| COMPONENT | ATTRIBUTE | VALUE |
| RSSM | INTERNAL STATE DIMENSION | 256 |
| MLP LAYER 1 UNITS | 256 |
| MLP LAYER 2 UNITS | 64 |
| EMBEDDING FUNCTION | CONV LAYER 1 NUMBER OF FILTERS | 32 |
| CONV LAYER 1 KERNEL SIZE | 4 |
| CONV LAYER 2 NUMBER OF FILTERS | 64 |
| CONV LAYER 2 KERNEL SIZE | 4 |
| CONV LAYER 3 NUMBER OF FILTERS | 128 |
| CONV LAYER 3 KERNEL SIZE | 4 |
| CONV LAYER 4 NUMBER OF FILTERS | 256 |
| CONV LAYER 4 KERNEL SIZE | 4 |
| FINAL OUTPUT OPERATION | CONCATENATION |
| GENERATIVE MODEL | MLP LAYER 1 NUMBER OF UNITS | 1024 |
| DECONVOLUTION LAYER 1 NUMBER OF FILTERS | 128 |
| DECONVOLUTION LAYER 1 KERNEL SIZE | 5 |
| DECONVOLUTION LAYER 2 NUMBER OF FILTERS | 64 |
| DECONVOLUTION LAYER 2 KERNEL SIZE | 5 |
| DECONVOLUTION LAYER 3 NUMBER OF FILTERS | 32 |
| DECONVOLUTION LAYER 3 KERNEL SIZE | 6 |
| DECONVOLUTION LAYER 4 NUMBER OF FILTERS | 3 |
| DECONVOLUTION LAYER 4 KERNEL SIZE | 6 |
| REWARD MODEL | MLP LAYER 1 NUMBER OF UNITS | 512 |
| MLP LAYER 2 NUMBER OF UNITS | 512 |
| MLP LAYER 3 NUMBER OF UNITS | 1 |
| VALUE MODEL | MLP LAYER 1 NUMBER OF UNITS | 512 |
| MLP LAYER 2 NUMBER OF UNITS | 512 |
| MLP LAYER 3 NUMBER OF UNITS | 512 |
| MLP LAYER 4 NUMBER OF UNITS | 1 |
| POLICY MODEL | ALL FULLY CONNECTED LAYERS ACTIVATION | ELU |
| MLP LAYER 1 NUMBER OF UNITS | 512 |
| MLP LAYER 2 NUMBER OF UNITS | 512 |
| MLP LAYER 3 NUMBER OF UNITS | 512 |
| MLP LAYER 4 NUMBER OF UNITS | 512 |
| MLP LAYER 5 NUMBER OF UNITS | 2 |
| ADAM OPTIMISER | LEARNING RATE FOR MODEL LEARNING | 6 × 10-4 |
| LEARNING RATE FOR VALUE MODEL | 8 × 10-5 |
| LEARNING RATE FOR POLICY MODEL | 8 × 10-5 |
+
+Table 2: Configurations for the neural network implementations of CCWM.
+
+For our implementation of the adaptive truncation technique (Section 3.3), the associated parameters that require pre-specification are the detection threshold, $\eta$ , adaptive truncation sliding window size, $S$ . There are also the optional parameters, the number of states preceding the detected changes to be omitted from "learning via retracing" (which is set to equal to $S$ by default, see Line 9 in Algorithm 1 and Section 3.3), $\tau$ , and the number of initial steps to be omitted from adaptive truncation (due to under-training of the critic, which is set to 0 by default), $\xi$ . The default values for the parameters we used for the empirical evaluation shown in Figure 5 are: $\eta = 0.10$ , $S = 10$ , $\tau = 5$ , $\xi = 1 \times 10^5$ . The hyperparameters are determined through standard grid search, and we leave the automatic determination of the parameters associated with adaptive truncation to future work.
+
+The python implementation of CCWM can be found at https://github.com/changmin-yu/CCWM_code.
+
+
+(a)
+
+
+Figure 7: Visualisation of similarity between the ground-truth states and retraced states using tSNE. The two-dimensional embeddings for a 500-step trajectory (a) and a 76-step subtrajectory (b) is shown. Color of the scatters indicates the temporal ordering of the states within the trajectories.
+
+
+(b)
+
+
+
+
+Figure 8: Visualisation of state representations along the same trajectory in Hopper Stand task. We show the TSNE-embeddings of the state representations over a trajectory of length 500 steps in Hopper Stand task, with a standard CCWM agent (without adaptive truncation) that has received $2 \times 10^{5}$ training steps. Clear clustered structured can be observed in the visualisation, which indicates: (a) the existence of "irreversible" transitions, and (b) the majority of the transitions are "reversible", hence "learning via retracing" should still apply to facilitate improved sample efficiency.
+
+
+Figure 9: Visualisation of the structure of the state representations with respect to the Q-values. We show the TSNE-embeddings of the state representations of 10000 randomly sampled states (from the cached replay buffer) of CCWM (left) and Dreamer (right) in the Walker Walk task. Visual inspections show that CCWM leads to more disentangled state representations with respect to the state-values, hence potentially supports stronger generalisability (Higgins et al., 2016).
+
+
+
+
+Figure 10: Qualitative comparison of long-range predictive reconstruction of CCWM and Dreamer. Predictive rollouts over 45 time-steps given the actions are computed using the representation models. Comparing to Dreamer, CCWM consistently generates more accurate predictive reconstructions over longer time spans. CCWM generates accurate predictions up to $\sim 18$ steps on Walker task, and $\sim 35$ steps on Cheetah task; Dreamer generates accurate predictions up to $\sim 8$ steps on Walker task, and $\sim 15$ steps on Cheetah task (evaluated over 5 randomly sampled trajectories).
+
+# E FURTHER INVESTIGATION OF THE EFFECTS OF “LEARNING VIA RETRACING” IN THE SAMPLE EFFICIENCY OF LEARNING
+
+In order to demonstrate that "learning via retracing" indeed brings additional "valid" supervision signals for model learning, we need to examine whether the retracing operation is correctly learned. In Figure 7 we show the t-SNE embeddings of the true and retracted latent states over a trajectory of 500 steps of the trained agent on the cheetah run task (Van der Maaten and Hinton, 2008). We observe a similar dominant ring structure in both embeddings. This matches our intuition, since during running, the simulated cheetah repeats its states periodically to resume running at a high speed. This is more clearly observed from a sub-trajectory towards the end of the entire episode in Figure 7b, where most states are densely distributed on both sides of the ring and more loosely distributed on the paths connecting the two sides for both the original and retracted states. Moreover, during the early stage of the episode (darker dots in Figure 7a), the agent starts to accelerate from stationary and the states of the agent are periodic in nature but differs from the states when the agent is running at full speed. Such early-phase structure is correctly captured by the retracing operation. Collectively these evidence support that the retracing is correctly learned hence provides useful supervision signals for model training.
+
+# F LONG-RANGE PREDICTION RECONSTRUCTION
+
+In Figure 10, we show the complete 45-step predictive reconstructions for CCWM and Dreamer on "walker walk" and "cheetah run" tasks. We see that CCWM is able to retain accurate predictions over the entire 45 predictive time-steps in the cheetah task, whereas Dreamer fails to do so. By looking more carefully at Figure 10, latent rollouts in the embedding space learned by CCWM yield less accurate predictions for the "irrecoverable" states (e.g., when the agent is falling, see Section 5.4 for further discussion). This property of the learned representation enables "task-aware planning", i.e., the behaviour learning agent is trained to acquire more precise control policy such that the "irrecoverable" states that might hinder the overall learning process are minimally encountered, leading to improved efficiency and quality of learning.
+
+
+
+
+
+
+
+
+
+
+Figure 11: Full ablation studies of the retracing loss function. CCWM with bisimulation metrics as the retracing loss function consistently outperforms the alternatives using the $L2$ and reconstruction retracing error.
+
+
+L2 CCWM-A3C reconstruction
+
+
+
+
+
+| method | FCN | Mem (GB) | \( \mathcal{R} \)Oxford med | hard | \( \mathcal{R} \)Oxford + \( \mathcal{R} \)1M med | hard | \( \mathcal{R} \)Paris med | hard | \( \mathcal{R} \)Paris + \( \mathcal{R} \)1M med | hard |
| Global descriptors |
| RMAC (Tolias et al., 2016) | R101 | 7.6 | 60.9 | 32.4 | 39.3 | 12.5 | 78.9 | 59.4 | 54.8 | 28.0 |
| \( AP-GeM^{\ddagger}_{\text{(Revaud et al., 2019a)}} \) | R101 | 7.6 | 67.1 | 42.3 | 47.8 | 22.5 | 80.3 | 60.9 | 51.9 | 24.6 |
| \( GeM+SOLAR_{\text{(Ng et al., 2020)}} \) | R101 | 7.6 | 69.9 | 47.9 | 53.5 | 29.9 | 81.6 | 64.5 | 59.2 | 33.4 |
| Global descriptors + reranking with local features |
| DELG (Cao et al., 2020) | R50 | 7.6 | 75.1 | 54.2 | 61.1 | 36.8 | 82.3 | 64.9 | 60.5 | 34.8 |
| DELG (Cao et al., 2020) | R101 | 7.6 | 78.5 | 59.3 | 62.7 | 39.3 | 82.9 | 65.5 | 62.6 | 37.0 |
| Local features + ASMK matching (max. 1000 features per image) |
| DELF (Noh et al., 2017) | \( R50^{-} \) | 9.2 | 67.8 | 43.1 | 53.8 | 31.2 | 76.9 | 55.4 | 57.3 | 26.4 |
| SELF-R-ASMK (Teichmann et al., 2019) | \( R50^{-} \) | 27.4 | 76.0 | 52.4 | 64.0 | 38.1 | 80.2 | 58.6 | 59.7 | 29.4 |
| HOW (Tolias et al., 2020) | \( R50^{-} \) | 7.9 | 78.3 | 55.8 | 63.6 | 36.8 | 80.1 | 60.1 | 58.4 | 30.7 |
| FIRE (ours) | \( R50^{-} \) | 6.4 | 81.8 | 61.2 | 66.5 | 40.1 | 85.3 | 70.0 | 67.6 | 42.9 |
| (standard deviation over 5 runs) | | | (±0.6) | (±1.0) | (±0.8) | (±1.1) | (±0.4) | (±0.6) | (±0.7) | (±0.8) |
| (mAP gains over HOW) | | | (↑3.5) | (↑5.4) | (↑2.9) | (↑3.3) | (↑5.2) | (↑9.9) | (↑9.2) | (↑12.2) |
+
+Table 3: Comparison to the state of the art. All models are trained on SfM-120k. FCN denotes the fully-convolutional network backbone, with $\mathbb{R}50^{-}$ denoting a ResNet-50 without the last block. Memory is reported for the image representation of the full R1M set (without counting local features for the global descriptors + reranking methods). $^{\ddagger}$ result from (Tolias et al., 2020). Bold denotes best performance, underlined second best among methods using ASMK, italics second best overall.
+
+at different resolutions (Jiang et al., 2013), aggregating SIFT descriptors with coding and pooling schemes (Boureau et al., 2010), or mining frequent patterns in sets of SIFT features (Singh et al., 2012). More recently, Hausler et al. (2021) introduced a multi-scale fusion of patch features.
+
+Supervision for local features. Several works provide supervision at the level of local features in the context of contrastive learning. Xie et al. (2021) and Chen et al. (2021) obtain several views of an input image using data augmentations with known pixel displacements. Similarly, Liu et al. (2020) train a model to predict the probability for a pair of local features to match, evaluated using known displacements. Wang et al. (2020) and Zhou et al. (2021) obtain local supervision by relying on epipolar coordinates and relative camera poses. Positive pairs in image retrieval depart from these setups, as pixel-level correspondences cannot be known. To build matches, Wang et al. (2021) use a standard nearest neighbor algorithm to build pairs of features, similarly to our approach, but without the use of filtering which is critical to our final performance. Using the current model predictions to define targets is reminiscent of modern self-supervised learning approaches which learn without any label (Caron et al., 2021; Grill et al., 2020). The additional filtering step can be seen as a way to keep only the most reliable model predictions, similar to self-distillation as in e.g. Sohn et al. (2020).
+
+Attention modules. Our LIT module is an iterative variant of standard attention (Bahdanau et al., 2015; Vaswani et al., 2017), adapted to map a variable number of input features to an ordered set of $N$ output features, similar to Lee et al. (2019). The Perceiver model (Jaegle et al., 2021) has demonstrated the flexibility of such constructions by using it to scale attention-based deep networks to large inputs. Our design was heavily inspired by slot-attention (Locatello et al., 2020), but has some key differences that enable us to achieve high performance in more complex visual environments: a) unlike the slot attention which initializes its slots with i.i.d sampling, we learn the initial templates and therefore define an ordering on the output set, a crucial property for selecting promising matches; b) we replace the recurrent network gates with a residual connection across iterations. These modifications, together with the attention decorrelation loss enable our module to go from a handful of object-oriented slots to a much larger set of output features. For object detection, Carion et al. (2020) rely on a set of learned object queries to initialize a stack of transformer layers. Unlike ours, their module is not recurrent; Appendix A.1 experimentally shows substantial benefits from applying LIT $T$ times, with weight sharing, to increase model flexibility without extra parameters.
+
+# 6 CONCLUSIONS
+
+We present an approach that aggregates local features into Super-features for image retrieval, a task that has up to now been dominated by approaches that work at the local feature level. We design an attention mechanism that outputs an ordered set of such features that are more discriminative and expressive than local features. Exploiting their ordered nature and without any extra supervision, we present a loss working directly on the proposed features. Our method not only significantly improves performance, but also requires less memory, a crucial requirement for scalable image retrieval.
+
+# REFERENCES
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+
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+
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+
+# Appendix
+
+# Table of Contents
+
+A Additional ablations 14
+
+A.1 Impact of the iteration and the number of templates hyper-parameters 14
+A.2 Impact of hard negatives 14
+A.3 Impact of the update function 15
+A.4 Extended ablation on matching constraints 15
+A.5 Impact of the template initialization 15
+
+B Analysis and discussions 16
+
+B.1 Single-scale results 16
+B.2 Consistency of attention maps across scales 16
+B.3 Redundancy in Super-features 16
+B.4 Are all Super-features trained? 17
+B.5 Using the Super-feature loss with local features 17
+B.6 Pretraining the backbone together with LIT 18
+B.7 Computational cost of Super-features extraction 18
+B.8 Is Google Landmarks v2 clean an appropriate training dataset for testing on Oxford and Paris? 18
+
+C Application to visual localization 18
+
+In this appendix, we present additional ablations (Appendix A), a deeper analysis of our framework (Appendix B) as well as results of the application of our FIRe framework for image retrieval to the task of visual localization (Appendix C). We briefly summarize the findings in the following paragraphs.
+
+Ablations. We study the impact of some hyper-parameters of our model, namely the size $N$ of the set of Super-features, and the number of iterations $T$ in the LIT module, in Appendix A.1. We show that 256 Super-features and 6 iterations offer the best trade-off between performance and computational cost. We also show in Appendix A.2 that we obtain further performance gains of over $1\%$ by increasing the number of negatives to 10 or 15. We then study the impact of replacing the residual connection inside the LIT module by a recurrent network (Appendix A.3). In Appendix A.4 we present an extended version of Table 1 with further matching constraints and we finally study the impact of the template initialization on performance in Appendix A.5.
+
+Properties of Super-features. We show single-scale results in Appendix B.1. In Appendix B.2, we display the attention maps of Super-features, at different scales for a fixed Super-feature ID. We further study the amount of redundancy in Super-features, compared to local features, in Appendix B.3. Next, we verify in Appendix B.4 that all Super-features receive training signal, as a sanity check. We discuss the case of applying a loss directly on local features in Appendix B.5 and give details about the pretraining on ImageNet in Appendix B.6. In Appendix B.7 we report the average extraction time for the proposed Super-features, while in Appendix B.8 we discuss the fact that there is an overlap between the queries from the common $\mathcal{R}$ Oxford and $\mathcal{R}$ Paris datasets and the Google Landmarks-v2-clean dataset that is commonly used as training set for retrieval on $\mathcal{R}$ Oxford and $\mathcal{R}$ Paris.
+
+Application to visual localization. We further evaluate our model using a visual localization setting on the Aachen Day-Night v1.1 dataset (Sattler et al., 2018) in Appendix C. To do so, we leverage a retrieval + local feature matching pipeline, and show that it is beneficial to use our method, especially in hard settings.
+
+# A ADDITIONAL ABLATIONS
+
+# A.1 IMPACT OF THE ITERATION AND THE NUMBER OFTEMPLATES HYPER-PARAMETERS
+
+In this section, we perform ablations on the number of Super-features $N$ and the number of iterations $T$ in the Local feature Integration Transformer. We first study the impact of the number $N$ of Super-features extracted for each scale of each image in Figure 7a. We observe that the best value is 256 on both the validation and test sets (R.Oxford and RParis). We then study the impact of the number of iterations T in Figure 7b. While the performance decreases when doing 2 or 3 iterations compared to just 1, a better performance is reached for 6 iterations, after which the performance saturates while requiring more computations. We thus use $T = 6$ .
+
+
+(a) Impact of the number of Super-features
+Figure 7: Varying the number of Super-features $N$ and of iterations $T$ in the Local feature Integration Transformer.
+
+
+$\text{一} =$ SfM-120k val. $\text{一} =$ ROxford (avg) $\text{一} =$ RParis (avg)
+(b) Impact of the number of iterations
+
+# A.2 IMPACT OF HARD NEGATIVES
+
+Each training tuple is composed of one image pair depicting the same landmark, and 5 negative images (i.e. from different landmarks). We plot the performance when varying this number in Figure 8. We observe that adding more negatives improves overall the performance on all datasets, i.e. by more than $1\%$ when increasing it to 10 or 15 negatives, but at the cost of a longer training time as more images need to be processed at each iteration. This is why we use 5 negatives in the rest of the paper as it offers a good compromise between performance and cost.
+
+
+Figure 8: Varying the number of hard negatives per training tuple.
+
+# A.3 IMPACT OF THE UPDATE FUNCTION
+
+The formula for $\psi$ in Equation (5) sums the previous $\mathcal{Q}$ with the output of the attention component $V(\mathcal{U})\cdot \alpha$ , i.e., with a residual connection. This is a different choice than the one made in the object-centric slot attention of Locatello et al. (2020), which proposes to use a Gated Recurrent Unit: $\psi (\mathcal{U};\mathcal{Q}) = \mathrm{GRU}(V(\mathcal{U})\cdot \alpha ,\mathcal{Q})$ . We thus compare the residual connection we use to a GRU function and report results in Table 4 with $T = 3$ . We observe that the residual connection reaches a better performance in all datasets while having the interest of not adding extra parameters.
