        "type": "text",
+        "text": "Deep learning has become essential in the biological species recognition task. However, a significant challenge is the ability to continuously learn new or mutated species with limited annotated samples. Since species within the same family typically share similar traits, distinguishing between new and existing (old) species during incremental learning often faces the issue of species confusion. This can result in \"catastrophic forgetting\" of old species and poor learning of new ones. To address this issue, we propose a Prototype Antithesis (PA) method, which leverages the hierarchical structures in biological taxa to reduce confusion between new and old species. PA operates in two steps: Residual Prototype Learning (RPL) and Residual Prototype Mixing (RPM). RPL enables the model to learn unique prototypes for each species alongside residual prototypes representing shared traits within families. RPM generates synthetic samples by blending features of new species with residual prototypes of old species, encouraging the model to focus on species-unique traits and minimize species confusion. By integrating RPL and RPM, the proposed PA method mitigates \"catastrophic forgetting\" while improving generalization to new species. Extensive experiments on CUB200, PlantVillage, and Tree-of-Life datasets demonstrate that PA significantly reduces inter-species confusion and achieves state-of-the-art performance, highlighting its potential for deep learning in biological data analysis.",
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+    {
+        "type": "text",
+        "text": "1 INTRODUCTION",
+        "text_level": 1,
+        "bbox": [
+            173,
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+            604
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "With advances in data science and computing, computer techniques are crucial in fields like biology, offering solutions for species recognition and evolutionary analysis. However, limited data on newly discovered or mutated species poses a challenge, as data annotation requires expertise and is costly. Additionally, similar traits among species within the same family complicate recognition. These issues—data scarcity and species similarity—can cause confusion between new and existing species during continual learning, leading to poor performance on new species and forgetting of old ones.",
+        "bbox": [
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+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "To overcome this, we introduce the few-shot class-incremental learning paradigm to biological research, allowing models to be trained on existing data and updated with only a few samples from new species. This approach enables continuous learning from new species while preserving knowledge of previously learned ones, addressing data limitations, and improving scalable species recognition. It underscores the vital role of advanced deep learning in advancing biological discovery.",
+        "bbox": [
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+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "Few-shot class-incremental learning (FSCIL) typically faces two key challenges: \"catastrophic forgetting\" and \"over-fitting\". These issues arise when models must learn new classes with few samples and no access to previous data. Early methods address forgetting through knowledge distillation, where the old model's outputs guide the new model. However, when training samples for new classes are limited, the model becomes more prone to over-fitting. Recent approaches Zhang et al. (2021); Zhou et al. (2022) mitigate forgetting by freezing the feature extractor and selectively updating classifier weights, preserving old class performance. However, in biological contexts, distinguishing between new and previously learned species presents an additional challenge due to their",
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+        "page_idx": 0
+    },
+    {
+        "type": "header",
+        "text": "Published as a conference paper at ICLR 2025",
+        "bbox": [
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+            47
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "page_footnote",
+        "text": "*Correspondence to: Fang Wan (wanfang@ucas.ac.cn), Binghao Liu (liubinghao.lbh@alibaba-inc.com).",
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+    {
+        "type": "page_number",
+        "text": "1",
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+        "page_idx": 0
+    },
+    {
+        "type": "image",
+        "img_path": "images/739be946002fbeb2d27c1059deed1c7e09d322010ad1500bf2ca8882ac7d4a2b.jpg",
+        "image_caption": [
+            "Figure 1: Comparison of prior works and the Prototype Antithesis (PA) method. In incremental learning, when similar new species resemble existing ones, their prototypes tend to be close, leading to confusion during classification (upper). In contrast, our proposed PA addresses this issue by decomposing the prototypes into two parts: the unique prototype specific to a species and the residual prototype shared within its family. The residual prototype is employed in the residual prototype mixing for new species learning, which maximizes the distance between similar species prototypes, thereby reducing classification confusion. (Best viewed in color)"
+        ],
+        "image_footnote": [],
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+    {
+        "type": "text",
+        "text": "close evolutionary relationships. The similar traits of species within the same family make it difficult for classifiers to differentiate between them. This complicates learning new classes and increases inter-species confusion, reducing performance on previously learned species (Fig. 1, prior works).",
+        "bbox": [
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+        "page_idx": 1
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+    {
+        "type": "text",
+        "text": "In this study, we propose a Prototype Antithesis (PA) method to address these challenges. PA encourages the model to learn both the most discriminative prototypes unique to individual species and the residual prototypes shared by species within the same family. During incremental learning, PA generates new feature samples for model learning by combining residual prototypes from previously learned species with features from new species. This process not only helps the classifier better distinguish between closely related new and old species, but also mitigates over-fitting caused by the limited data available for new species, Fig. 1 (PA).",
+        "bbox": [
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+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "The PA method operates in two key steps: Residual Prototype Learning (RPL) and Residual Prototype Mixing (RPM). RPL is conducted during base training and can be further divided into two phases. In phase one, images are assigned with \"species-level\" labels, and those images are used for classification training by calculating the cosine distance between the features and a set of randomly initialized prototypes. After training, those prototypes converge to represent the most discriminative semantics of the corresponding species. In phase two, the species-unique and family-shared semantics are decoupled by a bi-directional optimization process. Specifically, we first compute the residuals between the features and their corresponding prototypes. Then these residual features are assigned with \"family-level\" labels for training, while the original features are classified by species labels. This bi-directional classification procedure ensures the convergence of species-unique and family-shared traits into their respective prototypes, referred to as species prototypes and family (residual) prototypes. RPM combines the residual prototypes of previously learned species with the new species features using a feature-mixing strategy to generate synthetic feature samples. These synthetic feature samples are labeled as their corresponding new species and are used to facilitate model learning. To classify these synthetic features accurately, the model must capture distinctive traits that differentiate them from residual prototypes and previously learned species, resulting in a well-separated decision boundary. Additionally, the synthetic features act as augmented data, increasing the number of training samples and helping to reduce over-fitting on the new species. Comprehensive experiments conducted on Tree-of-Life Stevens et al. (2024), as well as on the PlantVillage Hughes & Salathé (2015) and CUB200 P. et al. (2010) datasets, demonstrate that our approach achieves state-of-the-art performance.",
+        "bbox": [
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+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "The contributions of this study are listed below:",
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+        "page_idx": 1
+    },
+    {
+        "type": "header",
+        "text": "Published as a conference paper at ICLR 2025",
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+    },
+    {
+        "type": "page_number",
+        "text": "2",
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+    {
+        "type": "list",
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+        "list_items": [
+            "- We propose a novel Prototype Antithesis (PA) method to address the challenges of species confusion and over-fitting in few-shot class-incremental learning (FSCIL), particularly within the biological domain.",
+            "- We introduce Residual Prototype Learning (RPL) and Residual Prototype Mixing (RPM) to enable the model to capture both species-unique and family-shared traits, focusing on distinguishing between similar new and old species while augmenting training samples.",
+            "- Our proposed PA method achieves state-of-the-art performance across both computer vision and biological benchmarks, demonstrating its superiority in addressing the unique challenges of FSCIL."
+        ],
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+    },
+    {
+        "type": "text",
+        "text": "2 RELATED WORKS",
+        "text_level": 1,
+        "bbox": [
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+            260,
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+            276
+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "Biological Recognition. Biological data, characterized by its hierarchical category structure, is frequently used in fine-grained classification tasks, such as those involving the CUB200 dataset P. et al. (2010). These tasks are particularly difficult because subspecies within the same family often display highly similar appearances, making it challenging for models to differentiate them, especially when new species are constantly being discovered. Moreover, these newly identified species often come with limited samples, hindering the model's ability to learn effectively. While recent work like Bio-CLIP Stevens et al. (2024) has introduced foundation models for biological research, it struggles with continual learning and tend to misclassify closely related species.",
+        "bbox": [
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+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "Few-shot Learning. The goal is to learn new classes with limited training samples. Existing methods have made significant progress through metric learning, meta-learning, and data augmentation. Metric learning Vinyals et al. (2016); Snell et al. (2017); Sung et al. (2018); Zhang et al. (2020); Liu et al. (2021); Yang et al. (2021a); Liu et al. (2021); Li et al. (2021) maps new class features into a representation space learned from base classes with sufficient data. Meta-learning Finn et al. (2017); Elsken et al. (2020); Sun et al. (2019) develops optimization strategies to enable models to generalize to new classes with limited data. Data augmentation Zhang et al. (2019); Li et al. (2020); Kim et al. (2020); Yang et al. (2021b) generates synthetic samples to reduce overfitting and support new class learning. However, these methods mainly focus on learning new classes and often overlook the problem of forgetting old classes during incremental learning. GFSL Schonfeld et al. (2019) addresses this by aligning old and new class distributions to mitigate forgetting, but it struggles with large distribution gaps and risks class confusion when old and new classes are highly similar.",
+        "bbox": [
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+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "Incremental Learning. Incremental learning can be divided into task-incremental and class-incremental learning, with the key difference being that task-incremental learning requires task IDs, while class-incremental does not Masana et al. (2020). Existing methods fall into three categories: memory replay, model regularization, and architecture configuration methods. Memory replay Rebuffi et al. (2017); Chaudhry et al. (2018) methods store old class samples to prevent forgetting. Model regularization Li & Hoiem (2018); Dhar et al. (2019) methods add terms to the loss function to reduce forgetting. Architecture configuration Serrà et al. (2018); Mallya & Lazebnik (2018) methods use techniques like model pruning to adjust the feature space. However, these approaches still face over-fitting issues, particularly with limited training samples for new classes.",
+        "bbox": [
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+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "Few-shot Class-Incremental Learning. Under few-shot class-incremental learning (FSCIL) settings, the model is first trained on base classes with sufficient data and then required to generalize to new classes with only a few samples. The primary challenges of FSCIL include not only \"catastrophic forgetting\" but also the issue of \"over-fitting\" due to limited training data. Feature topology preservation methods Tao et al. (2020); Zhang et al. (2021) maintain the feature topology of old classes using techniques like neural gas or decoupled training strategies to prevent forgetting. Network regularization methods Akyurek et al. (2021); Kang et al. (2022) align the weight vectors of new classes with those of old classes or selectively update model weights to learn new classes without degrading the performance of old classes. Data generation methods Cheraghian et al. (2021); Liu et al. (2023) generate synthetic feature samples to alleviate over-fitting during incremental learning. However, the confusion between new and old classes remains an unsolved problem in FSCIL. Since the model can only access data from individual incremental sessions, it struggles to learn well-separated decision boundaries between old and new classes, especially when these classes are similar. This confusion can lead to both forgetting of old classes and poor learning of new ones, which is the problem this study aims to address.",
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+        "page_idx": 2
+    },
+    {
+        "type": "header",
+        "text": "Published as a conference paper at ICLR 2025",
+        "bbox": [
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+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "page_number",
+        "text": "3",
+        "bbox": [
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+        "page_idx": 2
+    },
+    {
+        "type": "image",
+        "img_path": "images/02492e9b54882bd54929b68b58b6dc51f8a19e30c1d65ec349b2ce1e06080908.jpg",
+        "image_caption": [
+            "Figure 2: The proposed Prototype Antithesis (PA) method consists of two components: RPL and RPM. RPL has two phases: in phase one, images are encoded by the feature extractor  $\\theta_{b}$  and classified using \"species-level\" labels; in phase two, residuals between the base (old) species features and species prototypes are calculated and classified using \"family-level\" labels, whereas the original features continue to be classified by \"species-level\" labels. In RPM, these residual prototypes are mixed with new species features to generate pseudo-samples, optimizing species prototypes and refining the decision boundary. (Best viewed in color)"
+        ],
+        "image_footnote": [],
+        "bbox": [
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+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "image",
+        "img_path": "images/1630ed7bd5b8e87a22c4ca82dc4c47878c8727f17d7b66e1fb9dfd9fda60d1e9.jpg",
+        "image_caption": [],
+        "image_footnote": [],
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+    },
+    {
+        "type": "text",
+        "text": "3 PRELIMINARY",
+        "text_level": 1,
+        "bbox": [
+            171,
+            401,
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+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "Few-shot class-incremental learning can be broadly divided into two stages: the base training stage and the incremental learning stage. During the base training stage, the model is trained using base classes  $\\mathcal{C}^{(0)}$ , which have sufficient training data. In the incremental learning stage, the new classes are divided into  $T$  groups, denoted as  $\\mathcal{C}^{(i)}$ , where  $i = 1,2,\\dots,T$ , corresponding to the  $T$  incremental sessions. In the  $i^{\\mathrm{th}}$  incremental session, only the dataset  $\\mathcal{D}^{(i)}$  for classes  $\\mathcal{C}^{(i)}$  is available for model learning. However, during evaluation, the model is required to classify both old and new classes from the combined set  $\\{\\mathcal{C}^{(0)},\\ldots ,\\mathcal{C}^{(i)}\\}$ . Importantly, for any  $i_1\\neq i_2$ , we have  $\\mathcal{C}^{(i_1)}\\cap \\mathcal{C}^{(i_2)} = \\emptyset$ , meaning that the classes in different incremental sessions are mutually exclusive. This setup imposes the challenge that the model must learn new classes without forgetting previously learned ones, despite not having access to the data from older classes during the incremental sessions.",
+        "bbox": [
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+    },
+    {
+        "type": "text",
+        "text": "4 METHODOLOGY",
+        "text_level": 1,
+        "bbox": [
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+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "In this section, we introduce the proposed PA method. First, we outline the flowchart of the base training and incremental learning procedures. Next, we provide detailed explanations of the residual prototype learning and residual prototype mixing techniques. Finally, we present a theoretical analysis of the proposed prototype antithesis method.",
+        "bbox": [
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+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "4.1 OVERVIEW",
+        "text_level": 1,
+        "bbox": [
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+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "The structure of the proposed PA method is illustrated in Fig. 2. The model consists of a feature extractor  $\\theta_{b}$ , species prototypes  $\\theta_{c} = \\{\\pmb{\\mu}_{c_{0}},\\pmb{\\mu}_{c_{1}},\\pmb{\\mu}_{c_{2}},\\dots \\}$ , and residual prototypes  $\\theta_r = \\{\\pmb{\\mu}_{r_0},\\pmb{\\mu}_{r_1},\\pmb{\\mu}_{r_2},\\dots \\}$ . We employ ResNet-18 as the feature extractor, which encodes images into feature vectors. The prototypes,  $\\theta_{c}$  and  $\\theta_{r}$ , act as classifiers, using cosine distance between the features and prototypes for classification. During base training, the feature extractor and prototypes are optimized using the residual prototype learning strategy. In the incremental learning phase, the species prototypes are further refined through residual prototype mixing, ensuring a well-separated decision boundary between new and existing species. During the inference stage, the features of both old and new species are classified solely by the prototypes, without incorporating the residual.",
+        "bbox": [
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+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "4.2 RESIDUAL PROTOTYPE LEARNING",
+        "text_level": 1,
+        "bbox": [
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+    },
+    {
+        "type": "text",
+        "text": "In standard classification tasks, most existing methods focus on learning the most discriminative representations, allowing classifiers to establish well-separated decision boundaries when all classes",
+        "bbox": [
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+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "header",
+        "text": "Published as a conference paper at ICLR 2025",
+        "bbox": [
+            171,
+            32,
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+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "page_number",
+        "text": "4",
+        "bbox": [
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+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "belong to a closed set. However, in few-shot class-incremental learning (FSCIL), where new classes are introduced incrementally, previously learned decision boundaries may struggle to distinguish between new and old classes. For example, when new and old species belong to the same family (e.g., apple leaf and peach leaf), the classifier may confuse them due to their shared \"leaf\" characteristics.",
+        "bbox": [
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+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "To address this issue, the model needs to differentiate between \"shared\" and \"unique\" traits of species, which is the goal of the proposed Residual Prototype Learning (RPL), Fig. 2 (left). RPL operates in two phases, in phase one, given training samples  $\\mathbf{X}^{(0)}$  from the base species, the feature extractor  $\\theta_{b}$  and species prototypes  $\\theta_{c}$  are jointly trained to classify the base species. The training objective for phase one is formulated as follows:",
+        "bbox": [
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+        "page_idx": 4
+    },
+    {
+        "type": "equation",
+        "text": "\n$$\n\\arg \\min  \\mathcal {L} (\\mathbf {Y} ^ {(0)}, \\hat {\\mathbf {Y}} ^ {(0)}; \\boldsymbol {\\theta} _ {b}, \\boldsymbol {\\theta} _ {c})\n$$\n",
+        "text_format": "latex",
+        "bbox": [
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+        "page_idx": 4
+    },
+    {
+        "type": "equation",
+        "text": "\n$$\n\\begin{array}{l} \\boldsymbol {\\theta} _ {b}, \\boldsymbol {\\theta} _ {c} \\\\ \\text {s . t .} \\quad \\hat {\\mathbf {Y}} ^ {(0)} = \\cos (\\mathbf {F} ^ {(0)}; \\boldsymbol {\\theta} _ {c}), \\quad \\mathbf {F} ^ {(0)} = f (\\mathbf {X} ^ {(0)}; \\boldsymbol {\\theta} _ {b}), \\end{array} \\tag {1}\n$$\n",
+        "text_format": "latex",
+        "bbox": [
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+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "where  $\\mathcal{L}$  is the classification loss,  $f$  represents the feature extraction process,  $\\cos$  calculates the cosine similarity between the feature and each prototype individually, resulting in a set of scores, and  $\\mathbf{Y}^{(0)}$  are the \"species-level\" labels. After training, the learned species prototypes capture the most discriminative features of the base species. In phase two, for any base species  $i$ , given the features  $\\mathbf{F}_i^{(0)}$  and the species prototype  $\\pmb{\\mu}_{c_i}$  (where  $\\pmb{\\mu}_{c_i} \\in \\theta_c$ ), the residual features are computed as  $\\mathbf{R}_i = \\mathbf{F}_i^{(0)} - \\pmb{\\mu}_{c_i}$ . Here,  $\\mathbf{R}_i$  contains both secondary discriminative features and background features, with the secondary discriminative features likely representing traits shared across the family. To decouple \"family-level\" traits from background noise and the species prototypes (since the most discriminative features might also include some \"family-level\" characteristics), we assign \"family-level\" labels to the residual features and classify them by calculating the cosine distance between the residual features and the prototypes. Meanwhile, the original features  $\\mathbf{F}_i^{(0)}$  are classified based on their \"species-level\" labels. The optimization objective for base species  $i$  in phase two is as:",
+        "bbox": [
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+        "page_idx": 4
+    },
+    {
+        "type": "equation",
+        "text": "\n$$\n\\begin{array}{l} \\underset {\\pmb {\\theta} _ {r}, \\pmb {\\theta} _ {c}} {\\arg \\min} \\left[ \\mathcal {L} (\\mathbf {Y} _ {i} ^ {r}, \\hat {\\mathbf {Y}} _ {i} ^ {r}; \\pmb {\\theta} _ {r}, \\pmb {\\theta} _ {c}) + \\mathcal {L} (\\mathbf {Y} _ {i} ^ {(0)}, \\hat {\\mathbf {Y}} _ {i} ^ {(0)}; \\pmb {\\theta} _ {r}, \\pmb {\\theta} _ {c}) \\right] \\\\ \\begin{array}{l l} \\text {s . t .} & \\hat {\\mathbf {Y}} _ {i} ^ {r} = \\cos (\\mathbf {R} _ {i}; \\boldsymbol {\\theta} _ {c}, \\boldsymbol {\\theta} _ {r}), \\quad \\hat {\\mathbf {Y}} _ {i} ^ {(0)} = \\cos (\\mathbf {F} _ {i} ^ {(0)}; \\boldsymbol {\\theta} _ {c}, \\boldsymbol {\\theta} _ {r}), \\end{array} \\\\ \\end{array}\n$$\n",
+        "text_format": "latex",
+        "bbox": [
+            308,
+            479,
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+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "where  $\\mathbf{Y}_i^r$  and  $\\mathbf{Y}_i^{(0)}$  denote the \"family-level\" and \"species-level\" labels, respectively. The bidirectional optimization process defined in Eq. 2 ensures that the residual prototypes  $\\theta_r$  converge towards \"family-level\" traits, while simultaneously driving the species prototypes  $\\theta_c$  to focus on \"species-unique\" traits. These prototypes are jointly optimized to refine the decision boundary, effectively distinguishing between traits that are unique to a species and those shared across the family. Please note that if a single species is restricted to a specific environment during base training, environmental features might be misinterpreted as species-specific traits. This issue can be resolved by enhancing the diversity of the data used during the base training phase.",
+        "bbox": [
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+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "4.3 RESIDUAL PROTOTYPE MIXING",
+        "text_level": 1,
+        "bbox": [
+            171,
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+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "Inspired by human's reference-based learning Kriegeskorte & Douglas (2018), we propose Residual Prototype Mixing (RPM), which drives the model to learn new species by referencing existing knowledge, Fig. 2 (right). The residual prototypes, optimized by Eq. 2, represent family-shared traits likely to appear in closely related species during new species learning. To address this potential confusion, we mix the residual prototypes of old species with new species features to encourage the model to capture the unique characteristics of the new species.",
+        "bbox": [
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+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "Specifically, in the  $t^{\\mathrm{th}}$  incremental session, given the residual prototypes  $\\pmb{\\theta}_r$  and features  $\\mathbf{F}^{(t)}$ , the features of the  $k^{\\mathrm{th}}$  new species are represented as  $\\mathbf{F}_k^{(t)}$ , where this species belongs to the  $j^{\\mathrm{th}}$  family. We generate synthetic feature samples as  $\\mathbf{S}_k^{(t)} = \\mathbf{F}_k^{(t)} + \\pmb{\\mu}_{r_j}$ , and these synthetic feature samples are assigned their original \"species-level\" labels for classification, with the optimization process for the  $k^{\\mathrm{th}}$  species formulated as:",
+        "bbox": [
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+        "page_idx": 4
+    },
+    {
+        "type": "equation",
+        "text": "\n$$\n\\underset {\\boldsymbol {\\theta} _ {c}} {\\arg \\min} \\mathcal {L} \\left(\\mathbf {Y} _ {k} ^ {(t)}, \\hat {\\mathbf {Y}} _ {k} ^ {(t)}; \\boldsymbol {\\theta} _ {c}, \\boldsymbol {\\theta} _ {r}\\right)\n$$\n",
+        "text_format": "latex",
+        "bbox": [
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+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "equation",
+        "text": "\n$$\n\\mathrm {s . t .} \\hat {\\mathbf {Y}} _ {k} ^ {(t)} = \\cos \\left(\\mathbf {S} _ {k} ^ {(t)}; \\pmb {\\theta} _ {c}, \\pmb {\\theta} _ {r}\\right),\n$$\n",
+        "text_format": "latex",
+        "bbox": [
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+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "header",
+        "text": "Published as a conference paper at ICLR 2025",
+        "bbox": [
+            171,
+            32,
+            478,
+            47
+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "page_number",
+        "text": "5",
+        "bbox": [
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+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "where  $\\mathbf{Y}_k^{(t)}$  denotes the \"species-level\" labels. To explicitly explain the optimization process of Eq. 3, we decompose the new species features into species-unique and the other (family-shared and background) features:  $\\mathbf{F}_k^{(t)} = \\mathbf{F}_{ku}^{(t)} + \\mathbf{F}_{ko}^{(t)}$ , where  $\\mathbf{F}_{ku}^{(t)}$  and  $\\mathbf{F}_{ko}^{(t)}$  are orthogonal, meaning they represent non-overlapping features. Consequently, the synthetic features are expressed as:  $\\mathbf{S}_k^{(t)} = \\left(\\mathbf{F}_{ko}^{(t)} + \\boldsymbol{\\mu}_{r_j}\\right) + \\mathbf{F}_{ku}^{(t)} = \\mathbf{F}_{ko}^{\\prime (t)} + \\mathbf{F}_{ku}^{(t)}$ , where  $\\mathbf{F}_{ko}^{\\prime (t)} = \\mathbf{F}_{ko}^{(t)} + \\boldsymbol{\\mu}_{r_j}$  represents the other features enhanced with family-shared traits. For the new species prototypes, the limited training data makes it challenging for the model to learn accurate prototypes. Consequently, these prototypes often contain coarse semantic information, combining both the most and second most discriminative features, which are decomposed into family-shared (residual) and species-unique components:  $\\boldsymbol{\\mu}_{c_k} = \\boldsymbol{\\mu}_{r_j} + \\boldsymbol{\\mu}_{c_ku}$ . Thus, the cosine similarity between  $\\mathbf{S}_k^{(t)}, \\boldsymbol{\\mu}_{c_k}$ , and  $\\boldsymbol{\\mu}_{r_j}$  are calculated as:",
+        "bbox": [
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+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "equation",
+        "text": "\n$$\n\\cos \\left(\\mathbf {S} _ {k} ^ {(t)}; \\boldsymbol {\\mu} _ {c _ {k}}\\right) = \\frac {\\left(\\mathbf {F} _ {k o} ^ {\\prime (t)} + \\mathbf {F} _ {k u} ^ {(t)}\\right) \\cdot \\left(\\boldsymbol {\\mu} _ {r _ {j}} + \\boldsymbol {\\mu} _ {c _ {k} u}\\right)}{\\left\\| \\mathbf {S} _ {k} ^ {(t)} \\right\\| \\cdot \\left\\| \\boldsymbol {\\mu} _ {c _ {k}} \\right\\|}, \\cos \\left(\\mathbf {S} _ {k} ^ {(t)}; \\boldsymbol {\\mu} _ {r _ {j}}\\right) = \\frac {\\left(\\mathbf {F} _ {k o} ^ {\\prime (t)} + \\mathbf {F} _ {k u} ^ {(t)}\\right) \\cdot \\boldsymbol {\\mu} _ {r _ {j}}}{\\left\\| \\mathbf {S} _ {k} ^ {(t)} \\right\\| \\cdot \\left\\| \\boldsymbol {\\mu} _ {r _ {j}} \\right\\|}. \\tag {4}\n$$\n",
+        "text_format": "latex",
+        "bbox": [
+            192,
+            272,
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+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "To avoid misclassifying  $\\mathbf{S}_k^{(t)}$  as the  $j^{\\mathrm{th}}$  family label, the difference term of Eq. 4  $\\Delta \\cos = \\cos (\\mathbf{S}_k^{(t)};\\pmb {\\mu}_{c_k}) - \\cos (\\mathbf{S}_k^{(t)};\\pmb {\\mu}_{r_j})$  must be maximized. As cosine similarity depends only on the angle between vectors, we assume  $\\pmb{\\mu}_{c_k}$  and  $\\pmb{\\mu}_{r_j}$  are unit vectors, and the difference term simplifies to*:  $\\Delta \\cos = \\frac{\\mathbf{F}_{ku}^{(t)}\\cdot\\pmb{\\mu}_{c_ku}}{\\left\\|\\mathbf{S}_k^{(t)}\\right\\|}$ . Since only  $\\pmb{\\mu}_{c_ku}$  is the optimization target, maximizing  $\\Delta$  cos equals maximizing  $\\mathbf{F}_{ku}^{(t)}\\cdot \\pmb{\\mu}_{c_ku}$ , which enhances the alignment between the unique features of the new species and its prototype, improving the model's ability to focus on distinctive features. Furthermore, we mix new species features with all previously learned residual prototypes to generate additional synthetic features. The features mixed with unrelated family (residual) prototypes are then used as augmented training data to help reduce model over-fitting.",
+        "bbox": [
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+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "4.4 THEORETICAL ANALYSIS",
+        "text_level": 1,
+        "bbox": [
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+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "We analyze Prototype Antithesis (PA) from the perspective of prototype independence and sample concentration. Given prototypes  $\\pmb{\\theta}_{c} = \\{\\pmb{\\mu}_{c_{0}},\\pmb{\\mu}_{c_{1}},\\pmb{\\mu}_{c_{2}},\\dots \\}$  including both old and new species, we define the class separability (CS) as:",
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+        "page_idx": 5
+    },
+    {
+        "type": "equation",
+        "text": "\n$$\n\\mathrm {C S} = \\frac {\\left\\| \\boldsymbol {\\mu} _ {c _ {i}} - \\boldsymbol {\\mu} _ {c _ {j}} \\right\\| _ {2} ^ {2}}{\\sum_ {m = 1} ^ {N _ {i}} \\left(1 - \\cos \\left(\\mathbf {f} _ {m} ; \\boldsymbol {\\mu} _ {c _ {i}}\\right)\\right) + \\sum_ {v = 1} ^ {N _ {j}} \\left(1 - \\cos \\left(\\mathbf {f} _ {v} ; \\boldsymbol {\\mu} _ {c _ {j}}\\right)\\right)}, \\tag {5}\n$$\n",
+        "text_format": "latex",
+        "bbox": [
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+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "where  $i$  and  $j$  denote any two different species, while  $\\mathbf{f}_m$  and  $\\mathbf{f}_v$  represent their feature samples. In this formulation, the numerator measures the independence or separation of the species prototypes  $\\pmb{\\mu}_{c_i}$  and  $\\pmb{\\mu}_{c_j}$  (unit vectors), while the denominator reflects the concentration of samples around their respective species prototype, quantifying how well the samples align with their class center. The goal of a classification model is to maximize class separability, where the large separation between species prototypes and tight concentration of samples around their respective prototypes leads to better classification performance. Without loss of generality, let us assume that the prototypes of an old species  $\\pmb{\\mu}_{c_i}$  and a new species  $\\pmb{\\mu}_{c_j}$  are related as  $\\pmb{\\mu}_{c_j} = \\hat{\\pmb{\\mu}}_{c_i} + \\delta$ ,  $\\|\\hat{\\pmb{\\mu}}_{c_i}\\| \\in [0,1]$ , where  $\\hat{\\pmb{\\mu}}_{c_i}$  represents the similar parts with  $\\pmb{\\mu}_{c_i}$ , and  $\\delta$  is orthogonal to  $\\pmb{\\mu}_{c_i}$  and  $\\hat{\\pmb{\\mu}}_{c_i}$ , representing the unique parts. Substituting this into Equation 5, we can rewrite the class separability as:",
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+        "page_idx": 5
+    },
+    {
+        "type": "equation",
+        "text": "\n$$\n\\mathrm {C S} = \\frac {1 + \\delta^ {2} - \\hat {\\boldsymbol {\\mu}} _ {c _ {i}} \\cdot (2 - \\hat {\\boldsymbol {\\mu}} _ {c _ {i}})}{\\sum_ {m = 1} ^ {N _ {i}} (1 - \\cos (\\mathbf {f} _ {m} ; \\boldsymbol {\\mu} _ {c _ {i}})) + \\sum_ {v = 1} ^ {N _ {j}} (1 - \\cos (\\mathbf {f} _ {v} ; \\boldsymbol {\\mu} _ {c _ {j}}))}. \\tag {6}\n$$\n",
+        "text_format": "latex",
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+    {
+        "type": "text",
+        "text": "While all methods minimize the denominator using classification loss, the key difference lies in the numerator. The term  $\\hat{\\mu}_{c_i}(2 - \\hat{\\mu}_{c_i})$  is monotonically increasing over the interval [0, 1]. In conventional approaches, when the new species prototype  $\\pmb{\\mu}_{c_j}$  is similar to the old one  $\\pmb{\\mu}_{c_i}$ ,  $\\hat{\\mu}_{c_i}$  becomes large, indicating prototype overlap and reducing the numerator in Eq. 6. However, PA reduces this overlap by emphasizing species-unique features ( $\\delta$ ), decreasing  $\\hat{\\mu}_{c_i}$ , thereby increasing the numerator in Eq. 6 and improving class separability.",
+        "bbox": [
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+        "page_idx": 5
+    },
+    {
+        "type": "header",
+        "text": "Published as a conference paper at ICLR 2025",
+        "bbox": [
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+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "page_footnote",
+        "text": "*The proof is included in the \"Appendix\".",
+        "bbox": [
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+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "page_footnote",
+        "text": "$\\dagger$  Please refer to the \"Appendix\" for the detailed derivation.",
+        "bbox": [
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+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "page_number",
+        "text": "6",
+        "bbox": [
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+    },
+    {
+        "type": "table",
+        "img_path": "images/be3aa088fa28175aacb21e7d7df08a2fbc59b92e90c133de9a102d8e9c72f3fb.jpg",
+        "table_caption": [
+            "Table 1: Ablation experiments of the proposed \"RPL\" and \"RPM\". The baseline method is a standard classification network consisting of a feature extractor and a classifier. \"Meanold\" and \"Meannew\" denote the average accuracy of old and new species across all incremental sessions."
