Daily Papers

Tables are ubiquitous across various domains for concisely representing structured information. Empowering large language models (LLMs) to reason over tabular data represents an actively explored direction. However, since typical LLMs only support one-dimensional~(1D) inputs, existing methods often flatten the two-dimensional~(2D) table structure into a sequence of tokens, which can severely disrupt the spatial relationships and result in an inevitable loss of vital contextual information. In this paper, we first empirically demonstrate the detrimental impact of such flattening operations on the performance of LLMs in capturing the spatial information of tables through two elaborate proxy tasks. Subsequently, we introduce a simple yet effective positional encoding method, termed ``2D-TPE'' (Two-Dimensional Table Positional Encoding), to address this challenge. 2D-TPE enables each attention head to dynamically select a permutation order of tokens within the context for attending to them, where each permutation represents a distinct traversal mode for the table, such as column-wise or row-wise traversal. 2D-TPE effectively mitigates the risk of losing essential spatial information while preserving computational efficiency, thus better preserving the table structure. Extensive experiments across five benchmarks demonstrate that 2D-TPE outperforms strong baselines, underscoring the importance of preserving the table structure for accurate table comprehension. Comprehensive analysis further reveals the substantially better scalability of 2D-TPE to large tables than baselines.

Unsupervised object-centric learning aims to decompose scenes into interpretable object entities, termed slots. Slot-based auto-encoders stand out as a prominent method for this task. Within them, crucial aspects include guiding the encoder to generate object-specific slots and ensuring the decoder utilizes them during reconstruction. This work introduces two novel techniques, (i) an attention-based self-training approach, which distills superior slot-based attention masks from the decoder to the encoder, enhancing object segmentation, and (ii) an innovative patch-order permutation strategy for autoregressive transformers that strengthens the role of slot vectors in reconstruction. The effectiveness of these strategies is showcased experimentally. The combined approach significantly surpasses prior slot-based autoencoder methods in unsupervised object segmentation, especially with complex real-world images. We provide the implementation code at https://github.com/gkakogeorgiou/spot .

The Segment Anything Model (SAM) achieves strong open-vocabulary segmentation, but its ViT-based image encoders dominate inference latency and memory. Existing activation compression methods, such as token merging, reduce the token length to process, yet introduce non-trivial runtime overhead and encounter catastrophic quality drop under high compression. Other methods applying Sparse Attention focus on attention alone, leaving the MLP fully dense and capping achievable speedup. We propose SparseSAM, a (i) training-free structured sparsification framework that jointly accelerates attention and MLP layers while preserving token identity. SparseSAM introduces (ii) Stripe-Sort Attention, which uses a deterministic Z-order permutation to transform dense attention into static hardware-friendly sparse patterns, eliminating dynamic masking overhead. SparseSAM further introduces a (iii) Residual-Consistency MLP that routes only informative tokens through the MLP while propagating remaining tokens through the residual pathway. Across four segmentation benchmarks, SparseSAM loses only 0.004 mIoU at a 0.4 density and 0.021 mIoU at 0.3, a 2.10x reduction in accuracy loss versus token merging advances, while achieving 2x faster inference and 2.8x memory reduction.

Transformer tends to overallocate attention to irrelevant context. In this work, we introduce Diff Transformer, which amplifies attention to the relevant context while canceling noise. Specifically, the differential attention mechanism calculates attention scores as the difference between two separate softmax attention maps. The subtraction cancels noise, promoting the emergence of sparse attention patterns. Experimental results on language modeling show that Diff Transformer outperforms Transformer in various settings of scaling up model size and training tokens. More intriguingly, it offers notable advantages in practical applications, such as long-context modeling, key information retrieval, hallucination mitigation, in-context learning, and reduction of activation outliers. By being less distracted by irrelevant context, Diff Transformer can mitigate hallucination in question answering and text summarization. For in-context learning, Diff Transformer not only enhances accuracy but is also more robust to order permutation, which was considered as a chronic robustness issue. The results position Diff Transformer as a highly effective and promising architecture to advance large language models.

Graph generation has been dominated by autoregressive models due to their simplicity and effectiveness, despite their sensitivity to ordering. Yet diffusion models have garnered increasing attention, as they offer comparable performance while being permutation-invariant. Current graph diffusion models generate graphs in a one-shot fashion, but they require extra features and thousands of denoising steps to achieve optimal performance. We introduce PARD, a Permutation-invariant Auto Regressive Diffusion model that integrates diffusion models with autoregressive methods. PARD harnesses the effectiveness and efficiency of the autoregressive model while maintaining permutation invariance without ordering sensitivity. Specifically, we show that contrary to sets, elements in a graph are not entirely unordered and there is a unique partial order for nodes and edges. With this partial order, PARD generates a graph in a block-by-block, autoregressive fashion, where each block's probability is conditionally modeled by a shared diffusion model with an equivariant network. To ensure efficiency while being expressive, we further propose a higher-order graph transformer, which integrates transformer with PPGN. Like GPT, we extend the higher-order graph transformer to support parallel training of all blocks. Without any extra features, PARD achieves state-of-the-art performance on molecular and non-molecular datasets, and scales to large datasets like MOSES containing 1.9M molecules.

We tackle the problem of sparse novel view synthesis (NVS) using video diffusion models; given K (approx 5) multi-view images of a scene and their camera poses, we predict the view from a target camera pose. Many prior approaches leverage generative image priors encoded via diffusion models. However, models trained on single images lack multi-view knowledge. We instead argue that video models already contain implicit multi-view knowledge and so should be easier to adapt for NVS. Our key insight is to formulate sparse NVS as a low frame-rate video completion task. However, one challenge is that sparse NVS is defined over an unordered set of inputs, often too sparse to admit a meaningful order, so the models should be invariant to permutations of that input set. To this end, we present FrameCrafter, which adapts video models (naturally trained with coherent frame orderings) to permutation-invariant NVS through several architectural modifications, including per-frame latent encodings and removal of temporal positional embeddings. Our results suggest that video models can be easily trained to "forget" about time with minimal supervision, producing competitive performance on sparse-view NVS benchmarks. Project page: https://frame-crafter.github.io/