+
+| Hyperparameter | Default value |
| DDQN & Dueling DDQN initial episodes | 1 |
| DDQN & Dueling DDQN batch size | 32 |
| DDQN & Dueling DDQN learning rate | 0.00025 |
| DDQN & Dueling DDQN target network update rate | 0.01 |
| DDQN & Dueling DDQN discount factor | 0.99 |
| DDQN & Dueling DDQN initial epsilon | 1.0 |
| DDQN & Dueling DDQN minimal epsilon | 0.1 |
| DDQN & Dueling DDQN epsilon decay factor | 0.9 |
| DDQN & Dueling DDQN number of hidden layers | 1 |
| DDQN & Dueling DDQN hidden layer size | 64 |
| DDQN & Dueling DDQN activation function | ReLU |
| DDQN & Dueling DDQN replay buffer size | 1000000 |
| Dueling DDQN feature dimension | 128 |
| Dueling DDQN value and adv. stream no. hidden layers | 1 |
| Dueling DDQN value and adv. stream hidden layer size | 128 |
+
+# B.3.5 AGENT AND NES HYPERPARAMETERS FOR MOUNTAINCARCONTINUOUS-V0 AND HALFCHEETAH-V3
+
+In Table 16 we list the NES and TD3 hyperparameter configuration spaces that we optimized for learning RNs on MountainCarContinuous (CMC) and HalfCheetah (HC). In Table 17 we list the default TD3 hyperparameters that we used once RNs have been learned. Table 18 shows the PPO agent hyperparameters we used for showing transfer of RNs to train PPO agents. In Table 19 we list the hyperparameter configuration spaces that we used (and optimized over) for the ICM baseline (Pathak et al., 2017). We note that, despite optimizing the ICM HPs, we did not find that the optimized ICM HPs yielded better results than the default ones which is why we use the default ICM HPs in all experiments that involve ICM.
+
+Table 15: Used ICM (Pathak et al., 2017) baseline hyperparameter configuration spaces optimized with CartPole and DDQN. We additionally report the default values taken from the ICM reference implementation.
+
+| Index |
| t | Time index |
| i, j | Variable element (channel) index |
| τ | Time lag index |
| perm | Random permuted variable index across the data batch |
| Variable |
| xt | Observation data |
| xt | Reconstructed observation |
| u | Auxiliary nonstationary regime variable |
| zt | Underlying sources |
| zHx | Time-delayed latent causal variables |
| Pa(zit) | Set of direct cause nodes/parents of node zit |
| et | Measurement error |
| H, H | Forward or backward embeddings in bidirectional RNN |
| σ(γ) | Soft mask vector |
| w | Modulation parameter vector |
| B | State transition matrix |
| εit | Process noise term |
| ˆit | Estimated process noise term |
| ẑit | Estimated sources |
| zit | True underlying sources |
| Function and Hyperparameter |
| p | Distribution function (e.g., pεit is the distribution of εit.) |
| g | Arbitrary nonlinear and injective mixing function |
| fi | Nonlinear transition function for zit |
| h | Indeterminancy mappings between zt andẑt |
| ri | Learned inverse transition function for residualˆi |
| si,u | Spline flow function for residualˆi in regime u |
| β, γ, σ | Weights in the augmented ELBO objective |
| n | Latent size |
| L | Maximum time lag |
| T | Total length of time series |
| π | Permutation operation |
| T | Component-wise invertible nonlinearities |
| q | Log density function |
| λ, α | Parameters of Laplacian distribution |
+
+# A.2 DISCUSSION OF OUR ASSUMED CONDITIONS
+
+We first explain and justify each critical assumption in the proposed conditions. We then discuss how restrictive or mild the conditions are in real applications.
+
+# A.2.1 INDEPENDENT NOISE (IN) CONDITION
+
+The IN condition was introduced in the Structural Equation Model (SEM), which represents effect $Y$ as a function of direct causes $X$ and noise $E$ :
+
+$$
+Y = f (X, E) \quad \text {w i t h} \quad \underbrace {X \perp E} _ {\text {}}. \tag {10}
+$$
+
+$$
+\text {I N}
+$$
+
+If $X$ and $Y$ do not have a common cause, as seen from the causal sufficiency assumption of structural equation models in Chapter 1.4.1 of Pearl's book (Pearl et al., 2000), the IN condition states that the unexplained noise variable $E$ is statistically independent of cause $X$ . IN is a direct result of assuming causal sufficiency in SEM. The main idea for the proof is that if IN is violated, then by the common cause principle (Reichenbach, 1956), there exist hidden confounders that cause their dependence, thus violating the causal sufficiency assumption. Furthermore, for a causally sufficient system with acyclic causal relations, the noise terms in different variables are mutually independent. The main idea is that when the noise terms are dependent, it is customary to encode such dependencies by augmenting the graph with hidden confounder variables (Pearl et al., 2000), which means that the system is not causally sufficient.
+
+In this paper, we assume the underlying latent processes form a casually-sufficient system without latent causal confounders. Then, the process noise terms $\epsilon_{it}$ are mutually independent, and moreover, the process noise terms $\epsilon_{it}$ are independent of direct cause/parent nodes $\mathbf{P}\mathbf{a}(z_{it})$ because of time information (the causal graph is acyclic because of the temporal precedence constraint).
+
+Applicability Loosely speaking, if there are no latent causal confounders in the (latent) causal processes and the sampling frequency is high enough to observe the underlying dynamics, then the IN condition assumed in this paper is satisfied in a causally-sufficient system and, moreover, there is no instantaneous causal influence (because of the high enough resolution). At the same time, we acknowledge that there exist situations where the resolution is low and there appears to be instantaneous dependence. However, there are several pieces of work dealing with causal discovery from measured time series in such situations; see. e.g., Granger (1987); Gong* et al. (2015); Danks & Plis (2013); Gong et al. (2017). In case there are instantaneous causal relations among latent causal processes, one would need additional sparsity or minimality conditions to recover the latent processes and their relations, as demonstrated in Silva et al. (2006); Adams et al. (2021). How to address the issue of instantaneous dependency or instantaneous causal relations in the latent processes will be one line of our future work.
+
+# A.2.2 NONSTATIONARY NOISE AND SUFFICIENT VARIABILITY CONDITION
+
+Nonstationary Noise For nonparametric processes, temporal constraints are not sufficient for the identification of latent causal transition dynamics whose functional or distributional form is not constrained. Otherwise, there is no need for Theorem 2 to assume the generalized Laplacian noise and the full-rankness of state transitions at all. In this paper, an alternative way is to exploit the (temporal) nonstationarity of the data caused by changing noise distribution (hence called nonstationary noise condition). We assume the functions of the temporal causal influences denoted by $f_{i}$ remain the same across across the $|\mathbf{u}|$ regimes or domains of data we have observed, but the distributions $p_{\epsilon_i|\mathbf{u}}$ , of noise terms that serve as arguments to the structural equation models, may change. One special case of this principle uses nonstationary variances, i.e., the noise variances change across nonstationary regimes. This kind of perturbation has been widely used in linear ICA (Matsuoka et al., 1995). Additionally, the nonstationary noise condition in this paper allows for any kinds of modulation of noise distribution by nonstationary regimes $\mathbf{u}$ , such as changing distributional forms, scale, and location by $\mathbf{u}$ , as long as the modulated sources satisfy sufficient variability condition described below.
+
+Sufficient Variability The sufficient variability condition was introduced in GCL (Hyvarinen et al., 2019) to extend the modulated exponential families (Hyvarinen & Morioka, 2016) to general modulated distributions. Essentially, the condition says that the nonstationary regimes $\mathbf{u}$ must have a sufficiently complex and diverse effect on the transition distributions. In other words, if the underlying distributions are composed of relatively many domains of data, the condition generally holds true. For instance, in the linear Auto-Regressive (AR) model with Gaussian innovations where only the noise variance changes, the condition reduces to the statement in (Matsuoka et al., 1995) that the variance of each noise term fluctuates somewhat independently of each other in different nonstationary regimes. Then the condition is easily attained if the variance vector of noise terms in any regime is not a linear combination of variance vectors of noise terms in other regimes.
+
+We further illustrate the condition using the example of modulated conditional exponential families in (Hyvarinen et al., 2019). Let the log-pdf $q(\mathbf{z}_t|\{\mathbf{z}_{t - \tau}\} ,\mathbf{u})$ be a conditional exponential family
+
+distribution of order $k$ given nonstationary regime $\mathbf{u}$ and history $\mathbf{z}_{\mathrm{Hx}} = \{\mathbf{z}_{t - \tau}\}$ :
+
+$$
+q \left(z _ {i t} \mid \mathbf {z} _ {\mathrm {H x}}, \mathbf {u}\right) = q _ {i} \left(z _ {i t}\right) + \sum_ {j = 1} ^ {k} q _ {i j} \left(z _ {i t}\right) \lambda_ {i j} \left(\mathbf {z} _ {\mathrm {H x}}, \mathbf {u}\right) - \log Z \left(\mathbf {z} _ {\mathrm {H x}}, \mathbf {u}\right), \tag {11}
+$$
+
+where $q_{i}$ is the base measure, $q_{ij}$ is the function of the sufficient statistic, $\lambda_{ij}$ is the natural parameter, and $\log Z$ is the log-partition. Loosely speaking, the sufficient variability holds if the modulation of by $\mathbf{u}$ on the conditional distribution $q(z_{it}|\mathbf{z}_{\mathrm{Hx}},\mathbf{u})$ is not too simple in the following sense:
+
+1. Higher order of $k$ ( $k > 1$ ) is required. If $k = 1$ , the sufficient variability cannot hold;
+2. The modulation impacts $\lambda_{ij}$ by $\mathbf{u}$ must be linearly independent across regimes $\mathbf{u}$ . The sufficient statistics functions $q_{ij}$ cannot be all linear, i.e., we require higher-order statistics.
+
+Further details of this example can be found in Appendix B of (Hyvarinen et al., 2019). In summary, we need the modulation by $\mathbf{u}$ to have diverse (i.e., distinct influences) and complex impacts on the underlying data generation process.
+
+Applicability The nonstationarity of process noise seems to be prominent in many kinds of temporal data. For example, nonstationary variances are seen in EEG/MEG, natural video, and closely related to changes in volatility in financial time series (Hyvarinen & Morioka, 2016). As we assume the transition functions $f_{i}$ are fixed across regimes, the data that most likely satisfy the proposed condition is a collection of multiple trials/segments of data with slightly different temporal dynamics in between, where the differences can be well modeled by different noise distributions. For instance, in MEG data, temporal nonstationarity can be modeled by segmenting the measured data into different sessions (e.g., stimuli, rest, etc.) where the session index modulates the noise variance.
+
+# A.2.3 GENERALIZED LAPLACIAN NOISE CONDITION
+
+In the parametric (VAR) processes in Theorem 2, we exploit the non-Gaussianity of noise perturbations to achieve identifiability. Specifically, we constrain the process noise distribution to be within the generalized Laplacian distribution family in this paper. This L1-sparse temporal prior is motivated by the natural statistics of video data, where the uncertainty could have sharp impacts on some latent factors, but most other factors are not perturbed between two adjacent frames. This transition prior has strong connections with slow feature analysis (Sprekeler et al., 2014; Klindt et al., 2020) which measures slowness in terms of the L2 distance between temporally adjacent encodings as temporal constraints for nonlinear ICA. Note that although the Laplacian-like distributional form is pre-defined, the generalized Laplacian distribution can still be used to fit a broad family of perturbations with different shapes by changing $\alpha$ and $\lambda$ of the distribution.
+
+Applicability L1-sparse transition priors are widely used to model video datasets and natural scene measurements. This condition is applicable to video datasets where the external factors have sharp effects on some but not all latent factors in two adjacent frames.
+
+# A.2.4 NONSINGULAR STATE TRANSITIONS CONDITION
+
+Nonsingularity is a standard assumption made in the previous studies (Zhang & Hyvarinen, 2011) to achieve identifiability of linear state-space models. We give a two-dimensional low-rank vector autoregressive (VAR) process example below to illustrate the concept. Define a low-rank VAR process with time lag $L = 1$ below:
+
+$$
+\mathbf {z} _ {t} = \mathbf {B} \mathbf {z} _ {t - 1} + \epsilon_ {t} \quad w i t h \quad \mathbf {B} = \left[ \begin{array}{c c} a & b \\ \alpha \times a & \alpha \times b \end{array} \right] = \left[ \begin{array}{l} 1 \\ \alpha \end{array} \right] \times [ a \quad b ], \tag {12}
+$$
+
+where $\mathbf{z}_t = [z_{1t}, z_{2t}]^\top$ and $\epsilon_t = [\epsilon_{1t}, \epsilon_{2t}]^\top$ . Multiplying both sides of the VAR process with a row vector $[a, b]$ , we have:
+
+$$
+\begin{array}{l} \underbrace {\left[ \begin{array}{l l} a & b \end{array} \right] \times \mathbf {z} _ {t}} _ {\tilde {\mathbf {z}} _ {t}} = \left[ \begin{array}{l l} a & b \end{array} \right] \times \left[ \begin{array}{l} 1 \\ \alpha \end{array} \right] \times \left[ \begin{array}{l l} a & b \end{array} \right] \mathbf {z} _ {t - 1} + \left[ \begin{array}{l l} a & b \end{array} \right] \times \epsilon_ {t} (13) \\ = (a + b \alpha) \underbrace {\left[ \begin{array}{l l} a & b \end{array} \right] \times \mathbf {z} _ {t - 1}} _ {\tilde {\mathbf {z}} _ {t - 1}} + \underbrace {\left[ \begin{array}{l l} a & b \end{array} \right] \times \epsilon_ {t}} _ {\tilde {\epsilon} _ {t}}. (14) \\ \end{array}
+$$
+
+Hence, the two-dimensional VAR process reduces to a single linear AR process with $\tilde{\mathbf{z}}_t = [a\quad b]\times$ $\mathbf{z}_t = az_{1t} + bz_{2t}$ and $\tilde{\epsilon}_{t} = [a\quad b]\times \epsilon_{t} = a\epsilon_{1t} + b\epsilon_{2t}$ , which is a linear combination of the original two processes. In this case, we cannot recover $\mathbf{z}_t$ at all, but only the linear combination $\tilde{\mathbf{z}}_t$ .
+
+In summary, when the state transition matrices are not of full rank, there exist low-dimensional projections of the underlying latent processes that satisfy the observational equivalence everywhere. By assuming the nonsingular state transitions, one could prevent from recovering low-dimensional projections of latent causal factors and time-delayed relations.
+
+# A.3 PROOF OF IDENTIFIABILITY THEORY
+
+# A.3.1 PRELIMINARIES
+
+Equivalent Relations on Latent Space Our proof of identifiability starts from deriving relations on estimated latent space from observational equivalence: the joint distribution $p_{\hat{g},\hat{f},\hat{p}_{\epsilon}}(\mathbf{x}_{\mathrm{Hx}},\mathbf{x}_t)$ matches $p_{g,f,p_{\epsilon}}(\mathbf{x}_{\mathrm{Hx}},\mathbf{x}_t)$ everywhere. Note that we consider only one future time step $\mathbf{x}_t$ for simplicity as the joint probability of the whole sequence can be decomposed into product of these terms. Since the learned mixing function $\mathbf{x}_t = \hat{g} (\mathbf{z}_t)$ can be written as $\mathbf{x}_t = (g\circ (g)^{-1}\circ \hat{g})(\mathbf{z}_t)$ because of injective properties of $(g,\hat{g})$ , we can see that $\hat{g} = g\circ ((g)^{-1}\circ \hat{g}) = g\circ h$ for some function $h = (g)^{-1}\circ \hat{g}$ on the latent space. Our goal here is really to show that this function $h$ , which represents the indeterminacy of the learned latent space, is a permutation with component-wise nonlinearities. It has been proved in (Klindt et al., 2020) that:
+
+1. Indeterminancy $h$ on latent space can only be a bijection on the latent space if both $g$ and $\hat{g}$ are injective functions, and $h$ preserves the prior distribution in the latent space. The proofs are in Appendix A.1 of (Klindt et al., 2020) on Page 18;
+2. Eq. 15 can be directly derived from observational equivalence using the injective properties of $(g,\hat{g})$ . The proofs are in Appendix A.1 of (Klindt et al., 2020) on Page 19:
+
+$$
+p \left(\mathbf {z} _ {t} \mid \left\{\mathbf {z} _ {t - \tau} \right\}\right) = p \left(h ^ {- 1} \left(\mathbf {z} _ {t}\right) \mid \left\{h ^ {- 1} \left(\mathbf {z} _ {t - \tau}\right) \right\}\right) \frac {p \left(\mathbf {z} _ {t}\right)}{p \left(h ^ {- 1} \left(\mathbf {z} _ {t}\right)\right)} \quad \forall \left(\mathbf {z} _ {t}, \left\{\mathbf {z} _ {t - \tau} \right\}\right). \tag {15}
+$$
+
+Identifiability of Linear Non-Gaussian State-Space Model Linear State Space Model (SSM) defined below has been proved to be fully identifiable in (Zhang & Hyvarinen, 2011) when both the process noise $\epsilon_{t}$ and measurement error $\mathbf{e}_t$ are temporally white and independent of each other, and at most one component of the process noise $\epsilon_{t}$ is Gaussian. The observation error $\mathbf{e}_t$ can be either Gaussian or non-Gaussian:
+
+$$
+\mathbf {x} _ {t} = \mathbf {A} \mathbf {z} _ {t} + \mathbf {e} _ {t}, \tag {16}
+$$
+
+$$
+\mathbf {z} _ {t} = \sum_ {\tau = 1} ^ {L} \mathbf {B} _ {\tau} \mathbf {z} _ {t - \tau} + \epsilon_ {t}. \tag {17}
+$$
+
+We will make use of this property of linear non-Gaussian SSM for deriving Theorem 2. The main idea is if we can prove $h$ of parametric conditions (which also has vector autoregressive processes as in Eq. 17 with non-Gaussian noise) is within affine transformations, the componentwise identifiability of true latent variables can be directly derived because we can treat the affine indeterminacy as a "high-level" affine mixing of sources same as $\mathbf{A}$ in Eq. 16 without measurement error.