+        ],
+        "table_footnote": [],
+        "table_body": "<table><tr><td rowspan=\"2\">Baseline</td><td rowspan=\"2\">RPL</td><td rowspan=\"2\">RPM</td><td colspan=\"5\">Accuracy in each session (%)</td><td rowspan=\"2\">Meanold</td><td rowspan=\"2\">Meannew</td><td rowspan=\"2\">PD↓</td></tr><tr><td>0</td><td>1</td><td>2</td><td>3</td><td>4</td></tr><tr><td>✓</td><td></td><td></td><td>99.47</td><td>93.83</td><td>90.76</td><td>90.47</td><td>86.21</td><td>95.63</td><td>81.58</td><td>13.26</td></tr><tr><td>✓</td><td>✓</td><td></td><td>99.47</td><td>94.21</td><td>93.56</td><td>92.12</td><td>88.86</td><td>96.89</td><td>84.17</td><td>10.61</td></tr><tr><td>✓</td><td>✓</td><td>✓</td><td>99.50</td><td>95.83</td><td>93.45</td><td>93.24</td><td>90.51</td><td>98.47</td><td>86.59</td><td>8.99</td></tr></table>",
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+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "5 EXPERIMENTS",
+        "text_level": 1,
+        "bbox": [
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+        ],
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+    },
+    {
+        "type": "text",
+        "text": "We conduct comprehensive experiments on several public datasets for model analysis, and compare our proposed method with the state-of-the-art methods under FSCIL settings.",
+        "bbox": [
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+    },
+    {
+        "type": "text",
+        "text": "5.1 EXPERIMENT SETTINGS",
+        "text_level": 1,
+        "bbox": [
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+    },
+    {
+        "type": "text",
+        "text": "Datasets. We adopt CUB200 P. et al. (2010), PlantVillage Hughes & Salathé (2015), and Tree-of-Life Stevens et al. (2024) as benchmarks. In the FSCIL task, classes are split into base and incremental categories. Under the  $N$ -way  $K$ -shot setting, the incremental classes are divided into  $T$  groups, corresponding to the  $T$  incremental sessions. In each session,  $N$  classes are presented for training, with each class containing  $K$  samples. For the CUB200 dataset, 100 classes are used as base classes, and the remaining 100 classes are divided into 10 sessions ( $T = 10$ ), with each session containing 10 classes (10-way) and 5 samples per class (5-shot). In the PlantVillage dataset, 19 classes serve as base classes, while the remaining 20 classes form incremental classes under a 5-way 5-shot setting. For the Tree-of-Life dataset, we randomly sample 100 classes for evaluation, where 60 are base classes, and the remaining 40 are split into groups under the 5-way 5-shot setting.",
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+    },
+    {
+        "type": "text",
+        "text": "Model Training. To maintain consistency with prior work, we use a ResNet-18 pretrained on ImageNet as the feature extractor for the CUB-200 dataset, while for the PlantVillage and Tree-of-Life datasets, we also use ResNet-18 as the backbone. The optimizer is SGD, with a \"milestone\" learning rate decay strategy. For data pre-processing, we apply image normalization, random cropping, random resizing, and horizontal flipping for data augmentation. During base training, we use a batch size of 128 and an initial learning rate of 0.004, training for 100 epochs. In the incremental learning phase, the classifier is optimized over 100 epochs with a learning rate of 0.01. We conducted 5 experiments with different seeds on the fixed datasets and reported the averaged results to minimize randomness. All experiments were performed using Pytorch 1.11.0 on Nvidia A800 GPUs.",
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+    {
+        "type": "text",
+        "text": "Evaluation. We adopt \"classification accuracy\" and \"performance drop (PD)\" as the evaluation metrics. \"Classification accuracy\" validates the discriminating ability of the model, and it is defined as accuracy =  $\\frac{\\mathrm{TP} + \\mathrm{TN}}{\\mathrm{TP} + \\mathrm{TN} + \\mathrm{FP} + \\mathrm{FN}}$ , where TP, TN, FP and FN denote the numbers of true positives, true negatives, false positives, and false negatives, respectively. \"PD\" measures the model forgetting and is defined as PD = accuracy(0) - accuracy(T), where accuracy(0) and accuracy(T) denote the model performance in the 0-th and T-th sessions, respectively.",
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+    {
+        "type": "text",
+        "text": "5.2 ABLATION STUDY",
+        "text_level": 1,
+        "bbox": [
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+    },
+    {
+        "type": "text",
+        "text": "To rigorously assess the effectiveness of our proposed approach, we perform ablation experiments on the PlantVillage dataset.",
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+    },
+    {
+        "type": "text",
+        "text": "Residual Prototype Learning. As seen in Table 1, the proposed RPL increases performance on old species by  $1.26\\%$  and on new species by  $2.59\\%$  over the baseline. This is attributed to the model's ability to classify both unique and shared features of the base species, resulting in",
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+    {
+        "type": "text",
+        "text": "a more compact decision boundary that supports learning in both old and new species.",
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+    },
+    {
+        "type": "table",
+        "img_path": "images/4daffcc5ab49a7c662e043040c586f402efecb9952c4d76326500680a2cbab35.jpg",
+        "table_caption": [
+            "Table 2: Comparison of \"CAM\", \"Grad-CAM\" and the proposed \"RPL\"."
+        ],
+        "table_footnote": [],
+        "table_body": "<table><tr><td>Method</td><td>Meanold</td><td>Meannew</td><td>Similarity</td></tr><tr><td>CAM</td><td>95.31%</td><td>81.25%</td><td>6.53%</td></tr><tr><td>Grad-CAM</td><td>95.32%</td><td>79.82%</td><td>8.15%</td></tr><tr><td>RPL</td><td>98.47%</td><td>86.59%</td><td>1.89%</td></tr></table>",
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+        "page_idx": 6
+    },
+    {
+        "type": "header",
+        "text": "Published as a conference paper at ICLR 2025",
+        "bbox": [
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+        ],
+        "page_idx": 6
+    },
+    {
+        "type": "page_number",
+        "text": "7",
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+        "page_idx": 6
+    },
+    {
+        "type": "image",
+        "img_path": "images/2c6a074a1f14517512112067efc6314147e1b798844598d734e493429807ccbd.jpg",
+        "image_caption": [
+            "Without Prototype Antithesis (Baseline)"
+        ],
+        "image_footnote": [],
+        "bbox": [
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+        ],
+        "page_idx": 7
+    },
+    {
+        "type": "image",
+        "img_path": "images/ab0bafd92c6fc7e75ca61ec48cb19cbf8365212220fc2f3bebe35aff32a45154.jpg",
+        "image_caption": [
+            "With Prototype Antithesis (Ours)",
+            "Figure 3: t-SNE visualization of feature distributions comparing models with and without Prototype Antithesis (PA). Without PA, the features of new species overlap with those of similar old species. In contrast, with PA, the new species features are pushed away from the residual features, effectively distinguishing the new and old species and creating a well-separated decision boundary. (Best viewed in color)"
+        ],
+        "image_footnote": [],
+        "bbox": [
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+        "page_idx": 7
+    },
+    {
+        "type": "text",
+        "text": "Residual Prototype Mixing. Table 1 demonstrates that combining RPL with RPM boosts old and new species performance by  $2.84\\%$  and  $5.01\\%$ , respectively, and decreases \"PD\" by  $4.37\\%$ . RPM reduces the overlap of shared components between new and old species, effectively minimizing species confusion. Together, RPL and RPM work synergistically to reduce \"old species forgetting\" while enhancing \"new species generalization\".",
+        "bbox": [
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+        "page_idx": 7
+    },
+    {
+        "type": "text",
+        "text": "Feature decomposition strategy. To assess the effectiveness of different feature decomposition strategies, we use CAM Zhou et al. (2016) and Grad-CAM Selvaraju et al. (2017) to distinguish between unique and common features. Specifically, we set a threshold to extract the highest-response regions from the feature maps as unique features, while the secondary-high-response regions represent common features. As shown in Table 2, RPL achieves the highest mean accuracy and the lowest similarity between unique and common features, which can be attributed to its bi-directional optimization that more effectively decouples unique features from shared ones.",
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+        "page_idx": 7
+    },
+    {
+        "type": "text",
+        "text": "5.3 MODEL ANALYSIS",
+        "text_level": 1,
+        "bbox": [
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+        "page_idx": 7
+    },
+    {
+        "type": "text",
+        "text": "In this section, we further conduct comprehensive experiments for detailed statistical results and visualization.",
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+    },
+    {
+        "type": "text",
+        "text": "Feature distribution. We employ t-SNE to visualize the feature distributions of similar old and new species, as well as the residual feature distributions. As depicted in Fig.3(left), without the proposed prototype antithesis, feature samples of the old and new species overlap at the edges of the distribution. However, with the introduction of the prototype antithesis, the distance between the new species and the residual features is maximized, making the features of the new and old species more distinguishable (Fig.3(right)). This adjustment optimizes the decision boundary and reduces species confusion.",
+        "bbox": [
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+        "page_idx": 7
+    },
+    {
+        "type": "image",
+        "img_path": "images/266f23ee92ba95161d97aac2b83751063ae51606331ebee022c8ea69a8305429.jpg",
+        "image_caption": [
+            "Figure 4: Confusion matrix of the baseline method and our proposed PA approach. Compared with the baseline method, PA significantly reduces the false prediction among old and new species, alleviating species confusion."
+        ],
+        "image_footnote": [],
+        "bbox": [
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+        "page_idx": 7
+    },
+    {
+        "type": "text",
+        "text": "Confusion Matrix. We plot the con",
+        "bbox": [
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+        ],
+        "page_idx": 7
+    },
+    {
+        "type": "text",
+        "text": "fusion matrix for both the baseline method and our proposed PA method. As shown in Fig. 4, the PA method significantly reduces false classifications between old and new species. These results demonstrate that PA effectively minimizes confusion between new and old species, mitigating forgetting of old species while enhancing learning for new species.",
+        "bbox": [
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+        ],
+        "page_idx": 7
+    },
+    {
+        "type": "header",
+        "text": "Published as a conference paper at ICLR 2025",
+        "bbox": [
+            173,
+            32,
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+            47
+        ],
+        "page_idx": 7
+    },
+    {
+        "type": "page_number",
+        "text": "8",
+        "bbox": [
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+    },
+    {
+        "type": "table",
+        "img_path": "images/6770ae4af57598d737caa6cffda60680d59f264604350f935e038eecc7aaff0e.jpg",
+        "table_caption": [
+            "Table 3: Performance comparison on the CUB200 dataset. “*” indicates results that were re-implemented using the official code."
+        ],
+        "table_footnote": [],
+        "table_body": "<table><tr><td rowspan=\"2\">Method</td><td colspan=\"12\">Accuracy in each session (%)</td><td>PD ↓</td></tr><tr><td>0</td><td>1</td><td>2</td><td>3</td><td>4</td><td>5</td><td>6</td><td>7</td><td>8</td><td>9</td><td>10</td><td></td><td></td></tr><tr><td>TOPIC Tao et al. (2020)</td><td>68.68</td><td>62.49</td><td>54.81</td><td>49.99</td><td>45.25</td><td>41.40</td><td>38.35</td><td>35.36</td><td>32.22</td><td>28.31</td><td>26.28</td><td>42.40</td><td></td></tr><tr><td>SPPR Zhu et al. (2021)</td><td>68.68</td><td>61.85</td><td>57.43</td><td>52.68</td><td>50.19</td><td>46.88</td><td>44.65</td><td>43.07</td><td>40.17</td><td>39.63</td><td>37.33</td><td>31.35</td><td></td></tr><tr><td>SFMS Cheraghian et al. (2021)</td><td>68.78</td><td>59.37</td><td>59.32</td><td>54.96</td><td>52.58</td><td>49.81</td><td>48.09</td><td>46.32</td><td>44.33</td><td>43.43</td><td>43.23</td><td>25.55</td><td></td></tr><tr><td>FSLL Mazumder et al. (2021)</td><td>72.77</td><td>69.33</td><td>65.51</td><td>62.66</td><td>61.10</td><td>58.65</td><td>57.78</td><td>57.26</td><td>55.59</td><td>55.39</td><td>54.21</td><td>17.38</td><td></td></tr><tr><td>CEC Zhang et al. (2021)</td><td>75.85</td><td>71.94</td><td>68.50</td><td>63.50</td><td>62.43</td><td>58.27</td><td>57.73</td><td>55.81</td><td>54.83</td><td>53.52</td><td>52.28</td><td>23.57</td><td></td></tr><tr><td>Meta-FSCIL Chi et al. (2022)</td><td>75.90</td><td>72.41</td><td>68.78</td><td>64.78</td><td>62.96</td><td>59.99</td><td>58.30</td><td>56.85</td><td>54.78</td><td>53.82</td><td>52.64</td><td>23.26</td><td></td></tr><tr><td>FACT Zhou et al. (2022)</td><td>75.90</td><td>73.23</td><td>70.84</td><td>66.13</td><td>65.56</td><td>62.15</td><td>61.74</td><td>59.83</td><td>58.41</td><td>57.89</td><td>56.94</td><td>18.96</td><td></td></tr><tr><td>SOFTNet Kang et al. (2023)</td><td>78.07</td><td>74.58</td><td>71.37</td><td>67.54</td><td>65.37</td><td>62.60</td><td>61.07</td><td>59.37</td><td>57.53</td><td>57.21</td><td>56.75</td><td>21.32</td><td></td></tr><tr><td>WaRP Kim et al. (2023)</td><td>77.74</td><td>74.15</td><td>70.82</td><td>66.90</td><td>65.01</td><td>62.64</td><td>61.40</td><td>59.86</td><td>57.95</td><td>57.77</td><td>57.01</td><td>20.73</td><td></td></tr><tr><td>BiDist Zhao et al. (2023)</td><td>79.12</td><td>74.99</td><td>70.87</td><td>67.30</td><td>65.89</td><td>63.45</td><td>61.40</td><td>60.11</td><td>58.61</td><td>58.23</td><td>57.48</td><td>21.64</td><td></td></tr><tr><td>NC-FSCIL Yang et al. (2023)</td><td>80.45</td><td>75.98</td><td>72.30</td><td>70.28</td><td>68.17</td><td>65.16</td><td>64.43</td><td>63.25</td><td>60.66</td><td>60.01</td><td>59.44</td><td>21.01</td><td></td></tr><tr><td>OrCo* Ahmed et al. (2024)</td><td>74.72</td><td>66.06</td><td>64.72</td><td>62.88</td><td>61.89</td><td>59.69</td><td>58.95</td><td>59.03</td><td>57.24</td><td>57.79</td><td>57.18</td><td>17.54</td><td></td></tr><tr><td>TEEN Wang et al. (2024)</td><td>77.26</td><td>76.13</td><td>72.81</td><td>68.16</td><td>67.77</td><td>64.40</td><td>63.25</td><td>62.29</td><td>61.19</td><td>60.32</td><td>59.31</td><td>18.13</td><td></td></tr><tr><td>PA(ours)</td><td>78.69</td><td>75.59</td><td>72.71</td><td>68.71</td><td>68.37</td><td>65.77</td><td>64.75</td><td>63.59</td><td>62.76</td><td>62.02</td><td>61.19</td><td>17.50</td><td></td></tr></table>",
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+    },
+    {
+        "type": "image",
+        "img_path": "images/8fdcf1eecc46d60dd8c6154229b0c8e08ac50e7b4658fc1f826f457cd80c2f2b.jpg",
+        "image_caption": [
+            "(a) Performance on PlantVillage",
+            "Figure 5: Performance on the PlantVillage and Tree-of-Life datasets."
+        ],
+        "image_footnote": [],
+        "bbox": [
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+    },
+    {
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+        "img_path": "images/146fbee789e208d044539feb6e284df564619d77e172d2a4ef312cb4516c055f.jpg",
+        "image_caption": [
+            "(b) Performance on Tree-of-Life"
+        ],
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+        "bbox": [
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+    },
+    {
+        "type": "text",
+        "text": "5.4 COMPARISON WITH THE STATE-OF-THE-ART METHODS",
+        "text_level": 1,
+        "bbox": [
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+        ],
+        "page_idx": 8
+    },
+    {
+        "type": "text",
+        "text": "We compare PA against state-of-the-art methods on both computer vision benchmarks (e.g., CUB200) and biological datasets (e.g., PlantVillage and Tree-of-Life).",
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+    {
+        "type": "text",
+        "text": "CUB200: As shown in Table 3, our proposed PA outperforms the TEEN and OrCo by  $1.88\\%$  and  $4.01\\%$ , as well as achieving a lower \"PD\" compared to most other methods. PlantVillage: From Fig. 5(a) we can see that PA outperforms other methods with significant margins. The categories in the PlantVillage dataset (different leaves with disease) share many common traits, and the significant improvement in performance shows PA's superiority in distinguishing closely related old and new species. Tree-of-Life: PA outperforms OrCo Ahmed et al. (2024) and BiDist Zhao et al. (2023) by  $6.26\\%$  and  $1.28\\%$  on this challenging dataset, respectively, Fig. 5(b). This result confirms PA's robustness in handling high-diversity biological data.",
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+    {
+        "type": "text",
+        "text": "6 CONCLUSION",
+        "text_level": 1,
+        "bbox": [
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+        "page_idx": 8
+    },
+    {
+        "type": "text",
+        "text": "In this paper, we propose a Prototype Antithesis (PA) method to address the issue of species confusion in biological few-shot class-incremental learning. The PA method comprises Residual Prototype Learning (RPL) and Residual Prototype Mixing (RPM). RPL decomposes species-unique and family-shared (residual) prototypes, while RPM generates synthetic samples by blending new species features with residual prototypes. Together, RPL and RPM optimize the decision boundary between old and new species, mitigating \"old species forgetting\" and enhancing \"new species learning\". PA offers a novel perspective for recognizing biological species and traits using few-shot class-incremental learning.",
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+        ],
+        "page_idx": 8
+    },
+    {
+        "type": "header",
+        "text": "Published as a conference paper at ICLR 2025",
+        "bbox": [
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+        "page_idx": 8
+    },
+    {
+        "type": "page_number",
+        "text": "9",
+        "bbox": [
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+        "page_idx": 8
+    },
+    {
+        "type": "text",
+        "text": "AUTHOR CONTRIBUTIONS",
+        "text_level": 1,
+        "bbox": [
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+        ],
+        "page_idx": 9
+    },
+    {
+        "type": "text",
+        "text": "Fei Gu served as the project manager. Binghao Liu proposed the idea of the paper, designed the methodology, conducted the experiments, and wrote the manuscript. Han Yang participated in discussions on experimental settings. Fang Wan participated in idea discussions and provided suggestions for figure visualization.",
+        "bbox": [
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+    },
+    {
+        "type": "text",
+        "text": "ACKNOWLEDGMENTS",
+        "text_level": 1,
+        "bbox": [
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+            200,
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+            214
+        ],
+        "page_idx": 9
+    },
+    {
+        "type": "text",
+        "text": "This work was financially supported in part by grants from the Biological Breeding - Major Projects (2023ZD04076).",
+        "bbox": [
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+    },
+    {
+        "type": "text",
+        "text": "REFERENCES",
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+        "page_idx": 9
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+        "text": "Published as a conference paper at ICLR 2025",
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+        ],
+        "bbox": [
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+        ],
+        "page_idx": 11
+    },
+    {
+        "type": "text",
+        "text": "A APPENDIX",
+        "text_level": 1,
+        "bbox": [
+            171,
+            580,
+            299,
+            594
+        ],
+        "page_idx": 11
+    },
+    {
+        "type": "text",
+        "text": "A.1 DERIVATION OF  $\\Delta$  cos.",
+        "text_level": 1,
+        "bbox": [
+            171,
+            609,
+            375,
+            625
+        ],
+        "page_idx": 11
+    },
+    {
+        "type": "text",
+        "text": "The  $\\Delta$  cos in section 4.3 is calculated as below:",
+        "bbox": [
+            171,
+            636,
+            485,
+            650
+        ],
+        "page_idx": 11
+    },
+    {
+        "type": "equation",
+        "text": "\n$$\n\\begin{array}{l} \\Delta \\cos = \\cos \\left(\\mathbf {S} _ {k} ^ {(t)}, \\mu_ {c _ {k}}\\right) - \\cos \\left(\\mathbf {S} _ {k} ^ {(t)}, \\mu_ {r _ {j}}\\right) \\\\ = \\frac {\\left(\\mathbf {F} _ {k o} ^ {\\prime (t)} + \\mathbf {F} _ {k u} ^ {(t)}\\right) \\cdot \\left(\\mu_ {r _ {j}} + \\mu_ {c _ {k} u}\\right)}{\\left\\| \\mathbf {S} _ {k} ^ {(t)} \\right\\| \\cdot \\left\\| \\mu_ {c _ {k}} \\right\\|} - \\frac {\\left(\\mathbf {F} _ {k o} ^ {\\prime (t)} + \\mathbf {F} _ {k u} ^ {(t)}\\right) \\cdot \\mu_ {r _ {j}}}{\\left\\| \\mathbf {S} _ {k} ^ {(t)} \\right\\| \\cdot \\left\\| \\mu_ {r _ {j}} \\right\\|} \\\\ = \\frac {\\mathbf {F} _ {k o} ^ {\\prime (t)} \\cdot \\mu_ {r j} + \\mathbf {F} _ {k o} ^ {\\prime (t)} \\cdot \\mu_ {c k u} + \\mathbf {F} _ {k u} ^ {(t)} \\cdot \\mu_ {r j} + \\mathbf {F} _ {k u} ^ {(t)} \\cdot \\mu_ {c k u} - \\mathbf {F} _ {k o} ^ {\\prime (t)} \\cdot \\mu_ {r j} - \\mathbf {F} _ {k u} ^ {(t)} \\cdot \\mu_ {r j}}{\\left\\| \\mathbf {S} _ {k} ^ {(t)} \\right\\| \\cdot \\left\\| \\mu_ {r j} \\right\\|} \\tag {7} \\\\ = \\frac {\\mathbf {F} _ {k o} ^ {\\prime (t)} \\cdot \\mu_ {c _ {k} u} + \\mathbf {F} _ {k u} ^ {(t)} \\cdot \\mu_ {c _ {k} u}}{\\left\\| \\mathbf {S} _ {k} ^ {(t)} \\right\\| \\cdot \\left\\| \\mu_ {r _ {j}} \\right\\|}. \\\\ \\end{array}\n$$\n",
+        "text_format": "latex",
+        "bbox": [
+            215,
+            652,
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+            821
+        ],
+        "page_idx": 11
+    },
+    {
+        "type": "text",
+        "text": "Since  $\\mathbf{F}_{ko}^{\\prime(t)}$  are orthogonal to  $\\mu_{c_k u}$  (there are no overlap between them),  $\\mathbf{F}_{ko}^{\\prime(t)} \\cdot \\mu_{c_k u}$  is equal to 0, and  $\\mu_{r_j}$  is unit vector, thus Eq. 7 can be rewrite as:",
+        "bbox": [
+            171,
+            824,
+            823,
+            856
+        ],
+        "page_idx": 11
+    },
+    {
+        "type": "equation",
+        "text": "\n$$\n\\begin{array}{l} \\Delta \\cos = \\cos (\\mathbf {S} _ {k} ^ {(t)}, \\mu_ {c _ {k}}) - \\cos (\\mathbf {S} _ {k} ^ {(t)}, \\mu_ {r _ {j}}) \\\\ = \\frac {\\mathbf {F} _ {k u} ^ {(t)} \\cdot \\mu_ {c k u}}{\\left\\| \\mathbf {S} _ {k} ^ {(t)} \\right\\|}. \\tag {8} \\\\ \\end{array}\n$$\n",
+        "text_format": "latex",
+        "bbox": [
+            367,
+            859,
+            823,
+            928
+        ],
+        "page_idx": 11
+    },
+    {
+        "type": "header",
+        "text": "Published as a conference paper at ICLR 2025",
+        "bbox": [
+            171,
+            32,
+            478,
+            47
+        ],
+        "page_idx": 11
+    },
+    {
+        "type": "page_number",
+        "text": "12",
+        "bbox": [
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+            946,
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+        ],
+        "page_idx": 11
+    },
+    {
+        "type": "table",
+        "img_path": "images/b1f9da2875a0faf15e82c4b8c3d1a3ba335b22b9b1b13c3f0a8d639b6b6ee59a.jpg",
+        "table_caption": [
+            "Table 4: Performance comparison on the PlantVillage dataset. “*” indicates results that were re-implemented using the official code."