We study retrieval over catalogs of structured metadata, where each record is a small schema whose fields answer different kinds of query. Embedding a record with a text encoder first serializes its fields into a string, which forces a choice of field order. We show this choice, usually treated as an implementation detail, silently controls retrieval quality once the encoder is fine-tuned. A standard fine-tune loses 7.4 nDCG@10 points when the index is rebuilt under a different field order, because it reads absolute position instead of the field labels. We propose permutation-invariant fine-tuning (PI-FT), which serializes each record under a freshly sampled field order with random field dropout, so meaning binds to the labels rather than to position. The change is about two lines in the data loader; it costs negligible in-distribution accuracy and cuts the order-change penalty to 0.2 points. We study this in the discovery of development statistics, a catalog of nearly 10,000 indicators that should be searchable in many languages by a model small enough to self-host. As AI assistants and agents increasingly mediate access to public data and statistics, this retrieval step decides whether an answer is grounded in the right indicator or series, making discoverability a precondition for disseminating data through AI. Because usage logs cannot provide training signal for indicators no one has searched, we generate the queries instead. DevDataBench is a fully LLM-generated benchmark of grounded, facet-targeted queries across 15 languages, covering every indicator for both training and evaluation. A fine-tuned 118M-parameter CPU encoder outperforms every zero-shot baseline, including text-embedding-3-large (0.707 vs.\ 0.556 nDCG@10), with the largest gains in low-resource languages. We release the benchmark, pipeline, models, and a reusable PI-FT framework.

We define a new class of set functions that in addition to being monotone and subadditive, also admit a very limited form of submodularity defined over a permutation of the ground set. We refer to this permutation as a submodular order. This class of functions includes monotone submodular functions as a sub-family. To understand the importance of this structure in optimization problems we consider the problem of maximizing function value under various types of constraints. To demonstrate the modeling power of submodular order functions we show applications in two different settings. First, we apply our results to the extensively studied problem of assortment optimization. While the objectives in assortment optimization are known to be non-submodular (and non-monotone) even for simple choice models, we show that they are compatible with the notion of submodular order. Consequently, we obtain new and in some cases the first constant factor guarantee for constrained assortment optimization in fundamental choice models. As a second application of submodular order functions, we show an intriguing connection to the maximization of monotone submodular functions in the streaming model. We recover some best known guarantees for this problem as a corollary of our results.

Providing Large Language Models with relevant contextual knowledge at inference time has been shown to greatly improve the quality of their generations. This is often achieved by prepending informative passages of text, or 'contexts', retrieved from external knowledge bases to their input. However, processing additional contexts online incurs significant computation costs that scale with their length. State Space Models (SSMs) offer a promising solution by allowing a database of contexts to be mapped onto fixed-dimensional states from which to start the generation. A key challenge arises when attempting to leverage information present across multiple contexts, since there is no straightforward way to condition generation on multiple independent states in existing SSMs. To address this, we leverage a simple mathematical relation derived from SSM dynamics to compose multiple states into one that efficiently approximates the effect of concatenating raw context tokens. Since the temporal ordering of contexts can often be uninformative, we enforce permutation-invariance by efficiently averaging states obtained via our composition algorithm across all possible context orderings. We evaluate our resulting method on WikiText and MSMARCO in both zero-shot and fine-tuned settings, and show that we can match the strongest performing baseline while enjoying on average 5.4x speedup.

Retrieval-Augmented Generation (RAG) has become a key paradigm for reducing factual hallucinations in large language models (LLMs), yet little is known about how the order of retrieved documents affects model behavior. We empirically show that under Top-5 retrieval with the gold document included, LLM answers vary substantially across permutations of the retrieved set, even when the gold document is fixed in the first position. This reveals a previously underexplored sensitivity to retrieval permutations. Although robust RAG methods primarily focus on enhancing LLM robustness to low-quality retrieval and mitigating positional bias to distribute attention fairly over long contexts, neither approach directly addresses permutation sensitivity. In this paper, we propose Stable-RAG, which exploits permutation sensitivity estimation to mitigate permutation-induced hallucinations. Stable-RAG runs the generator under multiple retrieval orders, clusters hidden states, and decodes from a cluster-center representation that captures the dominant reasoning pattern. It then uses these reasoning results to align hallucinated outputs toward the correct answer, encouraging the model to produce consistent and accurate predictions across document permutations. Experiments on three QA datasets show that Stable-RAG significantly improves answer accuracy, reasoning consistency and robust generalization across datasets, retrievers, and input lengths compared with baselines.

The assumption that response and predictor belong to the same statistical unit may be violated in practice. Unbiased estimation and recovery of true label ordering based on unlabeled data are challenging tasks and have attracted increasing attentions in the recent literature. In this paper, we present a relatively complete analysis of label permutation problem for the generalized linear model with multivariate responses. The theory is established under different scenarios, with knowledge of true parameters, with partial knowledge of underlying label permutation matrix and without any knowledge. Our results remove the stringent conditions required by the current literature and are further extended to the missing observation setting which has never been considered in the field of label permutation problem. On computational side, we propose two methods, "maximum likelihood estimation" algorithm and "two-step estimation" algorithm, to accommodate for different settings. When the proportion of permuted labels is moderate, both methods work effectively. Multiple numerical experiments are provided and corroborate our theoretical findings.

Large language models (LLMs) exhibit positional bias in how they use context, which especially complicates listwise ranking. To address this, we propose permutation self-consistency, a form of self-consistency over ranking list outputs of black-box LLMs. Our key idea is to marginalize out different list orders in the prompt to produce an order-independent ranking with less positional bias. First, given some input prompt, we repeatedly shuffle the list in the prompt and pass it through the LLM while holding the instructions the same. Next, we aggregate the resulting sample of rankings by computing the central ranking closest in distance to all of them, marginalizing out prompt order biases in the process. Theoretically, we prove the robustness of our method, showing convergence to the true ranking in the presence of random perturbations. Empirically, on five list-ranking datasets in sorting and passage reranking, our approach improves scores from conventional inference by up to 7-18% for GPT-3.5 and 8-16% for LLaMA v2 (70B), surpassing the previous state of the art in passage reranking. Our code is at https://github.com/castorini/perm-sc.

We introduce pi^3, a feed-forward neural network that offers a novel approach to visual geometry reconstruction, breaking the reliance on a conventional fixed reference view. Previous methods often anchor their reconstructions to a designated viewpoint, an inductive bias that can lead to instability and failures if the reference is suboptimal. In contrast, pi^3 employs a fully permutation-equivariant architecture to predict affine-invariant camera poses and scale-invariant local point maps without any reference frames. This design makes our model inherently robust to input ordering and highly scalable. These advantages enable our simple and bias-free approach to achieve state-of-the-art performance on a wide range of tasks, including camera pose estimation, monocular/video depth estimation, and dense point map reconstruction. Code and models are publicly available.