+
+# A.3.2 PROOF OF THEOREM 1
+
+Theorem A.1 (Nonparametric Processes) Assume nonparametric processes in Eq. 18, where the transition functions $f_{i}$ are third-order differentiable functions and mixing function $g$ is injective and differentiable almost everywhere; let $\mathbf{Pa}(z_{it})$ denote the set of (time-delayed) parent nodes of $z_{it}$ :
+
+$$
+\underbrace {\mathbf {x} _ {t} = g \left(\mathbf {z} _ {t}\right)} _ {\text {N o n l i n e a r m i x i n g}}, \quad \underbrace {z _ {i t} = f _ {i} \left(\left\{z _ {j , t - \tau} \mid z _ {j , t - \tau} \in \mathbf {P a} \left(z _ {i t}\right) \right\} , \epsilon_ {i t}\right)} _ {\text {N o n p a r a m e t r i c t r a n s i t i o n}} \text {w i t h} \underbrace {\epsilon_ {i t} \sim p _ {\epsilon_ {i} | \mathbf {u}}} _ {\text {N o n s t a t i o n a r y n o i s e}}. \tag {18}
+$$
+
+Here we assume:
+
+1. (Nonstationary Noise): Noise distribution $p_{\epsilon_z|\mathbf{u}}$ is modulated (in any way) by the observed categorical auxiliary variables $\mathbf{u}$ , which denotes nonstationary regimes or domain index;
+2. (Independent Noise): The noise terms $\epsilon_{it}$ are mutually independent (i.e., spatially and temporally independent) in each regime of $\mathbf{u}$ (note that this directly implies that $\epsilon_{it}$ are independent from $\mathbf{Pa}(z_{it})$ in each regime);
+3. (Sufficient Variability): For any $\mathbf{z}_t \in \mathbb{R}^n$ there exist $2n + 1$ values for $\mathbf{u}$ , i.e., $\mathbf{u}_j$ with $j = 0,1,\dots,2n$ , such that the $2n$ vectors $\mathbf{w}(\mathbf{z}_t, \mathbf{u}_{j+1}) - \mathbf{w}(\mathbf{z}_t, \mathbf{u}_j)$ , with $j = 0,1,\dots,2n$ , are linearly independent with $\mathbf{w}(\mathbf{z}_t, \mathbf{u})$ defined below, where $q_i$ is the log density of the conditional distribution and $\mathbf{z}_{Hx} = \{\mathbf{z}_{t-\tau}\}$ denotes history information up to maximum time lag $L$ :
+
+$$
+\mathbf {w} (\mathbf {z} _ {t}, \mathbf {u}) \triangleq \left(\frac {\partial q _ {1} (z _ {1 t} | \mathbf {z} _ {H x} , \mathbf {u})}{\partial z _ {1 t}}, \dots , \frac {\partial q _ {n} (z _ {n t} | \mathbf {z} _ {H x} , \mathbf {u})}{\partial z _ {n t}}, \frac {\partial^ {2} q _ {1} (z _ {1 t} | \mathbf {z} _ {H x} , \mathbf {u})}{\partial z _ {1 t} ^ {2}}, \dots , \frac {\partial^ {2} q _ {n} (z _ {n t} | \mathbf {z} _ {H x} , \mathbf {u})}{\partial z _ {n t} ^ {2}}\right). \tag {19}
+$$
+
+Then the componentwise identifiability property of temporally causal latent processes is ensured.
+
+Proof: We first extend Eq. 15 to include conditioning on the nonstationary regime $\mathbf{u}$ . We then show that if sufficient variability condition is satisfied, the indeterminancy function $h$ can only be permutation with component-wise nonlinearities.
+
+Step 1 We first derive equivalent relations on the latent space by conditioning on the nonstationary regime $\mathbf{u}$ . This can be directly achieved by applying the change of variable formula on the $L + 1$ invertible maps: $\mathbf{z}_t \Rightarrow h^{-1}(\mathbf{z}_t)$ , $\mathbf{z}_{t-1} \Rightarrow h^{-1}(\mathbf{z}_{t-1})$ , ..., $\mathbf{z}_{t-L} \Rightarrow h^{-1}(\mathbf{z}_{t-L})$ . W.l.o.g, let's assume $L = 1$ for now. We then have the following three equalities:
+
+$$
+p \left(\mathbf {z} _ {t}, \mathbf {z} _ {t - 1}, \mathbf {u}\right) = p \left(h ^ {- 1} \left(\mathbf {z} _ {t}\right), h ^ {- 1} \left(\mathbf {z} _ {t - 1}\right), \mathbf {u}\right) \left| \det \frac {\partial h ^ {- 1} \left(\mathbf {z} _ {t}\right)}{\partial \mathbf {z} _ {t}} \right| \left| \det \frac {\partial h ^ {- 1} \left(\mathbf {z} _ {t - 1}\right)}{\partial \mathbf {z} _ {t - 1}} \right|, \tag {20}
+$$
+
+$$
+p \left(\mathbf {z} _ {t}\right) = p \left(h ^ {- 1} \left(\mathbf {z} _ {t}\right)\right) \left| \det \frac {\partial h ^ {- 1} \left(\mathbf {z} _ {t}\right)}{\partial \mathbf {z} _ {t}} \right|, \tag {21}
+$$
+
+$$
+p \left(\mathbf {z} _ {t - 1}, \mathbf {u}\right) = p \left(h ^ {- 1} \left(\mathbf {z} _ {t - 1}\right), \mathbf {u}\right) \left| \det \frac {\partial h ^ {- 1} \left(\mathbf {z} _ {t - 1}\right)}{\partial \mathbf {z} _ {t - 1}} \right|. \tag {22}
+$$
+
+Solving for the determinant terms in Eq. 21 and Eq. 22 and plugging them into Eq. 20, we have:
+
+$$
+p \left(\mathbf {z} _ {t} \mid \mathbf {z} _ {t - 1}, \mathbf {u}\right) = p \left(h ^ {- 1} \left(\mathbf {z} _ {t}\right) \mid h ^ {- 1} \left(\mathbf {z} _ {t - 1}\right), \mathbf {u}\right) \frac {p \left(\mathbf {z} _ {t}\right)}{p \left(h ^ {- 1} \left(\mathbf {z} _ {t}\right)\right)}. \tag {23}
+$$
+
+It is straightforward to see that this relation holds for multiple time lags where $L > 1$ . We take logs on both sides, and we now define $\bar{q}(\mathbf{z}_t) \triangleq q(\mathbf{z}_t)$ as the marginal log-density of the components $\mathbf{z}_t$ when $\mathbf{u}$ is integrated out. We then have:
+
+$$
+q \left(\mathbf {z} _ {t} \mid \left\{\mathbf {z} _ {t - \tau} \right\}, \mathbf {u}\right) - q \left(h ^ {- 1} \left(\mathbf {z} _ {t}\right) \mid h ^ {- 1} \left(\left\{\mathbf {z} _ {t - \tau} \right\}, \mathbf {u}\right)\right) = \bar {q} \left(\mathbf {z} _ {t}\right) - \bar {q} \left(h ^ {- 1} \left(\mathbf {z} _ {t}\right)\right), \tag {24}
+$$
+
+and using the Independent Noise (IN) assumption, the conditional log-pdf $q(\mathbf{z}_t|\{\mathbf{z}_{t - \tau}\} ,\mathbf{u})$ and its estimated version $q(h^{-1}(\mathbf{z}_t)|\{h^{-1}(\mathbf{z}_{t - \tau})\} ,\mathbf{u})$ are conditional independent (note this has been enforced in causal process network as constraints) and LHS can be factorized as:
+
+$$
+\sum_ {i} q _ {i} \left(z _ {i t} \mid \left\{\mathbf {z} _ {t - \tau} \right\}, \mathbf {u}\right) - q _ {i} \left(\left[ h ^ {- 1} (\mathbf {z} _ {t}) \right] _ {i} \mid \left\{h ^ {- 1} \left(\mathbf {z} _ {t - \tau}\right) \right\}, \mathbf {u}\right) = \bar {q} (\mathbf {z} _ {t}) - \bar {q} \left(h ^ {- 1} \left(\mathbf {z} _ {t}\right)\right). \tag {25}
+$$
+
+where $\bar{q}$ is the marginal log-density of the components $\mathbf{z}_t$ when $\mathbf{u}$ is integrated out and it does not need to be factorial.
+
+Step 2 Now we do the following simplification of notations. Let $h_i^{-1}(\mathbf{z}_t) = \left[h^{-1}(\mathbf{z}_t)\right]_i$ . Denote the first-order and second-order derivatives by a superscript as:
+
+$$
+q _ {i} ^ {1} \left(z _ {i t} | \left\{\mathbf {z} _ {t - \tau} \right\}, \mathbf {u}\right) = \frac {\partial q _ {i} \left(z _ {i t} \mid \left\{\mathbf {z} _ {t - \tau} \right\} , \mathbf {u}\right)}{\partial z _ {i t}}, \tag {26}
+$$
+
+$$
+q _ {i} ^ {2} \left(z _ {i t} \mid \left\{\mathbf {z} _ {t - \tau} \right\}, \mathbf {u}\right) = \frac {\partial^ {2} q _ {i} \left(z _ {i t} \mid \left\{\mathbf {z} _ {t - \tau} \right\} , \mathbf {u}\right)}{\partial z _ {i t} ^ {2}}, \tag {27}
+$$
+
+and take derivatives of both sides of Eq. 25 with respect to $z_{jt}$ , we have:
+
+$$
+q _ {j} ^ {1} \left(z _ {j t} \mid \left\{\mathbf {z} _ {t - \tau} \right\}, \mathbf {u}\right) - \sum_ {i = 1} ^ {n} q _ {i} ^ {1} \left(h _ {i} ^ {- 1} \left(\mathbf {z} _ {t}\right) \mid \left\{h ^ {- 1} \left(\mathbf {z} _ {t - \tau}\right) \right\}, \mathbf {u}\right) \frac {\partial h _ {i} ^ {- 1} \left(\mathbf {z} _ {t}\right)}{\partial z _ {j t}} \tag {28}
+$$
+
+$$
+= \bar {q} ^ {j} (\mathbf {z} _ {t}) - \sum_ {i} \bar {q} ^ {j} \left(h _ {i} ^ {- 1} (\mathbf {z} _ {t})\right) \frac {\partial h _ {i} ^ {- 1} (\mathbf {z} _ {t})}{\partial z _ {j t}}. \tag {29}
+$$
+
+Denote the first order derivative of $h^{-1}$ as $v_{i}^{j}(\mathbf{z}_{t}) = \frac{\partial h_{i}^{-1}(\mathbf{z}_{t})}{\partial z_{j t}}$ and $v_{i}^{jj^{\prime}}(\mathbf{z}_{t})$ is the second-order derivative with respect to a different component $z_{j't}$ for any $j \neq j'$ . Taking another derivative with respect to $z_{j't}$ on both sides of Eq. 29, the first term on LHS vanishes and we have:
+
+$$
+\sum_ {i} q _ {i} ^ {1 1} \left(h _ {i} ^ {- 1} \left(\mathbf {z} _ {t}\right) \mid \left\{h ^ {- 1} \left(\mathbf {z} _ {t - \tau}\right) \right\}, \mathbf {u}\right) v _ {i} ^ {j} \left(\mathbf {z} _ {t}\right) v _ {i} ^ {j ^ {\prime}} \left(\mathbf {z} _ {t}\right) + q _ {i} ^ {1} \left(h _ {i} ^ {- 1} \left(\mathbf {z} _ {t}\right) \mid \left\{h ^ {- 1} \left(\mathbf {z} _ {t - \tau}\right) \right\}, \mathbf {u}\right) v _ {i} ^ {j j ^ {\prime}} \left(\mathbf {z} _ {t}\right) = c ^ {j j ^ {\prime}}. \tag {30}
+$$
+
+where $c^{jj'}$ denotes the derivatives of RHS of Eq. 29 which does not depend on $\mathbf{u}$ . Same as (Hyvarinen et al., 2019), we collect all these equations in vector form by defining $\mathbf{a}_i(y)$ as a vector collecting all entries $v_i^j(\mathbf{z}_t)v_i^{j'}(\mathbf{z}_t)$ for $j \in [1, n]$ and $j' \in [1, j-1]$ . We omit diagonal terms, and by symmetry, take only one half of the indices. Likewise, collect all the entries $v_i^{jj'}(\mathbf{z}_t)$ for $j \in [1, n]$ and $j' \in [1, j-1]$ in the vector $\mathbf{b}(\mathbf{z}_t)$ . All the entries of $c^{jj'}$ are in $\mathbf{c}(\mathbf{z}_t)$ . These $n(n-1)/2$ equations can be written a single system of equations:
+
+$$
+\sum_ {i} \mathbf {a} _ {i} (y) q _ {i} ^ {1 1} \left(h _ {i} ^ {- 1} \left(\mathbf {z} _ {t}\right) \mid \left\{h ^ {- 1} \left(\mathbf {z} _ {t - \tau}\right) \right\}, \mathbf {u}\right) + \mathbf {b} _ {i} \left(\mathbf {z} _ {t}\right) q _ {i} ^ {1} \left(h _ {i} ^ {- 1} \left(\mathbf {z} _ {t}\right) \mid \left\{h ^ {- 1} \left(\mathbf {z} _ {t - \tau}\right) \right\}, \mathbf {u}\right) = \mathbf {c} \left(\mathbf {z} _ {t}\right). \tag {31}
+$$
+
+Now, collect the a and b into a matrix M:
+
+$$
+\mathbf {M} (\mathbf {z} _ {t}) = \left(\mathbf {a} _ {1} \left(\mathbf {z} _ {t}\right), \dots , \mathbf {a} _ {n} \left(\mathbf {z} _ {t}\right), \mathbf {b} _ {i} \left(\mathbf {z} _ {t}\right), \dots , \mathbf {b} _ {n} \left(\mathbf {z} _ {t}\right)\right). \tag {32}
+$$
+
+Eq. 31 takes the form of the following linear system:
+
+$$
+\mathbf {M} (\mathbf {z} _ {t}) \mathbf {w} (\mathbf {z} _ {t}, \mathbf {u}) = \mathbf {c} (\mathbf {z} _ {t}), \tag {33}
+$$
+
+where $\mathbf{w}$ are the vectors defined in the sufficient variability assumption, and $\mathbf{w}$ is defined for any input $\mathbf{z}_t$ . Notice that the RHS of the linear system does not depend on $\mathbf{u}$ , so we fix $\mathbf{z}_t$ and consider the $2n + 1$ points $\mathbf{u}$ given for that $\mathbf{z}_t$ by the sufficient variability assumption.
+
+Collect Eq. 33 above for $2n$ points starting from index 1:
+
+$$
+\mathbf {M} \left(\mathbf {z} _ {t}\right) \left(\mathbf {w} \left(\mathbf {z} _ {t}, \mathbf {u} _ {1}\right), \dots , \mathbf {w} \left(\mathbf {z} _ {t}, \mathbf {u} _ {2 n}\right)\right) = \left(\mathbf {c} \left(\mathbf {z} _ {t}\right), \dots , \mathbf {w} \left(\mathbf {z} _ {t}, \mathbf {u} _ {1}\right)\right), \tag {34}
+$$
+
+$$
+\mathbf {M} (\mathbf {z} _ {t}) \left(\mathbf {w} \left(\mathbf {z} _ {t}, \mathbf {u} _ {0}\right), \dots , \mathbf {w} \left(\mathbf {z} _ {t}, \mathbf {u} _ {2 n - 1}\right)\right) = \left(\mathbf {c} \left(\mathbf {z} _ {t}\right), \dots , \mathbf {w} \left(\mathbf {z} _ {t}, \mathbf {u} _ {1}\right)\right). \tag {35}
+$$
+
+Subtract Eq. 35 from Eq. 34, we then have:
+
+$$
+\mathbf {M} \left(\mathbf {z} _ {t}\right) \underbrace {\left[ \mathbf {w} \left(\mathbf {z} _ {t} , \mathbf {u} _ {1}\right) - \mathbf {w} \left(\mathbf {z} _ {t} , \mathbf {u} _ {0}\right) , \dots , \mathbf {w} \left(\mathbf {z} _ {t} , \mathbf {u} _ {2 n}\right) - \mathbf {w} \left(\mathbf {z} _ {t} , \mathbf {u} _ {0}\right) \right]} _ {\mathbf {W}} = \mathbf {0}. \tag {36}
+$$
+
+By the suffienct variability assumption, the matrix $\mathbf{W}$ that has linearly independent columns and is a square matrix so is nonsingular. The only solution to the linear system above is thus:
+
+$$
+\mathbf {M} (\mathbf {z} _ {t}) = \left(\mathbf {a} _ {1} \left(\mathbf {z} _ {t}\right), \dots , \mathbf {a} _ {n} \left(\mathbf {z} _ {t}\right), \mathbf {b} _ {i} \left(\mathbf {z} _ {t}\right), \dots , \mathbf {b} _ {n} \left(\mathbf {z} _ {t}\right)\right) = \mathbf {0}. \tag {37}
+$$
+
+Following (Hyvarinen et al., 2019), $\mathbf{a}(\mathbf{z}_t)$ being zero implies no row of the Jacobian of $h^{-1}(\mathbf{z}_t)$ can have more than one non-zero entry. This holds for any $\mathbf{z}_t$ . By continuity of the Jacobian and its invertibility, the non-zero entries in the Jacobian must be in the same places for all $\mathbf{z}_t$ : If they switched places, there would have to be a point where the Jacobian is singular, which would contradict the bijection properties of $h^{-1}$ derived in Section A.3.1. This means that each $h_i^{-1}(\mathbf{z}_t)$ is a function of only one $z_{kt}$ for $k \in [1, n]$ . The bijection $h^{-1}$ also implies that each of the componentwise functions is invertible. Thus, we have proven that latent variables are identifiable up to permutation and componentwise invertible transformations and temporally causal latent processes with conditions required by Theorem 1 are proved to be identifiable from observed variables.