+        ],
+        "table_footnote": [],
+        "table_body": "<table><tr><td rowspan=\"2\">Method</td><td colspan=\"5\">Accuracy in each session (%)</td><td rowspan=\"2\">PD ↓</td></tr><tr><td>0</td><td>1</td><td>2</td><td>3</td><td>4</td></tr><tr><td>Ft-CNN</td><td>99.89</td><td>82.33</td><td>36.11</td><td>34.38</td><td>28.53</td><td>71.47</td></tr><tr><td>CEC*</td><td>99.53</td><td>92.46</td><td>90.13</td><td>88.35</td><td>85.01</td><td>14.52</td></tr><tr><td>FACT*</td><td>99.47</td><td>93.33</td><td>90.69</td><td>90.00</td><td>86.15</td><td>13.32</td></tr><tr><td>TEEN*</td><td>99.50</td><td>90.83</td><td>90.56</td><td>86.38</td><td>85.19</td><td>14.31</td></tr><tr><td>BiDist*</td><td>99.51</td><td>91.25</td><td>84.14</td><td>81.18</td><td>78.97</td><td>20.54</td></tr><tr><td>NC-FSCIL*</td><td>99.48</td><td>96.05</td><td>93.59</td><td>92.88</td><td>89.13</td><td>10.35</td></tr><tr><td>OrCo*</td><td>99.58</td><td>84.25</td><td>75.24</td><td>64.77</td><td>60.82</td><td>38.76</td></tr><tr><td>PA (ours)</td><td>99.50</td><td>95.83</td><td>93.45</td><td>93.24</td><td>90.51</td><td>8.99</td></tr></table>",
+        "bbox": [
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+        ],
+        "page_idx": 12
+    },
+    {
+        "type": "text",
+        "text": "This is consistent with the result presented in Section 4.3.",
+        "bbox": [
+            171,
+            321,
+            550,
+            335
+        ],
+        "page_idx": 12
+    },
+    {
+        "type": "text",
+        "text": "A.2 DERIVATION OF EQ. 6 IN THE PAPER.",
+        "text_level": 1,
+        "bbox": [
+            171,
+            353,
+            477,
+            367
+        ],
+        "page_idx": 12
+    },
+    {
+        "type": "text",
+        "text": "The class separability in section 4.4 is written as:",
+        "bbox": [
+            171,
+            378,
+            496,
+            393
+        ],
+        "page_idx": 12
+    },
+    {
+        "type": "equation",
+        "text": "\n$$\n\\begin{array}{l} \\mathbf {C S} = \\frac {\\| \\mu_ {c _ {i}} - (\\hat {\\mu} _ {c _ {i}} + \\delta) \\| _ {2} ^ {2}}{\\sum_ {m = 1} ^ {N _ {i}} (1 - \\cos (\\mathbf {f} _ {m} ; \\mu_ {c _ {i}})) + \\sum_ {v = 1} ^ {N _ {j}} (1 - \\cos (\\mathbf {f} _ {v} ; \\mu_ {c _ {j}}))}, \\\\ = \\frac {\\mu_ {c _ {i}} ^ {2} + \\hat {\\mu} _ {c _ {i}} ^ {2} + \\delta^ {2} + 2 \\cdot \\delta \\cdot \\hat {\\mu} _ {c _ {i}} - 2 \\cdot \\mu_ {c _ {i}} \\cdot (\\hat {\\mu} _ {c _ {i}} + \\delta)}{\\sum_ {m = 1} ^ {N _ {i}} (1 - \\cos (\\mathbf {f} _ {m} ; \\mu_ {c _ {i}})) + \\sum_ {v = 1} ^ {N _ {j}} (1 - \\cos (\\mathbf {f} _ {v} ; \\mu_ {c _ {j}}))}. \\\\ \\end{array}\n$$\n",
+        "text_format": "latex",
+        "bbox": [
+            294,
+            398,
+            699,
+            479
+        ],
+        "page_idx": 12
+    },
+    {
+        "type": "text",
+        "text": "Since  $\\delta$  is orthogonal to  $\\pmb{\\mu}_{c_i}$  and  $\\hat{\\pmb{\\mu}}_{c_i}$ , and  $\\mu_{c_i}$  is unit vector, Eq. 10 is rewritten as:",
+        "bbox": [
+            171,
+            484,
+            709,
+            500
+        ],
+        "page_idx": 12
+    },
+    {
+        "type": "equation",
+        "text": "\n$$\n\\begin{array}{l} \\mathbf {C S} = \\frac {1 + \\hat {\\mu} _ {c _ {i}} ^ {2} + \\delta^ {2} - 2 \\cdot \\mu_ {c _ {i}} \\cdot \\hat {\\mu} _ {c _ {i}}}{\\sum_ {m = 1} ^ {N _ {i}} (1 - \\cos (\\mathbf {f} _ {m} ; \\mu_ {c _ {i}})) + \\sum_ {v = 1} ^ {N _ {j}} (1 - \\cos (\\mathbf {f} _ {v} ; \\mu_ {c _ {j}}))}, \\\\ = \\frac {1 + \\delta^ {2} - \\hat {\\mu} _ {c _ {i}} \\cdot (2 - \\hat {\\mu} _ {c _ {i}})}{\\sum_ {m = 1} ^ {N _ {i}} (1 - \\cos (\\mathbf {f} _ {m} ; \\mu_ {c _ {i}})) + \\sum_ {v = 1} ^ {N _ {j}} (1 - \\cos (\\mathbf {f} _ {v} ; \\mu_ {c _ {j}}))}, \\\\ \\end{array}\n$$\n",
+        "text_format": "latex",
+        "bbox": [
+            294,
+            506,
+            699,
+            584
+        ],
+        "page_idx": 12
+    },
+    {
+        "type": "text",
+        "text": "which aligns with Eq. 6 in the paper.",
+        "bbox": [
+            171,
+            590,
+            415,
+            604
+        ],
+        "page_idx": 12
+    },
+    {
+        "type": "text",
+        "text": "A.3 PERFORMANCE ON THE PLANTVILLAGE DATASET",
+        "text_level": 1,
+        "bbox": [
+            171,
+            621,
+            565,
+            635
+        ],
+        "page_idx": 12
+    },
+    {
+        "type": "text",
+        "text": "We present the detailed classification accuracy on the PlantVillage dataset in Table 4. The results show that our method significantly outperforms other methods in the final session and achieves the smallest performance drop.",
+        "bbox": [
+            169,
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+        ],
+        "page_idx": 12
+    },
+    {
+        "type": "text",
+        "text": "A.4 PERFORMANCE ON THE TREE-OF-LIFE DATASET",
+        "text_level": 1,
+        "bbox": [
+            171,
+            705,
+            553,
+            720
+        ],
+        "page_idx": 12
+    },
+    {
+        "type": "text",
+        "text": "The detailed results on the Tree-of-Life dataset are shown in Table 5. Our proposed PA method outperforms most methods while delivering comparable performance (slightly lower) than NC-FSCIL. Unlike NC-FSCIL, however, our method does not require prior knowledge of the total number of unknown classes, making it more scalable and practical.",
+        "bbox": [
+            169,
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+        ],
+        "page_idx": 12
+    },
+    {
+        "type": "text",
+        "text": "B DATASETS STATISTICS",
+        "text_level": 1,
+        "bbox": [
+            171,
+            809,
+            395,
+            824
+        ],
+        "page_idx": 12
+    },
+    {
+        "type": "text",
+        "text": "CUB200. The CUB-200-2011 dataset contains 11,788 images of birds, categorized into 200 fine-grained subclasses (species) and 37 broader classes (families). PlantVillage. The PlantVillage dataset contains 54,306 images of plant leaves, categorized into 38 fine-grained subclasses (species) and 14 broader classes (families). The images include healthy and diseased leaves, making it a widely used benchmark for plant disease classification and recognition tasks. Tree-of-Life. We randomly sampled 13,487 images from 100 species in the Tree-of-Life dataset, spanning 42 families.",
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+        "page_idx": 12
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+    {
+        "type": "header",
+        "text": "Published as a conference paper at ICLR 2025",
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+    },
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+        "table_caption": [
+            "Table 5: Performance comparison on the Tree-of-Life dataset. “*” indicates results that were re-implemented using the official code."
+        ],
+        "table_footnote": [],
+        "table_body": "<table><tr><td rowspan=\"2\">Method</td><td colspan=\"10\">Accuracy in each session (%)</td><td>PD ↓</td></tr><tr><td>0</td><td>1</td><td>2</td><td>3</td><td>4</td><td>5</td><td>6</td><td>7</td><td>8</td><td></td><td></td></tr><tr><td>Ft-CNN</td><td>49.98</td><td>41.25</td><td>36.53</td><td>34.42</td><td>31.64</td><td>31.20</td><td>29.15</td><td>28.32</td><td>27.91</td><td>22.07</td><td></td></tr><tr><td>CEC*</td><td>49.68</td><td>43.35</td><td>39.02</td><td>37.71</td><td>35.56</td><td>34.87</td><td>32.41</td><td>32.21</td><td>29.33</td><td>20.35</td><td></td></tr><tr><td>FACT*</td><td>48.33</td><td>45.25</td><td>39.25</td><td>39.00</td><td>35.20</td><td>35.33</td><td>32.17</td><td>32.43</td><td>29.71</td><td>18.62</td><td></td></tr><tr><td>TEEN*</td><td>49.33</td><td>43.00</td><td>38.25</td><td>37.60</td><td>34.40</td><td>33.83</td><td>32.00</td><td>33.00</td><td>30.57</td><td>18.76</td><td></td></tr><tr><td>BiDist*</td><td>49.67</td><td>45.43</td><td>41.25</td><td>38.44</td><td>35.80</td><td>33.45</td><td>33.33</td><td>32.15</td><td>30.86</td><td>18.81</td><td></td></tr><tr><td>NC-FSCIL*</td><td>49.89</td><td>46.52</td><td>42.08</td><td>41.23</td><td>37.56</td><td>36.49</td><td>34.45</td><td>34.21</td><td>32.42</td><td>17.47</td><td></td></tr><tr><td>OrCo*</td><td>49.33</td><td>45.14</td><td>39.25</td><td>35.56</td><td>33.20</td><td>30.91</td><td>30.83</td><td>27.54</td><td>25.86</td><td>23.47</td><td></td></tr><tr><td>PA (ours)</td><td>49.67</td><td>46.75</td><td>41.75</td><td>41.00</td><td>37.68</td><td>36.67</td><td>34.33</td><td>34.57</td><td>32.14</td><td>17.53</td><td></td></tr></table>",
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+        "text": "This dataset exhibits high diversity, making it a challenging benchmark for the few-shot incremental learning task.",
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2025/Prototype antithesis for biological few-shot class-incremental learning/4e47f23f-6c01-419d-aa9c-fb65c25d5341_origin.pdf

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2025/Prototype antithesis for biological few-shot class-incremental learning/full.md

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+# PROTOTYPE ANTITHESIS FOR BIOLOGICAL FEW-SHOT CLASS-INCREMENTAL LEARNING
+
+Binghao Liu $^{1,2*}$ , Han Yang $^{1,2}$ , Fang Wan $^{3*}$ , Fei Gu $^{1,2}$
+
+$^{1}$ DAMO Academy, Alibaba Group, Hangzhou, 310023, China  
+$^{2}$ Hupan Lab, Hangzhou, 310023, China  
+<sup>3</sup>University of Chinese Academy of Sciences, Beijing, 101408, China
+
+# ABSTRACT
+
+Deep learning has become essential in the biological species recognition task. However, a significant challenge is the ability to continuously learn new or mutated species with limited annotated samples. Since species within the same family typically share similar traits, distinguishing between new and existing (old) species during incremental learning often faces the issue of species confusion. This can result in "catastrophic forgetting" of old species and poor learning of new ones. To address this issue, we propose a Prototype Antithesis (PA) method, which leverages the hierarchical structures in biological taxa to reduce confusion between new and old species. PA operates in two steps: Residual Prototype Learning (RPL) and Residual Prototype Mixing (RPM). RPL enables the model to learn unique prototypes for each species alongside residual prototypes representing shared traits within families. RPM generates synthetic samples by blending features of new species with residual prototypes of old species, encouraging the model to focus on species-unique traits and minimize species confusion. By integrating RPL and RPM, the proposed PA method mitigates "catastrophic forgetting" while improving generalization to new species. Extensive experiments on CUB200, PlantVillage, and Tree-of-Life datasets demonstrate that PA significantly reduces inter-species confusion and achieves state-of-the-art performance, highlighting its potential for deep learning in biological data analysis.
+
+# 1 INTRODUCTION
+
+With advances in data science and computing, computer techniques are crucial in fields like biology, offering solutions for species recognition and evolutionary analysis. However, limited data on newly discovered or mutated species poses a challenge, as data annotation requires expertise and is costly. Additionally, similar traits among species within the same family complicate recognition. These issues—data scarcity and species similarity—can cause confusion between new and existing species during continual learning, leading to poor performance on new species and forgetting of old ones.
+
+To overcome this, we introduce the few-shot class-incremental learning paradigm to biological research, allowing models to be trained on existing data and updated with only a few samples from new species. This approach enables continuous learning from new species while preserving knowledge of previously learned ones, addressing data limitations, and improving scalable species recognition. It underscores the vital role of advanced deep learning in advancing biological discovery.
+
+Few-shot class-incremental learning (FSCIL) typically faces two key challenges: "catastrophic forgetting" and "over-fitting". These issues arise when models must learn new classes with few samples and no access to previous data. Early methods address forgetting through knowledge distillation, where the old model's outputs guide the new model. However, when training samples for new classes are limited, the model becomes more prone to over-fitting. Recent approaches Zhang et al. (2021); Zhou et al. (2022) mitigate forgetting by freezing the feature extractor and selectively updating classifier weights, preserving old class performance. However, in biological contexts, distinguishing between new and previously learned species presents an additional challenge due to their
+
+![](images/739be946002fbeb2d27c1059deed1c7e09d322010ad1500bf2ca8882ac7d4a2b.jpg)  
+Figure 1: Comparison of prior works and the Prototype Antithesis (PA) method. In incremental learning, when similar new species resemble existing ones, their prototypes tend to be close, leading to confusion during classification (upper). In contrast, our proposed PA addresses this issue by decomposing the prototypes into two parts: the unique prototype specific to a species and the residual prototype shared within its family. The residual prototype is employed in the residual prototype mixing for new species learning, which maximizes the distance between similar species prototypes, thereby reducing classification confusion. (Best viewed in color)
+
+close evolutionary relationships. The similar traits of species within the same family make it difficult for classifiers to differentiate between them. This complicates learning new classes and increases inter-species confusion, reducing performance on previously learned species (Fig. 1, prior works).
+
+In this study, we propose a Prototype Antithesis (PA) method to address these challenges. PA encourages the model to learn both the most discriminative prototypes unique to individual species and the residual prototypes shared by species within the same family. During incremental learning, PA generates new feature samples for model learning by combining residual prototypes from previously learned species with features from new species. This process not only helps the classifier better distinguish between closely related new and old species, but also mitigates over-fitting caused by the limited data available for new species, Fig. 1 (PA).
+
+The PA method operates in two key steps: Residual Prototype Learning (RPL) and Residual Prototype Mixing (RPM). RPL is conducted during base training and can be further divided into two phases. In phase one, images are assigned with "species-level" labels, and those images are used for classification training by calculating the cosine distance between the features and a set of randomly initialized prototypes. After training, those prototypes converge to represent the most discriminative semantics of the corresponding species. In phase two, the species-unique and family-shared semantics are decoupled by a bi-directional optimization process. Specifically, we first compute the residuals between the features and their corresponding prototypes. Then these residual features are assigned with "family-level" labels for training, while the original features are classified by species labels. This bi-directional classification procedure ensures the convergence of species-unique and family-shared traits into their respective prototypes, referred to as species prototypes and family (residual) prototypes. RPM combines the residual prototypes of previously learned species with the new species features using a feature-mixing strategy to generate synthetic feature samples. These synthetic feature samples are labeled as their corresponding new species and are used to facilitate model learning. To classify these synthetic features accurately, the model must capture distinctive traits that differentiate them from residual prototypes and previously learned species, resulting in a well-separated decision boundary. Additionally, the synthetic features act as augmented data, increasing the number of training samples and helping to reduce over-fitting on the new species. Comprehensive experiments conducted on Tree-of-Life Stevens et al. (2024), as well as on the PlantVillage Hughes & Salathé (2015) and CUB200 P. et al. (2010) datasets, demonstrate that our approach achieves state-of-the-art performance.
+
+The contributions of this study are listed below:
+
+- We propose a novel Prototype Antithesis (PA) method to address the challenges of species confusion and over-fitting in few-shot class-incremental learning (FSCIL), particularly within the biological domain.  
+- We introduce Residual Prototype Learning (RPL) and Residual Prototype Mixing (RPM) to enable the model to capture both species-unique and family-shared traits, focusing on distinguishing between similar new and old species while augmenting training samples.  
+- Our proposed PA method achieves state-of-the-art performance across both computer vision and biological benchmarks, demonstrating its superiority in addressing the unique challenges of FSCIL.
+
+# 2 RELATED WORKS
+
+Biological Recognition. Biological data, characterized by its hierarchical category structure, is frequently used in fine-grained classification tasks, such as those involving the CUB200 dataset P. et al. (2010). These tasks are particularly difficult because subspecies within the same family often display highly similar appearances, making it challenging for models to differentiate them, especially when new species are constantly being discovered. Moreover, these newly identified species often come with limited samples, hindering the model's ability to learn effectively. While recent work like Bio-CLIP Stevens et al. (2024) has introduced foundation models for biological research, it struggles with continual learning and tend to misclassify closely related species.
+
+Few-shot Learning. The goal is to learn new classes with limited training samples. Existing methods have made significant progress through metric learning, meta-learning, and data augmentation. Metric learning Vinyals et al. (2016); Snell et al. (2017); Sung et al. (2018); Zhang et al. (2020); Liu et al. (2021); Yang et al. (2021a); Liu et al. (2021); Li et al. (2021) maps new class features into a representation space learned from base classes with sufficient data. Meta-learning Finn et al. (2017); Elsken et al. (2020); Sun et al. (2019) develops optimization strategies to enable models to generalize to new classes with limited data. Data augmentation Zhang et al. (2019); Li et al. (2020); Kim et al. (2020); Yang et al. (2021b) generates synthetic samples to reduce overfitting and support new class learning. However, these methods mainly focus on learning new classes and often overlook the problem of forgetting old classes during incremental learning. GFSL Schonfeld et al. (2019) addresses this by aligning old and new class distributions to mitigate forgetting, but it struggles with large distribution gaps and risks class confusion when old and new classes are highly similar.
+
+Incremental Learning. Incremental learning can be divided into task-incremental and class-incremental learning, with the key difference being that task-incremental learning requires task IDs, while class-incremental does not Masana et al. (2020). Existing methods fall into three categories: memory replay, model regularization, and architecture configuration methods. Memory replay Rebuffi et al. (2017); Chaudhry et al. (2018) methods store old class samples to prevent forgetting. Model regularization Li & Hoiem (2018); Dhar et al. (2019) methods add terms to the loss function to reduce forgetting. Architecture configuration Serrà et al. (2018); Mallya & Lazebnik (2018) methods use techniques like model pruning to adjust the feature space. However, these approaches still face over-fitting issues, particularly with limited training samples for new classes.
+
+Few-shot Class-Incremental Learning. Under few-shot class-incremental learning (FSCIL) settings, the model is first trained on base classes with sufficient data and then required to generalize to new classes with only a few samples. The primary challenges of FSCIL include not only "catastrophic forgetting" but also the issue of "over-fitting" due to limited training data. Feature topology preservation methods Tao et al. (2020); Zhang et al. (2021) maintain the feature topology of old classes using techniques like neural gas or decoupled training strategies to prevent forgetting. Network regularization methods Akyurek et al. (2021); Kang et al. (2022) align the weight vectors of new classes with those of old classes or selectively update model weights to learn new classes without degrading the performance of old classes. Data generation methods Cheraghian et al. (2021); Liu et al. (2023) generate synthetic feature samples to alleviate over-fitting during incremental learning. However, the confusion between new and old classes remains an unsolved problem in FSCIL. Since the model can only access data from individual incremental sessions, it struggles to learn well-separated decision boundaries between old and new classes, especially when these classes are similar. This confusion can lead to both forgetting of old classes and poor learning of new ones, which is the problem this study aims to address.
+
+![](images/02492e9b54882bd54929b68b58b6dc51f8a19e30c1d65ec349b2ce1e06080908.jpg)  
+Figure 2: The proposed Prototype Antithesis (PA) method consists of two components: RPL and RPM. RPL has two phases: in phase one, images are encoded by the feature extractor  $\theta_{b}$  and classified using "species-level" labels; in phase two, residuals between the base (old) species features and species prototypes are calculated and classified using "family-level" labels, whereas the original features continue to be classified by "species-level" labels. In RPM, these residual prototypes are mixed with new species features to generate pseudo-samples, optimizing species prototypes and refining the decision boundary. (Best viewed in color)
+
+![](images/1630ed7bd5b8e87a22c4ca82dc4c47878c8727f17d7b66e1fb9dfd9fda60d1e9.jpg)
+
+# 3 PRELIMINARY
+
+Few-shot class-incremental learning can be broadly divided into two stages: the base training stage and the incremental learning stage. During the base training stage, the model is trained using base classes  $\mathcal{C}^{(0)}$ , which have sufficient training data. In the incremental learning stage, the new classes are divided into  $T$  groups, denoted as  $\mathcal{C}^{(i)}$ , where  $i = 1,2,\dots,T$ , corresponding to the  $T$  incremental sessions. In the  $i^{\mathrm{th}}$  incremental session, only the dataset  $\mathcal{D}^{(i)}$  for classes  $\mathcal{C}^{(i)}$  is available for model learning. However, during evaluation, the model is required to classify both old and new classes from the combined set  $\{\mathcal{C}^{(0)},\ldots ,\mathcal{C}^{(i)}\}$ . Importantly, for any  $i_1\neq i_2$ , we have  $\mathcal{C}^{(i_1)}\cap \mathcal{C}^{(i_2)} = \emptyset$ , meaning that the classes in different incremental sessions are mutually exclusive. This setup imposes the challenge that the model must learn new classes without forgetting previously learned ones, despite not having access to the data from older classes during the incremental sessions.
+
+# 4 METHODOLOGY
+
+In this section, we introduce the proposed PA method. First, we outline the flowchart of the base training and incremental learning procedures. Next, we provide detailed explanations of the residual prototype learning and residual prototype mixing techniques. Finally, we present a theoretical analysis of the proposed prototype antithesis method.
+
+# 4.1 OVERVIEW
+
+The structure of the proposed PA method is illustrated in Fig. 2. The model consists of a feature extractor  $\theta_{b}$ , species prototypes  $\theta_{c} = \{\pmb{\mu}_{c_{0}},\pmb{\mu}_{c_{1}},\pmb{\mu}_{c_{2}},\dots \}$ , and residual prototypes  $\theta_r = \{\pmb{\mu}_{r_0},\pmb{\mu}_{r_1},\pmb{\mu}_{r_2},\dots \}$ . We employ ResNet-18 as the feature extractor, which encodes images into feature vectors. The prototypes,  $\theta_{c}$  and  $\theta_{r}$ , act as classifiers, using cosine distance between the features and prototypes for classification. During base training, the feature extractor and prototypes are optimized using the residual prototype learning strategy. In the incremental learning phase, the species prototypes are further refined through residual prototype mixing, ensuring a well-separated decision boundary between new and existing species. During the inference stage, the features of both old and new species are classified solely by the prototypes, without incorporating the residual.
+
+# 4.2 RESIDUAL PROTOTYPE LEARNING
+
+In standard classification tasks, most existing methods focus on learning the most discriminative representations, allowing classifiers to establish well-separated decision boundaries when all classes
+
+belong to a closed set. However, in few-shot class-incremental learning (FSCIL), where new classes are introduced incrementally, previously learned decision boundaries may struggle to distinguish between new and old classes. For example, when new and old species belong to the same family (e.g., apple leaf and peach leaf), the classifier may confuse them due to their shared "leaf" characteristics.
+
+To address this issue, the model needs to differentiate between "shared" and "unique" traits of species, which is the goal of the proposed Residual Prototype Learning (RPL), Fig. 2 (left). RPL operates in two phases, in phase one, given training samples  $\mathbf{X}^{(0)}$  from the base species, the feature extractor  $\theta_{b}$  and species prototypes  $\theta_{c}$  are jointly trained to classify the base species. The training objective for phase one is formulated as follows:
+
+$$
+\arg \min  \mathcal {L} (\mathbf {Y} ^ {(0)}, \hat {\mathbf {Y}} ^ {(0)}; \boldsymbol {\theta} _ {b}, \boldsymbol {\theta} _ {c})
+$$
+
+$$
+\begin{array}{l} \boldsymbol {\theta} _ {b}, \boldsymbol {\theta} _ {c} \\ \text {s . t .} \quad \hat {\mathbf {Y}} ^ {(0)} = \cos (\mathbf {F} ^ {(0)}; \boldsymbol {\theta} _ {c}), \quad \mathbf {F} ^ {(0)} = f (\mathbf {X} ^ {(0)}; \boldsymbol {\theta} _ {b}), \end{array} \tag {1}
+$$
+
+where  $\mathcal{L}$  is the classification loss,  $f$  represents the feature extraction process,  $\cos$  calculates the cosine similarity between the feature and each prototype individually, resulting in a set of scores, and  $\mathbf{Y}^{(0)}$  are the "species-level" labels. After training, the learned species prototypes capture the most discriminative features of the base species. In phase two, for any base species  $i$ , given the features  $\mathbf{F}_i^{(0)}$  and the species prototype  $\pmb{\mu}_{c_i}$  (where  $\pmb{\mu}_{c_i} \in \theta_c$ ), the residual features are computed as  $\mathbf{R}_i = \mathbf{F}_i^{(0)} - \pmb{\mu}_{c_i}$ . Here,  $\mathbf{R}_i$  contains both secondary discriminative features and background features, with the secondary discriminative features likely representing traits shared across the family. To decouple "family-level" traits from background noise and the species prototypes (since the most discriminative features might also include some "family-level" characteristics), we assign "family-level" labels to the residual features and classify them by calculating the cosine distance between the residual features and the prototypes. Meanwhile, the original features  $\mathbf{F}_i^{(0)}$  are classified based on their "species-level" labels. The optimization objective for base species  $i$  in phase two is as:
+
+$$
+\begin{array}{l} \underset {\pmb {\theta} _ {r}, \pmb {\theta} _ {c}} {\arg \min} \left[ \mathcal {L} (\mathbf {Y} _ {i} ^ {r}, \hat {\mathbf {Y}} _ {i} ^ {r}; \pmb {\theta} _ {r}, \pmb {\theta} _ {c}) + \mathcal {L} (\mathbf {Y} _ {i} ^ {(0)}, \hat {\mathbf {Y}} _ {i} ^ {(0)}; \pmb {\theta} _ {r}, \pmb {\theta} _ {c}) \right] \\ \begin{array}{l l} \text {s . t .} & \hat {\mathbf {Y}} _ {i} ^ {r} = \cos (\mathbf {R} _ {i}; \boldsymbol {\theta} _ {c}, \boldsymbol {\theta} _ {r}), \quad \hat {\mathbf {Y}} _ {i} ^ {(0)} = \cos (\mathbf {F} _ {i} ^ {(0)}; \boldsymbol {\theta} _ {c}, \boldsymbol {\theta} _ {r}), \end{array} \\ \end{array}
+$$
+
+where  $\mathbf{Y}_i^r$  and  $\mathbf{Y}_i^{(0)}$  denote the "family-level" and "species-level" labels, respectively. The bidirectional optimization process defined in Eq. 2 ensures that the residual prototypes  $\theta_r$  converge towards "family-level" traits, while simultaneously driving the species prototypes  $\theta_c$  to focus on "species-unique" traits. These prototypes are jointly optimized to refine the decision boundary, effectively distinguishing between traits that are unique to a species and those shared across the family. Please note that if a single species is restricted to a specific environment during base training, environmental features might be misinterpreted as species-specific traits. This issue can be resolved by enhancing the diversity of the data used during the base training phase.
+
+# 4.3 RESIDUAL PROTOTYPE MIXING
+
+Inspired by human's reference-based learning Kriegeskorte & Douglas (2018), we propose Residual Prototype Mixing (RPM), which drives the model to learn new species by referencing existing knowledge, Fig. 2 (right). The residual prototypes, optimized by Eq. 2, represent family-shared traits likely to appear in closely related species during new species learning. To address this potential confusion, we mix the residual prototypes of old species with new species features to encourage the model to capture the unique characteristics of the new species.