Many machine learning tasks such as multiple instance learning, 3D shape recognition, and few-shot image classification are defined on sets of instances. Since solutions to such problems do not depend on the order of elements of the set, models used to address them should be permutation invariant. We present an attention-based neural network module, the Set Transformer, specifically designed to model interactions among elements in the input set. The model consists of an encoder and a decoder, both of which rely on attention mechanisms. In an effort to reduce computational complexity, we introduce an attention scheme inspired by inducing point methods from sparse Gaussian process literature. It reduces the computation time of self-attention from quadratic to linear in the number of elements in the set. We show that our model is theoretically attractive and we evaluate it on a range of tasks, demonstrating the state-of-the-art performance compared to recent methods for set-structured data.

Instructing large language models (LLMs) to solve elementary school math problems has shown great success using Chain of Thought (CoT). However, the CoT approach relies on an LLM to generate a sequence of arithmetic calculations which can be prone to cascaded calculation errors. We hypothesize that an LLM should focus on extracting predicates and generating symbolic formulas from the math problem description so that the underlying calculation can be done via an external code interpreter. We investigate using LLM to generate Prolog programs to solve mathematical questions. Experimental results show that our Prolog-based arithmetic problem-solving outperforms CoT generation in the GSM8K benchmark across three distinct LLMs. In addition, given the insensitive ordering of predicates and symbolic formulas in Prolog, we propose to permute the ground truth predicates for more robust LLM training via data augmentation.

Positional embeddings (PE) play a crucial role in Vision Transformers (ViTs) by providing spatial information otherwise lost due to the permutation invariant nature of self attention. While absolute positional embeddings (APE) have shown theoretical advantages over relative positional embeddings (RPE), particularly due to the ability of sinusoidal functions to preserve spatial inductive biases like monotonicity and shift invariance, a fundamental challenge arises when mapping a 2D grid to a 1D sequence. Existing methods have mostly overlooked or never explored the impact of patch ordering in positional embeddings. To address this, we propose LOOPE, a learnable patch-ordering method that optimizes spatial representation for a given set of frequencies, providing a principled approach to patch order optimization. Empirical results show that our PE significantly improves classification accuracy across various ViT architectures. To rigorously evaluate the effectiveness of positional embeddings, we introduce the "Three Cell Experiment", a novel benchmarking framework that assesses the ability of PEs to retain relative and absolute positional information across different ViT architectures. Unlike standard evaluations, which typically report a performance gap of 4 to 6% between models with and without PE, our method reveals a striking 30 to 35% difference, offering a more sensitive diagnostic tool to measure the efficacy of PEs. Our experimental analysis confirms that the proposed LOOPE demonstrates enhanced effectiveness in retaining both relative and absolute positional information.

The success of deep learning is due in large part to our ability to solve certain massive non-convex optimization problems with relative ease. Though non-convex optimization is NP-hard, simple algorithms -- often variants of stochastic gradient descent -- exhibit surprising effectiveness in fitting large neural networks in practice. We argue that neural network loss landscapes often contain (nearly) a single basin after accounting for all possible permutation symmetries of hidden units a la Entezari et al. 2021. We introduce three algorithms to permute the units of one model to bring them into alignment with a reference model in order to merge the two models in weight space. This transformation produces a functionally equivalent set of weights that lie in an approximately convex basin near the reference model. Experimentally, we demonstrate the single basin phenomenon across a variety of model architectures and datasets, including the first (to our knowledge) demonstration of zero-barrier linear mode connectivity between independently trained ResNet models on CIFAR-10. Additionally, we identify intriguing phenomena relating model width and training time to mode connectivity. Finally, we discuss shortcomings of the linear mode connectivity hypothesis, including a counterexample to the single basin theory.

Bayesian Optimization (BO) is widely and successfully adopted for solving optimization problems having an expensive-to-evaluate, black-box, and non-convex objective function. However, the vanilla BO algorithm is not able to exploit possible symmetries characterizing the target problem. An intuitive case is given by optimal location problems, whose decision variables refer to a finite set of points within a continuous space, with the order of points not affecting the value of the objective function. We refer to this setting as optimization over layouts to distinguish from optimization over point-clouds where, instead, the order of points counts. As an instance of optimization over layouts we consider a real-life industrial-relevant application, that is the optimization of the layout of an offshore wind farm: given identical wind turbines, switching any pair of them has not any effect on the annual energy production. Based on Optimal Transport theory, we propose a Permutation-Invariant BO approach, namely PIBO, proved to provide better wind farm layouts when compared to the vanilla BO approach while cutting computation time roughly in half.

Existing cross-encoder models can be categorized as pointwise, pairwise, or listwise. Pairwise and listwise models allow passage interactions, which typically makes them more effective than pointwise models but less efficient and less robust to input passage order permutations. To enable efficient permutation-invariant passage interactions during re-ranking, we propose a new cross-encoder architecture with inter-passage attention: the Set-Encoder. In experiments on TREC Deep Learning and TIREx, the Set-Encoder is as effective as state-of-the-art listwise models while being more efficient and invariant to input passage order permutations. Compared to pointwise models, the Set-Encoder is particularly more effective when considering inter-passage information, such as novelty, and retains its advantageous properties compared to other listwise models. Our code is publicly available at https://github.com/webis-de/ECIR-25.

In complex systems, we often observe complex global behavior emerge from a collection of agents interacting with each other in their environment, with each individual agent acting only on locally available information, without knowing the full picture. Such systems have inspired development of artificial intelligence algorithms in areas such as swarm optimization and cellular automata. Motivated by the emergence of collective behavior from complex cellular systems, we build systems that feed each sensory input from the environment into distinct, but identical neural networks, each with no fixed relationship with one another. We show that these sensory networks can be trained to integrate information received locally, and through communication via an attention mechanism, can collectively produce a globally coherent policy. Moreover, the system can still perform its task even if the ordering of its inputs is randomly permuted several times during an episode. These permutation invariant systems also display useful robustness and generalization properties that are broadly applicable. Interactive demo and videos of our results: https://attentionneuron.github.io/

Contrastive learning has become an important tool in learning representations from unlabeled data mainly relying on the idea of minimizing distance between positive data pairs, e.g., views from the same images, and maximizing distance between negative data pairs, e.g., views from different images. This paper proposes a new variation of the contrastive learning objective, Group Ordering Constraints (GroCo), that leverages the idea of sorting the distances of positive and negative pairs and computing the respective loss based on how many positive pairs have a larger distance than the negative pairs, and thus are not ordered correctly. To this end, the GroCo loss is based on differentiable sorting networks, which enable training with sorting supervision by matching a differentiable permutation matrix, which is produced by sorting a given set of scores, to a respective ground truth permutation matrix. Applying this idea to groupwise pre-ordered inputs of multiple positive and negative pairs allows introducing the GroCo loss with implicit emphasis on strong positives and negatives, leading to better optimization of the local neighborhood. We evaluate the proposed formulation on various self-supervised learning benchmarks and show that it not only leads to improved results compared to vanilla contrastive learning but also shows competitive performance to comparable methods in linear probing and outperforms current methods in k-NN performance.