+
+# A.3.3 PROOF OF THEOREM 2
+
+Theorem A.2 (Parametric Processes) Assume the vector autoregressive process in Eq. 38, where the state transition functions are linear and additive and mixing function $g$ is injective and differentiable almost everywhere. Let $\mathbf{B}_{\tau} \in \mathbb{R}^{n \times n}$ be the state transition matrix at lag $\tau$ . The process noises $\epsilon_{it}$ are assumed to be stationary and both spatially and temporally independent:
+
+$$
+\underbrace {\mathbf {x} _ {t} = g \left(\mathbf {z} _ {t}\right)} _ {\text {N o n l i n e a r m i x i n g}}, \quad \underbrace {\mathbf {z} _ {t} = \sum_ {\tau = 1} ^ {L} \mathbf {B} _ {\tau} \mathbf {z} _ {t - \tau} + \epsilon_ {t}} _ {\text {L i n e a r a d d i t i v e t r a n s i t i o n}} \text {w i t h} \underbrace {\epsilon_ {i t} \sim p _ {\epsilon_ {i}}} _ {\text {I n d e p e n d e n t n o i s e}}. \tag {38}
+$$
+
+Here we assume:
+
+1. (Generalized Laplacian Noise): Process noises $\epsilon_{it} \sim p_{\epsilon_i}$ are mutually independent and follow the generalized Laplacian distribution $p_{\epsilon_i} = \frac{\alpha_i \lambda_i}{2 \Gamma(1 / \alpha_i)} \exp \left(-\lambda_i |\epsilon_i|^\alpha_i\right)$ with $\alpha_i < 2$ ;
+2. (Nonsingular State Transitions): For at least one $\tau$ , the state transition matrix $\mathbf{B}_{\tau}$ is of full rank.
+
+Then the componentwise identifiability property of temporally causal latent processes is ensured.
+
+Proof: The following proof is inspired by Theorem 1 in (Klindt et al., 2020). The key differences are (i) allowing temporal causal relations $\mathbf{B}_{\tau}$ among the sources instead of independent sources assumption, (ii) extending the single time lag restriction to multiple time lags case.
+
+Identifiability on Causally-Related Sources Let us start from the simple case where the time lag $\tau = 1$ . In this case, the transition dynamic in Eq. 38 can be simplified as
+
+$$
+\mathbf {x} _ {t} = g \left(\mathbf {z} _ {t}\right), \quad \mathbf {z} _ {t} = \mathbf {B} \mathbf {z} _ {t - 1} + \epsilon_ {t}. \tag {39}
+$$
+
+Using Eq. 15 and by applying the distributional forms of generalized Laplacian noise, we have:
+
+$$
+\begin{array}{l} p \left(\mathbf {z} _ {t} \mid \mathbf {z} _ {t - 1}\right) = p \left(h ^ {- 1} \left(\mathbf {z} _ {t}\right) \mid h ^ {- 1} \left(\mathbf {z} _ {t - 1}\right)\right) \frac {p \left(\mathbf {z} _ {t}\right)}{p \left(h ^ {- 1} \left(\mathbf {z} _ {t}\right)\right)} \tag {40} \\ \Longrightarrow \quad M | | \mathbf {z} _ {t} - \mathbf {B z} _ {t - 1} | | _ {\alpha} ^ {\alpha} - N | | h ^ {- 1} (\mathbf {z} _ {t}) - \mathbf {B} h ^ {- 1} (\mathbf {z} _ {t - 1}) | | _ {\alpha} ^ {\alpha} = \log \frac {p (\mathbf {z} _ {t})}{p (h ^ {- 1} (\mathbf {z} _ {t}))}, \\ \end{array}
+$$
+
+where $M$ and $N$ are the constants appearing in the exponentials in $p(\mathbf{z}_t|\mathbf{z}_{t - 1})$ and $p(h^{-1}(\mathbf{z}_t)|h^{-1}(\mathbf{z}_{t - 1}))$ .
+
+Taking the derivative w.r.t $\mathbf{z}_{t - 1}$ on both sides, we obtain
+
+$$
+\frac {\partial \left\| \mathbf {z} _ {t} - \mathbf {B} \mathbf {z} _ {t - 1} \right\| _ {\alpha} ^ {\alpha}}{\partial \mathbf {z} _ {t - 1}} = \frac {\partial \left\| h ^ {- 1} \left(\mathbf {z} _ {t}\right) - \mathbf {B} h ^ {- 1} \left(\mathbf {z} _ {t - 1}\right) \right\| _ {\alpha} ^ {\alpha}}{\partial \mathbf {z} _ {t - 1}}. \tag {41}
+$$
+
+For the left hand of the Eq. 41, we can derive
+
+$$
+\begin{array}{l} \frac {\partial \left\| \mathbf {z} _ {t} - \mathbf {B} \mathbf {z} _ {t - 1} \right\| _ {\alpha} ^ {\alpha}}{\partial \left\| \mathbf {z} _ {t} - \mathbf {B} \mathbf {z} _ {t - 1} \right\| _ {\alpha}} \frac {\partial \left\| \mathbf {z} _ {t} - \mathbf {B} \mathbf {z} _ {t - 1} \right\| _ {\alpha}}{\partial \mathbf {z} _ {t - 1}} = \frac {\partial \left\| h ^ {- 1} (\mathbf {z} _ {t}) - \mathbf {B} h ^ {- 1} (\mathbf {z} _ {t - 1}) \right\| _ {\alpha} ^ {\alpha}}{\partial \mathbf {z} _ {t - 1}} \\ \Longrightarrow \quad \alpha \| \mathbf {z} _ {t} - \mathbf {B} \mathbf {z} _ {t - 1} \| _ {\alpha} ^ {\alpha - 1} \frac {\partial \| \mathbf {z} _ {t} - \mathbf {B} \mathbf {z} _ {t - 1} \| _ {\alpha}}{\partial \mathbf {z} _ {t - 1}} = \frac {\partial \| h ^ {- 1} (\mathbf {z} _ {t}) - \mathbf {B} h ^ {- 1} (\mathbf {z} _ {t - 1}) \| _ {\alpha} ^ {\alpha}}{\partial \mathbf {z} _ {t - 1}} \\ \Longrightarrow \quad \alpha \| \mathbf {z} _ {t} - \mathbf {B} \mathbf {z} _ {t - 1} \| _ {\alpha} ^ {\alpha - 1} \frac {\partial \| \epsilon_ {t} \| _ {\alpha}}{\partial \epsilon_ {t}} \frac {\partial \mathbf {z} _ {t} - \mathbf {B} \mathbf {z} _ {t - 1}}{\partial \mathbf {z} _ {t - 1}} = \frac {\partial \| h ^ {- 1} (\mathbf {z} _ {t}) - \mathbf {B} h ^ {- 1} (\mathbf {z} _ {t - 1}) \| _ {\alpha} ^ {\alpha}}{\partial \mathbf {z} _ {t - 1}} \tag {42} \\ \Longrightarrow \quad - \alpha | | \mathbf {z} _ {t} - \mathbf {B} \mathbf {z} _ {t - 1} | | _ {\alpha} ^ {\alpha - 1} \frac {\partial | | \epsilon_ {t} | | _ {\alpha}}{\partial \epsilon_ {t}} \mathbf {B} = \frac {\partial | | h ^ {- 1} (\mathbf {z} _ {t}) - \mathbf {B} h ^ {- 1} (\mathbf {z} _ {t - 1}) | | _ {\alpha} ^ {\alpha}}{\partial \mathbf {z} _ {t - 1}} \\ \Rightarrow - \alpha (\mathbf {z} _ {t} - \mathbf {B z} _ {t - 1}) \odot | \mathbf {z} _ {t} - \mathbf {B z} _ {t - 1} | ^ {\alpha - 2} \mathbf {B} = \frac {\partial | | h ^ {- 1} (\mathbf {z} _ {t}) - \mathbf {B h} ^ {- 1} (\mathbf {z} _ {t - 1}) | | _ {\alpha} ^ {\alpha}}{\partial \mathbf {z} _ {t - 1}}. \\ \end{array}
+$$
+
+Making the same derivation process on the right hand, we obtain
+
+$$
+\begin{array}{l} - \alpha (\mathbf {z} _ {t} - \mathbf {B} \mathbf {z} _ {t - 1}) \odot | \mathbf {z} _ {t} - \mathbf {B} \mathbf {z} _ {t - 1} | ^ {\alpha - 2} \mathbf {B} \\ = - \alpha \left(h ^ {- 1} \left(\mathbf {z} _ {t}\right) - \mathbf {B} h ^ {- 1} \left(\mathbf {z} _ {t - 1}\right)\right) \odot | h ^ {- 1} \left(\mathbf {z} _ {t}\right) - \mathbf {B} h ^ {- 1} \left(\mathbf {z} _ {t - 1}\right) | ^ {\alpha - 2} \mathbf {B} \frac {\partial h ^ {- 1} \left(\mathbf {z} _ {t - 1}\right)}{\mathbf {z} _ {t - 1}}. \tag {43} \\ \end{array}
+$$
+
+For any $\mathbf{z}_t$ we can choose $\mathbf{z}_t = \mathbf{B}\mathbf{z}_{t - 1}$ and thus the Eq. 43 can be written as:
+
+$$
+\left(h ^ {- 1} \left(\mathbf {z} _ {t}\right) - \mathbf {B} h ^ {- 1} \left(\mathbf {z} _ {t - 1}\right)\right) \odot | h ^ {- 1} \left(\mathbf {z} _ {t}\right) - \mathbf {B} h ^ {- 1} \left(\mathbf {z} _ {t - 1}\right) | ^ {\alpha - 2} \mathbf {B} \frac {\partial h ^ {- 1} \left(\mathbf {z} _ {t - 1}\right)}{\mathbf {z} _ {t - 1}} = \mathbf {0}. \tag {44}
+$$
+
+Considering the nonsingularity of matrix $\mathbf{B}$ (by assumption) and bijection of $h$ , we can derive
+
+$$
+\begin{array}{l} \left(h ^ {- 1} \left(\mathbf {z} _ {t}\right) - \mathbf {B} h ^ {- 1} \left(\mathbf {z} _ {t - 1}\right)\right) \odot \left| h ^ {- 1} \left(\mathbf {z} _ {t}\right) - \mathbf {B} h ^ {- 1} \left(\mathbf {z} _ {t - 1}\right) \right| ^ {\alpha - 2} = \mathbf {0} \\ \Rightarrow \quad (h _ {i} ^ {- 1} (\mathbf {z} _ {t}) - \mathbf {B} h _ {i} ^ {- 1} (\mathbf {z} _ {t - 1})) | h _ {i} ^ {- 1} (\mathbf {z} _ {t}) - \mathbf {B} h _ {i} ^ {- 1} (\mathbf {z} _ {t - 1}) | ^ {\alpha - 2} = 0. \\ \end{array}
+$$
+
+for all $i = 1,\dots ,d$ . Apparently $h^{-1}(\mathbf{z}_t) = \mathbf{B}h^{-1}(\mathbf{z}_{t - 1})$ is the only solution, thus
+
+$$
+h ^ {- 1} \left(\mathbf {B} \mathbf {z} _ {t - 1}\right) = \mathbf {B} h ^ {- 1} \left(\mathbf {z} _ {t - 1}\right). \tag {46}
+$$
+
+Substitute Eq. 46 to the right hand of Eq. 40, we have
+
+$$
+\begin{array}{l} \left| \left| z _ {t} - \mathbf {B} z _ {t - 1} \right| \right| _ {\alpha} ^ {\alpha} = \left| \left| h ^ {- 1} (z _ {t}) - \mathbf {B} h ^ {- 1} (z _ {t - 1}) \right| \right| _ {\alpha} ^ {\alpha} \\ \Rightarrow \quad | | z _ {t} - \mathbf {B} z _ {t - 1} | | _ {\alpha} ^ {\alpha} = | | h ^ {- 1} (z _ {t}) - h ^ {- 1} (\mathbf {B} z _ {t - 1}) | | _ {\alpha} ^ {\alpha}. \\ \end{array}
+$$
+
+This indicates that $h^{-1}$ preserves the $\alpha$ -distances between points. Since $h$ is bijective, then by Mazur-Ulam theorem Mazur & Ulam (1932), $h$ must be an affine transform. According to Theorem 2 in (Zhang & Hyvarinen, 2011), the model is identifiable, which proves the theorem.
+
+Extension to Multiple Time Lags We can extend the result in Eq. 41 and have
+
+$$
+\frac {\partial \left| \left| \mathbf {z} _ {t} - \sum_ {\tau = 1} ^ {L} \mathbf {B} _ {\tau} \mathbf {z} _ {t - \tau} \right| \right| _ {\alpha} ^ {\alpha}}{\partial \mathbf {z} _ {t - i}} = \frac {\partial \left| \left| h ^ {- 1} (\mathbf {z} _ {t}) - \sum_ {\tau = 1} ^ {L} \mathbf {B} _ {\tau} h ^ {- 1} (\mathbf {z} _ {t - \tau}) \right| \right| _ {\alpha} ^ {\alpha}}{\partial \mathbf {z} _ {t - i}}, \tag {48}
+$$
+
+where $\mathbf{z}_{t - i}$ is any lag latents with the limitation that the $\mathbf{B}_i$ corresponding to $\mathbf{z}_{t - i}$ is of full rank. Following the same above-mentioned derivation process, we can obtain
+
+$$
+\begin{array}{l} \left(\mathbf {z} _ {t} - \sum_ {\tau = 1} ^ {L} \mathbf {B} _ {\tau} \mathbf {z} _ {t - \tau}\right) \odot | \mathbf {z} _ {t} - \sum_ {\tau = 1} ^ {L} \mathbf {B} _ {\tau} \mathbf {z} _ {t - \tau} | ^ {\alpha - 2} \mathbf {B} _ {i} \\ = (h ^ {- 1} (\mathbf {z} _ {t}) - \sum_ {\tau = 1} ^ {L} \mathbf {B} _ {\tau} h ^ {- 1} (\mathbf {z} _ {t - \tau})) \odot | h ^ {- 1} (\mathbf {z} _ {t}) - \sum_ {\tau = 1} ^ {L} \mathbf {B} _ {\tau} h ^ {- 1} (\mathbf {z} _ {t - \tau}) | ^ {\alpha - 2} \mathbf {B} _ {i} \frac {\partial h ^ {- 1} (\mathbf {z} _ {t - i})}{\mathbf {z} _ {t - i}}. \\ \end{array}
+$$
+
+For any $\mathbf{z}_t$ we can choose $\mathbf{z}_t = \sum_{\tau=1}^{L} \mathbf{B}_{\tau} \mathbf{z}_{t-\tau}$ , thus the Eq. 49 can be written as:
+
+$$
+\left(h ^ {- 1} \left(\mathbf {z} _ {t}\right) - \sum_ {\tau = 1} ^ {L} \mathbf {B} _ {\tau} h ^ {- 1} \left(\mathbf {z} _ {t - \tau}\right)\right) \odot | h ^ {- 1} \left(\mathbf {z} _ {t}\right) - \sum_ {\tau = 1} ^ {L} \mathbf {B} _ {\tau} h ^ {- 1} \left(\mathbf {z} _ {t - \tau}\right) | ^ {\alpha - 2} \mathbf {B} _ {i} \frac {\partial h ^ {- 1} \left(\mathbf {z} _ {t - i}\right)}{\mathbf {z} _ {t - i}} = \mathbf {0}. \tag {50}
+$$
+
+As mentioned above, $h^{-1}(\mathbf{z}_t) = \sum_{\tau = 1}^{L}\mathbf{B}_{\tau}h^{-1}(\mathbf{z}_{t - \tau})$ is the only solution, thus
+
+$$
+h ^ {- 1} \left(\sum_ {\tau = 1} ^ {L} \mathbf {B} _ {\tau} \mathbf {z} _ {t - \tau}\right) = \sum_ {\tau = 1} ^ {L} \mathbf {B} _ {\tau} h ^ {- 1} \left(\mathbf {z} _ {t - \tau}\right). \tag {51}
+$$
+
+Following the same procedure in the simple case, the theorem is proven.
+
+# A.4 COMPARISONS WITH EXISTING THEORIES
+
+The closest work to ours includes (1) PCL (Hyvarinen & Morioka, 2017), which exploited temporal constraints to separate independent sources, (2) SlowVAE (Klindt et al., 2020), which leveraged sparse transition of adjacent video frames to separate independent sources, and (3) iVAE (Khemakhem et al., 2020), which leveraged the nonstationarity by the modulation of side information $\mathbf{u}$ on the prior distribution $p(\mathbf{z}|\mathbf{u})$ of conditional factorial latent variables. Our work extends the theories to the discovery of the conditional independent sources with time-delayed causal relations in between by leveraging nonstationarity, or functional and distribution forms of temporal statistics. To the best of our knowledge, this is one of the first works that successfully recover time-delayed latent processes from their nonlinear mixtures without using sparsity or minimality assumptions.
+
+PCL The sources $z_{it}$ in PCL were assumed to be mutually independent (see Assumption 1 of Theorem 1 in PCL). In contrast, we allow the sources to have time-delayed causal relations in between, which is much more realistic in real-world applications. They further assumed the sources are stationary, while we allow nonstationarity in the nonparametric setting (the nonstationary noise assumption). The underlying processes of PCL are described by Eq. 52:
+
+$$
+\log p \left(z _ {i, t} \mid z _ {i, t - 1}\right) = G \left(z _ {i, t} - \rho z _ {i, t - 1}\right) \quad \text {o r} \quad \log p \left(z _ {i, t} \mid z _ {i, t - 1}\right) = - \lambda \left(z _ {i, t} - r \left(z _ {i, t - 1}\right)\right) ^ {2} + \text {c o n s t .} \tag {52}
+$$
+
+where $G$ is some non-quadratic function corresponding to the log-pdf of innovations, $\rho < 1$ is regression coefficient, $r$ is some nonlinear, strictly monotonic regression, and $\lambda$ is a positive precision parameter. Both theorems of our work extend the theory to the discovery of the conditional independent sources with time-delayed causal relations in between. Furthermore, in the nonparametric process condition, we do not restrict the functional and distributional forms of underlying transitions. Our proposed nonparametric condition naturally includes PCL as a special case.
+
+SlowVAE Inspired by slow feature analysis, SlowVAE assumes the underlying sources to have identity transitions with generalized Laplacian innovations described in Eq. 53:
+
+$$
+p \left(\mathbf {z} _ {t} \mid \mathbf {z} _ {t - 1}\right) = \prod_ {i = 1} ^ {d} \frac {\alpha \lambda}{2 \Gamma (1 / \alpha)} \exp - (\lambda \left| z _ {i, t} - z _ {i, t - 1} \right| ^ {\alpha}) \quad w i t h \quad \alpha < 2. \tag {53}
+$$
+
+Our proposed parametric condition (Theorem 2) in Eq. 4 is a natural extension to the Laplacian innovation model above by allowing time-delayed vector autoregressive transitions in the latent process with multiple time lags. Consequently, temporally causally-related latent processes with linear transition dynamics can thus be modeled and recovered from their nonlinear mixtures with our parametric condition.