+
+Specifically, in the  $t^{\mathrm{th}}$  incremental session, given the residual prototypes  $\pmb{\theta}_r$  and features  $\mathbf{F}^{(t)}$ , the features of the  $k^{\mathrm{th}}$  new species are represented as  $\mathbf{F}_k^{(t)}$ , where this species belongs to the  $j^{\mathrm{th}}$  family. We generate synthetic feature samples as  $\mathbf{S}_k^{(t)} = \mathbf{F}_k^{(t)} + \pmb{\mu}_{r_j}$ , and these synthetic feature samples are assigned their original "species-level" labels for classification, with the optimization process for the  $k^{\mathrm{th}}$  species formulated as:
+
+$$
+\underset {\boldsymbol {\theta} _ {c}} {\arg \min} \mathcal {L} \left(\mathbf {Y} _ {k} ^ {(t)}, \hat {\mathbf {Y}} _ {k} ^ {(t)}; \boldsymbol {\theta} _ {c}, \boldsymbol {\theta} _ {r}\right)
+$$
+
+$$
+\mathrm {s . t .} \hat {\mathbf {Y}} _ {k} ^ {(t)} = \cos \left(\mathbf {S} _ {k} ^ {(t)}; \pmb {\theta} _ {c}, \pmb {\theta} _ {r}\right),
+$$
+
+where  $\mathbf{Y}_k^{(t)}$  denotes the "species-level" labels. To explicitly explain the optimization process of Eq. 3, we decompose the new species features into species-unique and the other (family-shared and background) features:  $\mathbf{F}_k^{(t)} = \mathbf{F}_{ku}^{(t)} + \mathbf{F}_{ko}^{(t)}$ , where  $\mathbf{F}_{ku}^{(t)}$  and  $\mathbf{F}_{ko}^{(t)}$  are orthogonal, meaning they represent non-overlapping features. Consequently, the synthetic features are expressed as:  $\mathbf{S}_k^{(t)} = \left(\mathbf{F}_{ko}^{(t)} + \boldsymbol{\mu}_{r_j}\right) + \mathbf{F}_{ku}^{(t)} = \mathbf{F}_{ko}^{\prime (t)} + \mathbf{F}_{ku}^{(t)}$ , where  $\mathbf{F}_{ko}^{\prime (t)} = \mathbf{F}_{ko}^{(t)} + \boldsymbol{\mu}_{r_j}$  represents the other features enhanced with family-shared traits. For the new species prototypes, the limited training data makes it challenging for the model to learn accurate prototypes. Consequently, these prototypes often contain coarse semantic information, combining both the most and second most discriminative features, which are decomposed into family-shared (residual) and species-unique components:  $\boldsymbol{\mu}_{c_k} = \boldsymbol{\mu}_{r_j} + \boldsymbol{\mu}_{c_ku}$ . Thus, the cosine similarity between  $\mathbf{S}_k^{(t)}, \boldsymbol{\mu}_{c_k}$ , and  $\boldsymbol{\mu}_{r_j}$  are calculated as:
+
+$$
+\cos \left(\mathbf {S} _ {k} ^ {(t)}; \boldsymbol {\mu} _ {c _ {k}}\right) = \frac {\left(\mathbf {F} _ {k o} ^ {\prime (t)} + \mathbf {F} _ {k u} ^ {(t)}\right) \cdot \left(\boldsymbol {\mu} _ {r _ {j}} + \boldsymbol {\mu} _ {c _ {k} u}\right)}{\left\| \mathbf {S} _ {k} ^ {(t)} \right\| \cdot \left\| \boldsymbol {\mu} _ {c _ {k}} \right\|}, \cos \left(\mathbf {S} _ {k} ^ {(t)}; \boldsymbol {\mu} _ {r _ {j}}\right) = \frac {\left(\mathbf {F} _ {k o} ^ {\prime (t)} + \mathbf {F} _ {k u} ^ {(t)}\right) \cdot \boldsymbol {\mu} _ {r _ {j}}}{\left\| \mathbf {S} _ {k} ^ {(t)} \right\| \cdot \left\| \boldsymbol {\mu} _ {r _ {j}} \right\|}. \tag {4}
+$$
+
+To avoid misclassifying  $\mathbf{S}_k^{(t)}$  as the  $j^{\mathrm{th}}$  family label, the difference term of Eq. 4  $\Delta \cos = \cos (\mathbf{S}_k^{(t)};\pmb {\mu}_{c_k}) - \cos (\mathbf{S}_k^{(t)};\pmb {\mu}_{r_j})$  must be maximized. As cosine similarity depends only on the angle between vectors, we assume  $\pmb{\mu}_{c_k}$  and  $\pmb{\mu}_{r_j}$  are unit vectors, and the difference term simplifies to*:  $\Delta \cos = \frac{\mathbf{F}_{ku}^{(t)}\cdot\pmb{\mu}_{c_ku}}{\left\|\mathbf{S}_k^{(t)}\right\|}$ . Since only  $\pmb{\mu}_{c_ku}$  is the optimization target, maximizing  $\Delta$  cos equals maximizing  $\mathbf{F}_{ku}^{(t)}\cdot \pmb{\mu}_{c_ku}$ , which enhances the alignment between the unique features of the new species and its prototype, improving the model's ability to focus on distinctive features. Furthermore, we mix new species features with all previously learned residual prototypes to generate additional synthetic features. The features mixed with unrelated family (residual) prototypes are then used as augmented training data to help reduce model over-fitting.
+
+# 4.4 THEORETICAL ANALYSIS
+
+We analyze Prototype Antithesis (PA) from the perspective of prototype independence and sample concentration. Given prototypes  $\pmb{\theta}_{c} = \{\pmb{\mu}_{c_{0}},\pmb{\mu}_{c_{1}},\pmb{\mu}_{c_{2}},\dots \}$  including both old and new species, we define the class separability (CS) as:
+
+$$
+\mathrm {C S} = \frac {\left\| \boldsymbol {\mu} _ {c _ {i}} - \boldsymbol {\mu} _ {c _ {j}} \right\| _ {2} ^ {2}}{\sum_ {m = 1} ^ {N _ {i}} \left(1 - \cos \left(\mathbf {f} _ {m} ; \boldsymbol {\mu} _ {c _ {i}}\right)\right) + \sum_ {v = 1} ^ {N _ {j}} \left(1 - \cos \left(\mathbf {f} _ {v} ; \boldsymbol {\mu} _ {c _ {j}}\right)\right)}, \tag {5}
+$$
+
+where  $i$  and  $j$  denote any two different species, while  $\mathbf{f}_m$  and  $\mathbf{f}_v$  represent their feature samples. In this formulation, the numerator measures the independence or separation of the species prototypes  $\pmb{\mu}_{c_i}$  and  $\pmb{\mu}_{c_j}$  (unit vectors), while the denominator reflects the concentration of samples around their respective species prototype, quantifying how well the samples align with their class center. The goal of a classification model is to maximize class separability, where the large separation between species prototypes and tight concentration of samples around their respective prototypes leads to better classification performance. Without loss of generality, let us assume that the prototypes of an old species  $\pmb{\mu}_{c_i}$  and a new species  $\pmb{\mu}_{c_j}$  are related as  $\pmb{\mu}_{c_j} = \hat{\pmb{\mu}}_{c_i} + \delta$ ,  $\|\hat{\pmb{\mu}}_{c_i}\| \in [0,1]$ , where  $\hat{\pmb{\mu}}_{c_i}$  represents the similar parts with  $\pmb{\mu}_{c_i}$ , and  $\delta$  is orthogonal to  $\pmb{\mu}_{c_i}$  and  $\hat{\pmb{\mu}}_{c_i}$ , representing the unique parts. Substituting this into Equation 5, we can rewrite the class separability as:
+
+$$
+\mathrm {C S} = \frac {1 + \delta^ {2} - \hat {\boldsymbol {\mu}} _ {c _ {i}} \cdot (2 - \hat {\boldsymbol {\mu}} _ {c _ {i}})}{\sum_ {m = 1} ^ {N _ {i}} (1 - \cos (\mathbf {f} _ {m} ; \boldsymbol {\mu} _ {c _ {i}})) + \sum_ {v = 1} ^ {N _ {j}} (1 - \cos (\mathbf {f} _ {v} ; \boldsymbol {\mu} _ {c _ {j}}))}. \tag {6}
+$$
+
+While all methods minimize the denominator using classification loss, the key difference lies in the numerator. The term  $\hat{\mu}_{c_i}(2 - \hat{\mu}_{c_i})$  is monotonically increasing over the interval [0, 1]. In conventional approaches, when the new species prototype  $\pmb{\mu}_{c_j}$  is similar to the old one  $\pmb{\mu}_{c_i}$ ,  $\hat{\mu}_{c_i}$  becomes large, indicating prototype overlap and reducing the numerator in Eq. 6. However, PA reduces this overlap by emphasizing species-unique features ( $\delta$ ), decreasing  $\hat{\mu}_{c_i}$ , thereby increasing the numerator in Eq. 6 and improving class separability.
+
+Table 1: Ablation experiments of the proposed "RPL" and "RPM". The baseline method is a standard classification network consisting of a feature extractor and a classifier. "Meanold" and "Meannew" denote the average accuracy of old and new species across all incremental sessions.  
+
+<table><tr><td rowspan="2">Baseline</td><td rowspan="2">RPL</td><td rowspan="2">RPM</td><td colspan="5">Accuracy in each session (%)</td><td rowspan="2">Meanold</td><td rowspan="2">Meannew</td><td rowspan="2">PD↓</td></tr><tr><td>0</td><td>1</td><td>2</td><td>3</td><td>4</td></tr><tr><td>✓</td><td></td><td></td><td>99.47</td><td>93.83</td><td>90.76</td><td>90.47</td><td>86.21</td><td>95.63</td><td>81.58</td><td>13.26</td></tr><tr><td>✓</td><td>✓</td><td></td><td>99.47</td><td>94.21</td><td>93.56</td><td>92.12</td><td>88.86</td><td>96.89</td><td>84.17</td><td>10.61</td></tr><tr><td>✓</td><td>✓</td><td>✓</td><td>99.50</td><td>95.83</td><td>93.45</td><td>93.24</td><td>90.51</td><td>98.47</td><td>86.59</td><td>8.99</td></tr></table>
+
+# 5 EXPERIMENTS
+
+We conduct comprehensive experiments on several public datasets for model analysis, and compare our proposed method with the state-of-the-art methods under FSCIL settings.
+
+# 5.1 EXPERIMENT SETTINGS
+
+Datasets. We adopt CUB200 P. et al. (2010), PlantVillage Hughes & Salathé (2015), and Tree-of-Life Stevens et al. (2024) as benchmarks. In the FSCIL task, classes are split into base and incremental categories. Under the  $N$ -way  $K$ -shot setting, the incremental classes are divided into  $T$  groups, corresponding to the  $T$  incremental sessions. In each session,  $N$  classes are presented for training, with each class containing  $K$  samples. For the CUB200 dataset, 100 classes are used as base classes, and the remaining 100 classes are divided into 10 sessions ( $T = 10$ ), with each session containing 10 classes (10-way) and 5 samples per class (5-shot). In the PlantVillage dataset, 19 classes serve as base classes, while the remaining 20 classes form incremental classes under a 5-way 5-shot setting. For the Tree-of-Life dataset, we randomly sample 100 classes for evaluation, where 60 are base classes, and the remaining 40 are split into groups under the 5-way 5-shot setting.
+
+Model Training. To maintain consistency with prior work, we use a ResNet-18 pretrained on ImageNet as the feature extractor for the CUB-200 dataset, while for the PlantVillage and Tree-of-Life datasets, we also use ResNet-18 as the backbone. The optimizer is SGD, with a "milestone" learning rate decay strategy. For data pre-processing, we apply image normalization, random cropping, random resizing, and horizontal flipping for data augmentation. During base training, we use a batch size of 128 and an initial learning rate of 0.004, training for 100 epochs. In the incremental learning phase, the classifier is optimized over 100 epochs with a learning rate of 0.01. We conducted 5 experiments with different seeds on the fixed datasets and reported the averaged results to minimize randomness. All experiments were performed using Pytorch 1.11.0 on Nvidia A800 GPUs.
+
+Evaluation. We adopt "classification accuracy" and "performance drop (PD)" as the evaluation metrics. "Classification accuracy" validates the discriminating ability of the model, and it is defined as accuracy =  $\frac{\mathrm{TP} + \mathrm{TN}}{\mathrm{TP} + \mathrm{TN} + \mathrm{FP} + \mathrm{FN}}$ , where TP, TN, FP and FN denote the numbers of true positives, true negatives, false positives, and false negatives, respectively. "PD" measures the model forgetting and is defined as PD = accuracy(0) - accuracy(T), where accuracy(0) and accuracy(T) denote the model performance in the 0-th and T-th sessions, respectively.
+
+# 5.2 ABLATION STUDY
+
+To rigorously assess the effectiveness of our proposed approach, we perform ablation experiments on the PlantVillage dataset.
+
+Residual Prototype Learning. As seen in Table 1, the proposed RPL increases performance on old species by  $1.26\%$  and on new species by  $2.59\%$  over the baseline. This is attributed to the model's ability to classify both unique and shared features of the base species, resulting in
+
+a more compact decision boundary that supports learning in both old and new species.
+
+Table 2: Comparison of "CAM", "Grad-CAM" and the proposed "RPL".  
+
+<table><tr><td>Method</td><td>Meanold</td><td>Meannew</td><td>Similarity</td></tr><tr><td>CAM</td><td>95.31%</td><td>81.25%</td><td>6.53%</td></tr><tr><td>Grad-CAM</td><td>95.32%</td><td>79.82%</td><td>8.15%</td></tr><tr><td>RPL</td><td>98.47%</td><td>86.59%</td><td>1.89%</td></tr></table>
+
+![](images/2c6a074a1f14517512112067efc6314147e1b798844598d734e493429807ccbd.jpg)  
+Without Prototype Antithesis (Baseline)
+
+![](images/ab0bafd92c6fc7e75ca61ec48cb19cbf8365212220fc2f3bebe35aff32a45154.jpg)  
+With Prototype Antithesis (Ours)  
+Figure 3: t-SNE visualization of feature distributions comparing models with and without Prototype Antithesis (PA). Without PA, the features of new species overlap with those of similar old species. In contrast, with PA, the new species features are pushed away from the residual features, effectively distinguishing the new and old species and creating a well-separated decision boundary. (Best viewed in color)
+
+Residual Prototype Mixing. Table 1 demonstrates that combining RPL with RPM boosts old and new species performance by  $2.84\%$  and  $5.01\%$ , respectively, and decreases "PD" by  $4.37\%$ . RPM reduces the overlap of shared components between new and old species, effectively minimizing species confusion. Together, RPL and RPM work synergistically to reduce "old species forgetting" while enhancing "new species generalization".
+
+Feature decomposition strategy. To assess the effectiveness of different feature decomposition strategies, we use CAM Zhou et al. (2016) and Grad-CAM Selvaraju et al. (2017) to distinguish between unique and common features. Specifically, we set a threshold to extract the highest-response regions from the feature maps as unique features, while the secondary-high-response regions represent common features. As shown in Table 2, RPL achieves the highest mean accuracy and the lowest similarity between unique and common features, which can be attributed to its bi-directional optimization that more effectively decouples unique features from shared ones.
+
+# 5.3 MODEL ANALYSIS
+
+In this section, we further conduct comprehensive experiments for detailed statistical results and visualization.
+
+Feature distribution. We employ t-SNE to visualize the feature distributions of similar old and new species, as well as the residual feature distributions. As depicted in Fig.3(left), without the proposed prototype antithesis, feature samples of the old and new species overlap at the edges of the distribution. However, with the introduction of the prototype antithesis, the distance between the new species and the residual features is maximized, making the features of the new and old species more distinguishable (Fig.3(right)). This adjustment optimizes the decision boundary and reduces species confusion.
+
+![](images/266f23ee92ba95161d97aac2b83751063ae51606331ebee022c8ea69a8305429.jpg)  
+Figure 4: Confusion matrix of the baseline method and our proposed PA approach. Compared with the baseline method, PA significantly reduces the false prediction among old and new species, alleviating species confusion.
+
+Confusion Matrix. We plot the con
+
+fusion matrix for both the baseline method and our proposed PA method. As shown in Fig. 4, the PA method significantly reduces false classifications between old and new species. These results demonstrate that PA effectively minimizes confusion between new and old species, mitigating forgetting of old species while enhancing learning for new species.
+
+Table 3: Performance comparison on the CUB200 dataset. “*” indicates results that were re-implemented using the official code.  
+
+<table><tr><td rowspan="2">Method</td><td colspan="12">Accuracy in each session (%)</td><td>PD ↓</td></tr><tr><td>0</td><td>1</td><td>2</td><td>3</td><td>4</td><td>5</td><td>6</td><td>7</td><td>8</td><td>9</td><td>10</td><td></td><td></td></tr><tr><td>TOPIC Tao et al. (2020)</td><td>68.68</td><td>62.49</td><td>54.81</td><td>49.99</td><td>45.25</td><td>41.40</td><td>38.35</td><td>35.36</td><td>32.22</td><td>28.31</td><td>26.28</td><td>42.40</td><td></td></tr><tr><td>SPPR Zhu et al. (2021)</td><td>68.68</td><td>61.85</td><td>57.43</td><td>52.68</td><td>50.19</td><td>46.88</td><td>44.65</td><td>43.07</td><td>40.17</td><td>39.63</td><td>37.33</td><td>31.35</td><td></td></tr><tr><td>SFMS Cheraghian et al. (2021)</td><td>68.78</td><td>59.37</td><td>59.32</td><td>54.96</td><td>52.58</td><td>49.81</td><td>48.09</td><td>46.32</td><td>44.33</td><td>43.43</td><td>43.23</td><td>25.55</td><td></td></tr><tr><td>FSLL Mazumder et al. (2021)</td><td>72.77</td><td>69.33</td><td>65.51</td><td>62.66</td><td>61.10</td><td>58.65</td><td>57.78</td><td>57.26</td><td>55.59</td><td>55.39</td><td>54.21</td><td>17.38</td><td></td></tr><tr><td>CEC Zhang et al. (2021)</td><td>75.85</td><td>71.94</td><td>68.50</td><td>63.50</td><td>62.43</td><td>58.27</td><td>57.73</td><td>55.81</td><td>54.83</td><td>53.52</td><td>52.28</td><td>23.57</td><td></td></tr><tr><td>Meta-FSCIL Chi et al. (2022)</td><td>75.90</td><td>72.41</td><td>68.78</td><td>64.78</td><td>62.96</td><td>59.99</td><td>58.30</td><td>56.85</td><td>54.78</td><td>53.82</td><td>52.64</td><td>23.26</td><td></td></tr><tr><td>FACT Zhou et al. (2022)</td><td>75.90</td><td>73.23</td><td>70.84</td><td>66.13</td><td>65.56</td><td>62.15</td><td>61.74</td><td>59.83</td><td>58.41</td><td>57.89</td><td>56.94</td><td>18.96</td><td></td></tr><tr><td>SOFTNet Kang et al. (2023)</td><td>78.07</td><td>74.58</td><td>71.37</td><td>67.54</td><td>65.37</td><td>62.60</td><td>61.07</td><td>59.37</td><td>57.53</td><td>57.21</td><td>56.75</td><td>21.32</td><td></td></tr><tr><td>WaRP Kim et al. (2023)</td><td>77.74</td><td>74.15</td><td>70.82</td><td>66.90</td><td>65.01</td><td>62.64</td><td>61.40</td><td>59.86</td><td>57.95</td><td>57.77</td><td>57.01</td><td>20.73</td><td></td></tr><tr><td>BiDist Zhao et al. (2023)</td><td>79.12</td><td>74.99</td><td>70.87</td><td>67.30</td><td>65.89</td><td>63.45</td><td>61.40</td><td>60.11</td><td>58.61</td><td>58.23</td><td>57.48</td><td>21.64</td><td></td></tr><tr><td>NC-FSCIL Yang et al. (2023)</td><td>80.45</td><td>75.98</td><td>72.30</td><td>70.28</td><td>68.17</td><td>65.16</td><td>64.43</td><td>63.25</td><td>60.66</td><td>60.01</td><td>59.44</td><td>21.01</td><td></td></tr><tr><td>OrCo* Ahmed et al. (2024)</td><td>74.72</td><td>66.06</td><td>64.72</td><td>62.88</td><td>61.89</td><td>59.69</td><td>58.95</td><td>59.03</td><td>57.24</td><td>57.79</td><td>57.18</td><td>17.54</td><td></td></tr><tr><td>TEEN Wang et al. (2024)</td><td>77.26</td><td>76.13</td><td>72.81</td><td>68.16</td><td>67.77</td><td>64.40</td><td>63.25</td><td>62.29</td><td>61.19</td><td>60.32</td><td>59.31</td><td>18.13</td><td></td></tr><tr><td>PA(ours)</td><td>78.69</td><td>75.59</td><td>72.71</td><td>68.71</td><td>68.37</td><td>65.77</td><td>64.75</td><td>63.59</td><td>62.76</td><td>62.02</td><td>61.19</td><td>17.50</td><td></td></tr></table>
+
+![](images/8fdcf1eecc46d60dd8c6154229b0c8e08ac50e7b4658fc1f826f457cd80c2f2b.jpg)  
+(a) Performance on PlantVillage  
+Figure 5: Performance on the PlantVillage and Tree-of-Life datasets.
+
+![](images/146fbee789e208d044539feb6e284df564619d77e172d2a4ef312cb4516c055f.jpg)  
+(b) Performance on Tree-of-Life
+
+# 5.4 COMPARISON WITH THE STATE-OF-THE-ART METHODS
+
+We compare PA against state-of-the-art methods on both computer vision benchmarks (e.g., CUB200) and biological datasets (e.g., PlantVillage and Tree-of-Life).
+
+CUB200: As shown in Table 3, our proposed PA outperforms the TEEN and OrCo by  $1.88\%$  and  $4.01\%$ , as well as achieving a lower "PD" compared to most other methods. PlantVillage: From Fig. 5(a) we can see that PA outperforms other methods with significant margins. The categories in the PlantVillage dataset (different leaves with disease) share many common traits, and the significant improvement in performance shows PA's superiority in distinguishing closely related old and new species. Tree-of-Life: PA outperforms OrCo Ahmed et al. (2024) and BiDist Zhao et al. (2023) by  $6.26\%$  and  $1.28\%$  on this challenging dataset, respectively, Fig. 5(b). This result confirms PA's robustness in handling high-diversity biological data.
+
+# 6 CONCLUSION
+
+In this paper, we propose a Prototype Antithesis (PA) method to address the issue of species confusion in biological few-shot class-incremental learning. The PA method comprises Residual Prototype Learning (RPL) and Residual Prototype Mixing (RPM). RPL decomposes species-unique and family-shared (residual) prototypes, while RPM generates synthetic samples by blending new species features with residual prototypes. Together, RPL and RPM optimize the decision boundary between old and new species, mitigating "old species forgetting" and enhancing "new species learning". PA offers a novel perspective for recognizing biological species and traits using few-shot class-incremental learning.
+
+# AUTHOR CONTRIBUTIONS
+
+Fei Gu served as the project manager. Binghao Liu proposed the idea of the paper, designed the methodology, conducted the experiments, and wrote the manuscript. Han Yang participated in discussions on experimental settings. Fang Wan participated in idea discussions and provided suggestions for figure visualization.
+
+# ACKNOWLEDGMENTS
+
+This work was financially supported in part by grants from the Biological Breeding - Major Projects (2023ZD04076).
+
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+Da-Wei Zhou, Fu-Yun Wang, Han-Jia Ye, Liang Ma, Shiliang Pu, and De-Chuan Zhan. Forward compatible few-shot class-incremental learning. In IEEE CVPR, pp. 9046-9056, 2022.  
+Kai Zhu, Yang Cao, Wei Zhai, Jie Cheng, and Zheng-Jun Zha. Self-promoted prototype refinement for few-shot class-incremental learning. In IEEE CVPR, pp. 6801-6810, June 2021.
+
+# A APPENDIX
+
+# A.1 DERIVATION OF  $\Delta$  cos.
+
+The  $\Delta$  cos in section 4.3 is calculated as below:
+
+$$
+\begin{array}{l} \Delta \cos = \cos \left(\mathbf {S} _ {k} ^ {(t)}, \mu_ {c _ {k}}\right) - \cos \left(\mathbf {S} _ {k} ^ {(t)}, \mu_ {r _ {j}}\right) \\ = \frac {\left(\mathbf {F} _ {k o} ^ {\prime (t)} + \mathbf {F} _ {k u} ^ {(t)}\right) \cdot \left(\mu_ {r _ {j}} + \mu_ {c _ {k} u}\right)}{\left\| \mathbf {S} _ {k} ^ {(t)} \right\| \cdot \left\| \mu_ {c _ {k}} \right\|} - \frac {\left(\mathbf {F} _ {k o} ^ {\prime (t)} + \mathbf {F} _ {k u} ^ {(t)}\right) \cdot \mu_ {r _ {j}}}{\left\| \mathbf {S} _ {k} ^ {(t)} \right\| \cdot \left\| \mu_ {r _ {j}} \right\|} \\ = \frac {\mathbf {F} _ {k o} ^ {\prime (t)} \cdot \mu_ {r j} + \mathbf {F} _ {k o} ^ {\prime (t)} \cdot \mu_ {c k u} + \mathbf {F} _ {k u} ^ {(t)} \cdot \mu_ {r j} + \mathbf {F} _ {k u} ^ {(t)} \cdot \mu_ {c k u} - \mathbf {F} _ {k o} ^ {\prime (t)} \cdot \mu_ {r j} - \mathbf {F} _ {k u} ^ {(t)} \cdot \mu_ {r j}}{\left\| \mathbf {S} _ {k} ^ {(t)} \right\| \cdot \left\| \mu_ {r j} \right\|} \tag {7} \\ = \frac {\mathbf {F} _ {k o} ^ {\prime (t)} \cdot \mu_ {c _ {k} u} + \mathbf {F} _ {k u} ^ {(t)} \cdot \mu_ {c _ {k} u}}{\left\| \mathbf {S} _ {k} ^ {(t)} \right\| \cdot \left\| \mu_ {r _ {j}} \right\|}. \\ \end{array}
+$$
+
+Since  $\mathbf{F}_{ko}^{\prime(t)}$  are orthogonal to  $\mu_{c_k u}$  (there are no overlap between them),  $\mathbf{F}_{ko}^{\prime(t)} \cdot \mu_{c_k u}$  is equal to 0, and  $\mu_{r_j}$  is unit vector, thus Eq. 7 can be rewrite as:
+
+$$
+\begin{array}{l} \Delta \cos = \cos (\mathbf {S} _ {k} ^ {(t)}, \mu_ {c _ {k}}) - \cos (\mathbf {S} _ {k} ^ {(t)}, \mu_ {r _ {j}}) \\ = \frac {\mathbf {F} _ {k u} ^ {(t)} \cdot \mu_ {c k u}}{\left\| \mathbf {S} _ {k} ^ {(t)} \right\|}. \tag {8} \\ \end{array}
+$$
+
+Table 4: Performance comparison on the PlantVillage dataset. “*” indicates results that were re-implemented using the official code.  
+
+<table><tr><td rowspan="2">Method</td><td colspan="5">Accuracy in each session (%)</td><td rowspan="2">PD ↓</td></tr><tr><td>0</td><td>1</td><td>2</td><td>3</td><td>4</td></tr><tr><td>Ft-CNN</td><td>99.89</td><td>82.33</td><td>36.11</td><td>34.38</td><td>28.53</td><td>71.47</td></tr><tr><td>CEC*</td><td>99.53</td><td>92.46</td><td>90.13</td><td>88.35</td><td>85.01</td><td>14.52</td></tr><tr><td>FACT*</td><td>99.47</td><td>93.33</td><td>90.69</td><td>90.00</td><td>86.15</td><td>13.32</td></tr><tr><td>TEEN*</td><td>99.50</td><td>90.83</td><td>90.56</td><td>86.38</td><td>85.19</td><td>14.31</td></tr><tr><td>BiDist*</td><td>99.51</td><td>91.25</td><td>84.14</td><td>81.18</td><td>78.97</td><td>20.54</td></tr><tr><td>NC-FSCIL*</td><td>99.48</td><td>96.05</td><td>93.59</td><td>92.88</td><td>89.13</td><td>10.35</td></tr><tr><td>OrCo*</td><td>99.58</td><td>84.25</td><td>75.24</td><td>64.77</td><td>60.82</td><td>38.76</td></tr><tr><td>PA (ours)</td><td>99.50</td><td>95.83</td><td>93.45</td><td>93.24</td><td>90.51</td><td>8.99</td></tr></table>
+
+This is consistent with the result presented in Section 4.3.