While large language models (LLMs) have achieved impressive performance across diverse tasks, recent studies showcase that causal LLMs suffer from the "reversal curse". It is a typical example that the model knows "A's father is B", but is unable to reason "B's child is A". This limitation poses a challenge to the advancement of artificial general intelligence (AGI), as it suggests a gap in the models' ability to comprehend and apply bidirectional reasoning. In this paper, we first conduct substantial evaluation and identify that the root cause of the reversal curse lies in the different word order between the training and inference stage, namely, the poor ability of causal language models to predict antecedent words within the training data. Accordingly, permutation on the training data is considered as a potential solution, since this can make the model predict antecedent words or tokens. However, previous permutation methods may disrupt complete phrases or entities, thereby posing challenges for the model to comprehend and learn from training data. To address this issue, we propose Semantic-aware Permutation Training (SPT), which addresses this issue by segmenting the training sentences into semantic units (i.e., entities or phrases) with an assistant language model and permuting these units before feeding into the model. Extensive experiments demonstrate that SPT effectively mitigates the reversal curse since the performance on reversed questions approximates that on the forward ones, and significantly advances the performance of existing works.

Attention scales quadratically with sequence length, fundamentally limiting long-context inference. Existing block-granularity sparsification can reduce latency, but coarse blocks impose an intrinsic sparsity ceiling, making further improvements difficult even with carefully engineered designs. We present S2O, which performs early stopping for sparse attention via online permutation. Inspired by virtual-to-physical address mapping in memory systems, S2O revisits and factorizes FlashAttention execution, enabling inference to load non-contiguous tokens rather than a contiguous span in the original order. Motivated by fine-grained structures in attention heatmaps, we transform explicit permutation into an online, index-guided, discrete loading policy; with extremely lightweight preprocessing and index-remapping overhead, it concentrates importance on a small set of high-priority blocks. Building on this importance-guided online permutation for loading, S2O further introduces an early-stopping rule: computation proceeds from high to low importance; once the current block score falls below a threshold, S2O terminates early and skips the remaining low-contribution blocks, thereby increasing effective sparsity and reducing computation under a controlled error budget. As a result, S2O substantially raises the practical sparsity ceiling. On Llama-3.1-8B under a 128K context, S2O reduces single-operator MSE by 3.82times at matched sparsity, and reduces prefill compute density by 3.31times at matched MSE; meanwhile, it preserves end-to-end accuracy and achieves 7.51times attention and 3.81times end-to-end speedups.

Multi-hop Question Answering (MHQA) adds layers of complexity to question answering, making it more challenging. When Language Models (LMs) are prompted with multiple search results, they are tasked not only with retrieving relevant information but also employing multi-hop reasoning across the information sources. Although LMs perform well on traditional question-answering tasks, the causal mask can hinder their capacity to reason across complex contexts. In this paper, we explore how LMs respond to multi-hop questions by permuting search results (retrieved documents) under various configurations. Our study reveals interesting findings as follows: 1) Encoder-decoder models, such as the ones in the Flan-T5 family, generally outperform causal decoder-only LMs in MHQA tasks, despite being significantly smaller in size; 2) altering the order of gold documents reveals distinct trends in both Flan T5 models and fine-tuned decoder-only models, with optimal performance observed when the document order aligns with the reasoning chain order; 3) enhancing causal decoder-only models with bi-directional attention by modifying the causal mask can effectively boost their end performance. In addition to the above, we conduct a thorough investigation of the distribution of LM attention weights in the context of MHQA. Our experiments reveal that attention weights tend to peak at higher values when the resulting answer is correct. We leverage this finding to heuristically improve LMs' performance on this task. Our code is publicly available at https://github.com/hwy9855/MultiHopQA-Reasoning.

Standard representation probing for visual models relies on mathematically permutation-invariant operations like Global Average Pooling (GAP) or CLS tokens, treating patch representations as an unstructured bag-of-words. We challenge this paradigm by demonstrating that token order is a critical, exploitable dimension in frozen visual representations (e.g., MAE, BEiT, DINOv2, and ViT as CLS-ablation extreme). We propose SSMProbe, a probing framework driven by a State Space Model (SSM). Operating as discrete Linear Time-Invariant (LTI) dynamical systems, SSMs act as permutation-sensitive probes where sequence order strictly dictates the final state due to inherent memory decay. Formulating token ordering as an information scheduling problem, we compare fixed scan heuristics against a differentiable soft permutation (Sinkhorn-based) learned from downstream supervision. Evaluations on standard and fine-grained classification benchmarks reveal a striking order gap: while fixed scans fail dramatically on highly localized patch features, our learned soft permutation successfully extracts highly competitive performance from otherwise heavily localized patch sequences. We find that pre-training objectives fundamentally shape token structure: DINOv2 concentrates global semantics in optimized CLS tokens leaving patches hyperspecialized, pure MAE preserves distributed representations with heterogeneous patch informativeness, and ViT represents a supervised CLS-dominated extreme. BEiT occupies middle ground. This heterogeneity is order-dependent -- meaning the SSM probe's performance depends critically on which tokens are placed at which temporal positions -- and is not merely a topological property of the spatial grid. SSMProbe's learned routing effectively discovers and exploits this heterogeneity, offering a powerful new diagnostic lens for visual representation analysis.

Autoregressive models excel in modeling sequential dependencies by enforcing causal constraints, yet they struggle to capture complex bidirectional patterns due to their unidirectional nature. In contrast, mask-based models leverage bidirectional context, enabling richer dependency modeling. However, they often assume token independence during prediction, which undermines the modeling of sequential dependencies. Additionally, the corruption of sequences through masking or absorption can introduce unnatural distortions, complicating the learning process. To address these issues, we propose Bidirectional Autoregressive Diffusion (BAD), a novel approach that unifies the strengths of autoregressive and mask-based generative models. BAD utilizes a permutation-based corruption technique that preserves the natural sequence structure while enforcing causal dependencies through randomized ordering, enabling the effective capture of both sequential and bidirectional relationships. Comprehensive experiments show that BAD outperforms autoregressive and mask-based models in text-to-motion generation, suggesting a novel pre-training strategy for sequence modeling. The codebase for BAD is available on https://github.com/RohollahHS/BAD.