+
+iVAE Similar to TCL (Hyvarinen & Morioka, 2016) and GIN (Sorrenson et al., 2020), iVAE exploits the nonstationarity brought by the side information (i.e., class label) on the prior distribution of latent variables $\mathbf{z}_t$ . As one can see from Eq. 54, the latent variables are conditionally independent, without causal relations in between while both of our theorems consider (time-delayed) causal relations between latent variables. In addition, iVAE exploits the nonstationarity brought by side information (i.e., class label) on the prior distribution of latent variables $\mathbf{z}$ . On the contrary, our nonparametric condition, instead of relying on the change in the prior distribution of latent variables,
+
+exploits the nonstationarity in the noise distribution, which is more natural in real-world datasets. Finally, iVAE assumes modulated exponential families in Eq. 54 while our nonparametric condition (Theorem 1) allows any kinds of modulation by side information $\mathbf{u}$ without those strong assumptions on the transition functions or distributions.
+
+$$
+p _ {T, \lambda} (\mathbf {z} | \mathbf {u}) = \prod_ {i} \frac {Q _ {i} \left(z _ {i}\right)}{Z _ {i} (\mathbf {u})} \exp \left[ \sum_ {j = 1} ^ {k} T _ {i, j} \left(z _ {i}\right) \lambda_ {i, j} (\mathbf {u}) \right] \tag {54}
+$$
+
+In terms of architecture innovations, to remove distributional and functional form constraints of iVAE, we design a novel causal transition prior network for nonparametric transitions by injecting the IN condition inside reparameterization trick, resulting in an efficient scoring mechanism of transition prior which only needs to compute the determinant of a low-triangular Jacobian matrix. This module was never seen in previous work.
+
+# A.5 CONNECTING THEORIES TO MODEL
+
+As one can see from the proofs in Appendix A.3, what have been assumed for the estimation framework are the conditional factorial properties of $q(\hat{\mathbf{z}}_t|\{\hat{\mathbf{z}}_{t - \tau}\} ,\mathbf{u})$ where $\hat{\mathbf{z}}_t = h^{-1}(\mathbf{z}_t)$ and the model of temporal nonstationarities through nonstationary noises. The conditional factorial properties have been injected using the reparameterization trick (Eq. 7) with the IN condition in causal transition prior and the enforcing of spatiotemporal independence of estimated residuals through contrastive learning. The nonstationary noises are modeled with flow-based density estimators. We share the weights of the other modules (e.g., encoder, transition function, decoder, inference network, etc.) across nonstationary regimes while using separate flow models to estimate the density of residuals and evaluate the prior scores in each regime. We also use componentwise flow models so the learned residuals will not interact with each other in the estimation framework. Finally, in nonparametric processes, we warm-start the flow models to generate standard Gaussian noise. In parametric processes, the flow models are initialized to generate standard Laplacian noise. Note that the other assumed conditions in the two theorems, such as sufficient variability and nonsingular state transitions, are data properties and do not need to be encoded as constraints in the estimation framework.
+
+# B EXPERIMENT SETTINGS
+
+# B.1 SYNTHETIC DATASET
+
+Seven synthetic datasets, including two datasets (NP and VAR) which satisfy our assumptions, and five datasets, which violate each of the assumption in the proposed theorems, are used in this paper. We set the latent size $n = 8$ and the lag number of the process $L = 2$ . The mixing function $g$ is a random three-layer MLP with LeakyReLU units.
+
+Nonparametric (NP) Dataset For nonparametric processes, we generate 150,000 data points according to Eq. 2. In particular, we use a Gaussian additive noise model as the latent processes. The noises $\epsilon_{it}$ are sampled from i.i.d. Gaussian distribution with variance modulated by 20 different nonstationary regimes. In each regime, the variance entries are uniformly sampled between 0 and 1. A 2-layer MLP with LeakyReLU units is used as the state transition function $f_{i}$ . When we need sparse causal structure for visualization, a random binary mask is added to the input nodes.
+
+(Violation) Insufficient Variability For this dataset, we create datasets that violate the nonstationary noise condition and sufficient variability by restricting the number of nonstationary regimes observed in the NP dataset. When only one regime is observed, we violate the nonstationary noise condition by using stationary noise. Furthermore, we vary the number of the observed regimes $|\mathbf{u}| \in \{1, 5, 10, 15, 20\}$ to assess the impacts of variability on the recovery of nonparametric processes.
+
+Parametric (VAR) Dataset For parametric processes, we generate 50,000 data points according to Eq. 4. The noises $\epsilon_{it}$ are sampled from i.i.d. Laplacian distribution $(\sigma = 0.1)$ . The entries of state transition matrices $\mathbf{B}_{\tau}$ are uniformly distributed between $[-0.5, 0.5]$ .
+
+(Violation) Low-rank State Transition For this dataset, the transition matrix $\mathbf{B}_{\tau}$ in Eq. 4 is low-rank instead of full-rank. The datasets are created following the steps in the VAR dataset, but we restrict the rank of state transition matrix $\mathbf{B}_{\tau}$ to 4 and time lag $L = 1$ . The full matrix rank is 8.
+
+(Violation) Gaussian Noise Distribution For this dataset, the noise terms $\epsilon_{it}$ in Eq. 4 follow the Gaussian distribution $(\alpha_{i} = 2)$ instead of Generalized Laplacian distribution $(\alpha_{i} < 2)$ . In particular, the noise terms $\epsilon_{it}$ are sampled from i.i.d. Gaussian distribution $(\sigma = 0.1)$ .
+
+(Violation) Regime-Variant Causal Relations For regime-variant causal relations, we generate 240,000 data points according to Eq. 55:
+
+$$
+\mathbf {x} _ {t} = g \left(\mathbf {z} _ {t}\right), \quad \mathbf {z} _ {t} = \sum_ {\tau = 1} ^ {L} \mathbf {B} _ {\tau} ^ {\mathbf {u}} \mathbf {z} _ {t - \tau} + \epsilon_ {t} \quad \text {w i t h} \quad \epsilon_ {i t} \sim p _ {\epsilon_ {i}}. \tag {55}
+$$
+
+The noises $\epsilon_{it}$ are sampled from i.i.d. Laplace distribution $(\sigma = 0.1)$ . In each regime $\mathbf{u}$ , the entries of state transition matrices $\mathbf{B}_{\tau}^{\mathbf{u}}$ are uniformly distributed between $[-0.5, 0.5]$ .
+
+(Violation) Instantaneous Causal Relations For instantaneous causal relations, we generate 45,000 data points according to Eq. 56:
+
+$$
+\mathbf {x} _ {t} = g \left(\mathbf {z} _ {t}\right), \quad \mathbf {z} _ {t} = \mathbf {A} \mathbf {z} _ {t} + \sum_ {\tau = 1} ^ {L} \mathbf {B} _ {\tau} \mathbf {z} _ {t - \tau} + \epsilon_ {t} \quad w i t h \quad \epsilon_ {i t} \sim p _ {\epsilon_ {i}}, \tag {56}
+$$
+
+where matrix $\mathbf{A}$ is a random Directed Acyclic Graph (DAG) which contains the coefficients of the linear instantaneous relations. The noises $\epsilon_{it}$ are sampled from i.i.d. Laplacian distribution with $\sigma = 0.1$ . The entries of state transition matrices $\mathbf{B}_{\tau}$ are uniformly distributed between $[-0.5, 0.5]$ .
+
+# B.2 REAL-WORLD DATASET
+
+Three public datasets, including KiTTiMask, Mass-Spring System, and CMU MoCap database, are used. The observations together with the true temporally causal latent processes are showcased in Fig. B.1. For CMU MoCap, the true latent causal variables and time-delayed relations are unknown.
+
+
+(a)
+
+
+
+
+
+
+(b)
+
+
+(c)
+
+
+Figure B.1: Real-world datasets: (a) KiTTiMask is a video dataset of binary pedestrian masks, (b) Mass-Spring system is a video dataset with ball movement rendered in color and invisible springs, and (c) CMU MoCap is a 3D point cloud dataset of skeleton-based signals.
+
+KiTTiMask The KiTTiMask dataset consists of pedestrian segmentation masks sampled from the autonomous driving vision benchmark KiTTi-MOTS. For each given frame, the position (vertical and horizontal) and the scale of the pedestrian masks are set using measured values. The difference in the sample time (e.g., $\Delta t = 0.15s$ ) generates the sparse Laplacian innovations between frames.
+
+Mass-Spring System The Mass-Spring system is a classical physical system that several objects are connected by some visible/invisible spring, which follows Hooke's law. In this work, we considered the system with five degrees of freedom and made linearization on the state without calculating the Euclidian distance between objects. Thus, there are ten causal relations, six of which were set connected, and the other four were disconnected. The rest length of the spring was uniformly distributed between [1, 10], and the stiffness of the spring relation was set as 20. The action was $a_{t} = 300e_{t}$ , where $e_t$ followed the Laplacian distribution with mean $\mu = 0$ and variance $\sigma = 1$ . We assumed there was no damping in the system and randomly assigned the objects in different positions at the beginning of each episode.
+
+CMU MoCap CMU MoCap (http://mocap.cs.cmu.edu/) is an open-source human motion capture dataset with various motion capture recordings (e.g., walk, jump, basketball, etc.) performed by over 140 subjects. In this work, we fit our model on 12 trials of "walk" recordings (Subject 7). Skeleton-based measurements have 62 observed variables corresponding to the locations of joints (e.g., head, foot, shoulder, wrist, throat, etc.) of the human body at each time step.
+
+# B.3 EVALUATION METRICS
+
+SHD: Structural Hamming Distance We use SHD (de Jongh & Druzdzel, 2009) to measure the distance between two causal graphs. It computes the number of edge insertions, deletions, or flips in order to transform one graph to another graph. SHD is one variant of Minimum Edit Distance (MED) in causal discovery area by allowing only insertions, deletions, and flips of edges.
+
+MCC: Mean Correlation Coefficient MCC is a standard metric for evaluating the recovery of latent factors in ICA literature. MCC first calculates the absolute values of correlation coefficient between every ground-truth factor against every estimated latent variable. Depending on whether componentwise invertible nonlinearities exist in the recovered factors, Pearson correlation coefficients or Spearman's rank correlation coefficients can be used. The possible permutation is adjusted by solving a linear sum assignment problem in polynomial time on the computed correlation matrix.
+
+In this work, we use the Pearson correlation coefficient for the VAR processes and Spearman's correlation coefficient for the NP processes.
+
+# C IMPLEMENTATION DETAILS
+
+In this section, we first provide the network architecture details of LEAP. The hyperparameter selection criteria and sensitivity analysis results are presented. The training settings are summarized.
+
+# C.1 NETWORK ARCHITECTURE
+
+We summarize our network architecture below and describe it in detail in Table C.1 and Table C.2.
+
+(1,2) MLP-Encoder and MLP-Decoder: These modules are used for the synthetic and motion capture datasets. They are composed of a series of fully-connected neural networks with LeakyReLU as the activation function. The universal approximation theorem guarantees that our model can approximate the mixing function. The encoder maps the raw observations into features, while the decoder maps the latent variables back to the inputs.
+- (3,4) CNN-Encoder and CNN-Decoder: For the KiTTiMask dataset, vanilla CNNs are used for both the encoder and decoder. For the Mass-Spring system dataset, the time-delayed causal variables use objects as the building blocks to factorize the scene. The object-centric representations contain object locations and some other attributes (e.g., color, size, etc.). We thus use two separate CNNs, one for extracting visual features (see Feature Extractor in Table C.2) and the other for locating object locations with a spatial softmax unit (see Keypoint Predictor in Table C.2). The decoder retrieves object features from feature maps using object locations and reconstructs the scene (see Refiner in Table C.2).
+(5) Inference Network: We apply the bidirectional Gated Recurrent Unit (GRU) (Cho et al., 2014) to preserve both the past and the future information. It processes the input sequence $\mathbf{x}_t$ in both directions: one for the forward pass and one for the backward pass. We denote the forward and backward embeddings as $\vec{H}$ and $\vec{H}$ . The other inference module (see TemporalDynamics in Table C.1) uses the sampled/inferred past latent variables to compute the posterior of $\mathbf{z}_t$ . We insert skip-connections (He et al., 2016) between the two inference modules to avoid the vanishing gradient problem and obtain better model convergence performances. Note that for the first $L$ temporally-earliest latent variables in the sequence, there is no time-delayed information, and we use isotropic Gaussian $\mathcal{N}(0,1)$ as their prior distributions. The prior of the remaining sequence is evaluated with the learned transition prior network described below.
+- (6,7) Causal Process Prior Network This module contains three components. (i) Inverse transition functions. For the NP transitions, we use MLPs to compute the estimated noises. For
+
+the VAR transitions, a group of state transition matrices, one for each time lag, is used. (ii) $\log(|\det(\mathbf{J})|)$ computation. For the NP transitions, the Jacobian matrix entries $\frac{\partial r_i}{\partial \hat{z}_{it}}$ are computed using torch.autogradfunctional.jacobian method. The log determinant is then evaluated by summing over log transformations of absolute values of Jacobian terms for $\hat{z}_{it}$ . For the VAR transitions, because of the additive noise assumption, the log determinant is directly 0. (iii) Spline flow model. Componentwise spline flow models use monotonic linear rational splines to transform standard Gaussian distribution to the estimated noise distribution. We use eight bins for the linear splines and set the bound as five so data points lying outside $[-5, 5]$ are evaluated using $\mathcal{N}(0,1)$ directly while the data points within the region are evaluated by spline flow models. For the nonparametric processes, we always warm-start the spline flows by training it on a dataset of standard Gaussian noises with steps=5000 and learning rate=0.001. For the parametric processes, we warm-start it instead on a dataset of standard Laplacian noises. All the three components of the transition prior network are set to be learnable during the VAE updates.
+
+Table C.1: Architecture details. BS: batch size, T: length of time series, i_dim: input dimension, z_dim: latent dimension, LeakyReLU: Leaky Rectified Linear Unit.
+
+| Configuration | Description | Output |
| 1. MLP-Encoder | Encoder for Synthetic/MoCap Data | |
| Input: x1:T | Observed time series | BS × T × i_dim |
| Dense | 128 neurons, LeakyReLU | BS × T × 128 |
| Dense | 128 neurons, LeakyReLU | BS × T × 128 |
| Dense | 128 neurons, LeakyReLU | BS × T × 128 |
| Dense | Temporal embeddings | BS × T × z_dim |
| 2. MLP-Decoder | Decoder for Synthetic/MoCap Data | |
| Input: ŷ1:T | Sampled latent variables | BS × T × z_dim |
| Dense | 128 neurons, LeakyReLU | BS × T × 128 |
| Dense | 128 neurons, LeakyReLU | BS × T × 128 |
| Dense | i_dim neurons, reconstructed x1:T | BS × T × i_dim |
| 5. Inference Network | Bidirectional Inference Network | |
| Input | Sequential embeddings | BS × T × z_dim |
| GRUInference | Bidirectional inference | BS × T × 2*z_dim |
| TemporalDynamics | Use past {z(t-τ)} to infer posteriors of z_t | BS × T × 2*z_dim |
| ResidualBlock | Skip-connection of the two inferences | BS × T × 2*z_dim |
| Bottleneck | Compute mean and variance of posterior | μ1:T, σ1:T |
| Reparameterization | Sequential sampling | x̂1:T |
| 6. Causal Process Prior (VAR) | Linear Transition Prior Network | |
| Input | Sampled latent variable sequence ŷ1:T | BS × T × z_dim |
| InverseTransition | Compute estimated residuals εit | BS × T × z_dim |
| SplineFlow | Score the likelihood of residuals | BS |
| 7. Causal Process Prior (NP) | Nonlinear Transition Prior Network | |
| Input | Sampled latent variable sequence ŷ1:T | BS × T × z_dim |
| InverseTransition | Compute estimated residuals εit | BS × T × z_dim |
| JacobianCompute | Compute log (|det(J)|) | BS |
| SplineFlow | Score the likelihood of residuals | BS |
+
+Model Ablations We start with BetaVAE and add our proposed modules successively as model variants. When the causal process prior network is added to the baseline, this model variant is equipped with the inference network and learned inverse transition functions. However, during the estimation of KL divergence in the causal process prior network, we use Mean Squared Error (MSE) directly, which corresponds to stationary Gaussian noise distribution, to replace the flow density estimators that evaluate the prior likelihood scores across nonstationary regimes. This variant shows the contributions of the learned causal transition functions. Furthermore, when the nonstationary flow estimator is added to the variant, this variant is implemented by removing the noise discriminator from LEAP while not changing any training settings.
+
+Table C.2: Architecture details on CNN encoder and decoder. BS: batch size, T: length of time series, h_dim: hidden dimension, z_dim: latent dimension, F: number of filters, (Leaky)ReLU: (Leaky) Rectified Linear Unit.