+
+# A.2 DERIVATION OF EQ. 6 IN THE PAPER.
+
+The class separability in section 4.4 is written as:
+
+$$
+\begin{array}{l} \mathbf {C S} = \frac {\| \mu_ {c _ {i}} - (\hat {\mu} _ {c _ {i}} + \delta) \| _ {2} ^ {2}}{\sum_ {m = 1} ^ {N _ {i}} (1 - \cos (\mathbf {f} _ {m} ; \mu_ {c _ {i}})) + \sum_ {v = 1} ^ {N _ {j}} (1 - \cos (\mathbf {f} _ {v} ; \mu_ {c _ {j}}))}, \\ = \frac {\mu_ {c _ {i}} ^ {2} + \hat {\mu} _ {c _ {i}} ^ {2} + \delta^ {2} + 2 \cdot \delta \cdot \hat {\mu} _ {c _ {i}} - 2 \cdot \mu_ {c _ {i}} \cdot (\hat {\mu} _ {c _ {i}} + \delta)}{\sum_ {m = 1} ^ {N _ {i}} (1 - \cos (\mathbf {f} _ {m} ; \mu_ {c _ {i}})) + \sum_ {v = 1} ^ {N _ {j}} (1 - \cos (\mathbf {f} _ {v} ; \mu_ {c _ {j}}))}. \\ \end{array}
+$$
+
+Since  $\delta$  is orthogonal to  $\pmb{\mu}_{c_i}$  and  $\hat{\pmb{\mu}}_{c_i}$ , and  $\mu_{c_i}$  is unit vector, Eq. 10 is rewritten as:
+
+$$
+\begin{array}{l} \mathbf {C S} = \frac {1 + \hat {\mu} _ {c _ {i}} ^ {2} + \delta^ {2} - 2 \cdot \mu_ {c _ {i}} \cdot \hat {\mu} _ {c _ {i}}}{\sum_ {m = 1} ^ {N _ {i}} (1 - \cos (\mathbf {f} _ {m} ; \mu_ {c _ {i}})) + \sum_ {v = 1} ^ {N _ {j}} (1 - \cos (\mathbf {f} _ {v} ; \mu_ {c _ {j}}))}, \\ = \frac {1 + \delta^ {2} - \hat {\mu} _ {c _ {i}} \cdot (2 - \hat {\mu} _ {c _ {i}})}{\sum_ {m = 1} ^ {N _ {i}} (1 - \cos (\mathbf {f} _ {m} ; \mu_ {c _ {i}})) + \sum_ {v = 1} ^ {N _ {j}} (1 - \cos (\mathbf {f} _ {v} ; \mu_ {c _ {j}}))}, \\ \end{array}
+$$
+
+which aligns with Eq. 6 in the paper.
+
+# A.3 PERFORMANCE ON THE PLANTVILLAGE DATASET
+
+We present the detailed classification accuracy on the PlantVillage dataset in Table 4. The results show that our method significantly outperforms other methods in the final session and achieves the smallest performance drop.
+
+# A.4 PERFORMANCE ON THE TREE-OF-LIFE DATASET
+
+The detailed results on the Tree-of-Life dataset are shown in Table 5. Our proposed PA method outperforms most methods while delivering comparable performance (slightly lower) than NC-FSCIL. Unlike NC-FSCIL, however, our method does not require prior knowledge of the total number of unknown classes, making it more scalable and practical.
+
+# B DATASETS STATISTICS
+
+CUB200. The CUB-200-2011 dataset contains 11,788 images of birds, categorized into 200 fine-grained subclasses (species) and 37 broader classes (families). PlantVillage. The PlantVillage dataset contains 54,306 images of plant leaves, categorized into 38 fine-grained subclasses (species) and 14 broader classes (families). The images include healthy and diseased leaves, making it a widely used benchmark for plant disease classification and recognition tasks. Tree-of-Life. We randomly sampled 13,487 images from 100 species in the Tree-of-Life dataset, spanning 42 families.
+
+Table 5: Performance comparison on the Tree-of-Life dataset. “*” indicates results that were re-implemented using the official code.  
+
+<table><tr><td rowspan="2">Method</td><td colspan="10">Accuracy in each session (%)</td><td>PD ↓</td></tr><tr><td>0</td><td>1</td><td>2</td><td>3</td><td>4</td><td>5</td><td>6</td><td>7</td><td>8</td><td></td><td></td></tr><tr><td>Ft-CNN</td><td>49.98</td><td>41.25</td><td>36.53</td><td>34.42</td><td>31.64</td><td>31.20</td><td>29.15</td><td>28.32</td><td>27.91</td><td>22.07</td><td></td></tr><tr><td>CEC*</td><td>49.68</td><td>43.35</td><td>39.02</td><td>37.71</td><td>35.56</td><td>34.87</td><td>32.41</td><td>32.21</td><td>29.33</td><td>20.35</td><td></td></tr><tr><td>FACT*</td><td>48.33</td><td>45.25</td><td>39.25</td><td>39.00</td><td>35.20</td><td>35.33</td><td>32.17</td><td>32.43</td><td>29.71</td><td>18.62</td><td></td></tr><tr><td>TEEN*</td><td>49.33</td><td>43.00</td><td>38.25</td><td>37.60</td><td>34.40</td><td>33.83</td><td>32.00</td><td>33.00</td><td>30.57</td><td>18.76</td><td></td></tr><tr><td>BiDist*</td><td>49.67</td><td>45.43</td><td>41.25</td><td>38.44</td><td>35.80</td><td>33.45</td><td>33.33</td><td>32.15</td><td>30.86</td><td>18.81</td><td></td></tr><tr><td>NC-FSCIL*</td><td>49.89</td><td>46.52</td><td>42.08</td><td>41.23</td><td>37.56</td><td>36.49</td><td>34.45</td><td>34.21</td><td>32.42</td><td>17.47</td><td></td></tr><tr><td>OrCo*</td><td>49.33</td><td>45.14</td><td>39.25</td><td>35.56</td><td>33.20</td><td>30.91</td><td>30.83</td><td>27.54</td><td>25.86</td><td>23.47</td><td></td></tr><tr><td>PA (ours)</td><td>49.67</td><td>46.75</td><td>41.75</td><td>41.00</td><td>37.68</td><td>36.67</td><td>34.33</td><td>34.57</td><td>32.14</td><td>17.53</td><td></td></tr></table>
+
+This dataset exhibits high diversity, making it a challenging benchmark for the few-shot incremental learning task.

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2025/Provable Benefit of Annealed Langevin Monte Carlo for Non-log-concave Sampling/b3206b81-2a8c-4a00-906e-6fa0ea9939cf_origin.pdf

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2025/Provable Benefit of Annealed Langevin Monte Carlo for Non-log-concave Sampling/full.md

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+# PROVABLE BENEFIT OF ANNEALED LANGEVIN MONTE CARLO FOR NON-LOG-CONCAVE SAMPLING
+
+Wei Guo, Molei Tao, Yongxin Chen
+
+Georgia Institute of Technology
+
+{wei.guo,mtao,yongchen}@gatech.edu
+
+# ABSTRACT
+
+We consider the outstanding problem of sampling from an unnormalized density that may be non-log-concave and multimodal. To enhance the performance of simple Markov chain Monte Carlo (MCMC) methods, techniques of annealing type have been widely used. However, quantitative theoretical guarantees of these techniques are under-explored. This study takes a first step toward providing a non-asymptotic analysis of annealed MCMC. Specifically, we establish, for the first time, an oracle complexity of  $\widetilde{O}\left(\frac{d\beta^2\mathcal{A}^2}{\varepsilon^6}\right)$  for the simple annealed Langevin Monte Carlo algorithm to achieve  $\varepsilon^2$  accuracy in Kullback-Leibler divergence to the target distribution  $\pi \propto \mathrm{e}^{-V}$  on  $\mathbb{R}^d$  with  $\beta$ -smooth potential  $V$ . Here,  $\mathcal{A}$  represents the action of a curve of probability measures interpolating the target distribution  $\pi$  and a readily sampleable distribution.
+
+# 1 INTRODUCTION
+
+We study the task of efficient sampling from a probability distribution  $\pi \propto \mathrm{e}^{-V}$  on  $\mathbb{R}^d$ . This fundamental problem is pivotal across various fields including computational statistics (Liu, 2004; Brooks et al., 2011), Bayesian inference (Gelman et al., 2013), statistical physics (Newman & Barkema, 1999), and finance (Dagpunar, 2007), and has been extensively studied in the literature (Chewi, 2024). The most common approach to this problem is Markov Chain Monte Carlo (MCMC), among which Langevin Monte Carlo (LMC) (Durmus et al., 2019; Vempala & Wibisono, 2019; Chewi et al., 2022; Mousavi-Hosseini et al., 2023) is a particularly popular choice. LMC can be understood as a time-discretization of a diffusion process, known as Langevin diffusion (LD), whose stationary distribution is the target distribution  $\pi$ , and has been attractive partly due to its robust performance despite conceptual simplicity.
+
+Although LMC and its variants converge rapidly when the target distribution  $\pi$  is strongly log-concave or satisfies isoperimetric inequalities such as the log-Sobolev inequality (LSI) (Durmus et al., 2019; Vempala & Wibisono, 2019; Chewi et al., 2022), its effectiveness diminishes when dealing with target distributions that are not strongly log-concave or are multimodal, such as mixtures of Gaussians. In such scenarios, the sampler often becomes confined to a single mode, severely limiting its ability to explore the entire distribution effectively. This results in significant challenges in transitioning between modes, which can dramatically increase the mixing time, making it exponential in problem parameters such as dimension, distance between modes, etc. (Dong & Tong, 2022; Ma et al., 2019). Such limitations highlight the need for enhanced MCMC methodologies that can efficiently navigate the complex landscapes of multimodal distributions, thereby improving convergence rates and overall sampling efficiency.
+
+To address the challenges posed by multimodality, techniques around the notion of annealing have been widely employed (Gelfand et al., 1990; Neal, 2001). The general philosophy involves constructing a sequence of intermediate distributions  $\pi_0, \pi_1, \dots, \pi_M$  that bridge the gap between an easily samplable distribution  $\pi_0$  (e.g., Gaussian or Dirac-like), and the target distribution  $\pi_M = \pi$ . The process starts with sampling from  $\pi_0$  and progressively samples from each subsequent distribution until  $\pi_M$  is reached. When  $\pi_i$  and  $\pi_{i+1}$  are close enough, approximate samples from  $\pi_i$  can serve as a warm start for sampling from  $\pi_{i+1}$ , thereby facilitating this transition. Employing LMC within this framework gives rise to what is known as the annealed LMC algorithm, which is the focus of
+
+our study. Despite its empirical success (Song & Ermon, 2019; 2020; Zilberstein et al., 2023; 2024), a thorough theoretical understanding of annealed LMC, particularly its non-asymptotic complexity bounds, remains elusive.
+
+In this work, we take a first step toward developing a non-asymptotic analysis of annealed MCMC. Utilizing the Girsanov theorem to quantify the differences between the sampling dynamics and a reference process, we derive an upper bound on the error of annealed MCMC, which consists of two key terms. The first term is the ratio of an action functional in Wasserstein geometry, induced by optimal transport, to the duration of the process. This term decreases when the annealing schedule is sufficiently slow. The second term captures the discretization error inherent in practical implementations. Our approach challenges the traditional view that annealed MCMC is simply a series of warm starts. Analyses based on this perspective typically require assumptions of log-concavity or isoperimetric inequalities. In contrast, our theoretical framework for annealed MCMC dispenses with these assumptions, marking a significant shift in the understanding and analysis of annealed MCMC.
+
+Contributions. Our key technical contributions are summarized as follows.
+
+- We propose a novel strategy to analyze the non-asymptotic complexity bounds of annealed MCMC algorithms, bypassing the need for assumptions such as log-concavity or isoperimetry.  
+- In Sec. 4, we investigate the annealed LD, which involves running LD with a dynamically changing target distribution. We derive a notable bound on the time required to simulate the SDE for achieving  $\varepsilon^2$ -accuracy in KL divergence.  
+- Building on the insights from the analysis of the continuous dynamic and incorporating discretization errors, we establish a non-asymptotic oracle complexity bound for annealed LMC in Sec. 5, which is applicable to a wide range of annealing schemes.
+
+Comparison of quantitative results. The quantitative results are summarized and compared to other sampling algorithms in Tab. 1. For algorithms requiring isoperimetric assumptions, we include Langevin Monte Carlo (LMC, Vempala & Wibisono (2019)) and Proximal Sampler (PS, Fan et al. (2023)), which converges rapidly but do not have theoretical guarantees without isoperimetric assumptions. For isoperimetry-free samplers, we include Simulated Tempering Langevin Monte Carlo (STLMC, Ge et al. (2018b)), a tempering-based sampler converging rapidly for a specific family of non-log-concave distributions, and three algorithms inspired by score-based generative models: Reverse Diffusion Monte Carlo (RDMC, Huang et al. (2024a)), Recursive Score Diffusion-based Monte Carlo (RS-DMC, Huang et al. (2024b)), and Zeroth Order Diffusion-Monte Carlo (ZOD-MC, He et al. (2024)), which involve simulating the time-reversal of the Ornstein-Uhlenbeck (OU) process and require estimating the scores of the intermediate distributions. Notably, our approach operates under the least stringent assumptions and exhibits the most favorable  $\varepsilon$ -dependence among all isoperimetry-free sampling methods.
+
+Related works. We provide a brief overview of the literature, mainly focusing on the algorithms for non-log-concave sampling and their theoretical analysis.
+
+1. Samplers based on tempering. The fundamental concept of tempering involves sampling the system at various temperatures simultaneously: at higher temperatures, the distribution flattens, allowing particles to easily transition between modes, while at lower temperatures, particles can more effectively explore local structures. In simulated tempering (Marinari & Parisi, 1992; Woodard et al., 2009), the system's temperature is randomly switched, while in parallel tempering (also known as replica exchange) (Swendsen & Wang, 1986; Lee & Shen, 2023), the temperatures of two particles are swapped according to a specific rule. However, quantitative theoretical results for tempering are limited, and the existing results (e.g., Ge et al. (2018a;b); Dong & Tong (2022)) apply primarily to certain special classes of non-log-concave distributions.  
+2. Samplers based on general diffusions. Inspired by score-based diffusion models (Ho et al., 2020; Song et al., 2021a;b), recent advances have introduced sampling methods that reverse the OU process, as detailed in Huang et al. (2024a;b); He et al. (2024). These samplers exhibit reduced sensitivity to isoperimetric conditions, but rely on estimating score functions (gradients of log-density)
+
+Table 1: Comparison of oracle complexities in terms of  $d$ ,  $\varepsilon$ , and the LSI constant for sampling from  $\pi \propto \mathrm{e}^{-V}$ . "poly(\cdot)" indicates a polynomial dependence on the specified parameters.  
+
+<table><tr><td>Algorithm</td><td>Isoperimetric Assumptions</td><td>Other Assumptions</td><td>Criterion</td><td>Complexity</td></tr><tr><td>LMC</td><td>C-LSI</td><td>Potential smooth</td><td>ε2, KL(·||π)</td><td>O(C2dε-2)</td></tr><tr><td>PS</td><td>C-LSI</td><td>Potential smooth</td><td>ε, TV</td><td>O(Cd1/2 log ε-1)</td></tr><tr><td>STLMC</td><td>/</td><td>Translated mixture of a well-conditioned distribution</td><td>ε, TV</td><td>O(poly(d, ε-1))</td></tr><tr><td>RDMC</td><td>/</td><td>Potential smooth, nearly convex at ∞</td><td>ε, TV</td><td>O(poly(d)e poly(ε-1))</td></tr><tr><td>RS-DMC</td><td>/</td><td>Potential smooth</td><td>ε2, KL(π||·)</td><td>exp(O(log3 dε-2))</td></tr><tr><td>ZOD-MC</td><td>/</td><td>Potential growing at most quadratically</td><td>ε, TV + W2</td><td>exp(O(d)O(log ε-1))</td></tr><tr><td>ALMC (ours)</td><td>/</td><td>Potential smooth</td><td>ε2, KL(π||·)</td><td>O(dA(d)2 ε-6)</td></tr></table>
+
+via importance sampling, which poses significant challenges in high-dimensional settings. Concurrently, studies such as Zhang & Chen (2022); Vargas et al. (2023; 2024); Richter & Berner (2024) have employed neural networks to approximate unknown drift terms, enabling an SDE to transport a readily sampleable distribution to the target distribution. This approach has shown excellent performance in handling complex distributions, albeit at the expense of significant computational resources required for neural network training. In contrast, annealed LMC runs on a known interpolation of probability distributions, thus simplifying sampling by obviating the need for intensive score estimation or neural network training.
+
+3. Non-asymptotic analysis for non-log-concave sampling. Drawing upon the stationary-point analysis in non-convex optimization, the seminal work Balasubramanian et al. (2022) characterizes the convergence of non-log-concave sampling via Fisher divergence. Subsequently, Cheng et al. (2023) applies this methodology to examine the local mixing of LMC. However, Fisher divergence is a relatively weak criterion compared to more commonly employed metrics such as total-variational distance or Wasserstein distances. In contrast, our study provides a convergence guarantee in terms of KL divergence, which implies convergence in total-variation distance and offers a stronger result.
+
+Notations and definitions. For  $a, b \in \mathbb{R}$ , we define  $\llbracket a, b\rrbracket := [a, b] \cap \mathbb{Z}$ ,  $a \wedge b := \min(a, b)$ , and  $a \vee b := \max(a, b)$ . For  $a, b > 0$ , the notations  $a \lesssim b$ ,  $a = O(b)$ , and  $b = \Omega(a)$  indicate that  $a \leq C b$  for some universal constant  $C > 0$ , and the notations  $a \asymp b$  and  $a = \Theta(b)$  stand for both  $a = O(b)$  and  $b = O(a)$ .  $\widetilde{O}(\cdot)$  hides logarithmic dependence in  $O(\cdot)$ . A function  $U \in C^2(\mathbb{R}^d)$  is  $\alpha (> 0)$ -strongly-convex if  $\nabla^2 U \succeq \alpha I$ , and is  $\beta (> 0)$ -smooth if  $-\beta I \preceq \nabla^2 U \preceq \beta I$ . The total-variation (TV) distance is defined as  $\mathrm{TV}(\mu, \nu) = \sup_{A \subset \mathbb{R}^d} |\mu(A) - \nu(A)|$ , and the Kullback-Leibler (KL) divergence is defined as  $\mathrm{KL}(\mu \| \nu) = \mathbb{E}_{\mu} \log \frac{\mathrm{d}\mu}{\mathrm{d}\nu}$ .  $\|\cdot\|$  represents the  $\ell^2$  norm on  $\mathbb{R}^d$ . For  $f: \mathbb{R}^d \to \mathbb{R}^{d'}$  and a probability measure  $\mu$  on  $\mathbb{R}^d$ ,  $\|f\|_{L^2(\mu)} := \left( \int \|f\|^2 \mathrm{d}\mu \right)^{\frac{1}{2}}$ , and the second-order moment of  $\mu$  is defined as  $\mathbb{E}_{\mu} \| \cdot\|^2$ .
+
+# 2 PRELIMINARIES
+
+# 2.1 STOCHASTIC DIFFERENTIAL EQUATIONS AND GIRSANOV THEOREM
+
+A stochastic differential equation (SDE)  $X = (X_{t})_{t\in [0,T]}$  is a stochastic process on  $\Omega = C([0,T];\mathbb{R}^d)$ , the space of continuous functions from  $[0,T]$  to  $\mathbb{R}^d$ . The dynamics of  $X$  are typically represented by the equation  $\mathrm{d}X_{t} = b_{t}(X)\mathrm{d}t + \sigma_{t}(X)\mathrm{d}B_{t}$ ,  $t\in [0,T]$ , where  $(B_{t})_{t\in [0,T]}$  is a standard Brownian motion in  $\mathbb{R}^d$ , and  $b_{t}(X)\in \mathbb{R}^{d}$ ,  $\sigma_t(X)\in \mathbb{R}^{d\times d}$  depends on  $(X_{s})_{s\in [0,t]}$ .
+
+The path measure of  $X$ , denoted  $\mathbb{P}^X$ , characterizes the distribution of  $X$  over  $\Omega$  and is defined by  $\mathbb{P}^X(A) = \operatorname*{Pr}(X \in A)$  for all measurable subset  $A$  of  $\Omega$ . The following lemma, as a corollary of the Girsanov theorem (Ustunel & Zakai, 2013), provides a methodology for computing the KL divergence between two path measures and serves as a crucial technical tool in our proof.
+
+Lemma 1. Assume we have the following two SDEs on  $\Omega$ :
+
+$$
+\mathrm {d} X _ {t} = a _ {t} (X) \mathrm {d} t + \sqrt {2} \mathrm {d} B _ {t}, X _ {0} \sim \mu ; \quad \mathrm {d} Y _ {t} = b _ {t} (Y) \mathrm {d} t + \sqrt {2} \mathrm {d} B _ {t}, Y _ {0} \sim \nu .
+$$
+
+Let  $\mathbb{P}^X$  and  $\mathbb{P}^Y$  denote the path measures of  $X$  and  $Y$ , respectively. Then
+
+$$
+\mathrm {K L} (\mathbb {P} ^ {X} \| \mathbb {P} ^ {Y}) = \mathrm {K L} (\mu \| \nu) + \frac {1}{4} \mathbb {E} _ {X \sim \mathbb {P} ^ {X}} \int_ {0} ^ {T} \| a _ {t} (X) - b _ {t} (X) \| ^ {2} \mathrm {d} t.
+$$
+
+# 2.2 LANGEVIN DIFFUSION AND LANGEVIN MONTE CARLO
+
+The Langevin diffusion (LD) with target distribution  $\pi \propto \mathrm{e}^{-V}$  is the solution to the SDE
+
+$$
+\mathrm {d} X _ {t} = - \nabla V (X _ {t}) \mathrm {d} t + \sqrt {2} \mathrm {d} B _ {t}, t \in [ 0, \infty); X _ {0} \sim \mu_ {0}. \tag {1}
+$$
+
+It is well-known that under mild conditions,  $\pi$  is the unique stationary distribution this SDE, and when  $\pi$  has good regularity properties, the marginal distribution of  $X_{t}$  converges to  $\pi$  as  $t\to +\infty$  so we can sample from  $\pi$  by simulating Eq. (1) for a long time. However, in most of the cases, LD is intractable to simulate exactly, and the Euler-Maruyama discretization of Eq. (1) leads to the Langevin Monte Carlo (LMC) algorithm. LMC with step size  $h > 0$  and target distribution  $\pi \propto \mathrm{e}^{-V}$  is a Markov chain  $\{X_{kh}\}_{k = 0,1,\ldots}$  constructed by iterating the following update rule:
+
+$$
+X _ {(k + 1) h} = X _ {k h} - h \nabla V \left(X _ {k h}\right) + \sqrt {2} \left(B _ {(k + 1) h} - B _ {k h}\right), k = 0, 1, \dots ; X _ {0} \sim \mu_ {0}, \tag {2}
+$$
+
+where  $\{B_{(k + 1)h} - B_{kh}\}_{k = 0,1,\ldots}\stackrel {\mathrm{i.i.d.}}{\sim}\mathcal{N}(0,hI)$
+
+# 2.3 ISOPERIMETRIC INEQUALITIES
+
+A probability measure  $\pi$  on  $\mathbb{R}^d$  satisfies a log-Sobolev inequality (LSI) with constant  $C$ , or  $C$ -LSI, if for all  $f \in C^1(\mathbb{R}^d)$  with  $\mathbb{E}_{\pi} f^2 > 0$ ,
+
+$$
+\mathbb {E} _ {\pi} f ^ {2} \log \frac {f ^ {2}}{\mathbb {E} _ {\pi} f ^ {2}} \leq 2 C \mathbb {E} _ {\pi} \| \nabla f \| ^ {2}.
+$$
+
+A probability measure  $\pi$  on  $\mathbb{R}^d$  satisfies a Poincaré inequality (PI) with constant  $C$ , or  $C$ -PI, if for all  $f \in C^1(\mathbb{R}^d)$ ,
+
+$$
+\operatorname {V a r} _ {\pi} f \leq C \mathbb {E} _ {\pi} \| \nabla f \| ^ {2}.
+$$
+
+It is worth noting that  $\alpha$ -strongly-log-concave distributions satisfy  $\frac{1}{\alpha}$ -LSI, and  $C$ -LSI implies  $C$ -PI (Bakry et al., 2014). It is established in Vempala & Wibisono (2019) that when  $\pi \propto \mathrm{e}^{-V}$  satisfies  $C$ -LSI, the LD converges exponentially fast in KL divergence; furthermore, when the potential  $V$  is  $\beta$ -smooth, the LMC also converges exponentially with a bias that vanishes when the step size approaches 0.
+
+# 2.4 WASSERSTEIN DISTANCE AND CURVES OF PROBABILITY MEASURES
+
+We briefly introduce several fundamental concepts in optimal transport, and direct readers to authoritative textbooks (Villani, 2008; 2021; Ambrosio et al., 2008; 2021) for an in-depth exploration.
+
+For two probability measures  $\mu, \nu$  on  $\mathbb{R}^d$  with finite second-order moments, the Wasserstein-2  $(\mathbf{W}_2)$  distance between  $\mu$  and  $\nu$  is defined as
+
+$$
+W _ {2} (\mu , \nu) = \inf  _ {\gamma \in \Pi (\mu , \nu)} \left(\int \| x - y \| ^ {2} \gamma (\mathrm {d} x, \mathrm {d} y)\right) ^ {\frac {1}{2}},
+$$
+
+where  $\Pi (\mu ,\nu)$  is the set of all couplings of  $(\mu ,\nu)$ , i.e., probability measure  $\gamma$  on  $\mathbb{R}^d\times \mathbb{R}^d$  with  $\gamma (A\times \mathbb{R}^d) = \mu (A)$  and  $\gamma (\mathbb{R}^d\times A) = \nu (A)$ , for all measurable set  $A\subset \mathbb{R}^d$ .
+
+Given a vector field  $v = (v_{t} : \mathbb{R}^{d} \to \mathbb{R}^{d})_{t \in [a, b]}$  and a curve of probability measures  $\rho = (\rho_{t})_{t \in [a, b]}$  on  $\mathbb{R}^{d}$  with finite second-order moments, we say that  $v$  generates  $\rho$  if the continuity equation  $\partial_{t} \rho_{t} + \nabla \cdot (\rho_{t} v_{t}) = 0, t \in [a, b]$  holds. The metric derivative of  $\rho$  at  $t \in [a, b]$  is defined as
+
+$$
+| \dot {\rho} | _ {t} := \lim  _ {\delta \rightarrow 0} \frac {W _ {2} (\rho_ {t + \delta} , \rho_ {t})}{| \delta |},
+$$
+
+which can be interpreted as the "speed" of this curve. If  $|\dot{\rho}|_t$  exists and is finite for all  $t \in [a, b]$ , we say that  $\rho$  is absolutely continuous (AC). AC is a fairly weak regularity condition on the curve of probability measures. In Lem. 4, we present a sufficient condition for AC as a formal statement.
+
+The metric derivative and the continuity equation are closely related by the following important fact from Ambrosio et al. (2008, Theorems 8.3.1 and 8.4.5):
+
+Lemma 2. For an AC curve of probability measures  $(\rho_{t})_{t\in [a,b]}$ , any vector field  $(v_{t})_{t\in [a,b]}$  that generates  $(\rho_{t})_{t\in [a,b]}$  satisfies  $|\dot{\rho} |_t\leq \| v_t\|_{L^2 (\rho_t)}$  for a.e.  $t\in [a,b]$ . Moreover, there exists a unique vector field  $(v_{t}^{*})_{t\in [a,b]}$  generating  $(\rho_{t})_{t\in [a,b]}$  that satisfies  $|\dot{\rho} |_t = \| v_t^*\|_{L^2 (\rho_t)}$  for a.e.  $t\in [a,b]$ .
+
+Finally, we define the action of an AC curve of probability measures  $(\rho_{t})_{t\in [a,b]}$  as  $\int_{a}^{b}|\dot{\rho} |_{t}^{2}\mathrm{d}t$  . As will be shown in the next section, the action is a key property characterizing the effectiveness of a curve in annealed sampling. The following lemma provides additional intuition about the action and may be helpful for interested readers.