Inductive Logic Programming (ILP) learns interpretable logical rules from data. Existing methods are transductive: their learned parameters are bound to specific predicates and require retraining for each new task. We introduce Neural Rule Inducer (NRI), a pretrained model for zero-shot rule induction. Rather than encoding literal identities, NRI represents literals using domain-agnostic statistical properties such as class-conditional rates, entropy, and co-occurrence, which generalize across variable identities and counts without retraining. The model consists of a statistical encoder and a parallel slot-based decoder. Parallel decoding preserves the permutation invariance of logical disjunction; an autoregressive decoder would instead impose an arbitrary clause order. Product T-norm relaxation makes rule execution differentiable, allowing end-to-end training on prediction accuracy alone. We evaluate NRI on rule recovery, robustness to label noise and spurious correlations, and zero-shot transfer to real-world benchmarks, and we believe this work opens up the possibility of foundation models for symbolic reasoning. Code and the reference checkpoint are available at https://github.com/phuayj/neural-rule-inducer.

Large language models perform near-Bayesian inference yet violate permutation invariance on exchangeable data. We resolve this by showing transformers minimize expected conditional description length (cross-entropy) over orderings, E_pi[ell(Y mid Gamma_pi(X))], which admits a Kolmogorov-complexity interpretation up to additive constants, rather than the permutation-invariant description length ell(Y mid X). This makes them Bayesian in expectation, not in realization. We derive (i) a Quantified Martingale Violation bound showing order-induced deviations scale as O(log n) with constants; (ii) the Expectation-level Decompression Law linking information budgets to reliability for Bernoulli predicates; and (iii) deployable planners (B2T/RoH/ISR) for answer/abstain decisions. Empirically, permutation dispersion follows a+bln n (Qwen2-7B b approx 0.377, Llama-3.1-8B b approx 0.147); permutation mixtures improve ground-truth likelihood/accuracy; and randomized dose-response shows hallucinations drop by sim 0.13 per additional nat. A pre-specified audit with a fixed ISR=1.0 achieves near-0\% hallucinations via calibrated refusal at 24\% abstention. The framework turns hallucinations into predictable compression failures and enables principled information budgeting.

Reinforcement learning based post-training has recently emerged as a powerful paradigm for enhancing the alignment and reasoning capabilities of multimodal large language models (MLLMs). While vision-centric post-training is crucial for enhancing MLLMs' intrinsic understanding of visual signals, current post-training paradigms are predominantly text-centric, where dense visual inputs are only leveraged to extract sparse cues for text-based reasoning. There exist a few approaches in this direction, however, they often still rely on text as an intermediate mediator or introduce additional visual generative designs. In this work, we introduce Visual Jigsaw, a generic self-supervised post-training framework designed to strengthen visual understanding in MLLMs. Visual Jigsaw is formulated as a general ordering task: visual inputs are partitioned, shuffled, and the model must reconstruct the visual information by producing the correct permutation in natural language. This naturally aligns with reinforcement learning from verifiable rewards (RLVR), requires no additional visual generative components, and derives its supervisory signal automatically without any annotations. We instantiate Visual Jigsaw across three visual modalities, including images, videos, and 3D data. Extensive experiments demonstrate substantial improvements in fine-grained perception, temporal reasoning, and 3D spatial understanding. Our findings highlight the potential of self-supervised vision-centric tasks in post-training MLLMs and aim to inspire further research on vision-centric pretext designs. Project Page: https://penghao-wu.github.io/visual_jigsaw/

Sequence models such as transformers require inputs to be represented as one-dimensional sequences. In vision, this typically involves flattening images using a fixed row-major (raster-scan) order. While full self-attention is permutation-equivariant, modern long-sequence transformers increasingly rely on architectural approximations that break this invariance and introduce sensitivity to patch ordering. We show that patch order significantly affects model performance in such settings, with simple alternatives like column-major or Hilbert curves yielding notable accuracy shifts. Motivated by this, we propose REOrder, a two-stage framework for discovering task-optimal patch orderings. First, we derive an information-theoretic prior by evaluating the compressibility of various patch sequences. Then, we learn a policy over permutations by optimizing a Plackett-Luce policy using REINFORCE. This approach enables efficient learning in a combinatorial permutation space. REOrder improves top-1 accuracy over row-major ordering on ImageNet-1K by up to 3.01% and Functional Map of the World by 13.35%.

This paper presents the key algorithmic techniques behind CatBoost, a new gradient boosting toolkit. Their combination leads to CatBoost outperforming other publicly available boosting implementations in terms of quality on a variety of datasets. Two critical algorithmic advances introduced in CatBoost are the implementation of ordered boosting, a permutation-driven alternative to the classic algorithm, and an innovative algorithm for processing categorical features. Both techniques were created to fight a prediction shift caused by a special kind of target leakage present in all currently existing implementations of gradient boosting algorithms. In this paper, we provide a detailed analysis of this problem and demonstrate that proposed algorithms solve it effectively, leading to excellent empirical results.

The output structure of database-like tables, consisting of values structured in horizontal rows and vertical columns identifiable by name, can cover a wide range of NLP tasks. Following this constatation, we propose a framework for text-to-table neural models applicable to problems such as extraction of line items, joint entity and relation extraction, or knowledge base population. The permutation-based decoder of our proposal is a generalized sequential method that comprehends information from all cells in the table. The training maximizes the expected log-likelihood for a table's content across all random permutations of the factorization order. During the content inference, we exploit the model's ability to generate cells in any order by searching over possible orderings to maximize the model's confidence and avoid substantial error accumulation, which other sequential models are prone to. Experiments demonstrate a high practical value of the framework, which establishes state-of-the-art results on several challenging datasets, outperforming previous solutions by up to 15%.