+
+| Configuration | Description | Output |
| 3.1.1 CNN-Encoder | Feature Extractor | |
| Input: x1:TConv2DConv2DConv2DConv2DConv2DConv2DConv2DConv2DConv2DConv2DConv2DConv2DConv2DConv2DConv2DConv2DConv2DConv2DConv2DConv2DConv2DConv2DConv2DConv2DConv2DConv2DConv2DConv2DConv2DConv2DConv2DConv2DConv2DConv2D Conv2DConv2DConv2DConv2DConv2DConv2DConv2DConv2DConv2DConv2DConv2DConv2DConv2DConv2DConv2DConv2DConv2DConv2DConv2DConv2DConv2DConv2DConv2DConv2DConv2DConv2DConv2DConv2DConv2DConv2DConv2DConv2DConv2DConv |
| 3.1.2 CNN-Encoder | Keypoint Predictor | |
| Input: x1:TConv2DConv2DConv2DConv2DConv2DConv2DConv2DConv2DConv2DConv2DConv2DConv2DConv2DConv2DConv2DConv2DConv2DConv2DConv2DConv2DConv2DConv2DConv2DConv2DConv2DConv2DConv2DConv2DConv2DConv2D Conv2DConv2DConv2D Conv2DConv2DConv2DConv2DConv2DConv2DConv2DConv2DConv2DConv2DConv2DConv2DConv2DConv2DConv2DConv2DConv2DConv2DConv2DConv2DConv2DConv2DConv2DConv2DConv2DConv2DConv2DConv2DConv2DConv2DConv |
| 3.2 KiTTiMask-Encoder | Mask Encoder | |
| Input: x1:TConv2DConv2DConv2DConv2DConv2DConv2DConv2DConv2DConv2DConv2DConv2DConv2DConv2DConv2DConv2DConv2DConv2DConv2DConv2DConv2DConv2DConv2DConv2DConv2DConv2DConv2DConv2DConv2DConv2DConv2D |
| 4.1 CNN-Decoder | Refiner | |
| Input: z1:TConvTranspose2DConv2DConvTranspose2DConv2DConv2DConv2DConv2DConv2DConv2DConv2DConv2DConv2DConv2DConv2DConv2DConv2DConv2DConv2DConv2DConv2DConv2DConv2DConv2DConv2DConv2DConv2DConv2DConv2DConv2DConv2DConv2DConv2DConv2DConv2DConv2DConv | Sampled latent variable sequenceF: 64, BatchNorm2D, LeakyReLUF: 64, BatchNorm2D, LeakyReLUF: 32, BatchNorm2D, LeakyReLUF: 32, BatchNorm2D, LeakyReLUF: 32, BatchNorm2D, LeakyReLUF: 32, BatchNorm2D, LeakyReLUF: 32, BatchNorm2D, LeakyReLUF: 32, BatchNorm2D, LeakyReLUF: 32, BatchNorm2D, LeakyReLUF: 32,BatchNorm2D, LeakyReLUF: 32,BatchNorm2D, LeakyReLUF: 32,BatchNorm2D, LeakyReLUF: 32,BatchNorm2D, LeakyReLUF: 32,BatchNorm2D, LeakyReLUF: 32,BatchNorm2D, LeakyReLUF: 32,BatchNorm2D, LeakyReLUF: 32,BatchNorm 2D, LeakyReLUF: 32,BatchNorm 2D, LeakyReLUF: 32,BatchNorm 2D, LeakyReLUF: 32,BatchNorm 2D, LeakyReLUF: 32,BatchNorm 2D, LeakyReLUF: 32,BatchNorm 2D, LeakyReLUF: 32,BatchNorm 2D, LeakyReLUF: 0 | BS × T × 64 × 2BS × T × 64 × 3 × 1 × 64 × 64 × 3 × 1 × 64 × 64 × 3 × 1 × 64 × 64 × 3 × 1 × 64 × 64 × 3 × 1 × 64 × 64 × 3 × 1 × 64 × 64 × 3 × 1 × 64 × 64 × 3 × 1 × 64 ×1 × 64 × 64 × 3 × 1 × 64 × 64 × 3 × 1 × 64 × 64 × 3 × 1 × 64 × 64 × 3 × 1 × 64 × 64 × 3 × 1 × 64 × 64 × 3 × 1 × 64 × 64 × 3 × 1× 64 × 64 × 3 × 1 × 64 × 64 × 3 × 1 × 64 × 64 × 3 × 1 × 64 × 64 × 3 × 1 × 64 × 64 × 3 × 1 × 64 × 64 × 3 × 1 × 64 × 64 × 3 × 1 × 1 × 64 × 64 × 3 × 1 × 64 × 64 × 3 × 1 × 64 × 64 × 3 × 1 × 64 × 64 × 3 × 1 × 64 × 64 × 3 × 1 × 64 × 64 × 3 × 1 × 64 × 64 × 3 ×1 × 64 × 64 × 3 × 1 × 64 × 64 × 3 × 1 × 64 × 64 × 3 × 1 × 64 × 64 × 3 × 1 × 64 × 64 × 3 × 1 × 64 × 64 × 3 × 1 × 64 × 64 × 3 × 0 |
| 4.2 KiTTiMask-Decoder | Mask Decoder | |
| Input: z1:TDenseConvTranspose2DConvTranspose2DConvTranspose2DConvTranspose2DConvTranspose2DConvTranspose2DConvTranspose2DConvTranspose2DConvTranspose2DConvTranspose2DConvTranspose2DConvTranspose2DConvTranspose2DConvTranspose2DConvTranspose2DConvTranspose2DConvTranspose2DConvTranspose2DConvTranspose2DConvTranspose2DConvTranspose2DConvTranspose2DConvTranspose2DConvTranspose2DConvTranspose2DConvTranspose2DT | Sampled latent variable sequenceh_dim neuronsF: 64, BatchNorm2D, ReLUF: 64, BatchNorm2D, ReLUF: 32, BatchNorm2D, ReLUF: 32, BatchNorm2D, ReLUF: 32, BatchNorm2D, ReLUF: 32, BatchNorm2D, ReLUF: 32, BatchNorm2D, ReLUF: 32, BatchNorm2D, ReLUF: 32, BatchNorm2D, ReLUF: 32, BatchNorm2D, ReLU | BS × T × z_dimBS × T × h_dim × 1 × 1BS × T × 64 × 4 × 4BS × T × 64 × 8 × 8BS × T × 32 × 64 × 64BS × T × 32 × 64 × 64BS × T × 1 × 64 × 64 |
+
+# C.2 HYPERPARAMETERS AND TRAINING DETAILS
+
+We describe the hyperparameter selection criteria and discuss the impacts of hyperparameter values on the model performances. The training details are provided.
+
+# C.2.1 HYPERPARAMETER SELECTION AND SENSITIVITY ANALYSIS
+
+The hyperparameters of LEAP include $[\beta, \gamma, \sigma]$ , which are the weights of each term in the augmented ELBO objective, as well as the latent size $n$ and maximum time lag $L$ . We use the ELBO loss on the validation dataset to select the best pair of $[\beta, \gamma, \sigma]$ because low ELBO loss always leads to high
+
+
+(a) MCC trajectory with different $\beta$ and $\gamma$ .
+
+
+(b) Visualization of the final MCC.
+
+
+(c) MCC trajectory with different $\sigma$
+Figure C.1: Impacts of hyperparameters on VAR dataset.
+
+
+(d) MCC with different latent size and time lag.
+
+MCC. We always set a larger latent size than the true latent size. This is critical in video datasets because the image pixels contain more information than the annotated latent causal variables, and restricting the latent size will hurt the reconstruction performances. For the maximum time lag $L$ , we set it by the rule of thumb. For instance, we use $L = 2$ for temporal datasets with a latent physics process.
+
+However, it is known (Mita et al., 2021) that the performances of VAEs might change drastically as a function of the regularization strength. We thus conduct a sensitivity analysis on the impacts of hyperparameters on our identifiability performances. We report the MCC scores for the synthetic NP and VAR datasets on a hyperparameter grid. We found that the values of $\beta$ and $\gamma$ have a larger impact on the identifiability results, while the effects of $\sigma$ are relatively smaller. Furthermore, we have verified the robustness of our approach under different maximum time lags $L$ and latent size $n$ settings. The final MCC scores only show marginal differences, indicating that our approach is robust to the choices of $n$ and $L$ . In summary, the performance of our approach is robust to the values of some of the hyperparameters, and for the remaining hyperparameters, we use separate validation data to set their values.
+
+Parametric (VAR) Dataset We have performed a grid search of $\beta \in [3\mathrm{E - }4,3\mathrm{E - }3,3\mathrm{E - }2]$ and $\gamma \in [9\mathrm{E - }4,9\mathrm{E - }3,9\mathrm{E - }2]$ and reported the results in Fig. C.1(a). The best configuration is $[\beta ,\gamma ] =$ [3E-3, 9E-3]. We plot the final MCC score as a function of the value of the two hyperparameters in Fig. C.1(b). For $\sigma$ , we compare the MCC scores under different $\sigma \in [5\mathrm{E - }7,1\mathrm{E - }6,1\mathrm{E - }5]$ with the optimal $(\beta ,\gamma)$ value. The optimal configuration for $\sigma$ is 1E-6 as shown in Fig. C.1(c). Furthermore, we verify the robustness of our approach under different time lags $L\in [2,3,4]$ and latent dimensions $n\in [4,6,8]$ with $[\beta ,\gamma ,\sigma ] = [3\mathrm{E - }3,9\mathrm{E - }3,1\mathrm{E - }6]$ and the results are shown in Fig. C.1(d). We can see that the final MCC scores in all cases are around 0.9 with marginal differences, indicating that latent recovery performances of our approach is robust to the choices of $n$ and $L$
+
+Nonparametric (NP) Dataset Similarly, we have performed a grid search of $\beta \in [2\mathrm{E - 4},2\mathrm{E - 3},2\mathrm{E - 2}]$ and $\gamma \in [2\mathrm{E - 3},2\mathrm{E - 2},2\mathrm{E - 1}]$ on the NP dataset and reported the results in Fig. C.2(a). The best configuration is $[\beta ,\gamma ] = [2\mathrm{E - 3},2\mathrm{E - 2}]$ . The final MCC scores under parameter grids of $[\beta ,\gamma ]$ are shown in Fig. C.2(b). The optimal configuration for $\sigma$ is 1E-6 in the search space $\sigma \in [1\mathrm{E - 7},1\mathrm{E - 6},1\mathrm{E - 5}]$ with the optimal $[\beta ,\gamma ]$ value, as shown in Fig. C.2(c). We verify the robust
+
+ness in terms of latent size $n \in [6,7,8]$ and the maximum time lags $L \in [1,2,3]$ with the optimal configuration $[\beta ,\gamma ,\sigma ] = [2\mathrm{E} - 3,2\mathrm{E} - 2,1\mathrm{E} - 6]$ and the results are shown in Fig. C.2(d). The final MCC scores in all cases are around or higher than 0.85 with marginal differences.
+
+
+(a) MCC trajectory with different $\beta$ and $\gamma$ .
+
+
+(b) Visualization of the final MCC.
+
+
+(c) MCC trajectory with different $\sigma$
+Figure C.2: Impacts of hyperparameters on NP dataset.
+
+
+(d) MCC with different latent size and time lag.
+
+# C.2.2 TRAINING
+
+Training Details The models were implemented in PyTorch 1.8.1. The VAE network is trained using AdamW optimizer for a maximum of 200 epochs and early stops if the validation ELBO loss does not decrease for five epochs. A learning rate of 0.002 and a mini-batch size of 32 are used. For the noise discriminator, we use SGD optimizer with a learning rate of 0.001. We have used four random seeds in each experiment and reported the mean performance with standard deviation averaged across random seeds.
+
+Computing Hardware We used a machine with the following CPU specifications: Intel(R) Core(TM) i7-7700K CPU @ 4.20GHz; 8 CPUs, four physical cores per CPU, a total of 32 logical CPU units. The machine has two GeForce GTX 1080 Ti GPUs with 11GB GPU memory.
+
+Training Stability We have used several standard tricks to improve training stability: (1) we use a slightly larger latent size than the true latent size for real-world datasets in order to make sure the meaningful latent variables are among the recovered latents; (2) we use AdamW optimizer as a regularizer to prevent training from being interrupted by overflow or underflow of variance terms of VAE; (3) we use a larger learning rate for the VAE than for the noise discriminator to prevent extreme extrapolation behavior of discriminator.
+
+# D ADDITIONAL EXPERIMENT RESULTS
+
+# D.1 COMPARISONS BETWEEN LEAP AND BASELINESON CMU-MOCAP DATASET
+
+Because the true latent variables for CMU-Mocap are unknown, we visualize the latent traversals and the recovered skeletons, qualitatively comparing our nonparametric method with baselines in terms of the how intuitively sensible the recovered processes and skeletons are.
+
+
+Figure D.1: Latent traversal comparisons between LEAP and the baselines. LEAP represents the data with causally-related factors, thus can represent the data with only much fewer latent variables (three vs eight) with smooth transitions dynamics. Video demonstrations are in: https://bit.ly/3kEVQhf.
+
+Latent Traversal We fit LEAP and the baseline models using the same latent size $n = 8$ and the maximum time lags $L = 2$ . As shown in Fig. D.1, LEAP represents the data with causally-related factors, thus explaining the data with fewer latent variables and smooth transitions dynamics. Only three latent variables are in fact used by LEAP and while the other five latent variables only encode random noise as seen from the video demonstration. BetaVAE and SlowVAE, however, need to use all the latent variables to represent the data. Furthermore, we find the three latent variables discovered by LEAP encode pitch, yaw, and roll rotations of walking cycles, which is close to how human beings perceive walking movement.
+
+Recovered Skeleton As shown in Fig. D.2, LEAP recovers the cross relations between causal variables while BetaVAE and SlowVAE can only recover independent relations. The latent traversals of LEAP have shown that the three recovered latent variables may be the pitch, yaw, and roll rotations of the walk cycles. Therefore, the results of our approach indicate that the pitch (e.g., limb movement) and roll (e.g., shoulder movement) are causally-related while yaw has independent dynamics, which is closer to reality than the independent transitions discovered by BetaVAE and SlowVAE.
+
+# D.2 MASS-SPRING SYSTEMS
+
+We render the recovered latent variables using keypoint heatmaps in Fig. D.3(a). The learned representation successfully disentangles five objects in the scene, and the latent variables represent the horizontal and vertical locations of the balls. We further visualize the recovered skeletons from the estimated state transition matrices in Fig. D.3(b). The recovered skeleton is consistent with the underlying processes described in Fig. B.1(b) with $\mathrm{SHD} = 0$ .
+
+
+Figure D.2: Comparisons between LEAP and the baselines in terms of skeleton recovery. LEAP recovers cross relations between causal variables while baselines can only recover independent relations.
+
+
+Figure D.3: Visualization of recovered latent variables and the estimated skeletons for Mass-Spring system dataset.
+
+# E EXTENDED RELATED WORK
+
+Temporal dependencies and nonstationarities were recently used as side information $\mathbf{u}$ to achieve identifiability of nonlinear ICA on latent space $\mathbf{z}$ . Hyvarinen & Morioka (2016) proposed time-contrastive learning (TCL) based on the independent sources assumption. It gave the very first identifiability results for a nonlinear mixing model with nonstationary data segmentation. Hyvarinen & Morioka (2017) developed a permutation-based contrastive (PCL) learning framework to separate independent sources using temporal dependencies. Their approach learns to discriminate between true time series and permuted time series, and the model is identifiable under the uniformly dependent assumption. Halvä & Hyvarinen (2020) combined nonlinear ICA with a Hidden Markov Model (HMM) to automatically model nonstationarity without the need for manual data segmentation. Khemakhem et al. (2020) introduced VAEs to approximate the true joint distribution over observed and auxiliary nonstationary regimes. The conditional distribution in their work $p(\mathbf{z}|\mathbf{u})$ is assumed to be within exponential families to achieve identifiability on the latent space. A more recent study in causally-related nonlinear ICA was given by (Yang et al., 2021), which introduced a linear causal layer to transform independent exogenous factors into endogenous causal variables.
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+# LEARNING TOWARDS THE LARGEST MARGINS
+
+Xiong Zhou $^{1*}$ , Xianming Liu $^{1,2\dagger}$ , Deming Zhai $^{1}$ , Junjun Jiang $^{1,2}$ , Xin Gao $^{3,2,4}$ , Xiangyang Ji $^{5}$
+
+$^{1}$ Harbin Institute of Technology $^{2}$ Peng Cheng Laboratory $^{3}$ King Abdullah University of Science and Technology
+
+$^{4}$ Gaoling School of Artificial Intelligence, Renmin University of China $^{5}$ Tsinghua University
+
+# ABSTRACT
+
+One of the main challenges for feature representation in deep learning-based classification is the design of appropriate loss functions that exhibit strong discriminative power. The classical softmax loss does not explicitly encourage discriminative learning of features. A popular direction of research is to incorporate margins in well-established losses in order to enforce extra intra-class compactness and interclass separability, which, however, were developed through heuristic means, as opposed to rigorous mathematical principles. In this work, we attempt to address this limitation by formulating the principled optimization objective as learning towards the largest margins. Specifically, we firstly define the class margin as the measure of inter-class separability, and the sample margin as the measure of intra-class compactness. Accordingly, to encourage discriminative representation of features, the loss function should promote the largest possible margins for both classes and samples. Furthermore, we derive a generalized margin softmax loss to draw general conclusions for the existing margin-based losses. Not only does this principled framework offer new perspectives to understand and interpret existing margin-based losses, but it also provides new insights that can guide the design of new tools, including sample margin regularization and largest margin softmax loss for the class-balanced case, and zero-centroid regularization for the class-imbalanced case. Experimental results demonstrate the effectiveness of our strategy on a variety of tasks, including visual classification, imbalanced classification, person re-identification, and face verification.
+
+# 1 INTRODUCTION
+
+Recent years have witnessed the great success of deep neural networks (DNNs) in a variety of tasks, especially for visual classification (Simonyan & Zisserman, 2014; Szegedy et al., 2015; He et al., 2016; Howard et al., 2017; Zoph et al., 2018; Touvron et al., 2019; Brock et al., 2021; Dosovitskiy et al., 2021). The improvement in accuracy is attributed not only to the use of DNNs, but also to the elaborated losses encouraging well-separated features (Elsayed et al., 2018; Musgrave et al., 2020).
+
+In general, the loss is expected to promote the learned features to have maximized intra-class compactness and inter-class separability simultaneously, so as to boost the feature discriminativeness. Softmax loss, which is the combination of a linear layer, a softmax function, and cross-entropy loss, is the most commonly-used ingredient in deep learning-based classification. However, the softmax loss only learns separable features that are not discriminative enough (Liu et al., 2017). To remedy the limitation of softmax loss, many variants have been proposed. Liu et al. (2016) proposed a generalized large-margin softmax loss, which incorporates a preset constant $m$ multiplying with the angle between samples and the classifier weight of the ground truth class, leading to potentially larger angular separability between learned features. SphereFace (Liu et al., 2017) further improved the performance of L-Softmax by normalizing the prototypes in the last inner-product layer. Subsequently, Wang et al. (2017) exhibited the usefulness of feature normalization when using feature vector dot products in the softmax function. Coincidentally, in the field of contrastive learning, Chen et al. (2020) also showed that normalization of outputs leads to superior representations. Due to its effectiveness, normalization on either features or prototypes or both becomes a standard procedure in margin-based losses, such as SphereFace (Liu et al., 2017), CosFace/AM-Softmax (Wang et al., 2018b;a) and ArcFace (Deng et al., 2019). However, there is no theoretical guarantee provided yet.
+
+Despite their effectiveness and popularity, the existing margin-based losses were developed through heuristic means, as opposed to rigorous mathematical principles, modeling and analysis. Although they offer geometric interpretations, which are helpful to understand the underlying intuition, the theoretical explanation and analysis that can guide the design and optimization is still vague. Some critical issues are unclear, e.g., why is the normalization of features and prototypes necessary? How can the loss be further improved or adapted to new tasks? Therefore, it naturally raises a fundamental question: how to develop a principled mathematical framework for better understanding and design of margin-based loss functions? The goal of this work is to address these questions by formulating the objective as learning towards the largest margins and offering rigorously theoretical analysis as well as extensive empirical results to support this point.