+
+Lemma 3. Given an AC curve of probability measures  $(\rho_t)_{t \in [0,1]}$ , and let  $\mathcal{A}$  be its action. Then
+
+1.  $\mathcal{A} \geq W_2^2 (\rho_0, \rho_1)$ , and the equality is attained when  $(\rho_t)_{t \in [0,1]}$  is a constant-speed Wasserstein geodesic, i.e., let two random variables  $(X_0, X_1)$  follow the optimal coupling of  $(\rho_0, \rho_1)$  such that  $\mathbb{E} \| X_0 - X_1 \|^2 = W_2^2 (\rho_0, \rho_1)$ , and define  $\rho_t$  as the law of  $(1 - t)X_0 + tX_1$ .  
+2. If  $\rho_t$  satisfies  $C_{\mathrm{LSI}}(\rho_t)$ -LSI for all  $t$ , then  $\mathcal{A} \leq \int_0^1 C_{\mathrm{LSI}}(\rho_t)^2 \| \partial_t \nabla \log \rho_t \|_{L^2(\rho_t)}^2 \, \mathrm{d}t$ .  
+3. If  $\rho_t$  satisfies  $C_{\mathrm{PI}}(\rho_t)$ - $PI$  for all  $t$ , then  $\mathcal{A} \leq \int_0^1 2C_{\mathrm{PI}}(\rho_t) \| \partial_t \log \rho_t \|_{L^2(\rho_t)}^2 \, \mathrm{d}t$ .
+
+The proof of Lem. 3 can be found in App. A. The lower bound above can be derived by definition or via variational representations of action and  $\mathrm{W}_2$  distance, while the two upper bounds relate the action to the isoperimetric inequalities, indicating that finite action is a weaker assumption than requiring the LSI or PI constants along the trajectory. In fact, these bounds may not be tight, as demonstrated by the following surprising example, in which the action is polynomial with respect to certain problem parameters and even independent of some of them, whereas the LSI and PI constants exhibit exponential dependence. Its proof is detailed in App. A.
+
+Example 1. Consider the curve that bridges a target distribution  $\rho_0$  to a readily sampleable distribution  $\rho_{1}$ , in the form  $\rho_t = \rho_0*\mathcal{N}(0,2StI)$ ,  $t\in [0,1]$ , for some large positive number  $S$ . Let  $\mathcal{A}$  be the action of this curve. Then
+
+1.  $\mathcal{A} = S\int_0^S\| \nabla \log p_s\|_{L^2 (p_s)}^2\mathrm{d}s,$  where  $p_{s} = \rho_{s / S} = \rho_{0}*\mathcal{N}(0,2sI), s\in [0,S]$ .  
+2. In particular, when  $\rho_0$  is a mixture of Gaussian distribution  $\sum_{i=1}^{N} w_i \mathcal{N}(\mu_i, \sigma^2 I)$ , then  $\mathcal{A} \leq \frac{Sd}{2} \log \left(1 + \frac{2S}{\sigma^2}\right)$  is independent of the number of components  $N$ , the weights  $w_i$ , and the means  $\mu_i$ . However, in general, the LSI and PI constants may depend exponentially on  $\max_{i,j} \| \mu_i - \mu_j \|$ , the maximum distance between the means.
+
+# 3 PROBLEM SETUP
+
+Recall that the rationale behind annealing involves a gradual transition from  $\pi_0$ , a simple distribution that is easy to sample from, to  $\pi_1 = \pi$ , the more complex target distribution. Throughout this paper, we define a curve of probability measures  $(\pi_{\theta})_{\theta \in [0,1]}$ , along which we will apply annealing-based MCMC for sampling from  $\pi$ . For now, we do not specify the exact form of this curve, but instead introduce the following mild regularity assumption:
+
+Assumption 1. Each  $\pi_{\theta}$  has a finite second-order moment, and the curve  $(\pi_{\theta})_{\theta \in [0,1]}$  is AC with finite action  $\mathcal{A} = \int_0^1 |\dot{\pi}|_\theta^2\mathrm{d}\theta$
+
+For the purpose of non-asymptotic analysis, we further introduce the following mild assumption on the target distribution  $\pi \propto \mathrm{e}^{-V}$  on  $\mathbb{R}^d$ , which is widely used in the field of sampling (see Chewi (2024) for an overview):
+
+Assumption 2. The potential  $V$  is  $\beta$ -smooth, and there exists a global minimizer  $x_{*}$  of  $V$  such that  $\| x_{*}\| \leq R$ . Moreover,  $\pi$  has finite second-order moment.
+
+With this foundational setup, we now proceed to introduce the annealed LD and annealed LMC algorithms. Our goal is to characterize the non-asymptotic complexity of annealed MCMC algorithms for sampling from possibly non-log-concave distributions.
+
+# 4 ANALYSIS OF ANNEALED LANGEVIN DIFFUSION
+
+To elucidate the concept of annealing more clearly, we first consider the annealed Langevin diffusion (ALD) algorithm, which samples from the  $\pi \propto \mathrm{e}^{-V}$  by running LD with a dynamically changing target distribution. For the sake of this discussion, we assume the following: (i) we can exactly sample from  $\pi_0$ , (ii) the scores  $(\nabla \log \pi_{\theta})_{\theta \in [0,1]}$  are known in closed form, and (iii) we can exactly simulate any SDE with known drift and diffusion terms.
+
+Fix a sufficiently long time  $T$ . We define a reparametrized curve of probability measures  $(\widetilde{\pi}_t \coloneqq \pi_{t / T})_{t\in [0,T]}$ . Starting with an initial sample  $X_0 \sim \pi_0 = \widetilde{\pi}_0$ , we run the SDE
+
+$$
+\mathrm {d} X _ {t} = \nabla \log \widetilde {\pi} _ {t} (X _ {t}) \mathrm {d} t + \sqrt {2} \mathrm {d} B _ {t}, t \in [ 0, T ], \tag {3}
+$$
+
+and ultimately output  $X_{T} \sim \nu^{\mathrm{ALD}}$  as an approximate sample from the target distribution  $\pi$ . Intuitively, when  $\widetilde{\pi}_{t}$  is changing slowly, the distribution of  $X_{t}$  should closely resemble  $\widetilde{\pi}_{t}$ , leading to an output distribution  $\nu^{\mathrm{ALD}}$  that approximates the target distribution. This turns out to be true, as is confirmed by the following theorem, which provides a convergence guarantee for the ALD process.
+
+Theorem 1. When choosing  $T = \frac{\mathcal{A}}{4\varepsilon^2}$ , it follows that  $\mathrm{KL}(\pi \| \nu^{\mathrm{ALD}}) \leq \varepsilon^2$ .
+
+Proof. Let  $\mathbb{Q}$  be the path measure of ALD (Eq. (3)) initialized at  $X_0\sim \widetilde{\pi}_0$  , and define  $\mathbb{P}$  as the path measure corresponding to the following reference SDE:
+
+$$
+\mathrm {d} X _ {t} = \left(\nabla \log \widetilde {\pi} _ {t} + v _ {t}\right) \left(X _ {t}\right) \mathrm {d} t + \sqrt {2} \mathrm {d} B _ {t}, X _ {0} \sim \widetilde {\pi} _ {0}, t \in [ 0, T ]. \tag {4}
+$$
+
+The vector field  $v = (v_{t})_{t\in [0,T]}$  is designed such that  $X_{t}\sim \widetilde{\pi}_{t}$  for all  $t\in [0,T]$ . According to the Fokker-Planck equation<sup>2</sup>, this is equivalent to the following PDE:
+
+$$
+\partial_ {t} \widetilde {\pi} _ {t} = - \nabla \cdot \left(\widetilde {\pi} _ {t} \left(\nabla \log \widetilde {\pi} _ {t} + v _ {t}\right)\right) + \Delta \widetilde {\pi} _ {t} = - \nabla \cdot \left(\widetilde {\pi} _ {t} v _ {t}\right), t \in [ 0, T ],
+$$
+
+which means that  $v$  generates  $\left(\widetilde{\pi}_t \coloneqq \pi_{t / T}\right)_{t\in [0,T]}$ . We can compute  $\mathrm{KL}(\mathbb{P}\| \mathbb{Q})$  using Lem. 1:
+
+$$
+\mathrm {K L} (\mathbb {P} \| \mathbb {Q}) = \frac {1}{4} \mathbb {E} _ {\mathbb {P}} \int_ {0} ^ {T} \| v _ {t} (X _ {t}) \| ^ {2} \mathrm {d} t = \frac {1}{4} \int_ {0} ^ {T} \| v _ {t} \| _ {L ^ {2} (\widetilde {\pi} _ {t})} ^ {2} \mathrm {d} t.
+$$
+
+Leveraging Lem. 2, among all vector fields  $v$  that generate  $\left(\widetilde{\pi}_t \coloneqq \pi_{t / T}\right)_{t\in [0,T]}$ , we can choose the one that minimizes  $\| v_t\|_{L^2 (\widetilde{\pi}_t)}$ , thereby making  $\| v_t\|_{L^2 (\widetilde{\pi}_t)} = |\dot{\widetilde{\pi}} |_t$ , the metric derivative. With the reparameterization  $\widetilde{\pi}_t = \pi_{t / T}$ , we have the following relation by chain rule:
+
+$$
+| \dot {\widetilde {\pi}} | _ {t} = \lim _ {\delta \to 0} \frac {W _ {2} (\widetilde {\pi} _ {t + \delta} , \widetilde {\pi} _ {t})}{| \delta |} = \lim _ {\delta \to 0} \frac {W _ {2} (\pi_ {(t + \delta) / T} , \pi_ {t / T})}{T | \delta / T |} = \frac {1}{T} | \dot {\pi} | _ {t / T}.
+$$
+
+Employing the change-of-variable formula leads to
+
+$$
+\mathrm {K L} (\mathbb {P} | | \mathbb {Q}) = \frac {1}{4} \int_ {0} ^ {T} | \dot {\tilde {\pi}} | _ {t} ^ {2} \mathrm {d} t = \frac {1}{4 T} \int_ {0} ^ {1} | \dot {\pi} | _ {\theta} ^ {2} \mathrm {d} \theta = \frac {\mathcal {A}}{4 T}.
+$$
+
+Finally, using data-processing inequality (see, e.g., Chewi (2024, Theorem 1.5.3)), with  $T = \frac{\mathcal{A}}{4\varepsilon^2}$ ,
+
+$$
+\mathrm {K L} (\pi \| \nu^ {\mathrm {A L D}}) = \mathrm {K L} (\mathbb {P} _ {T} \| \mathbb {Q} _ {T}) \leq \mathrm {K L} (\mathbb {P} \| \mathbb {Q}) = \varepsilon^ {2},
+$$
+
+where  $\mathbb{P}_T$  and  $\mathbb{Q}_T$  stand for the marginal distributions of  $\mathbb{P}$  and  $\mathbb{Q}$  at time  $T$ , respectively.
+
+![](images/39a43f12e35962ac0ec0db42a1f0874925e2c73005cc9e67ed38877bd1edaa46.jpg)
+
+Let us delve deeper into the mechanics of the ALD. Although at time  $t$  the SDE (Eq. (3)) targets the distribution  $\widetilde{\pi}_t$ , the distribution of  $X_{t}$  does not precisely align with  $\widetilde{\pi}_t$ . Nevertheless, by choosing a sufficiently long time  $T$ , we actually move on the curve  $(\pi_{\theta})_{\theta \in [0,1]}$  sufficiently slowly, thus minimizing the discrepancy between the path measure of  $(X_{t})_{t\in [0,T]}$  and the reference curve  $(\widetilde{\pi}_t)_{t\in [0,T]}$ . Using the data-processing inequality, we can upper bound the error between the marginal distributions at time  $T$  by the joint distributions of the two paths. In essence, moving more slowly, sampling more precisely.
+
+Our analysis primarily addresses the global error across the entire curve of probability measures, rather than focusing solely on the local error at time  $T$ . This approach is inspired by Dalalyan & Tsybakov (2012); Chen et al. (2023), and stands in contrast to the isoperimetry-based analyses of LD (e.g., Vempala & Wibisono (2019); Chewi et al. (2022); Balasubramanian et al. (2022)), which focus on the decay of the KL divergence from the distribution of  $X_{t}$  to the target distribution, and require LSI to bound the time derivative of this quantity. Notably, the total time  $T$  needed to run the SDE depends solely on the action of the curve  $(\pi_{\theta})_{\theta \in [0,1]}$ , obviating the need for assumptions related to log-concavity or isoperimetry.
+
+It is also worth noting that ALD plays a critical role in the field of non-equilibrium stochastic thermodynamics (Seifert, 2012). Recently, a refinement of the fluctuation theorem was discovered in Chen et al. (2020); Fu et al. (2021), showing that the irreversible entropy production in a stochastic thermodynamic system is equal to the ratio of a similar action integral and the duration of the process  $T$ , closely resembling Thm. 1.
+
+# 5 ANALYSIS OF ANNEALED LANGEVIN MONTE CARLO
+
+It is crucial to recognize that, in practice, running the algorithm requires knowledge of the score functions  $(\nabla \log \pi_{\theta})_{\theta \in [0,1]}$  in closed form. Additionally, simulating ALD (Eq. (3)) necessitates discretization schemes, which introduces further errors. This section presents a detailed non-asymptotic convergence analysis for the annealed Langevin Monte Carlo (ALMC) algorithm, which is a practical approach for real-world implementations.
+
+The outline of this section is as follows. We will first introduce a family of the interpolation curve  $(\pi_{\theta})_{\theta \in [0,1]}$  that serves as the basis for our analysis. Next, we propose a discretization scheme tailored to the special structure of  $(\pi_{\theta})_{\theta \in [0,1]}$ . Finally, we state the complexity analysis in Thm. 2, and provide an example demonstrating the improvement of our bounds over existing results.
+
+# 5.1 CHOICE OF THE INTERPOLATION CURVE
+
+We consider the following curve of probability measures on  $\mathbb{R}^d$ :
+
+$$
+\pi_ {\theta} \propto \exp \left(- \eta (\theta) V - \frac {\lambda (\theta)}{2} \| \cdot \| ^ {2}\right), \theta \in [ 0, 1 ], \tag {5}
+$$
+
+where the functions  $\eta(\cdot)$  and  $\lambda(\cdot)$  are called the annealing schedule. These functions must be differentiable and monotonic, satisfying the boundary conditions  $\eta_0 = \eta(0) \nearrow \eta(1) = 1$  and  $\lambda_0 = \lambda(0) \searrow \lambda(1) = 0$ . The values of  $\eta_0 \in [0,1]$  and  $\lambda_0 \in (1, +\infty)$  are chosen so that  $\pi_0$  corresponds to a Gaussian distribution  $\mathcal{N}(0, \lambda_0^{-1}I)$  or can be efficiently sampled within  $\widetilde{O}(1)$  steps of rejection sampling (see Lem. 5 for details). We also remark that, under Assump. 2,  $\pi_\theta$  has a finite second-order moment (see Lem. 8 the proof), ensuring the  $\mathrm{W}_2$  distance between  $\pi_\theta$ 's is well-defined.
+
+This flexible interpolation scheme includes many general cases. For example, Brosse et al. (2018) and Ge et al. (2020) used the schedule  $\eta(\cdot) \equiv 1$ , while Neal (2001) used the schedule  $\lambda(\theta) = c(1 - \eta(\theta))$ . The key motivation for this interpolation is that when  $\theta = 0$ , the quadratic term predominates, making the potential of  $\pi_0$  strongly-convex with a moderate condition number, thus  $\pi_0$  is easy to sample from; on the other hand, when  $\theta = 1$ ,  $\pi_1$  is just the target distribution  $\pi$ . As
+
+$\theta$  gradually increases from 0 to 1, the readily sampleable distribution  $\pi_0$  slowly transforms into the target distribution  $\pi_1$ .
+
+# 5.2 THE DISCRETIZATION ALGORITHM
+
+With the interpolation curve  $(\pi_{\theta})_{\theta \in [0,1]}$  chosen as above, a straightforward yet non-optimal method to discretize Eq. (3) involves employing the Euler-Maruyama scheme, i.e.,
+
+$$
+X _ {t + \Delta t} \approx X _ {t} + \nabla \log \tilde {\pi} _ {t} (X _ {t}) \Delta t + \sqrt {2} \left(B _ {t + \Delta t} - B _ {t}\right), 0 \leq t <   t + \Delta t \leq T.
+$$
+
+However, considering that  $\nabla \log \widetilde{\pi}_t(x) = -\eta \left(\frac{t}{T}\right)\nabla V(x) - \lambda \left(\frac{t}{T}\right)x$ , the integral of the linear term can be computed in closed form, so we can use the exponential-integrator scheme (Zhang & Chen, 2023; Zhang et al., 2023) to further reduce the discretization error. Given the total time  $T$ , we define a sequence of points  $0 = \theta_0 < \theta_1 < \ldots < \theta_M = 1$ , and set  $T_{\ell} = T\theta_{\ell}$ ,  $h_\ell = T(\theta_\ell - \theta_{\ell-1})$ . The exponential-integrator scheme is then expressed as
+
+$$
+\mathrm {d} X _ {t} = \left(- \eta \left(\frac {t}{T}\right) \nabla V \left(X _ {t -}\right) - \lambda \left(\frac {t}{T}\right) X _ {t}\right) \mathrm {d} t + \sqrt {2} \mathrm {d} B _ {t}, t \in [ 0, T ], X _ {0} \sim \pi_ {0}, \tag {6}
+$$
+
+where  $t_{-} \coloneqq T_{\ell - 1}$  when  $t \in [T_{\ell - 1}, T_{\ell})$ ,  $\ell \in [[1, M]]$ . The explicit update rule is detailed in Alg. 1, with  $x_{\ell}$  denoting  $X_{T_{\ell}}$ , and the derivation of Eq. (7) is presented in App. C.1. Notably, replacing  $\nabla V(X_{t_{-}})$  with  $\nabla V(X_{t})$  recovers the ALD (Eq. (3)), and setting  $\eta \equiv 1$  and  $\lambda \equiv 0$  reduces to the classical LMC iterations.
+
+# Algorithm 1: Annealed LMC Algorithm
+
+Input: Target distribution  $\pi \propto \mathrm{e}^{-V}$ , total time  $T$ , annealing schedule  $\eta(\cdot)$  and  $\lambda(\cdot)$ , discrete points  $\theta_0, \dots, \theta_M$ .
+
+1 For  $0 \leq \theta < \theta' \leq 1$ , define  $\Lambda_0(\theta', \theta) = \exp \left(-T \int_{\theta}^{\theta'} \lambda(u) \, \mathrm{d}u\right)$ ,
+
+$$
+H \left(\theta^ {\prime}, \theta\right) = T \int_ {\theta} ^ {\theta^ {\prime}} \eta (u) \Lambda_ {0} \left(u, \theta^ {\prime}\right) \mathrm {d} u, \text {a n d} \Lambda_ {1} \left(\theta^ {\prime}, \theta\right) = \sqrt {2 T \int_ {\theta} ^ {\theta^ {\prime}} \Lambda_ {0} ^ {2} \left(u , \theta^ {\prime}\right) \mathrm {d} u};
+$$
+
+2 Obtain a sample  $x_0 \sim \pi_0$  (e.g., using rejection sampling (Alg. 2));  
+3 for  $\ell = 1,2,\dots,M$  do  
+4 Sample an independent Gaussian noise  $\xi_{\ell}\sim \mathcal{N}(0,I)$
+
+5 Update
+
+$$
+x _ {\ell} = \Lambda_ {0} \left(\theta_ {\ell}, \theta_ {\ell - 1}\right) x _ {\ell - 1} - H \left(\theta_ {\ell}, \theta_ {\ell - 1}\right) \nabla V \left(x _ {\ell - 1}\right) + \Lambda_ {1} \left(\theta_ {\ell}, \theta_ {\ell - 1}\right) \xi_ {\ell}. \tag {7}
+$$
+
+6 end
+
+Output:  $x_{M}\sim \nu^{\mathrm{ALMC}}$  , an approximate sample from  $\pi$
+
+We illustrate the ALMC algorithm in Fig. 1. The underlying intuition behind this algorithm is that by setting a sufficiently long total time  $T$ , the trajectory of the continuous dynamic (i.e., annealed LD) approaches the reference trajectory closely, as established in Thm. 1. Additionally, by adopting sufficiently small step sizes  $h_1, \ldots, h_M$ , the discretization error can be substantially reduced. Unlike traditional annealed LMC methods, which require multiple LMC update steps for each intermediate distribution  $\pi_1, \ldots, \pi_M$ , our approach views the annealed LMC as a discretization of a continuous-time process. Consequently, it is adequate to perform a single step of LMC towards each intermediate distribution  $\widetilde{\pi}_{T_1}, \ldots, \widetilde{\pi}_{T_M}$ , provided that  $T$  is sufficiently large and the step sizes are appropriately small.
+
+# 5.3 CONVERGENCE ANALYSIS OF THE ALGORITHM
+
+The subsequent theorem provides a convergence guarantee for the annealed LMC algorithm, with a detailed proof available in App. C.2.
+
+Theorem 2. Under Assumps. 1 and 2, Alg. 1 can generate a distribution  $\nu^{\mathrm{ALMC}}$  satisfying  $\mathrm{KL}(\pi ||\nu^{\mathrm{ALMC}})\leq \varepsilon^{2}$  within
+
+$$
+\widetilde {O} \left(\frac {d \beta^ {2} \mathcal {A} ^ {2}}{\varepsilon^ {6}}\right)
+$$
+
+calls to the oracle of  $V$  and  $\nabla V$  in expectation.
+
+![](images/63b42f87a6563474c57f58b1974d04193a888cb1590e4249047a83bd1fda291f.jpg)  
+Figure 1: Illustration of the ALMC algorithm. The  $\ell$ -th green arrow, proceeding from left to right, represents one step of LMC towards  $\widetilde{\pi}_{T_{\ell}}$  with step size  $h_{\ell}$ , while each red arrow corresponds to the application of the same transition kernel, initialized at  $\widetilde{\pi}_{T_{\ell-1}}$  on the reference trajectory  $\mathbb{P}$ , which is depicted in purple. To evaluate KL  $(\mathbb{P}||\mathbb{Q})$ , the Girsanov theorem implies that we only need to bound the aggregate KL divergence across each small interval (i.e., the sum of the blue "distances").
+
+Sketch of Proof. Let  $\mathbb{Q}$  be the path measure of ALMC (Eq. (3)), the time-discretized sampling process, whose marginal distribution at time  $T$  is the output distribution  $\nu^{\mathrm{ALMC}}$ . Again, let  $\mathbb{P}$  denote the reference path measure of Eq. (4) used in the proof of Thm. 1, in which the same vector field  $(v_{t})_{t\in [0,T]}$  ensures that  $X_{t}\sim \widetilde{\pi}_{t}$  under  $\mathbb{P}$  for all  $t\in [0,T]$ . Applying Girsanov theorem (Lem. 1) and carefully dealing with the discretization error, we can upper bound  $\mathrm{KL}(\mathbb{P}\| \mathbb{Q})$  by
+
+$$
+\mathrm {K L} (\mathbb {P} \| \mathbb {Q}) \lesssim \sum_ {\ell = 1} ^ {M} \left(\frac {1 + \eta \left(\theta_ {\ell}\right) ^ {2} \beta^ {2} h _ {\ell} ^ {2}}{T} \int_ {\theta_ {\ell - 1}} ^ {\theta_ {\ell}} | \dot {\pi} | _ {\theta} ^ {2} \mathrm {d} \theta + \eta \left(\theta_ {\ell}\right) ^ {2} \beta^ {2} d h _ {\ell} ^ {2} \left(1 + h _ {\ell} \left(\beta \eta \left(\theta_ {\ell}\right) + \lambda \left(\theta_ {\ell - 1}\right)\right)\right)\right).
+$$
+
+The first summation is governed by the total time  $T$ , which pertains to the convergence of the continuous dynamic (i.e., ALD). Setting  $T \asymp \frac{A}{\varepsilon^2}$  ensures that the first summation remains  $O(\varepsilon^2)$ , provided that the step size  $h_\ell$  is sufficiently small. The second summation addresses the discretization error, and it suffices to determine the appropriate value of  $\theta_\ell$  to minimize  $M$ , the total number of calls to the oracle of  $\nabla V$  for discretizing the SDE. Combining  $M$  with the complexity of sampling from  $\pi_0$  determines the overall complexity of the algorithm.
+
+Once again, our analysis relies on bounding the global error between two path measures by Girsanov theorem. The annealing schedule  $\eta(\cdot)$  and  $\lambda(\cdot)$  influences the complexity exclusively through the action  $\mathcal{A}$ . This crucial identity, based on the choice of interpolation curve  $(\pi_{\theta})_{\theta \in [0,1]}$ , significantly affects the effectiveness of ALMC. Our specific choice (Eq. (5)), among all interpolation curves with closed-form scores, aims at simplifying the discretization error analysis. However, alternative annealing schemes could be explored to find the optimal one in practice. For instance, a recent paper (Fan et al., 2024) proposed an annealing scheme  $\pi_{\theta}(x) \propto \pi_0((1 - a\theta)x)^{1 - \theta} \pi_1\left(\frac{x}{b + (1 - b)\theta}\right)^{\theta}$  for some  $a \in [0,1]$  and  $b \in (0,1]$ . The extra scaling factor  $\frac{1}{b + (1 - b)\theta}$  might improve mixing speed. We leave the choice of the interpolation curve as a topic for future research.
+
+We also note that our Assump. 2 encompasses strongly-log-concave target distributions. For sampling from these well-conditioned distributions via LMC, the complexity required to achieve  $\varepsilon$ -accuracy in TV distance scales as  $\widetilde{O}(\varepsilon^{-2})$  (Chewi, 2024). However, using Pinsker inequality  $\mathrm{KL} \geq 2\mathrm{TV}^2$ , our complexity to meet the same error criterion is  $O(\varepsilon^{-6})$ , indicating a significantly higher computational demand. While our discretization error is as sharp as existing works, the main reason for our worse  $\varepsilon$ -dependence is due to the time required for the continuous dynamics to converge, which is based on Girsanov theorem rather than LSI. While sharpening the  $\varepsilon$ -dependence based on Girsanov theorem remains a challenge, our  $O(\varepsilon^{-6})$  dependence still outperforms all the other bounds in Tab. 1 without isoperimetry assumptions.
+
+Finally, we conclude by demonstrating an example of a class of mixture of Gaussian distributions, which illustrates how our analysis can improve the complexity bound from exponential to polynomial. The detailed proof is provided in App. D.
+
+Example 2. Consider a mixture of Gaussian distributions in  $\mathbb{R}^d$  defined by  $\pi = \sum_{i=1}^{N} p_i \mathcal{N}(y_i, \beta^{-1}I)$ , where the weights  $p_i > 0$ ,  $\sum_{i=1}^{N} p_i = 1$ , and  $\|y_i\| = r$  for all  $i \in [1, N]$ . Consequently, the potential  $V = -\log \pi$  is  $B$ -smooth, where  $B = \beta (4r^2\beta + 1)$ . With an annealing
+
+schedule defined by  $\eta(\cdot) \equiv 1$  and  $\lambda(\theta) = dB(1 - \theta)^{\gamma}$  for some  $1 \leq \gamma = O(1)$ , it follows that  $\mathcal{A} = O\left(d(r^2\beta + 1)\left(r^2 + \frac{d}{\beta}\right)\right)$ .
+
+To show the superiority of our theoretical results, consider a simplified scenario with  $N = 2$ ,  $y_{1} = -y_{2}$ , and  $r^2 \gg \beta^{-1}$ . By applying Ex. 2 and the Pinsker inequality, the total complexity required to obtain an  $\varepsilon$ -accurate sample in TV distance is  $\widetilde{O}(d^3 \beta^2 r^4 (r^4 \beta^2 \vee d^2) \varepsilon^{-6})$ . In contrast, studies such as Schlichting (2019); Chen et al. (2021); Dong & Tong (2022) indicate that the LSI constant of  $\pi$  is  $\Omega(\mathrm{e}^{\Theta(\beta r^2)})$ , so existing bounds in Tab. 1 suggest that LMC, to achieve the same accuracy, can only provide an exponential complexity guarantee of  $\widetilde{O}(\mathrm{e}^{\Theta(\beta r^2)} d\varepsilon^{-2})$ .