World models for interactive video generation have largely focused on single-agent settings, where future observations are generated from a single control signal. However, many generated environments require multi-agent interaction: multiple players, robots, or embodied agents act simultaneously within a shared space. Scaling world models to such settings requires a principled multi-agent design: agents should remain independently controllable, permutation-symmetric, and support efficient inference while maintaining consistency across time and perspectives. In this paper, we present our generative multi-agent world model for interactive simulation. It introduces Simplex Rotary Agent Encoding, a parameter-free extension of 3D RoPE that represents agents as vertices of a regular simplex in rotary angle space. This gives each agent a distinct phase while making all agents permutation-equivalent, enabling scalable agent identity without learned per-slot identities or a fixed agent ordering. To avoid dense all-to-all attention across agents, we further propose Sparse Hub Attention, where learnable hub tokens mediate token interaction across agents, reducing cross-agent attention cost from quadratic to linear in the number of agents. For real-time rollout, we distill a full-context diffusion teacher into a causal student that generates temporal blocks sequentially with KV caching, enabling action-responsive generation at 24 FPS. Experiments in multiplayer virtual environments show that our model improves video fidelity, action controllability, and inter-agent consistency over slot-based and dense-attention baselines, while generalizing from two to four players without additional training.

In this paper, we propose Multiresolution Equivariant Graph Variational Autoencoders (MGVAE), the first hierarchical generative model to learn and generate graphs in a multiresolution and equivariant manner. At each resolution level, MGVAE employs higher order message passing to encode the graph while learning to partition it into mutually exclusive clusters and coarsening into a lower resolution that eventually creates a hierarchy of latent distributions. MGVAE then constructs a hierarchical generative model to variationally decode into a hierarchy of coarsened graphs. Importantly, our proposed framework is end-to-end permutation equivariant with respect to node ordering. MGVAE achieves competitive results with several generative tasks including general graph generation, molecular generation, unsupervised molecular representation learning to predict molecular properties, link prediction on citation graphs, and graph-based image generation.

Autoregressive point cloud generation has long lagged behind diffusion-based approaches in quality. The performance gap stems from the fact that autoregressive models impose an artificial ordering on inherently unordered point sets, forcing shape generation to proceed as a sequence of local predictions. This sequential bias emphasizes short-range continuity but undermines the model's capacity to capture long-range dependencies, hindering its ability to enforce global structural properties such as symmetry, consistent topology, and large-scale geometric regularities. Inspired by the level-of-detail (LOD) principle in shape modeling, we propose PointNSP, a coarse-to-fine generative framework that preserves global shape structure at low resolutions and progressively refines fine-grained geometry at higher scales through a next-scale prediction paradigm. This multi-scale factorization aligns the autoregressive objective with the permutation-invariant nature of point sets, enabling rich intra-scale interactions while avoiding brittle fixed orderings. Experiments on ShapeNet show that PointNSP establishes state-of-the-art (SOTA) generation quality for the first time within the autoregressive paradigm. In addition, it surpasses strong diffusion-based baselines in parameter, training, and inference efficiency. Finally, in dense generation with 8,192 points, PointNSP's advantages become even more pronounced, underscoring its scalability potential.

Linear Mode Connectivity (LMC) refers to the phenomenon that performance remains consistent for linearly interpolated models in the parameter space. For independently optimized model pairs from different random initializations, achieving LMC is considered crucial for validating the stable success of the non-convex optimization in modern machine learning models and for facilitating practical parameter-based operations such as model merging. While LMC has been achieved for neural networks by considering the permutation invariance of neurons in each hidden layer, its attainment for other models remains an open question. In this paper, we first achieve LMC for soft tree ensembles, which are tree-based differentiable models extensively used in practice. We show the necessity of incorporating two invariances: subtree flip invariance and splitting order invariance, which do not exist in neural networks but are inherent to tree architectures, in addition to permutation invariance of trees. Moreover, we demonstrate that it is even possible to exclude such additional invariances while keeping LMC by designing decision list-based tree architectures, where such invariances do not exist by definition. Our findings indicate the significance of accounting for architecture-specific invariances in achieving LMC.

Humans rapidly learn abstract knowledge when encountering novel environments and flexibly deploy this knowledge to guide efficient and intelligent action. Can modern AI systems learn and plan in a similar way? We study this question using a dataset of complex human gameplay with concurrent fMRI recordings, in which participants learn novel video games that require rule discovery, hypothesis revision, and multi-step planning. We jointly evaluate models by their ability to play the games, match human learning behavior, and predict brain activity during the same task, comparing a suite of frontier Large Reasoning Models (LRMs) against model-free and model-based deep reinforcement learning agents and a Bayesian theory-based agent. We find that frontier LRMs most closely match human behavioral patterns during game discovery and predict brain activity an order of magnitude better than both reinforcement learning alternatives across cortical and subcortical regions, with effects robust to permutation controls. Through targeted manipulations, we further show that brain alignment reflects the model's in-context representation of the game state rather than its downstream planning or reasoning. Our results establish LRMs as compelling computational accounts of human learning and decision making in complex, naturalistic environments. Project page with interactive replays: https://botcs.github.io/reason-to-play/

The sparse Mixture-of-Experts (Sparse-MoE) framework efficiently scales up model capacity in various domains, such as natural language processing and vision. Sparse-MoEs select a subset of the "experts" (thus, only a portion of the overall network) for each input sample using a sparse, trainable gate. Existing sparse gates are prone to convergence and performance issues when training with first-order optimization methods. In this paper, we introduce two improvements to current MoE approaches. First, we propose a new sparse gate: COMET, which relies on a novel tree-based mechanism. COMET is differentiable, can exploit sparsity to speed up computation, and outperforms state-of-the-art gates. Second, due to the challenging combinatorial nature of sparse expert selection, first-order methods are typically prone to low-quality solutions. To deal with this challenge, we propose a novel, permutation-based local search method that can complement first-order methods in training any sparse gate, e.g., Hash routing, Top-k, DSelect-k, and COMET. We show that local search can help networks escape bad initializations or solutions. We performed large-scale experiments on various domains, including recommender systems, vision, and natural language processing. On standard vision and recommender systems benchmarks, COMET+ (COMET with local search) achieves up to 13% improvement in ROC AUC over popular gates, e.g., Hash routing and Top-k, and up to 9% over prior differentiable gates e.g., DSelect-k. When Top-k and Hash gates are combined with local search, we see up to 100times reduction in the budget needed for hyperparameter tuning. Moreover, for language modeling, our approach improves over the state-of-the-art MoEBERT model for distilling BERT on 5/7 GLUE benchmarks as well as SQuAD dataset.