+
+To obtain an estimizable objective, firstly, we should define measures of intra-class compactness and inter-class separability. To this end, we propose to employ the class margin as the measure of inter-class separability, which is defined as the minimal pairwise angle distance between prototypes that reflects the angular margin of the two closest prototypes. Moreover, we define the sample margin following the classic approach in (Koltchinskii et al., 2002, Sec. 5), which denotes the similarity difference of a sample to the prototype of the class it belongs to and to the nearest prototype of other classes and thus measures the intra-class compactness. We provide a rigorous theoretical guarantee that maximizing the minimal sample margin over the entire dataset leads to maximizing the class margin regardless of feature dimension, class number, and class balancedness. It denotes that the sample margin also has the power of measuring inter-class separability.
+
+According to the defined measures, we can obtain categorical discriminativeness of features by the loss function promoting the largest margins for both classes and samples, which also meets to tighten the margin-based generalization bound in (Kakade et al., 2008; Cao et al., 2019). The main contributions of this work are highlighted as follows:
+
+- For a better understanding of margin-based losses, we provide a rigorous analysis about the necessity of normalization on prototypes and features. Moreover, we propose a generalized margin softmax loss (GM-Softmax), which can be derived to cover most of existing margin-based losses. We prove that, for the class-balance case, learning with the GM-Softmax loss leads to maximizing both class margin and sample margin under mild conditions.
+- We show that learning with existing margin-based loss functions, such as SphereFace, NormFace, CosFace, AM-Softmax and ArcFace, would share the same optimal solution. In other words, all of them attempt to learn towards the largest margins, even though they are tailored to obtain different desired margins with explicit decision boundaries. However, these losses do not always maximize margins under different hyper-parameter settings. Instead, we propose an explicit sample margin regularization term and a novel largest margin softmax loss (LM-Softmax) derived from the minimal sample margin, which significantly improves the class margin and the sample margin.
+- We consider the class-imbalanced case, in which the margins are severely affected. We provide a sufficient condition, which reveals that, if the centroid of prototypes is equal to zero, learning with GM-Softmax will provide the largest margins. Accordingly, we propose a simple but effective zero-centroid regularization term, which can be combined with commonly-used losses to mitigate class imbalance.
+- Extensive experimental results are offered to demonstrate that the strategy of learning towards the largest margins significantly can improve the performance in accuracy and class/sample margins for various tasks, including visual classification, imbalanced classification, person re-identification, and face verification.
+
+# 2 MEASURES OF INTRA-CLASS COMPACTNESS AND INTER-CLASS SEPARABILITY
+
+With a labeled dataset $D = \{(\pmb{x}_i, y_i)\}_{i=1}^N$ (where $\pmb{x}_i$ denotes a training example with label $y_i$ , and $y_i \in [1, k] = \{1, 2, \dots, k\}$ ), the softmax loss for a $k$ -classification problem is formulated as
+
+$$
+L = \frac {1}{N} \sum_ {i = 1} ^ {N} - \log \frac {\exp \left(\boldsymbol {w} _ {y _ {i}} ^ {\mathrm {T}} \boldsymbol {z} _ {i}\right)}{\sum_ {j = 1} ^ {k} \exp \left(\boldsymbol {w} _ {j} ^ {\mathrm {T}} \boldsymbol {z} _ {i}\right)} = \sum_ {i = 1} ^ {N} - \log \frac {\exp \left(\| \boldsymbol {w} _ {y _ {i}} \| _ {2} \| \boldsymbol {z} _ {i} \| _ {2} \cos \left(\theta_ {i y _ {i}}\right)\right)}{\sum_ {j = 1} ^ {k} \exp \left(\| \boldsymbol {w} _ {j} \| _ {2} \| \boldsymbol {z} _ {i} \| _ {2} \cos \left(\theta_ {i j}\right)\right)}, \tag {2.1}
+$$
+
+where $\boldsymbol{z}_i = \phi_{\Theta}(\boldsymbol{x}_i) \in \mathbb{R}^d$ (usually $k \leq d + 1$ ) is the learned feature representation vector; $\phi_{\Theta}$ denotes the feature extraction sub-network; $W = (\boldsymbol{w}_1, \dots, \boldsymbol{w}_k) \in \mathbb{R}^{d \times k}$ denotes the linear classifier which is implemented with a linear layer at the end of the network (some works omit the bias and use an inner-product layer); $\theta_{ij}$ denotes the angle between $\boldsymbol{z}_i$ and $\boldsymbol{w}_j$ ; and $\| \cdot \|_2$ denotes the Euclidean norm, where $\boldsymbol{w}_1, \dots, \boldsymbol{w}_k$ can be regarded as the class centers or prototypes (Mettes et al., 2019). For simplicity, we use prototypes to denote the weight vectors in the last inner-product layer.
+
+The softmax loss intuitively encourages the learned feature representation $\mathbf{z}_i$ to be similar to the corresponding prototype $\mathbf{w}_{y_i}$ , while pushing $\mathbf{z}_i$ away from the other prototypes. Recently, some works (Liu et al., 2016; 2017; Deng et al., 2019) aim to achieve better performance by modifying the softmax loss with explicit decision boundaries to enforce extra intra-class compactness and inter-class separability. However, they do not provide the theoretical explanation and analysis about the newly designed losses. In this paper, we claim that a loss function to obtain better inter-class separability and intra-class compactness should learn towards the largest class and sample margin, and offer rigorously theoretical analysis as support. All proofs can be found in the Appendix A.
+
+In the following, we define class margin and sample margin as the measures of inter-class separability and intra-class compactness, respectively, which serve as the base for our further derivation.
+
+# 2.1 CLASS MARGIN
+
+With prototypes $\pmb{w}_1, \dots, \pmb{w}_k \in \mathbb{R}^d$ , the class margin is defined as the minimal pairwise angle distance:
+
+$$
+m _ {c} \left(\left\{\boldsymbol {w} _ {i} \right\} _ {i = 1} ^ {k}\right) = \min _ {i \neq j} \angle \left(\boldsymbol {w} _ {i}, \boldsymbol {w} _ {j}\right) = \operatorname {a r c c o s} \left[ \max _ {i \neq j} \frac {\boldsymbol {w} _ {i} ^ {\mathrm {T}} \boldsymbol {w} _ {j}}{\| \boldsymbol {w} _ {i} \| _ {2} \| \boldsymbol {w} _ {j} \| _ {2}} \right], \tag {2.2}
+$$
+
+where $\angle (\pmb{w}_i, \pmb{w}_j)$ denotes the angle between the vectors $\pmb{w}_i$ and $\pmb{w}_j$ . Note that we omit the magnitudes of the prototypes in the definition, since the magnitudes tend to be very close according to the symmetry property. To verify this, we compute the ratio between the maximum and minimum magnitudes, which tends to be close to 1 on different datasets, as shown in Fig. 1.
+
+To obtain better inter-class separability, we seek the largest class margin, which can be formulated as
+
+$$
+\max_{\{\boldsymbol{w}_{i}\}_{i = 1}^{k}}m_{c}(\{\boldsymbol{w}_{i}\}_{i = 1}^{k}) = \max_{\{\boldsymbol{w}_{i}\}_{i = 1}^{k}}\min_{i\neq j}\angle (\boldsymbol{w}_{i},\boldsymbol{w}_{j}).
+$$
+
+Since magnitudes do not affect the solution of the max-min problem, we perform $\ell_2$ normalization for each $w_i$ to effectively restrict the prototypes on the unit sphere $\mathbb{S}^{d-1}$ with center at the origin. Under this constraint, the maximization of the class margin is equivalent to
+
+
+Figure 1: The curves of ratio between maximum and minimum magnitudes of prototypes on MNIST and CIFAR-10/-100 using the CE loss. The ratio is roughly close to 1 (< 1.3).
+
+the configuration of $k$ points on $\mathbb{S}^{d - 1}$ to maximize their minimum pairwise distance:
+
+$$
+\underset {\{\boldsymbol {w} _ {i} \} _ {i = 1} ^ {k} \subset \mathbb {S} ^ {d - 1}} {\arg \max } \min _ {i \neq j} \angle (\boldsymbol {w} _ {i}, \boldsymbol {w} _ {j}) = \underset {\{\boldsymbol {w} \} _ {i = 1} ^ {k} \subset \mathbb {S} ^ {d - 1}} {\arg \max } \underset {i \neq j} {\min } \| \boldsymbol {w} _ {i} - \boldsymbol {w} _ {j} \| _ {2}, \tag {2.3}
+$$
+
+The right-hand side is well known as the $k$ -points best-packing problem on spheres (often called the Tammes problem), whose solution leads to the optimal separation of points (Borodachov et al., 2019). The best-packing problem turns out to be the limiting case of the minimal Riesz energy:
+
+$$
+\underset {\{\boldsymbol {w} \} _ {i = 1} ^ {k} \subset \mathbb {S} ^ {d - 1}} {\arg \min } \underset {t \rightarrow \infty} {\lim } \sum_ {i \neq j} \frac {1}{\| \boldsymbol {w} _ {i} - \boldsymbol {w} _ {j} \| _ {2} ^ {t}} = \underset {\{\boldsymbol {w} \} _ {i = 1} ^ {k} \subset \mathbb {S} ^ {d - 1}} {\arg \max } \underset {i \neq j} {\min } \| \boldsymbol {w} _ {i} - \boldsymbol {w} _ {j} \| _ {2}. \tag {2.4}
+$$
+
+Interestingly, Liu et al. (2018) utilized the minimum hyperspherical energy as a generic regularization for neural networks to reduce undesired representation redundancy. When $\pmb{w}_1,\dots,\pmb{w}_k\in \mathbb{S}^{d - 1}$ , $k\leq d + 1$ , and $t > 0$ , the solution of the best-packing problem leads to the minimal Riesz $t$ -energy:
+
+Lemma 2.1. For any $\mathbf{w}_1, \dots, \mathbf{w}_k \in \mathbb{S}^{d-1}$ , $d \geq 2$ , and $2 \leq k \leq d+1$ , the solution of minimal Riesz t-energy and $k$ -points best-packing configurations are uniquely given by the vertices of regular $(k-1)$ -simplices inscribed in $\mathbb{S}^{d-1}$ . Furthermore, $\mathbf{w}_i^{\mathrm{T}}\mathbf{w}_j = \frac{-1}{k-1}, \forall i \neq j$ .
+
+This lemma shows that the maximum of $m_c(\{\pmb{w}_i\}_{i=1}^k)$ is $\arccos \left(\frac{-1}{k-1}\right)$ when $k \leq d+1$ , which is analytical and can be constructed artificially. However, when $k > d+1$ , the optimal $k$ -point configurations on the sphere $\mathbb{S}^{d-1}$ have no generic analytical solution, and are only known explicitly for a handful of cases, even for $d=3$ .
+
+# 2.2 SAMPLE MARGIN
+
+According to the definition in (Koltchinskii et al., 2002), for the network $\pmb{f}(\pmb{x};\Theta ,W) = W^{\mathrm{T}}\phi_{\Theta}(\pmb{x}):\mathbb{R}^{m}\to \mathbb{R}^{k}$ that outputs $k$ logits, the sample margin for $(\pmb{x},y)$ is defined as
+
+$$
+\gamma (\boldsymbol {x}, y) = \boldsymbol {f} (\boldsymbol {x}) _ {y} - \max _ {j \neq y} \boldsymbol {f} (\boldsymbol {x}) _ {j} = \boldsymbol {w} _ {y} ^ {\mathrm {T}} \boldsymbol {z} - \max _ {j \neq y} \boldsymbol {w} _ {j} ^ {\mathrm {T}} \boldsymbol {z}, \tag {2.5}
+$$
+
+where $\mathbf{z} = \phi_{\Theta}(\mathbf{x})$ denotes the corresponding feature. Let $n_j$ be the number of samples in class $j$ and $S_j = \{i : y_i = j\}$ denote the sample indices corresponding to class $j$ . We can further define the sample margin for samples in class $j$ as
+
+$$
+\gamma_ {j} = \min _ {i \in S _ {j}} \gamma \left(\boldsymbol {x} _ {i}, y _ {i}\right). \tag {2.6}
+$$
+
+Accordingly, the minimal sample margin over the entire dataset is $\gamma_{\mathrm{min}} = \min \{\gamma_1,\dots,\gamma_k\}$ . Intuitively, learning features and prototypes to maximize the minimum of all sample margins means making the feature embeddings close to their corresponding classes and far away from the others:
+
+Theorem 2.2. For $\pmb{w}_1, \dots, \pmb{w}_k, \pmb{z}_1, \dots, \pmb{z}_N \in \mathbb{S}^{d-1}$ (where $n_j > 0$ for each $j \in [1,k]$ ), the optimal solution $\{\pmb{w}_i^*\}_{i=1}^k$ , $\{\pmb{z}_i^*\}_{i=1}^N = \arg \max_{\{\pmb{w}_i\}_{i=1}^k, \{\pmb{z}_i\}_{i=1}^N} \gamma_{\min}$ is obtained if and only if $\{\pmb{w}_i^*\}_{i=1}^k$ maximizes the class margin $m_c(\{\pmb{w}_i\}_{i=1}^k)$ , and $\pmb{z}_i^* = \frac{\pmb{w}_{y_i}^* - \overline{\pmb{w}}_{y_i}^*}{\|\pmb{w}_{y_i}^* - \overline{\pmb{w}}_{y_i}^*\|_2}$ , where $\overline{\pmb{w}}_{y_i}^*$ denotes the centroid of the vectors $\{\pmb{w}_j : j \text{ maximizes } \pmb{w}_{y_i}^{\mathrm{T}}\pmb{w}_j, j \neq y_i\}$ .
+
+As shown in the proof A, Theorem 2.2 guarantees that maximizing $\gamma_{\mathrm{min}}$ will provide the solution of the Tammes problem with respect to any feature dimension $d$ , class number $k$ , and both class-balanced and class-imbalance cases. When $2 \leq k \leq d + 1$ , we can derive the following proposition:
+
+Proposition 2.3. For any $\pmb{w}_1, \dots, \pmb{w}_k, \pmb{z}_1, \dots, \pmb{z}_N \in \mathbb{S}^{d-1}$ , $d \geq 2$ , and $2 \leq k \leq d+1$ , the maximum of $\gamma_{\min}$ is $\frac{k}{k-1}$ , which is obtained if and only if $\forall i \neq j$ , $\pmb{w}_i^{\mathrm{T}}\pmb{w}_j = -\frac{1}{k-1}$ , and $\pmb{z}_i = \pmb{w}_{y_i}$ .
+
+Theorem 2.2 and Proposition 2.3 show that the best separation of prototypes is obtained when maximizing the minimal sample margin $\gamma_{\mathrm{min}}$ .
+
+On the other hand, let $L_{\gamma,j}[f] = \operatorname{Pr}_{\boldsymbol{x} \sim \mathcal{P}_j}[\max_{j' \neq j} f(\boldsymbol{x})_{j'} > f(\boldsymbol{x})_j - \gamma]$ denote the hard margin loss on samples from class $j$ , and $\widehat{L}_{\gamma,j}$ denote its empirical variant. When the training dataset is separable (which indicates that there exists $f$ such that $\gamma_{\min} > 0$ ), Cao et al. (2019) provided a fine-grained generalization error bound under the setting with balanced test distribution by considering the margin of each class, i.e., for $\gamma_j > 0$ and all $f \in \mathcal{F}$ , with a high probability, we have
+
+$$
+\Pr_ {(\boldsymbol {x}, y)} [ f (\boldsymbol {x}) _ {y} < \max _ {l \neq y} f (\boldsymbol {x}) _ {l} ] \leq \frac {1}{k} \sum_ {j = 1} ^ {k} \left(\widehat {L} _ {\gamma_ {j}, j} [ f ] + \frac {4}{\gamma_ {j}} \widehat {\Re} _ {j} (\mathcal {F}) + \varepsilon_ {j} (\gamma_ {j})\right). \tag {2.7}
+$$
+
+In the right-hand side, the empirical Rademacher complexity $\frac{4}{\gamma_j}\widehat{\mathfrak{R}}_j(\mathcal{F})$ has a big impact. From the perspective of our work, a straightforward way to tighten the generalization bound is to enlarge the minimal sample margin $\gamma_{\mathrm{min}}$ , which further leads to the larger margin $\gamma_{j}$ for each class $j$ .
+
+# 3 LEARNING TOWARDS THE LARGEST MARGINS
+
+# 3.1 CLASS-BALANCED CASE
+
+According to the above derivations, to encourage discriminative representation of features, the loss function should promote the largest possible margins for both classes and samples. In (Mettes et al., 2019), the pre-defined prototypes positioned through data-independent optimization are used to obtain a large class margin. As shown in Figure 2, although they keep the particularly large margin
+
+
+(a) Test Accuracy
+
+
+(b) Class Margin
+
+
+(c) Sample Margin
+Figure 2: Test accuracies, class margins and sample margins on CIFAR-10 and CIFAR-100 with and without fixed prototypes, where fixed prototypes are pre-trained for very large class margins.
+
+from the beginning, the sample margin is smaller than that optimized without fixed prototypes, leading to insignificant improvements in accuracy.
+
+In recent years, in the design of variants of softmax loss, one popular approach (Bojanowski & Joulin, 2017; Wang et al., 2017; Mettes et al., 2019; Wang & Isola, 2020) is to perform normalization on prototypes or/and features, leading to superior performance than unnormalized counterparts (Parkhi et al., 2015; Schroff et al., 2015; Liu et al., 2017). However, there is no theoretical guarantee provided yet. In the following, we provide a rigorous analysis about the necessity of normalization. Firstly, we prove that minimizing the original softmax loss without normalization for both features and prototypes may result in a very small class margin:
+
+Theorem 3.1. $\forall \varepsilon \in (0,\pi /2]$ , if the range of $w_{1},\dots,w_{k}$ or $z_{1},\dots,z_{N}$ is $\mathbb{R}^d$ ( $2\leq k\leq d + 1$ ), then there exists prototypes that achieve the infimum of the softmax loss and have the class margin $\varepsilon$ .
+
+This theorem reveals that, unless both features and prototypes are normalized, the original softmax loss may produce an arbitrary small class margin $\varepsilon$ . As a corroboration of this conclusion, L-Softmax (Liu et al., 2016) and A-Softmax (Liu et al., 2017) that do not perform any normalization or only do on prototypes, cannot guarantee to maximize the class margin. To remedy this issue, some works (Wang et al., 2017; 2018a;b; Deng et al., 2019) proposed to normalize both features and prototypes.