+
+# 6 EXPERIMENTS
+
+We conduct simple numerical experiments to verify the findings in Ex. 2, which demonstrate that for a certain class of mixture of Gaussian distributions, ALMC achieves polynomial convergence with respect to  $r$ . Specifically, we consider a target distribution comprising a mixture of 6 Gaussians with uniform weights  $\frac{1}{6}$ , means  $\left\{\left(r\cos \frac{k\pi}{3},r\sin \frac{k\pi}{3}\right):k\in [[0,5]]\right\}$ , and the same covariance  $0.1I$  (corresponding to  $\beta = 10$  in Ex. 2). We experimented  $r\in \{2,5,10,15,20,25,30\}$ , and compute the number of iterations required for the empirical KL divergence from the target distribution to the sampled distribution to fall below 0.2 (blue curve) and 0.1 (orange curve). The results, displayed in Fig. 2, use a log-log scale for both axes. The near-linear behavior of the curves in this plot confirms that the complexity depends polynomially on  $r$ , validating our theory. Further details of the experimental setup and implementation can be found in App. F.
+
+![](images/a5995d19828e454f27a4f8086272418239669862000d5e8fd0bc9f45ca8d1aad.jpg)  
+Figure 2: Relationship between norm of means  $r$  and the number of iterations to reach 0.2 and 0.1 accuracy in KL divergence, both in log scale.
+
+# 7 CONCLUSIONS AND FUTURE WORK
+
+In this paper, we have explored the complexity of ALMC for sampling from a non-log-concave probability measure, circumventing the reliance on log-concavity or isoperimetric inequalities. Central to our analysis are the Girsanov theorem and optimal transport techniques, providing a novel approach. While our proof primarily focuses on the annealing scheme as described in Eq. (5), it can potentially be adapted to more general interpolations. Further exploration of these applications will be a key direction for future research. Technically, our proof methodology could be expanded to a broader range of target distributions beyond those with smooth potentials, such as those with Hölder-continuous gradients (Chatterji et al., 2020; Liang & Chen, 2022; Fan et al., 2023). Eliminating the assumption of global minimizers of the potential function would further enhance the practical applicability of our algorithm. Finally, while this work emphasizes the upper bounds for non-log-concave sampling, exploring the tightness of these bounds and investigating potential lower bounds for this problem (Chewi et al., 2023a,b) are intriguing avenues for future research.
+
+# ACKNOWLEDGMENTS
+
+We would like to thank Ye He, Sinho Chewi, Matthew S. Zhang, Omar Chehab, Adrien Vacher, and Anna Korba for insightful discussions. MT is supported by the National Science Foundation under Award No. DMS-1847802, and WG and YC are supported by the National Science Foundation under Award No. ECCS-1942523 and DMS-2206576.
+
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+
+# A PROOF OF RESULTS ON ACTION
+
+Lemma 4 (Sufficient Condition for Absolute Continuity (Informal)). Assume that a curve of probability distributions  $(\pi_{\theta})_{\theta \in [0,1]}$  on  $\mathbb{R}^d$  has a density  $\pi :[0,1]\times \mathbb{R}^d\ni (\theta ,x)\mapsto \pi_{\theta}(x) > 0$  that is jointly  $C^1$ . Then, this curve is AC.
+
+Proof. The following proof is informal. We leave the task of formalizing this statement for future work.
+
+We first prove the result in one-dimension. Let  $F_{\theta}(x) = \int_{-\infty}^{x}\pi_{\theta}(u)\mathrm{d}u$  be the c.d.f. of  $\pi_{\theta}$ , which is strictly increasing as  $\pi_{\theta}(x) > 0$  everywhere. As a result, its inverse  $F_{\theta}^{-1}$  is well-defined, and  $F_{\cdot}^{-1}(\cdot)$  is also jointly  $C^1$ . We know from Ambrosio et al. (2008, Theorem 6.0.2) that
+
+$$
+W _ {2} ^ {2} \left(\pi_ {\theta}, \pi_ {\theta + \delta}\right) = \int_ {0} ^ {1} \left| F _ {\theta + \delta} ^ {- 1} (q) - F _ {\theta} ^ {- 1} (q) \right| ^ {2} \mathrm {d} q.
+$$
+
+The above quantity should be  $O(\delta^2)$  as  $\delta \to 0$  due to the  $C^1$  property, and hence the curve is AC.
+
+In  $d$ -dimensions, consider the one-dimensional subspace robust  $\mathrm{W}_2$  distance (Paty & Cuturi, 2019) defined as
+
+$$
+S_{2}(\mu ,\nu) = \sup_{v\in \mathbb{S}^{d - 1}}W_{2}(v_{\sharp}\mu ,v_{\sharp}\nu),
+$$
+
+where  $\mathbb{S}^{d - 1} = \{v\in \mathbb{R}^d:\| v\| = 1\}$  is the set of unit vectors in  $\mathbb{R}^d$ , and  $v_{\sharp}\mu$  (respectively,  $v_{\sharp}\nu$ ) is the law of  $\langle v,X\rangle$  when  $X\sim \mu$  (respectively,  $X\sim \nu$ ), which is a probability measure in  $\mathbb{R}$ . By Paty & Cuturi (2019, Proposition 2),  $W_{2}(\mu ,\nu)\leq \sqrt{d} S_{2}(\mu ,\nu)$ . Similar argument as above shows that  $W_{2}^{2}(v_{\sharp}\pi_{\theta},v_{\sharp}\pi_{\theta +\delta}) = O(\delta^{2})$ , and hence  $W_{2}^{2}(\pi_{\theta},\pi_{\theta +\delta}) = O(\delta^{2})$ .
+
+# Proof of Lem. 3.
+
+1. By Cauchy-Schwarz inequality,
+
+$$
+\mathcal {A} = \int_ {0} ^ {1} | \dot {\rho} | _ {t} ^ {2} \mathrm {d} t \int_ {0} ^ {1} 1 \mathrm {d} t \geq \left(\int_ {0} ^ {1} | \dot {\rho} | _ {t} \mathrm {d} t\right) ^ {2}.
+$$
+
+It remains to prove that  $W_{2}(\rho_{0},\rho_{1}) \leq \int_{0}^{1}|\dot{\rho}|_{t}\mathrm{d}t$ . In fact, take any sequence  $0 = t_0 < t_1 < \ldots < t_N = 1$ , and let  $\Delta t \coloneqq \max_{1\leq i\leq N}(t_i - t_{i - 1})$ . By triangle inequality of the  $\mathbf{W}_2$  distance,
+
+$$
+\begin{array}{l} W _ {2} (\rho_ {0}, \rho_ {1}) \leq \sum_ {i = 1} ^ {N} W _ {2} \left(\rho_ {t _ {i}}, \rho_ {t _ {i - 1}}\right) \\ = \sum_ {i = 1} ^ {N} \frac {W _ {2} (\rho_ {t _ {i}} , \rho_ {t _ {i - 1}})}{t _ {i} - t _ {i - 1}} (t _ {i} - t _ {i - 1}) \\ = \sum_ {i = 1} ^ {N} (| \dot {\rho} | _ {t _ {i}} + o (\Delta t)) (t _ {i} - t _ {i - 1}) \to \int_ {0} ^ {1} | \dot {\rho} | _ {t} \mathrm {d} t. \\ \end{array}
+$$
+
+Another derivation is by investigating the variational representations of these two quantities. First, by Lem. 2, one can write
+
+$$
+\mathcal {A} = \int_ {0} ^ {1} | \dot {\rho} | _ {t} ^ {2} \mathrm {d} t = \inf  _ {v _ {t}: \partial_ {t} \rho_ {t} + \nabla \cdot (\rho_ {t} v _ {t}) = 0} \int_ {0} ^ {1} \| v _ {t} \| _ {L ^ {2} (\rho_ {t})} ^ {2} \mathrm {d} t.
+$$
+
+On the other hand, the Benamou-Brenier formula (Ambrosio et al., 2008, Equation 8.0.3) implies
+
+$$
+W _ {2} ^ {2} (\rho_ {0}, \rho_ {1}) = \inf _ {(\rho_ {t}, v _ {t}): \partial_ {t} \rho_ {t} + \nabla \cdot (\rho_ {t} v _ {t}) = 0; \rho_ {t = 0} = \rho_ {0}, \rho_ {t = 1} = \rho_ {1}} \int_ {0} ^ {1} \| v _ {t} \| _ {L ^ {2} (\rho_ {t})} ^ {2} \mathrm {d} t.
+$$
+
+Thus, one can easily conclude the desired inequality.
+
+To show that the inequality is attained when  $(\rho_t)_{t \in [0,1]}$  is the constant-speed Wasserstein geodesic, note that the derivation implies that the inequality is attained when  $|\dot{\rho}|_t$  is a constant for all  $t \in [0,1]$ , and for any  $0 \leq t_1 < t_2 < t_3 \leq 1$ ,  $W_2(\rho_{t_1},\rho_{t_2}) + W_2(\rho_{t_2},\rho_{t_3}) = W_2(\rho_{t_1},\rho_{t_3})$ . We refer readers to Ambrosio et al. (2008, Chapter 7.2) for the construction of the constant-speed Wasserstein geodesic.
+
+2. It is known from Bakry et al. (2014) that  $\pi$  satisfies  $C$ -LSI implies
+
+$$
+\mathrm {K L} (\mu \| \pi) \leq \frac {C}{2} \mathbb {E} _ {\mu} \left\| \nabla \log \frac {\mathrm {d} \mu}{\mathrm {d} \pi} \right\| ^ {2}, \quad \mathrm {K L} (\mu \| \pi) \geq \frac {1}{2 C} W _ {2} ^ {2} (\mu , \pi), \forall \mu .
+$$
+
+Therefore,
+
+$$
+\frac {1}{\delta^ {2}} W _ {2} ^ {2} (\rho_ {t + \delta}, \rho_ {t}) \leq C _ {\mathrm {L S I}} (\rho_ {t}) ^ {2} \int \frac {\| \nabla \log \rho_ {t + \delta} - \nabla \log \rho_ {t} \| ^ {2}}{\delta^ {2}} \mathrm {d} \rho_ {t + \delta}.
+$$
+
+By letting  $\delta \to 0$  and assuming regularity conditions, we have
+
+$$
+| \dot {\rho} | _ {t} ^ {2} \leq C _ {\mathrm {L S I}} (\rho_ {t}) ^ {2} \int \| \partial_ {t} \nabla \log \rho_ {t} \| ^ {2} \mathrm {d} \rho_ {t} = C _ {\mathrm {L S I}} (\rho_ {t}) ^ {2} \| \partial_ {t} \nabla \log \rho_ {t} \| _ {L ^ {2} (\rho_ {t})} ^ {2}
+$$
+
+3. By Liu (2020), we know that  $\pi$  satisfies  $C$ -PI implies
+
+$$
+W _ {2} ^ {2} (\mu , \pi) \leq 2 C \left(\int \frac {\mathrm {d} \mu}{\mathrm {d} \pi} \mathrm {d} \mu - 1\right), \forall \mu .
+$$
+
+Therefore,
+
+$$
+\frac {1}{\delta^ {2}} W _ {2} ^ {2} \left(\rho_ {t + \delta}, \rho_ {t}\right) \leq 2 C _ {\mathrm {P I}} \left(\rho_ {t}\right) \int \frac {\left(\rho_ {t + \delta} - \rho_ {t}\right) ^ {2}}{\delta^ {2} \rho_ {t}} \mathrm {d} x.
+$$
+
+By letting  $\delta \to 0$ , we have
+
+$$
+| \dot {\rho} | _ {t} ^ {2} \leq 2 C _ {\mathrm {P I}} (\rho_ {t}) \int \frac {\left(\partial_ {t} \rho_ {t}\right) ^ {2}}{\rho_ {t}} \mathrm {d} x = 2 C _ {\mathrm {P I}} (\rho_ {t}) \| \partial_ {t} \log \rho_ {t} \| _ {L ^ {2} (\rho_ {t})} ^ {2}.
+$$
+
+![](images/09c1d42a2fabc43ceeee76fd32f597a1eba4a09b2fc57fc3adb09208c06a8e1d.jpg)
+
+# Proof of Ex. 1.
+
+1. The reparameterized curve  $(p_{s})_{s\in [0,S]}$  satisfies the standard heat equation:
+
+$$
+\partial_ {s} p _ {s} = \Delta p _ {s} \Rightarrow \partial_ {s} p _ {s} + \nabla \cdot (p _ {s} (- \nabla \log p _ {s})) = 0,
+$$
+
+which means  $(- \nabla \log p_{s})_{s \in [0, S]}$  generates  $(p_{s})_{s \in [0, S]}$ . According to the uniqueness result in Lem. 2 and also Ambrosio et al. (2008, Theorem 8.3.1), the Lebesgue-a.e. unique vector field  $(v_{t}^{*})_{t \in [a, b]}$  that generates  $(\rho_{t})_{t \in [a, b]}$  and satisfies  $|\dot{\rho}|_{t} = \| v_{t}^{*}\|_{L^{2}(\rho_{t})}$  for Lebesgue-a.e.  $t \in [a, b]$  can be written in a gradient field  $v_{t}^{*} = \nabla \phi_{t}$  for some  $\phi_{t}: \mathbb{R}^{d} \to \mathbb{R}$ . This implies that
+
+$$
+| \dot {p} | _ {s} = \| \nabla \log p _ {s} \| _ {L ^ {2} \left(p _ {s}\right)} ^ {2}.
+$$
+
+By the time change  $[0, S] \ni s \mapsto \theta = s / S \in [0, 1]$  and taking integral, we obtain the desired result.
+
+2. Since  $p_s = \sum_{i=1}^{N} w_i \mathcal{N}\left(\mu_i, (\sigma^2 + 2s)I\right)$ , by Lem. 7, we know that  $-\nabla^2 \log p_s \preceq \frac{1}{\sigma^2 + 2s} I$ . Thus Lem. 6 implies  $\| \nabla \log p_s \|_{L^2(p_s)}^2 \leq \frac{d}{\sigma^2 + 2s}$ , so
+
+$$
+\mathcal {A} \leq S \int_ {0} ^ {S} \frac {d}{\sigma^ {2} + 2 s} \mathrm {d} s = \frac {S d}{2} \log \left(1 + \frac {2 S}{\sigma^ {2}}\right).
+$$
+
+Finally, the LSI and PI constant of mixture of Gaussian distributions are, in general, exponential with respect to the maximum distance between the means. To illustrate this, consider a simpler case where there are only 2 Gaussian components:  $\rho = \frac{1}{2}\mathcal{N}(0,\sigma^2 I) + \frac{1}{2}\mathcal{N}(y,\sigma^2 I)$ . Schlichting (2019, Section 4.1) has shown that  $C_{\mathrm{PI}}(\rho)\leq C_{\mathrm{LSI}}(\rho)\leq \frac{\sigma^2}{2}\left(\mathrm{e}^{\| y\| ^2 /\sigma^2} + 3\right)$ , while Dong & Tong (2022, Proposition 1) has shown that when  $\| y\| \gtrsim \sigma \sqrt{d}$ ,  $C_\mathrm{LSI}(\rho)\geq C_\mathrm{PI}(\rho)\gtrsim \frac{\sigma^4}{\|y\|^2}\mathrm{e}^{\Omega (\| y\|^2 /\sigma^2)}$ .
+
+# B SAMPLING FROM  $\pi_0$
+
+Lemma 5. When  $\eta_0 = 0$ ,  $\pi_0 = \mathcal{N}\left(0,\lambda_0^{-1}I\right)$  can be sampled directly; when  $\eta_0\in (0,1]$ , we choose  $\lambda_0 = \eta_0d\beta$ , so that under Assump. 2, it takes  $O\left(1\vee \log \frac{\eta_0\beta R^2}{d^2}\right) = \widetilde{O} (1)$  queries to the oracle of  $V$  and  $\nabla V$  in expectation to precisely sample from  $\pi_0$  via rejection sampling.
+
+Proof. Our proof is inspired by Liang & Chen (2023). We only consider the nontrivial case  $\eta_0 \in (0,1]$ . The potential of  $\pi_0$  is  $V_0 := \eta_0 V + \frac{\lambda_0}{2} \| \cdot \|^2$ , which is  $(\lambda_0 - \eta_0 \beta)$ -strongly-convex and  $(\lambda_0 + \eta_0 \beta)$ -smooth. Note that for any fixed point  $x' \in \mathbb{R}^d$ ,
+
+$$
+\pi_ {0} (x) \propto \exp (- V _ {0} (x)) \leq \exp \left(- V _ {0} (x ^ {\prime}) - \langle \nabla V _ {0} (x ^ {\prime}), x - x ^ {\prime} \rangle - \frac {\lambda_ {0} - \eta_ {0} \beta}{2} \| x - x ^ {\prime} \| ^ {2}\right).
+$$
+
+The r.h.s. is the unnormalized density of  $\pi_0^{\prime} := \mathcal{N}\left(x^{\prime} - \frac{\nabla V_0(x^{\prime})}{\lambda_0 - \eta_0\beta}, \frac{1}{\lambda_0 - \eta_0\beta} I\right)$ . The rejection sampling algorithm is as follows: sample  $X \sim \pi_0^{\prime}$ , and accept  $X$  as a sample from  $\pi_0$  with probability
+
+$$
+\exp \left(- V _ {0} (X) + V _ {0} (x ^ {\prime}) + \langle \nabla V _ {0} (x ^ {\prime}), X - x ^ {\prime} \rangle + \frac {\lambda_ {0} - \eta_ {0} \beta}{2} \| x - x ^ {\prime} \| ^ {2}\right) \in (0, 1 ].
+$$
+
+By Chewi (2024, Theorem 7.1.1), the number of queries to the oracle of  $V$  until acceptance follows a geometric distribution with mean
+
+$$
+\begin{array}{l} \frac {\int \exp \left(- V _ {0} \left(x ^ {\prime}\right) - \langle \nabla V _ {0} \left(x ^ {\prime}\right) , x - x ^ {\prime} \rangle - \frac {\lambda_ {0} - \eta_ {0} \beta}{2} \| x - x ^ {\prime} \| ^ {2}\right) \mathrm {d} x}{\int \exp \left(- V _ {0} (x)\right) \mathrm {d} x} \\ \leq \frac {\int \exp \left(- V _ {0} \left(x ^ {\prime}\right) - \left\langle \nabla V _ {0} \left(x ^ {\prime}\right) , x - x ^ {\prime} \right\rangle - \frac {\lambda_ {0} - \eta_ {0} \beta}{2} \| x - x ^ {\prime} \| ^ {2}\right) \mathrm {d} x}{\int \exp \left(- V _ {0} \left(x ^ {\prime}\right) - \left\langle \nabla V _ {0} \left(x ^ {\prime}\right) , x - x ^ {\prime} \right\rangle - \frac {\lambda_ {0} + \eta_ {0} \beta}{2} \| x - x ^ {\prime} \| ^ {2}\right) \mathrm {d} x} \\ = \frac {\int \exp \left(- \langle \nabla V _ {0} (x ^ {\prime}) , x \rangle - \frac {\lambda_ {0} - \eta_ {0} \beta}{2} \| x \| ^ {2}\right) \mathrm {d} x}{\int \exp \left(- \langle \nabla V _ {0} (x ^ {\prime}) , x \rangle - \frac {\lambda_ {0} + \eta_ {0} \beta}{2} \| x \| ^ {2}\right) \mathrm {d} x} \\ = \frac {\int \exp \left(- \frac {\lambda_ {0} - \eta_ {0} \beta}{2} \left\| x + \frac {\nabla V _ {0} (x ^ {\prime})}{\lambda_ {0} - \eta_ {0} \beta} \right\| ^ {2} + \frac {\| \nabla V _ {0} (x ^ {\prime}) \| ^ {2}}{2 (\lambda_ {0} - \eta_ {0} \beta)}\right) \mathrm {d} x}{\int \exp \left(- \frac {\lambda_ {0} + \eta_ {0} \beta}{2} \left\| x + \frac {\nabla V _ {0} (x ^ {\prime})}{\lambda_ {0} + \eta_ {0} \beta} \right\| ^ {2} + \frac {\| \nabla V _ {0} (x ^ {\prime}) \| ^ {2}}{2 (\lambda_ {0} + \eta_ {0} \beta)}\right) \mathrm {d} x} \\ = \left(\frac {\lambda_ {0} + \eta_ {0} \beta}{\lambda_ {0} - \eta_ {0} \beta}\right) ^ {\frac {d}{2}} \exp \left(\frac {\eta_ {0} \beta}{\lambda_ {0} ^ {2} - \eta_ {0} ^ {2} \beta^ {2}} \| \nabla V _ {0} (x ^ {\prime}) \| ^ {2}\right) \\ \leq \exp \left(\frac {\eta_ {0} \beta d}{\lambda_ {0} - \eta_ {0} \beta}\right) \exp \left(\frac {\eta_ {0} \beta}{\lambda_ {0} ^ {2} - \eta_ {0} ^ {2} \beta^ {2}} \| \nabla V _ {0} (x ^ {\prime}) \| ^ {2}\right) \\ \end{array}
+$$
+
+We choose  $\lambda_0 = \eta_0\beta d$  such that  $\exp \left(\frac{\eta_0\beta d}{\lambda_0 - \eta_0\beta}\right) \lesssim 1$ . With this  $\lambda_0$ ,  $\exp \left(\frac{\eta_0\beta}{\lambda_0^2 - \eta_0^2\beta^2} \| \nabla V_0(x')\|^2\right)$  is also  $O(1)$  as long as  $\| \nabla V_0(x')\|^2 \lesssim \eta_0\beta d^2$ .
+
+Let  $x''$  be the global minimizer of the strongly convex potential function  $V_{0}$ , which satisfies
+
+$$
+0 = \nabla V _ {0} \left(x ^ {\prime \prime}\right) = \eta_ {0} \nabla V \left(x ^ {\prime \prime}\right) + \lambda_ {0} x ^ {\prime \prime} = \eta_ {0} \nabla V \left(x ^ {\prime \prime}\right) + \eta_ {0} \beta d x ^ {\prime \prime}.
+$$
+
+Given the smoothness of  $V_{0}$
+
+$$
+\| \nabla V _ {0} (x ^ {\prime}) \| ^ {2} = \| \nabla V _ {0} (x ^ {\prime}) - \nabla V _ {0} (x ^ {\prime \prime}) \| ^ {2} \leq (\eta_ {0} \beta d + \eta_ {0} \beta) ^ {2} \| x ^ {\prime} - x ^ {\prime \prime} \| ^ {2}.
+$$
+
+Therefore, to guarantee  $\| \nabla V_0(x')\|^2 \lesssim \eta_0\beta d^2$ , it suffices to find an  $x'$  such that  $\| x' - x''\| \lesssim \frac{1}{\sqrt{\eta_0\beta}}$ .
+
+We first derive an upper bound of  $\| x'' \|$  under Assump. 2. Since  $x_*$  is a global minimizer of  $V$ ,
+
+$$
+\begin{array}{l} \beta d \| x ^ {\prime \prime} \| = \| \nabla V (x ^ {\prime \prime}) \| = \| \nabla V (x ^ {\prime \prime}) - \nabla V (x _ {*}) \| \\ \leq \beta \| x ^ {\prime \prime} - x _ {*} \| \leq \beta (\| x ^ {\prime \prime} \| + R), \\ \Rightarrow \| x ^ {\prime \prime} \| \leq \frac {R}{d - 1} \lesssim \frac {R}{d}. \\ \end{array}
+$$
+
+When  $R = 0$ , i.e., 0 is a known global minimizer of both  $V$  and  $V_{0}$ , we can directly set  $x^{\prime} = 0$ ; otherwise, we need to run optimization algorithms to find such an  $x^{\prime}$ . According to standard results in convex optimization (see, e.g., Garrigos & Gower (2023, Theorem 3.6)), running gradient descent on function  $V_{0}$  with step size  $\frac{1}{\lambda_0 + \eta_0\beta}$  yields the following convergence rate: starting from  $x_{0} = 0$ , the  $t$ -th iterate  $x_{t}$  satisfies
+
+$$
+\| x _ {t} - x ^ {\prime \prime} \| ^ {2} \leq \left(1 - \frac {\lambda_ {0} - \eta_ {0} \beta}{\lambda_ {0} + \eta_ {0} \beta}\right) ^ {t} \| 0 - x ^ {\prime \prime} \| ^ {2} \lesssim \left(\frac {2}{d}\right) ^ {t} \frac {R ^ {2}}{d ^ {2}} \leq \frac {R ^ {2}}{\mathrm {e} ^ {t} d ^ {2}},
+$$
+
+where the last inequality holds when  $d \geq 6$ . Thus,  $\log \frac{\eta_0\beta R^2}{d^2} + O(1)$  iterations are sufficient to find a desired  $x'$ . We summarize the rejection sampling in Alg. 2. In conclusion, precisely sampling from  $\pi_0$  requires
+
+$$
+O \left(1 \vee \log \frac {\eta_ {0} \beta R ^ {2}}{d ^ {2}}\right) = \widetilde {O} (1)
+$$
+
+calls to the oracle of  $V$  and  $\nabla V$  in expectation.
+
+![](images/be29de22fb3e5330ca9d2d718dc6d8f367aca3918d83c783ac2c35e213042512.jpg)
+
+Algorithm 2: Rejection Sampling for  $\pi_0$ .
+
+Input:  $\pi \propto \exp (-V)$ $\eta_0\in (0,1],\lambda_0 = \eta_0\beta d$
+
+1 Let  $V_0 \coloneqq \eta_0 V + \frac{\lambda_0}{2} \| \cdot \|^2$ ;  
+2 Use gradient descent to find an  $x^{\prime}$  that is  $O\left(\frac{1}{\sqrt{\eta_0\beta}}\right)$ -close to the global minimizer of  $V_{0}$ ;  
+3 Let  $\pi_0^{\prime}:= \mathcal{N}\left(x^{\prime} - \frac{\nabla V_0(x^{\prime})}{\lambda_0 - \eta_0\beta},\frac{1}{\lambda_0 - \eta_0\beta} I\right)$  
+4 while True do  
+5 Sample  $X\sim \pi_0^{\prime}$  and  $U\in [0,1]$  independently;
+
+Accept  $X$  as a sample of  $\pi_0$  when
+
+$$
+U \leq \exp \left(- V _ {0} (X) + V _ {0} \left(x ^ {\prime}\right) + \left\langle \nabla V _ {0} \left(x ^ {\prime}\right), X - x ^ {\prime} \right\rangle + \frac {\lambda_ {0} - \eta_ {0} \beta}{2} \| x - x ^ {\prime} \| ^ {2}\right).
+$$
+
+7 end
+
+Output:  $X\sim \pi_0$
+
+Remark. The parameter  $R$  reflects prior knowledge about global minimizer(s) of the potential function  $V$ . Unless it is exceptionally large, indicating scarce prior information about the global minimizer(s) of  $V$ , this  $\widetilde{O}(1)$  complexity is negligible compared to the overall complexity of sampling. In particular, when the exact location of a global minimizer of  $V$  is known, we can adjust the potential  $V$  so that 0 becomes a global minimizer, thereby reducing the complexity to  $O(1)$ .
+
+# C PROOFS FOR ANNEALED LMC
+
+# C.1 PROOF OF EQ.(7)
+
+Define  $\varLambda(t):=\int_0^t\lambda\left(\frac{\tau}{T}\right)\mathrm{d}\tau$ , whose derivative is  $\varLambda'(t)=\lambda\left(\frac{t}{T}\right)$ . By Ito's formula, we have
+
+$$
+\mathrm {d} \left(\mathrm {e} ^ {\Lambda (t)} X _ {t}\right) = \mathrm {e} ^ {\Lambda (t)} \left(\lambda \left(\frac {t}{T}\right) X _ {t} \mathrm {d} t + \mathrm {d} X _ {t}\right) = \mathrm {e} ^ {\Lambda (t)} \left(- \eta \left(\frac {t}{T}\right) \nabla V (X _ {t -}) \mathrm {d} t + \sqrt {2} \mathrm {d} B _ {t}\right).