Recently, there has been a surge of Transformer-based solutions for the long-term time series forecasting (LTSF) task. Despite the growing performance over the past few years, we question the validity of this line of research in this work. Specifically, Transformers is arguably the most successful solution to extract the semantic correlations among the elements in a long sequence. However, in time series modeling, we are to extract the temporal relations in an ordered set of continuous points. While employing positional encoding and using tokens to embed sub-series in Transformers facilitate preserving some ordering information, the nature of the permutation-invariant self-attention mechanism inevitably results in temporal information loss. To validate our claim, we introduce a set of embarrassingly simple one-layer linear models named LTSF-Linear for comparison. Experimental results on nine real-life datasets show that LTSF-Linear surprisingly outperforms existing sophisticated Transformer-based LTSF models in all cases, and often by a large margin. Moreover, we conduct comprehensive empirical studies to explore the impacts of various design elements of LTSF models on their temporal relation extraction capability. We hope this surprising finding opens up new research directions for the LTSF task. We also advocate revisiting the validity of Transformer-based solutions for other time series analysis tasks (e.g., anomaly detection) in the future. Code is available at: https://github.com/cure-lab/LTSF-Linear.

The whole-brain connectome of a fruit fly comprises over 130K neurons connected with a probability of merely 0.02%, yet achieves an average shortest path of only 4.4 hops. Despite being highly structured at the circuit level, the network's long-range connections are broadly distributed across brain regions, functioning as stochastic shortcuts that enable efficient global communication. Inspired by this observation, we propose Stochastic Attention (SA), a drop-in enhancement for sliding-window attention (SWA) that applies a random permutation to the token sequence before windowed attention and restores the original order afterward. This transforms the fixed local window into a stochastic global one within the same O(nw) per-layer budget. Through depth, independently sampled permutations yield exponentially growing receptive fields, achieving full sequence coverage in O(log_w n) layers versus O(n/w) for SWA. We validate SA in two settings: pre-training language models from scratch, where a gated SA + SWA combination achieves the best average zero-shot accuracy, and training-free inference on Qwen3-8B and Qwen3-30B-A3B, where SA consistently outperforms SWA and matches or exceeds Mixture of Block Attention at comparable compute budgets. These results suggest that connectome-inspired stochastic routing is a practical primitive for improving the expressivity of efficient attention, complementary to existing linear and sparse approaches.

A universal word (u-word) for d-dimensional permutations of length n is a 2-dimensional word with d-1 rows, any size n window of which is order-isomorphic to exactly one permutation of length n, and all permutations of length n are covered. It is known that u-words (in fact, even u-cycles, a stronger claim) for d-dimensional permutations exist. In this paper, we use the idea of incomparable elements to prove that u-words of length (n!)^{d-1}+n-1-i(n-1), for dgeq 2 and $0leq ileq 2^{d-1}{n-1}left[(1+(n-1)!)^{d-1}-left(1+(n-1)!{2}right)^{d-1}right], for d-dimensional permutations of length n exist, which generalizes the respective result of Kitaev, Potapov and Vajnovszki for ``usual'' permutations (d=2$).

Active matter systems, from self-propelled colloids to motile bacteria, are characterized by the conversion of free energy into useful work at the microscopic scale. These systems generically involve physics beyond the reach of equilibrium statistical mechanics, and a persistent challenge has been to understand the nature of their nonequilibrium states. The entropy production rate and the magnitude of the steady-state probability current provide quantitative ways to do so by measuring the breakdown of time-reversal symmetry and the strength of nonequilibrium transport of measure. Yet, their efficient computation has remained elusive, as they depend on the system's unknown and high-dimensional probability density. Here, building upon recent advances in generative modeling, we develop a deep learning framework that estimates the score of this density. We show that the score, together with the microscopic equations of motion, gives direct access to the entropy production rate, the probability current, and their decomposition into local contributions from individual particles, spatial regions, and degrees of freedom. To represent the score, we introduce a novel, spatially-local transformer-based network architecture that learns high-order interactions between particles while respecting their underlying permutation symmetry. We demonstrate the broad utility and scalability of the method by applying it to several high-dimensional systems of interacting active particles undergoing motility-induced phase separation (MIPS). We show that a single instance of our network trained on a system of 4096 particles at one packing fraction can generalize to other regions of the phase diagram, including systems with as many as 32768 particles. We use this observation to quantify the spatial structure of the departure from equilibrium in MIPS as a function of the number of particles and the packing fraction.

Designing machine learning architectures for processing neural networks in their raw weight matrix form is a newly introduced research direction. Unfortunately, the unique symmetry structure of deep weight spaces makes this design very challenging. If successful, such architectures would be capable of performing a wide range of intriguing tasks, from adapting a pre-trained network to a new domain to editing objects represented as functions (INRs or NeRFs). As a first step towards this goal, we present here a novel network architecture for learning in deep weight spaces. It takes as input a concatenation of weights and biases of a pre-trained MLP and processes it using a composition of layers that are equivariant to the natural permutation symmetry of the MLP's weights: Changing the order of neurons in intermediate layers of the MLP does not affect the function it represents. We provide a full characterization of all affine equivariant and invariant layers for these symmetries and show how these layers can be implemented using three basic operations: pooling, broadcasting, and fully connected layers applied to the input in an appropriate manner. We demonstrate the effectiveness of our architecture and its advantages over natural baselines in a variety of learning tasks.

Implicit neural representations (INRs) enable fast video compression and effective video processing, but a single model rarely offers scalable decoding across rates and resolutions. In practice, multi-resolution typically relies on retraining or multi-branch designs, and structured pruning failed to provide a permutation-invariant progressive transmission order. Motivated by the explicit structure and efficiency of Gaussian splatting, we propose D2GV-AR, a deformable 2D Gaussian video representation that enables arbitrary-scale rendering and any-ratio progressive coding within a single model. We partition each video into fixed-length Groups of Pictures and represent each group with a canonical set of 2D Gaussian primitives, whose temporal evolution is modeled by a neural ordinary differential equation. During training and rendering, we apply scale-aware grouping according to Nyquist sampling theorem to form a nested hierarchy across resolutions. Once trained, primitives can be pruned via a D-optimal subset objective to enable any-ratio progressive coding. Extensive experiments show that D2GV-AR renders at over 250 FPS while matching or surpassing recent INR baselines, enabling multiscale continuous rate--quality adaptation.

We propose masked particle modeling (MPM) as a self-supervised method for learning generic, transferable, and reusable representations on unordered sets of inputs for use in high energy physics (HEP) scientific data. This work provides a novel scheme to perform masked modeling based pre-training to learn permutation invariant functions on sets. More generally, this work provides a step towards building large foundation models for HEP that can be generically pre-trained with self-supervised learning and later fine-tuned for a variety of down-stream tasks. In MPM, particles in a set are masked and the training objective is to recover their identity, as defined by a discretized token representation of a pre-trained vector quantized variational autoencoder. We study the efficacy of the method in samples of high energy jets at collider physics experiments, including studies on the impact of discretization, permutation invariance, and ordering. We also study the fine-tuning capability of the model, showing that it can be adapted to tasks such as supervised and weakly supervised jet classification, and that the model can transfer efficiently with small fine-tuning data sets to new classes and new data domains.