+
+A unified framework (Deng et al., 2019) that covers A-Softmax (Liu et al., 2017) with feature normalization, NormFace (Wang et al., 2017), CosFace/AM-Softmax (Wang et al., 2018b;a), ArcFace (Deng et al., 2019) as special cases can be formulated with hyper-parameters $m_{1}$ , $m_{2}$ and $m_{3}$ :
+
+$$
+L _ {i} ^ {\prime} = - \log \frac {\exp \left(s \left(\cos \left(m _ {1} \theta_ {i y _ {i}} + m _ {2}\right) - m _ {3}\right)\right)}{\exp \left(s \left(\cos \left(m _ {1} \theta_ {i y _ {i}} + m _ {2}\right) - m _ {3}\right)\right) + \sum_ {j \neq y _ {i}} \exp \left(s \cos \theta_ {i j}\right)}, \tag {3.1}
+$$
+
+where $\theta_{ij} = \angle (\pmb{w}_j, \pmb{z}_i)$ . The hyper-parameters setting usually guarantees that $\cos(m_1\theta_{iy_i} + m_2) - m_3 \leq \cos m_2\cos\theta_{iy_i} - m_3$ , and $m_2$ is usually set to satisfy $\cos m_2 \geq \frac{1}{2}$ . Let $\alpha = \cos m_2$ and $\beta = -m_3 < 0$ , then we have
+
+$$
+L _ {i} ^ {\prime} \geq - \log \frac {\exp (s (\alpha \cos \theta_ {i y _ {i}} + \beta))}{\exp (s (\alpha \cos \theta_ {i y _ {i}} + \beta)) + \sum_ {j \neq y _ {i}} \exp (s \cos \theta_ {i j})}, \tag {3.2}
+$$
+
+which indicates that the existing well-designed normalized softmax loss functions are all considered as the upper bound of the right-hand side, and the equality holds if and only if $\theta_{iy_i} = 0$ .
+
+Generalized Margin Softmax Loss. Based on the right-hand side of (3.2), we can derive a more general formulation, called Generalized Margin Softmax (GM-Softmax) loss:
+
+$$
+L _ {i} = - \log \frac {\exp \left(s \left(\alpha_ {i 1} \cos \theta_ {i y _ {i}} + \beta_ {i 1}\right)\right)}{\exp \left(s \left(\alpha_ {i 2} \cos \theta_ {i y _ {i}} + \beta_ {i 2}\right)\right) + \sum_ {j \neq y _ {i}} \exp (s \cos \theta_ {i j})}, \tag {3.3}
+$$
+
+where $\alpha_{i1},\alpha_{i2},\beta_{i1}$ and $\beta_{i2}$ are hyper-parameters to handle the margins in training, which are set specifically for each sample instead of the same in (3.2). We also require that $\alpha_{i1}\geq \frac{1}{2}$ , $\alpha_{i2}\leq \alpha_{i1}$ , $s > 0$ , $\beta_{i1},\beta_{i2}\in \mathbb{R}$ . For class-balanced case, each sample is treated equally, thus setting $\alpha_{i1} = \alpha_{1}$
+
+$\alpha_{i2} = \alpha_{2}$ , $\beta_{i1} = \beta_{1}$ and $\beta_{i2} = \beta_{2}$ , $\forall i$ . For class-imbalanced case, the setting relies on the data distribution, e.g., the LDAM loss (Cao et al., 2019) achieves the trade-off of margins with $\alpha_{i1} = \alpha_{i2} = 1$ and $\beta_{i1} = \beta_{i2} = -Cn_{y_i}^{-1/4}$ . It is worth noting that we merely use the GM-Softmax loss as a theoretical formulation and will derive a more efficient form for the practical implementation.
+
+Wang et al. (2017) provided a lower bound for normalized softmax loss, which relies on the assumption that all samples are well-separated, i.e., each sample's feature is exactly the same as its corresponding prototype. However, this assumption could be invalid during training, e.g., for binary classification, the best feature of the first class $\mathbf{z}$ obtained by minimizing $-\log \frac{\exp(s\mathbf{w}_1^{\mathrm{T}}\mathbf{z})}{\exp(s\mathbf{w}_1^{\mathrm{T}}\mathbf{z}) + \exp(s\mathbf{w}_2^{\mathrm{T}}\mathbf{z})}$ is $\frac{\mathbf{w}_1 - \mathbf{w}_2}{\|\mathbf{w}_1 - \mathbf{w}_2\|_2}$ rather than $\mathbf{w}_1$ . In the following, we provide a more general theorem, which does not rely on such a strong assumption. Moreover, we prove that the solutions $\{\mathbf{w}_j^*\}_{j=1}^k$ , $\{\mathbf{z}_i^*\}_{i=1}^N$ minimizing the GM-Softmax loss will maximize both class margin and sample margin.
+
+Theorem 3.2. For class-balanced datasets, $\mathbf{w}_1, \dots, \mathbf{w}_k, \mathbf{z}_1, \dots, \mathbf{z}_N \in \mathbb{S}^{d-1}$ , $d \geq 2$ , and $2 \leq k \leq d+1$ , learning with GM-Softmax (where $\alpha_{i1} = \alpha_1$ , $\alpha_{i2} = \alpha_2$ , $\beta_{i1} = \beta_1$ and $\beta_{i2} = \beta_2$ ) leads to maximizing both class margin and sample margin.
+
+As can be seen, for any $\alpha_{1} \geq \frac{1}{2}$ , $\alpha_{2} \leq \alpha_{1}$ , $s > 0$ , and $\beta_{1}, \beta \in \mathbb{R}$ , minimizing the GM-Softmax loss produces the same optimal solution or leads to neural collapse (Papyan et al., 2020)), even though they are intuitively designed to obtain different decision boundaries. Moreover, we have
+
+Proposition 3.3. For class-balanced datasets, $\mathbf{w}_1, \dots, \mathbf{w}_k, \mathbf{z}_1, \dots, \mathbf{z}_N \in \mathbb{S}^{d-1}$ , $d \geq 2$ , and $2 \leq k \leq d+1$ , learning with the loss functions A-Softmax (Liu et al., 2017) with feature normalization, NormFace (Wang et al., 2017), CosFace (Wang et al., 2018b) or AM-Softmax (Wang et al., 2018a), and ArcFace (Deng et al., 2019) share the same optimal solution.
+
+Although these losses theoretically share the same optimal solution, in practice they usually meet sub-optimal solutions under different hyper-parameter settings when optimizing a neural network, which is demonstrated in Table 1. Moreover, these losses are complicated and possibly redundantly designed, leading to difficulties in practical implementation. Instead, we suggest a concise and easily implemented regularization term and a loss function in the following.
+
+Sample Margin Regularization. In order to encourage learning towards the largest margins, we may explicitly leverage the sample margin (2.5) as the loss, which is defined as:
+
+$$
+R _ {\mathrm {s m}} (\boldsymbol {x}, y) = - \left(\boldsymbol {w} _ {y} ^ {\mathrm {T}} \boldsymbol {z} - \max _ {j \neq y} \boldsymbol {w} _ {j} ^ {\mathrm {T}} \boldsymbol {z}\right). \tag {3.4}
+$$
+
+Noticeably, the empirical risk $\frac{1}{N}\sum_{i=1}^{N}R_{\mathrm{sm}}(\pmb{x}_i,y_i)$ is a lower-bounded surrogate of $-\gamma_{\mathrm{min}}$ , i.e., $-\gamma_{\mathrm{min}}\geq \frac{1}{N}\sum_{i=1}^{N}R_{\mathrm{sm}}(\pmb{x}_i,y_i)$ , while directly minimizing $-\gamma_{\mathrm{min}}$ is too difficult to optimize neural networks. When $k\leq d + 1$ , learning with $R_{\mathrm{sm}}$ will promote the learning towards the largest margins:
+
+Theorem 3.4. For class-balanced datasets, $\mathbf{w}_1, \dots, \mathbf{w}_k, \mathbf{z}_1, \dots, \mathbf{z}_N \in \mathbb{S}^{d-1}$ , $d \geq 2$ , and $2 \leq k \leq d+1$ , learning with $R_{sm}$ leads to the maximization of both class margin and sample margin.
+
+Although learning with $R_{\mathrm{sm}}$ theoretically achieves the largest margins, in practical implementation, the optimization by the gradient-based methods shows unstable and non-convergent results for large scale datasets. Alternatively, we turn to combine $R_{\mathrm{sm}}$ as a regularization or complementary term with commonly-used losses, which is referred to as sample margin regularization. The empirical results demonstrate its superiority in learning towards the large margins, as depicted in Table 1.
+
+Largest Margin Softmax Loss (LM-Softmax). Theorem 2.2 provides a theoretical guarantee that maximizing $\gamma_{\mathrm{min}}$ will obtain the maximum of class margin regardless of feature dimension, class number, and class balancedness. It offers a straightforward approach to meet our purpose, i.e., learning towards the largest margins. However, directly maximizing $\gamma_{\mathrm{min}}$ is difficult to optimize a neural network with only one sample margin. As a consequence, we introduce a surrogate loss for balanced datasets, which is called the Largest Margin Softmax (LM-Softmax) loss:
+
+$$
+L (\boldsymbol {x}, y; s) = - \frac {1}{s} \log \frac {\exp \left(s \boldsymbol {w} _ {y} ^ {\mathrm {T}} \boldsymbol {z}\right)}{\sum_ {j \neq y} \exp \left(s \boldsymbol {w} _ {j} ^ {\mathrm {T}} \boldsymbol {z}\right)} = \frac {1}{s} \log \sum_ {j \neq y} \exp \left(s \left(\boldsymbol {w} _ {j} - \boldsymbol {w} _ {y}\right) ^ {\mathrm {T}} \boldsymbol {z}\right) \tag {3.5}
+$$
+
+which is derived by the limiting case of the logsumexp operator, i.e., we have $-\gamma_{\min} = \lim_{s\to \infty}\frac{1}{s}\log (\sum_{i = 1}^{N}\sum_{j\neq y_i}\exp (s(\pmb {w}_j^{\mathrm{T}}\pmb {z}_i - \pmb {w}_{y_i}^{\mathrm{T}}\pmb {z}_i)))$ . Moreover, since log is strictly concave, we
+
+can derive the following inequality
+
+$$
+\frac {1}{s} \log \left(\sum_ {i = 1} ^ {N} \sum_ {j \neq y _ {i}} \exp \left(s \left(\boldsymbol {w} _ {j} ^ {\mathrm {T}} \boldsymbol {z} _ {i} - \boldsymbol {w} _ {y _ {i}} ^ {\mathrm {T}} \boldsymbol {z} _ {i}\right)\right)\right) \geq \frac {1}{N} \sum_ {i = 1} ^ {N} L \left(\boldsymbol {x} _ {i}, y _ {i}; s\right) + \frac {1}{s} \log N. \tag {3.6}
+$$
+
+Minimizing the right-hand side of (3.6) usually leads to that $\sum_{j\neq y_i}\exp (s(\pmb {w}_j^{\mathrm{T}}\pmb {z} - \pmb{w}_{y_i}^{\mathrm{T}}\pmb {z}))$ is a constant, while the equality of (3.6) holds if and only if $\sum_{j\neq y_i}\exp (s(\pmb {w}_j^{\mathrm{T}}\pmb {z} - \pmb{w}_{y_i}^{\mathrm{T}}\pmb {z}))$ is a constant. Thus, we can achieve the maximum of $\gamma_{\mathrm{min}}$ by minimizing $L(\pmb {x},y;s)$ defined in (3.5).
+
+It can be found that, LM-Softmax can be regarded as a special case of the GM-Softmax loss when $\alpha_{2}$ or $\beta_{2}$ approaches $-\infty$ , which can be more efficiently implemented than the GM-Softmax loss. With respect to the original softmax loss, LM-Softmax removes the term $\exp(s w_{y}^{\mathrm{T}} z)$ in the denominator.
+
+# 3.2 CLASS-IMBALANCED CASE
+
+Class imbalance is ubiquitous and inherent in real-world classification problems (Buda et al., 2018; Liu et al., 2019). However, the performance of deep learning-based classification would drop significantly when the training dataset suffers from heavy class imbalance effect. According to (2.7), enlarging the sample margin can tighten the upper bound in case of class imbalance. To learn towards the largest margins on class-imbalanced datasets, we provide the following sufficient condition:
+
+Theorem 3.5. For class-balanced or -imbalanced datasets, $\pmb{w}_1,\dots,\pmb{w}_k,\pmb{z}_1,\dots,\pmb{z}_N\in \mathbb{S}^{d - 1}$ , $d\geq 2$ and $2\leq k\leq d + 1$ , if $\sum_{i = 1}^{K}\pmb{w}_{i} = 0$ , learning with GM-Softmax in (3.3) leads to maximizing both class margin and sample margin.
+
+This theorem reveals that, if the centroid of prototypes is equal to zero, learning with GM-Softmax will provide the largest margins.
+
+Zero-centroid Regularization. As a consequence, we propose a straight regularization term as follows, which can be combined with commonly-used losses to remedy the class imbalance effect:
+
+$$
+R _ {\mathrm {w}} \left\{\boldsymbol {w} _ {j} \right\} _ {j = 1} ^ {k} = \lambda \left\| \frac {1}{k} \sum_ {j = 1} ^ {k} \boldsymbol {w} _ {j} \right\| _ {2} ^ {2}. \tag {3.7}
+$$
+
+The zero-centroid regularization is only applied to prototypes at the last inner-product layer.
+
+# 4 EXPERIMENTS
+
+In this section, we provide extensive experimental results to show superiority of our method on a variety of tasks, including visual classification, imbalanced classification, person ReID, and face verification. More experimental analysis and implementation details can be found in the appendix.
+
+# 4.1 VISUAL CLASSIFICATION
+
+To verify the effectiveness of the proposed sample margin regularization in improving inter-class separability and intra-class compactness, we conduct experiments of classification on balanced datasets MNIST (LeCun et al., 1998), CIFAR-10 and CIFAR-100 (Krizhevsky & Hinton, 2009). We evaluate performance with three metrics: 1) top-1 validation accuracy acc; 2) the class margin $m_{cls}$ defined in (2.2); 3) the average of sample margins $m_{samp}$ . We use a 4-layer CNN, ResNet-18, and ResNet-34 on MNIST, CIFAR-10, and CIFAR-100, respectively. Moreover, some commonly-used neural units are considered, such as ReLU, BatchNorm, and cosine learning rate annealing. We use CE, CosFace, ArcFace, NormFace as the compared baseline methods. Note that CosFace, ArcFace, NormFace have one identical hyper-parameter $s$ , which is used for comprehensive study.
+
+Results. As shown in Table 1, all baseline losses fail in learning with large margins for all $s$ , in which the class margin decreases as $s$ increases. There is no significant performance difference among them. In contrast, by coupling with the proposed sample margin regularization $R_{\mathrm{sm}}$ , the losses turn to have larger margins. The results demonstrate that the proposed sample margin regularization is really beneficial to learn towards the possible largest margins. Moreover, the enlargement on class margin and sample margin means better inter-class separability and intra-class compactness, which further brings the improvement of classification accuracy in most cases.
+
+Table 1: Test accuracies (acc), class margins $(m_{cls})$ and sample margins $(m_{samp})$ on MNIST, CIFAR-10 and CIFAR-100 using loss functions with/without $R_{\mathrm{sm}}$ in (3.4). The results with positive gains are highlighted.
+
+| Method | Domain Order | Average |
| (i)DB-Amazon-Yahoo-AG (Avg.) | (ii)DB-Amazon-AG-Yahoo (Avg.) | (iii)Yahoo-Amazon-AG-DB (Avg.) | |
| Prompt tuning | 00.00, 00.00, 07.29, 81.71 (18.88) | 00.00, 00.00, 52.57, 64.57 (24.85) | 00.71, 00.00, 00.00, 97.86 (41.67) | 28.47±9.65 | |
| EWC-Prompt tuning | 00.00, 00.00, 10.86, 82.43 (19.79) | 00.00, 00.00, 56.14, 68.14 (26.36) | 00.00, 00.00, 00.00, 96.93 (41.12) | 29.09±8.92 | |
| MAS-Prompt tuning | 00.00, 00.00, 12.57, 83.86 (20.45) | 00.00, 00.00, 58.00, 65.71 (26.24) | 00.29, 00.00, 00.00, 97.86 (41.70) | 29.46±8.97 | |
| Fine-tuning | 26.71, 06.40, 09.43, 85.71 (32.48) | 21.36, 05.40, 57.00, 71.29 (37.09) | 04.43, 17.80, 25.86, 98.14 (50.76) | 40.11±7.76 | |
| EWC-Fine-tuning | 38.21, 00.20, 21.00, 86.29 (39.00) | 24.07, 04.00, 59.86, 68.14 (37.97) | 00.14, 12.80, 05.86, 98.07 (44.82) | 40.60±3.02 | |
| MAS-Fine-tuning | 37.29, 00.60, 15.71, 83.29 (36.91) | 20.79, 00.40, 61.14, 67.00 (36.06) | 01.00, 11.40, 27.57, 98.07 (49.39) | 40.79±6.09 | |
| Prompt tuning-Real | 85.79, 31.20, 43.14, 82.00 (67.67) | 82.57, 37.40, 67.29, 64.43 (68.64) | 43.71, 24.20, 58.71, 94.29 (65.39) | 67.23±1.36 | |
| LFPT5 | 48.57, 23.20, 32.43, 78.43 (47.64) | 54.93, 12.20, 61.86, 67.43 (52.58) | 10.57, 09.20, 59.86, 98.00 (57.91) | 52.71±4.19 | |
| MT-Prompt tuning | 95.00, 47.40, 62.29, 75.57 (76.73) | 93.57, 47.80, 73.86, 65.57 (76.52) | 61.71, 43.00, 74.71, 93.21 (75.00) | 76.08±0.77 | |
+
+# 5 ANALYSIS
+
+Influence of Domain Order To evaluate the influence of domain orders when LFPT5 is learning different task domains, we show the results of three runs with different domain order on the classification task in Table 6. We can see that the order of domains influences the performance of all methods a lot. For example, PT can achieve 41.67 accuracy on the third run while the accuracy of the first run is only 18.88. This phenomenon indicates that the difficulty of transferring knowledge from one domain to another might be quite different from that of the opposite transfer direction. Though the performance is affected by the order, LFPT5 outperforms previous regularization-based lifelong learning methods by a large margin for all different orders (see Appendix A.8 for more analysis).
+
+Table 6: Text classification accuracy on the whole test set for three runs with different domain order.
+
+