+$$
+
+Integrating over  $t \in [T_{\ell - 1}, T_{\ell}]$  (note that in this case  $t_{-} = T_{\ell - 1}$ ), we obtain
+
+$$
+\mathrm {e} ^ {\Lambda (T _ {\ell})} X _ {T _ {\ell}} - \mathrm {e} ^ {\Lambda (T _ {\ell - 1})} X _ {T _ {\ell - 1}} = - \left(\int_ {T _ {\ell - 1}} ^ {T _ {\ell}} \eta \left(\frac {t}{T}\right) \mathrm {e} ^ {\Lambda (t)} \mathrm {d} t\right) \nabla V \left(X _ {T _ {\ell - 1}}\right) + \sqrt {2} \int_ {T _ {\ell - 1}} ^ {T _ {\ell}} \mathrm {e} ^ {\Lambda (t)} \mathrm {d} B _ {t},
+$$
+
+i.e.,
+
+$$
+\begin{array}{l} X _ {T _ {\ell}} = \mathrm {e} ^ {- (\Lambda (T _ {\ell}) - \Lambda (T _ {\ell - 1}))} X _ {T _ {\ell - 1}} - \left(\int_ {T _ {\ell - 1}} ^ {T _ {\ell}} \eta \left(\frac {t}{T}\right) \mathrm {e} ^ {- (\Lambda (T _ {\ell}) - \Lambda (t))} \mathrm {d} t\right) \nabla V \left(X _ {T _ {\ell - 1}}\right) \\ + \sqrt {2} \int_ {T _ {\ell - 1}} ^ {T _ {\ell}} \mathrm {e} ^ {- (\Lambda (T _ {\ell}) - \Lambda (t))} \mathrm {d} B _ {t}. \\ \end{array}
+$$
+
+Since
+
+$$
+\Lambda (T _ {\ell}) - \Lambda (t) = \int_ {t} ^ {T _ {\ell}} \lambda \left(\frac {\tau}{T}\right) \mathrm {d} \tau = T \int_ {t / T} ^ {T _ {\ell} / T} \lambda (u) \mathrm {d} u,
+$$
+
+by defining  $\Lambda_0(\theta', \theta) = \exp\left(-T \int_{\theta}^{\theta'} \lambda(u) \, \mathrm{d}u\right)$ , we have  $\mathrm{e}^{-(\Lambda(T_\ell) - \Lambda(T_{\ell-1}))} = \Lambda_0(\theta_\ell, \theta_{\ell-1})$ . Similarly, we can show that
+
+$$
+\int_ {T _ {\ell - 1}} ^ {T _ {\ell}} \eta \left(\frac {t}{T}\right) \mathrm {e} ^ {- (\Lambda (T _ {\ell}) - \Lambda (t))} \mathrm {d} t = \int_ {T _ {\ell - 1}} ^ {T _ {\ell}} \eta \left(\frac {t}{T}\right) \Lambda_ {0} \left(\frac {T _ {\ell}}{T}, \frac {t}{T}\right) \mathrm {d} t = T \int_ {\theta_ {\ell - 1}} ^ {\theta_ {\ell}} \eta (u) \Lambda_ {0} (\theta_ {\ell}, u) \mathrm {d} u,
+$$
+
+and  $\sqrt{2}\int_{T_{\ell -1}}^{T_{\ell}}\mathrm{e}^{-(\varLambda(T_{\ell}) - \varLambda(t))}\mathrm{d}B_{t}$  is a zero-mean Gaussian random vector with covariance
+
+$$
+2 \int_ {T _ {\ell - 1}} ^ {T _ {\ell}} \mathrm {e} ^ {- 2 (\Lambda (T _ {\ell}) - \Lambda (t))} \mathrm {d} t \cdot I = 2 \int_ {T _ {\ell - 1}} ^ {T _ {\ell}} \Lambda_ {0} ^ {2} \left(\frac {T _ {\ell}}{T}, \frac {t}{T}\right) \mathrm {d} t \cdot I = 2 T \int_ {\theta_ {\ell - 1}} ^ {\theta_ {\ell}} \Lambda_ {0} ^ {2} (\theta_ {\ell}, u) \mathrm {d} u \cdot I.
+$$
+
+![](images/6803734b337cd783c1a8f9cce8872e2486274e4fad9efe1de1c5666d4c9d8768.jpg)
+
+# C.2 PROOF OF THM. 2
+
+We denote the path measure of ALMC (Eq. (6)) by  $\mathbb{Q}$ . Then,  $\mathbb{Q}_T$ , the marginal distribution of  $X_{T}$  serves as the output distribution  $\nu^{\mathrm{ALMC}}$ . Similar to the methodology in the proof of Thm. 1, we use  $\mathbb{P}$  to denote the reference path measure of Eq. (4), in which the vector field  $(v_{t})_{t\in [0,T]}$  generates the cure of probability distributions  $(\widetilde{\pi}_t)_{t\in [0,T]}$ .
+
+Using the data-processing inequality, it suffices to demonstrate that  $\mathrm{KL}(\mathbb{P}\| \mathbb{Q})\leq \varepsilon^2$ . By Girsanov theorem (Lem. 1) and triangle inequality, we have
+
+$$
+\begin{array}{l} \mathrm {K L} (\mathbb {P} | | \mathbb {Q}) = \frac {1}{4} \int_ {0} ^ {T} \mathbb {E} _ {\mathbb {P}} \left\| \eta \left(\frac {t}{T}\right) \left(\nabla V (X _ {t}) - \nabla V (X _ {t _ {-}})\right) - v _ {t} (X _ {t}) \right\| ^ {2} \mathrm {d} t \\ \lesssim \int_ {0} ^ {T} \mathbb {E} _ {\mathbb {P}} \left[ \eta \left(\frac {t}{T}\right) ^ {2} \| \nabla V (X _ {t}) - \nabla V (X _ {t _ {-}}) \| ^ {2} + \| v _ {t} (X _ {t}) \| ^ {2} \right] \mathrm {d} t \\ \leq \sum_ {\ell = 1} ^ {M} \eta \left(\frac {T _ {\ell}}{T}\right) ^ {2} \beta^ {2} \int_ {T _ {\ell - 1}} ^ {T _ {\ell}} \mathbb {E} _ {\mathbb {P}} \| X _ {t} - X _ {t -} \| ^ {2} \mathrm {d} t + \int_ {0} ^ {T} \| v _ {t} \| _ {L ^ {2} (\tilde {\pi} _ {t})} ^ {2} \mathrm {d} t. \\ \end{array}
+$$
+
+The last inequality arises from the smoothness of  $V$  and the increasing property of  $\eta(\cdot)$ . To bound  $\mathbb{E}_{\mathbb{P}}\|X_t - X_{t-}\|^2$ , note that under  $\mathbb{P}$ , for  $t \in [T_{\ell-1}, T_\ell)$ , we have
+
+$$
+X _ {t} - X _ {t _ {-}} = \int_ {T _ {\ell - 1}} ^ {t} \left(\nabla \log \widetilde {\pi} _ {\tau} + v _ {\tau}\right) (X _ {\tau}) \mathrm {d} \tau + \sqrt {2} (B _ {t} - B _ {T _ {\ell - 1}}).
+$$
+
+Thanks to the fact that  $X_{t}\sim \widetilde{\pi}_{t}$  under  $\mathbb{P}$
+
+$$
+\begin{array}{l} \mathbb {E} _ {\mathbb {P}} \| X _ {t} - X _ {t _ {-}} \| ^ {2} \lesssim \mathbb {E} _ {\mathbb {P}} \left\| \int_ {T _ {\ell - 1}} ^ {t} \left(\nabla \log \widetilde {\pi} _ {\tau} + v _ {\tau}\right) (X _ {\tau}) \mathrm {d} \tau \right\| ^ {2} + \mathbb {E} \| \sqrt {2} \left(B _ {t} - B _ {T _ {\ell - 1}}\right) \| ^ {2} \\ \lesssim (t - T _ {\ell - 1}) \int_ {T _ {\ell - 1}} ^ {t} \mathbb {E} _ {\mathbb {P}} \| (\nabla \log \widetilde {\pi} _ {\tau} + v _ {\tau}) (X _ {\tau}) \| ^ {2} \mathrm {d} \tau + d (t - T _ {\ell - 1}) \\ \lesssim (t - T _ {\ell - 1}) \int_ {T _ {\ell - 1}} ^ {t} \left(\| \nabla \log \widetilde {\pi} _ {\tau} \| _ {L ^ {2} (\widetilde {\pi} _ {\tau})} ^ {2} + \| v _ {\tau} \| _ {L ^ {2} (\widetilde {\pi} _ {\tau})} ^ {2}\right) d \tau + d (t - T _ {\ell - 1}) \\ \lesssim h _ {\ell} \int_ {T _ {\ell - 1}} ^ {T _ {\ell}} \left(\| \nabla \log \widetilde {\pi} _ {\tau} \| _ {L ^ {2} (\widetilde {\pi} _ {\tau})} ^ {2} + \| v _ {\tau} \| _ {L ^ {2} (\widetilde {\pi} _ {\tau})} ^ {2}\right) d \tau + d h _ {\ell}. \\ \end{array}
+$$
+
+The second inequality arises from the application of the Cauchy-Schwarz inequality, and the last inequality is due to the definition  $h_\ell = T_\ell - T_{\ell-1}$ . Taking integral over  $t \in [T_{\ell-1}, T_\ell]$ ,
+
+$$
+\int_ {T _ {\ell - 1}} ^ {T _ {\ell}} \mathbb {E} \| X _ {t} - X _ {t -} \| ^ {2} \mathrm {d} t \lesssim h _ {\ell} ^ {2} \int_ {T _ {\ell - 1}} ^ {T _ {\ell}} \left(\| \nabla \log \widetilde {\pi} _ {t} \| _ {L ^ {2} (\widetilde {\pi} _ {t})} ^ {2} + \| v _ {t} \| _ {L ^ {2} (\widetilde {\pi} _ {t})} ^ {2}\right) \mathrm {d} t + d h _ {\ell} ^ {2}.
+$$
+
+Recall that the potential of  $\widetilde{\pi}_t$  is  $\left(\eta\left(\frac{t}{T}\right)\beta + \lambda\left(\frac{t}{T}\right)\right)$ -smooth. By Lem. 6 and the monotonicity of  $\eta(\cdot)$  and  $\lambda(\cdot)$ , we have
+
+$$
+\begin{array}{l} \int_ {T _ {\ell - 1}} ^ {T _ {\ell}} \| \nabla \log \widetilde {\pi} _ {t} \| _ {L ^ {2} (\widetilde {\pi} _ {t})} ^ {2} \mathrm {d} t \leq \int_ {T _ {\ell - 1}} ^ {T _ {\ell}} d \left(\eta \left(\frac {t}{T}\right) \beta + \lambda \left(\frac {t}{T}\right)\right) \mathrm {d} t \\ \leq d h _ {\ell} \left(\beta \eta \left(\frac {T _ {\ell}}{T}\right) + \lambda \left(\frac {T _ {\ell - 1}}{T}\right)\right) \\ = d h _ {\ell} \left(\beta \eta \left(\theta_ {\ell}\right) + \lambda \left(\theta_ {\ell - 1}\right)\right). \\ \end{array}
+$$
+
+Therefore,  $\mathrm{KL}(\mathbb{P}\| \mathbb{Q})$  is upper bounded by
+
+$$
+\begin{array}{l} \lesssim \sum_ {\ell = 1} ^ {M} \eta (\theta_ {\ell}) ^ {2} \beta^ {2} \int_ {T _ {\ell - 1}} ^ {T _ {\ell}} \mathbb {E} _ {\mathbb {P}} \| X _ {t} - X _ {t -} \| ^ {2} \mathrm {d} t + \int_ {0} ^ {T} \| v _ {t} \| _ {L ^ {2} (\widetilde {\pi} _ {t})} ^ {2} \mathrm {d} t \\ \lesssim \sum_ {\ell = 1} ^ {M} \eta \left(\theta_ {\ell}\right) ^ {2} \beta^ {2} \left(d h _ {\ell} ^ {3} (\beta \eta (\theta_ {\ell}) + \lambda (\theta_ {\ell - 1})) + h _ {\ell} ^ {2} \int_ {T _ {\ell - 1}} ^ {T _ {\ell}} \| v _ {t} \| _ {L ^ {2} (\widetilde {\pi} _ {t})} ^ {2} d t + d h _ {\ell} ^ {2}\right) + \int_ {0} ^ {T} \| v _ {t} \| _ {L ^ {2} (\widetilde {\pi} _ {t})} ^ {2} d t \\ = \sum_ {\ell = 1} ^ {M} \left(\left(1 + \eta \left(\theta_ {\ell}\right) ^ {2} \beta^ {2} h _ {\ell} ^ {2}\right) \int_ {T _ {\ell - 1}} ^ {T _ {\ell}} \| v _ {t} \| _ {L ^ {2} \left(\widetilde {\pi} _ {t}\right)} ^ {2} \mathrm {d} t + \eta \left(\theta_ {\ell}\right) ^ {2} \beta^ {2} d h _ {\ell} ^ {2} \left(1 + h _ {\ell} \left(\beta \eta \left(\theta_ {\ell}\right) + \lambda \left(\theta_ {\ell - 1}\right)\right)\right)\right). \\ \end{array}
+$$
+
+For the remaining integral, given that  $(v_{t})_{t\in [0,T]}$  generates  $(\widetilde{\pi}_t)_{t\in [0,T]}$ , according to Lem. 2, we may choose  $v_{t}$  such that  $\| v_t\|_{L^2 (\widetilde{\pi}_t)} = |\dot{\widetilde{\pi}} |_t$ . Thus,
+
+$$
+\int_ {T _ {\ell - 1}} ^ {T _ {\ell}} \| v _ {t} \| _ {L ^ {2} (\widetilde {\pi} _ {t})} ^ {2} \mathrm {d} t = \int_ {T _ {\ell - 1}} ^ {T _ {\ell}} | \dot {\widetilde {\pi}} | _ {t} ^ {2} \mathrm {d} t = \int_ {T _ {\ell - 1}} ^ {T _ {\ell}} \frac {1}{T ^ {2}} | \dot {\pi} | _ {t / T} ^ {2} \mathrm {d} t = \frac {1}{T} \int_ {\theta_ {\ell - 1}} ^ {\theta_ {\ell}} | \dot {\pi} | _ {\theta} ^ {2} \mathrm {d} \theta ,
+$$
+
+through a change-of-variable analogous to that used in the proof of Thm. 1. Therefore,
+
+$$
+\mathrm {K L} (\mathbb {P} \| \mathbb {Q}) \lesssim \sum_ {\ell = 1} ^ {M} \left(\frac {1 + \eta (\theta_ {\ell}) ^ {2} \beta^ {2} h _ {\ell} ^ {2}}{T} \int_ {\theta_ {\ell - 1}} ^ {\theta_ {\ell}} | \dot {\pi} | _ {\theta} ^ {2} \mathrm {d} \theta + \eta (\theta_ {\ell}) ^ {2} \beta^ {2} d h _ {\ell} ^ {2} (1 + h _ {\ell} (\beta \eta (\theta_ {\ell}) + \lambda (\theta_ {\ell - 1})))\right).
+$$
+
+Assume  $h_{\ell} \lesssim \frac{1}{\beta d}$  (which will be verified later), so we can further simplify the above expression to
+
+$$
+\begin{array}{l} \operatorname {K L} (\mathbb {P} \| \mathbb {Q}) \lesssim \sum_ {\ell = 1} ^ {M} \left(\frac {1}{T} \int_ {\theta_ {\ell - 1}} ^ {\theta_ {\ell}} | \dot {\pi} | _ {\theta} ^ {2} \mathrm {d} \theta + \eta (\theta_ {\ell}) ^ {2} \beta^ {2} d h _ {\ell} ^ {2}\right) \\ = \frac {\mathcal {A}}{T} + \beta^ {2} d \sum_ {\ell = 1} ^ {M} \eta \left(\theta_ {\ell}\right) ^ {2} h _ {\ell} ^ {2} \\ = \frac {\mathcal {A}}{T} + \beta^ {2} d \sum_ {\ell = 1} ^ {M} \eta (\theta_ {\ell}) ^ {2} T ^ {2} \left(\theta_ {\ell} - \theta_ {\ell - 1}\right) ^ {2}. \\ \end{array}
+$$
+
+To bound the above expression by  $\varepsilon^2$ , we first select  $T \asymp \frac{A}{\varepsilon^2}$ , mirroring the total time  $T$  required for running annealed LD as specified in Thm. 1. This guarantees that the continuous dynamics closely approximate the reference path measure. Given that  $\eta(\cdot) \leq 1$ , it remains only to ensure
+
+$$
+\beta^ {2} d \sum_ {\ell = 1} ^ {M} \eta (\theta_ {\ell}) ^ {2} T ^ {2} (\theta_ {\ell} - \theta_ {\ell - 1}) ^ {2} \leq \beta^ {2} d \frac {\mathcal {A} ^ {2}}{\varepsilon^ {4}} \sum_ {\ell = 1} ^ {M} (\theta_ {\ell} - \theta_ {\ell - 1}) ^ {2} \lesssim \varepsilon^ {2},
+$$
+
+which is equivalent to
+
+$$
+\sum_ {\ell = 1} ^ {M} \left(\theta_ {\ell} - \theta_ {\ell - 1}\right) ^ {2} \lesssim \frac {\varepsilon^ {6}}{d \beta^ {2} \mathcal {A} ^ {2}}. \tag {8}
+$$
+
+To minimize  $M$ , we apply Cauchy-Schwarz inequality:
+
+$$
+\left(\sum_ {\ell = 1} ^ {M} 1\right) \left(\sum_ {\ell = 1} ^ {M} \left(\theta_ {\ell} - \theta_ {\ell - 1}\right) ^ {2}\right) \geq \left(\sum_ {\ell = 1} ^ {M} \left(\theta_ {\ell} - \theta_ {\ell - 1}\right)\right) ^ {2} = 1,
+$$
+
+which establishes a lower bound for  $M$ . The equality is achieved when  $\theta_{\ell} - \theta_{\ell -1} = \frac{1}{M}$  for all  $\ell \in [1,M]$ . Thus, selecting
+
+$$
+M \asymp \frac {d \beta^ {2} \mathcal {A} ^ {2}}{\varepsilon^ {6}}
+$$
+
+satisfies the constraint given in Eq. (8). In this case, the step size  $h_\ell = \frac{T}{M} \asymp \frac{\varepsilon^4}{d\beta^2} \lesssim \frac{1}{\beta d}$ . Combining this with the  $\widetilde{O}(1)$  complexity for sampling from  $\pi_0$ , we have completed the proof.
+
+# D PROOF OF Ex. 2
+
+The smoothness of  $V$  comes from Lem. 7.
+
+Note that  $\pi (x)\propto \sum_{i = 1}^{N}p_{i}\exp \left(-\frac{\beta}{2}\| x - y_{i}\|^{2}\right)$ , and define
+
+$$
+\begin{array}{l} \widehat {\pi} _ {\lambda} (x) \propto \pi (x) \exp \left(- \frac {\lambda}{2} \| x \| ^ {2}\right) \\ = \sum_ {i = 1} ^ {N} p _ {i} \exp \left(- \frac {\lambda \beta}{2 (\lambda + \beta)} \| y _ {i} \| ^ {2} - \frac {\lambda + \beta}{2} \left\| x - \frac {\beta}{\lambda + \beta} y _ {i} \right\| ^ {2}\right) \\ \propto \sum_ {i = 1} ^ {N} p _ {i} \exp \left(- \frac {\lambda + \beta}{2} \left\| x - \frac {\beta}{\lambda + \beta} y _ {i} \right\| ^ {2}\right) \\ = \sum_ {i = 1} ^ {N} p _ {i} \mathcal {N} \left(\frac {\beta}{\lambda + \beta} y _ {i}, \frac {1}{\lambda + \beta} I\right). \\ \end{array}
+$$
+
+We use coupling method to upper bound  $W_2^2(\widehat{\pi}_\lambda, \widehat{\pi}_{\lambda + \delta})$ . We first sample  $I$  following distribution  $\mathbb{P}(I = i) = p_i$ ,  $i \in [1, N]$ , and then independently sample  $\eta \sim \mathcal{N}(0, I)$ . We have
+
+$$
+X := \frac {\beta}{\lambda + \beta} y _ {I} + \frac {1}{\sqrt {\lambda + \beta}} \eta \sim \widehat {\pi} _ {\lambda},
+$$
+
+$$
+Y := \frac {\beta}{\lambda + \delta + \beta} y _ {I} + \frac {1}{\sqrt {\lambda + \delta + \beta}} \eta \sim \widehat {\pi} _ {\lambda + \delta}.
+$$
+
+By the definition of the  $\mathbf{W}_2$  distance, we have
+
+$$
+\begin{array}{l} W _ {2} ^ {2} \left(\widehat {\pi} _ {\lambda}, \widehat {\pi} _ {\lambda + \delta}\right) \leq \mathbb {E} \| X - Y \| ^ {2} \\ = \mathbb {E} _ {I} \mathbb {E} _ {\eta} \left\| \left(\frac {\beta}{\lambda + \beta} - \frac {\beta}{\lambda + \delta + \beta}\right) y _ {I} + \left(\frac {1}{\sqrt {\lambda + \beta}} - \frac {1}{\sqrt {\lambda + \delta + \beta}}\right) \eta \right\| ^ {2} \\ = \mathbb {E} _ {I} \left(\frac {\beta}{\lambda + \beta} - \frac {\beta}{\lambda + \delta + \beta}\right) ^ {2} \| y _ {I} \| ^ {2} + \left(\frac {1}{\sqrt {\lambda + \beta}} - \frac {1}{\sqrt {\lambda + \delta + \beta}}\right) ^ {2} d \\ = \left(\frac {\beta}{\lambda + \beta} - \frac {\beta}{\lambda + \delta + \beta}\right) ^ {2} r ^ {2} + \left(\frac {1}{\sqrt {\lambda + \beta}} - \frac {1}{\sqrt {\lambda + \delta + \beta}}\right) ^ {2} d. \\ \end{array}
+$$
+
+This implies
+
+$$
+| \dot {\hat {\pi}} | _ {\lambda} ^ {2} = \lim  _ {\delta \rightarrow 0} \frac {W _ {2} ^ {2} (\widehat {\pi} _ {\lambda} , \widehat {\pi} _ {\lambda + \delta})}{\delta^ {2}} \leq \frac {\beta^ {2} r ^ {2}}{(\lambda + \beta) ^ {4}} + \frac {d}{4 (\lambda + \beta) ^ {3}}.
+$$
+
+By time reparameterization  $\pi_{\theta} = \widehat{\pi}_{\lambda (\theta)}$ $|\dot{\pi} |_{\theta} = |\dot{\widehat{\pi}} |_{\lambda (\theta)}|\dot{\lambda} (\theta)|$  . With  $\lambda (\theta) = \lambda_0(1 - \theta)^\gamma$ $|\dot{\lambda} (\theta)| =$ $\gamma \lambda_0(1 - \theta)^{\gamma -1}\leq \gamma \lambda_0\lesssim \lambda_0$  . Therefore,
+
+$$
+\begin{array}{l} \mathcal {A} = \int_ {0} ^ {1} | \dot {\pi} | _ {\theta} ^ {2} \mathrm {d} \theta = \int_ {0} ^ {1} | \dot {\bar {\pi}} | _ {\lambda (\theta)} ^ {2} | \dot {\lambda} (\theta) | ^ {2} \mathrm {d} \theta \\ \lesssim \lambda_ {0} \int_ {0} ^ {1} | \dot {\bar {\pi}} | _ {\lambda (\theta)} ^ {2} | \dot {\lambda} (\theta) | d \theta = \lambda_ {0} \int_ {0} ^ {\lambda_ {0}} | \dot {\bar {\pi}} | _ {\lambda} ^ {2} d \lambda \\ = \lambda_ {0} \int_ {0} ^ {\lambda_ {0}} \left(\frac {\beta^ {2} r ^ {2}}{(\lambda + \beta) ^ {4}} + \frac {d}{4 (\lambda + \beta) ^ {3}}\right) d \lambda \\ \lesssim \lambda_ {0} \left(\beta^ {2} r ^ {2} \frac {1}{\beta^ {3}} + d \frac {1}{\beta^ {2}}\right) \\ \lesssim d \beta (r ^ {2} \beta + 1) \left(\frac {r ^ {2}}{\beta} + \frac {d}{\beta^ {2}}\right) \\ = d \left(r ^ {2} \beta + 1\right) \left(r ^ {2} + \frac {d}{\beta}\right). \\ \end{array}
+$$
+
+It follows from the proof that as long as  $\max_{\theta \in [0,1]}|\dot{\lambda} (\theta)|$  is bounded by a polynomial function of  $\beta ,d$  and  $r$  , so is the action  $\mathcal{A}$
+
+# E SUPPLEMENTARY LEMMAS
+
+Lemma 6 (Chewi (2024, Lemma 4.E.1)). Consider a probability measure  $\mu \propto \mathrm{e}^{-U}$  on  $\mathbb{R}^d$ . If  $\nabla^2 U \preceq \beta I$  for some  $\beta > 0$ , then  $\mathbb{E}_{\mu} \| \nabla U \|^2 \leq \beta d$ .
+
+Lemma 7 (Cheng et al. (2023, Lemma 4)). For a Gaussian mixture distribution  $\pi = \sum_{i} w_{i} \mathcal{N}\left(\mu_{i}, \sigma^{2} I\right)$ , we have
+
+$$
+\left(\frac {1}{\sigma^ {2}} - \frac {\max _ {i , j} \| \mu_ {i} - \mu_ {j} \| ^ {2}}{2 \sigma^ {4}}\right) I \preceq - \nabla^ {2} \log \pi \preceq \frac {1}{\sigma^ {2}} I.
+$$
+
+Remark. This is a slightly improved version, as in the original lemma the authors omitted the factor 2 in the denominator on the l.h.s.
+
+Lemma 8. Under Assump. 2,  $\pi_{\theta}$  defined in Eq. (5) has finite second-order moment when  $\eta (\theta)\in [0,1]$ .
+
+Proof. When  $\eta(\theta) = 1$ ,  $\pi_{\theta} \propto \exp \left(-V - \frac{\lambda(\theta)}{2} \| \cdot \|^2\right) \leq \exp (-V)$ , and the claim is straightforward. Otherwise, by convexity of  $u \mapsto \mathrm{e}^{-u}$ , we have
+
+$$
+\begin{array}{l} \exp \left(- \eta (\theta) V - \frac {\lambda (\theta)}{2} \| \cdot \| ^ {2}\right) = \exp \left(- \eta (\theta) V - (1 - \eta (\theta)) \frac {\lambda (\theta)}{2 (1 - \eta (\theta))} \| \cdot \| ^ {2}\right) \\ \leq \eta (\theta) \exp (- V) + (1 - \eta (\theta)) \exp \left(- \frac {\lambda (\theta)}{2 (1 - \eta (\theta))} \| \cdot \| ^ {2}\right). \\ \end{array}
+$$
+
+Multiplying both sides by  $\| \cdot \| ^2$  and taking integral over  $\mathbb{R}^d$ , we see that  $\mathbb{E}_{\pi_{\theta}}\| \cdot \| ^2 < + \infty$
+
+![](images/0d7723d9267cd2ff1e1438d47252acff3125896d6061e7cb6840b7f843c2f12e.jpg)
+
+# F FURTHER DETAILS OF EXPERIMENTS IN SEC. 6
+
+Hyperparameter settings. For all the experiments, we use the annealing schedule  $\lambda (\theta) = 5(1 - \theta)^{10}$  and  $\eta (\theta)\equiv 1$ . The step size for ALMC is designed to follow a quadratic schedule: the step size at the  $\ell$ -th iteration (out of  $M$  total iterations,  $\ell \in [1,M])$  is given by  $-\frac{s_{\mathrm{max}} - s_{\mathrm{min}}}{M^2 / 4}\left(\ell -\frac{M}{2}\right)^2 +s_{\mathrm{max}}$ , which increases on  $[0,\frac{M}{2} ]$  and decreases on  $[\frac{M}{2},M]$ , with a maximum of  $s_{\mathrm{max}}$  and a minimum of  $s_{\mathrm{min}}$ . For  $r = 2,5,10,15,20,25,30$ , the  $M$  we choose are 200, 500, 2500, 10000, 20000, 40000, 60000, respectively, and we set  $s_{\mathrm{max}} = 0.05$  and  $s_{\mathrm{min}} = 0.01$ , which achieve the best performance across all settings after tuning.
+
+KL divergence estimation. We use the Information Theoretical Estimators (ITE) toolbox (Szabó, 2014) to empirically estimate the KL divergence. In all cases, we generate 1000 samples using ALMC, and an additional 1000 samples from the target distribution. The KL divergence is estimated using the ite.cost.BDKL_KnnK() function, which leverages  $k$ -nearest-neighbor techniques to compute the divergence from the target distribution to the sampled distribution.
+
+Regression coefficients. We also compute the linear regression approximation of the curves in Fig. 2 via sklearn.linear_model.LinearRegression. The blue curve has a slope of 2.841 and an intercept of 1.257, while the orange one has a slope of 2.890 and an intercept of 0.904. The  $R^2$  scores, calculated using sklearn.metrics.r2_score, are 0.995 and 0.997, respectively.

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