Generating new molecules is fundamental to advancing critical applications such as drug discovery and material synthesis. Flows can generate molecules effectively by inverting the encoding process, however, existing flow models either require artifactual dequantization or specific node/edge orderings, lack desiderata such as permutation invariance, or induce discrepancy between the encoding and the decoding steps that necessitates post hoc validity correction. We circumvent these issues with novel continuous normalizing E(3)-equivariant flows, based on a system of node ODEs coupled as a graph PDE, that repeatedly reconcile locally toward globally aligned densities. Our models can be cast as message-passing temporal networks, and result in superlative performance on the tasks of density estimation and molecular generation. In particular, our generated samples achieve state-of-the-art on both the standard QM9 and ZINC250K benchmarks.

Prompt Tuning is emerging as a scalable and cost-effective method to fine-tune Pretrained Language Models (PLMs), which are often referred to as Large Language Models (LLMs). This study benchmarks the performance and computational efficiency of Prompt Tuning and baselines for multi-label text classification. This is applied to the challenging task of classifying companies into an investment firm's proprietary industry taxonomy, supporting their thematic investment strategy. Text-to-text classification is frequently reported to outperform task-specific classification heads, but has several limitations when applied to a multi-label classification problem where each label consists of multiple tokens: (a) Generated labels may not match any label in the label taxonomy; (b) The fine-tuning process lacks permutation invariance and is sensitive to the order of the provided labels; (c) The model provides binary decisions rather than appropriate confidence scores. Limitation (a) is addressed by applying constrained decoding using Trie Search, which slightly improves classification performance. All limitations (a), (b), and (c) are addressed by replacing the PLM's language head with a classification head, which is referred to as Prompt Tuned Embedding Classification (PTEC). This improves performance significantly, while also reducing computational costs during inference. In our industrial application, the training data is skewed towards well-known companies. We confirm that the model's performance is consistent across both well-known and less-known companies. Our overall results indicate the continuing need to adapt state-of-the-art methods to domain-specific tasks, even in the era of PLMs with strong generalization abilities. We release our codebase and a benchmarking dataset at https://github.com/EQTPartners/PTEC.

We present ManzaiSet, the first large scale multimodal dataset of viewer responses to Japanese manzai comedy, capturing facial videos and audio from 241 participants watching up to 10 professional performances in randomized order (94.6 percent watched >= 8; analyses focus on n=228). This addresses the Western centric bias in affective computing. Three key findings emerge: (1) k means clustering identified three distinct viewer types: High and Stable Appreciators (72.8 percent, n=166), Low and Variable Decliners (13.2 percent, n=30), and Variable Improvers (14.0 percent, n=32), with heterogeneity of variance (Brown Forsythe p < 0.001); (2) individual level analysis revealed a positive viewing order effect (mean slope = 0.488, t(227) = 5.42, p < 0.001, permutation p < 0.001), contradicting fatigue hypotheses; (3) automated humor classification (77 instances, 131 labels) plus viewer level response modeling found no type wise differences after FDR correction. The dataset enables culturally aware emotion AI development and personalized entertainment systems tailored to non Western contexts.

The question of whether there exists a finite group of order at least three in which every element except one is a commutator has remained unresolved in group theory. In this article, we address this open problem by developing an algorithmic approach that leverages several group theoretic properties of such groups. Specifically, we utilize a result of Frobenius and various necessary properties of such groups, combined with Plesken and Holt's extensive enumeration of finite perfect groups, to systematically examine all finite groups up to a certain order for the desired property. The computational core of our work is implemented using the computer system GAP (Groups, Algorithms, and Programming). We discover two nonisomorphic groups of order 368,640 that exhibit the desired property. Our investigation also establishes that this order is the minimum order for such a group to exist. As a result, this study provides a positive answer to Problem 17.76 in the Kourovka Notebook. In addition to the algorithmic framework, this paper provides a structural description of one of the two groups found.

The Markov numbers are positive integers appearing as solutions to the Diophantine equation x^2 + y^2 + z^2 = 3xyz. These numbers are very well-studied and have many combinatorial properties, as well as being the source of the long-standing unicity conjecture. In 2018, Canakc{\i} and Schiffler showed that the Markov number m_{a{b}} is the number of perfect matchings of a certain snake graph corresponding to the Christoffel path from (0,0) to (a,b). Based on this correspondence, Schiffler in 2023 introduced two orderings on lattice paths. For any path omega, associate a snake graph G(omega) and a continued fraction g(omega). The ordering <_M is given by the number of perfect matchings on G(omega), and the ordering <_L is given by the Lagrange number of g(omega). In this work, we settle two conjectures of Schiffler. First, we show that the path omega(a,b) = RRcdots R UU cdots U is the unique maximum over all lattice paths from (0,0) to (a,b) with respect to both orderings <_M and <_L. We then use this result to prove that sup L(omega) over all lattice paths is exactly 1+sqrt5.

Recent works show that ordering of the training data affects the model performance for Neural Machine Translation. Several approaches involving dynamic data ordering and data sharding based on curriculum learning have been analysed for the their performance gains and faster convergence. In this work we propose to empirically study several ordering approaches for the training data based on different metrics and evaluate their impact on the model performance. Results from our study show that pre-fixing the ordering of the training data based on perplexity scores from a pre-trained model performs the best and outperforms the default approach of randomly shuffling the training data every epoch.

We present a principled approach to uncover the structure of visual data by solving a novel deep learning task coined visual permutation learning. The goal of this task is to find the permutation that recovers the structure of data from shuffled versions of it. In the case of natural images, this task boils down to recovering the original image from patches shuffled by an unknown permutation matrix. Unfortunately, permutation matrices are discrete, thereby posing difficulties for gradient-based methods. To this end, we resort to a continuous approximation of these matrices using doubly-stochastic matrices which we generate from standard CNN predictions using Sinkhorn iterations. Unrolling these iterations in a Sinkhorn network layer, we propose DeepPermNet, an end-to-end CNN model for this task. The utility of DeepPermNet is demonstrated on two challenging computer vision problems, namely, (i) relative attributes learning and (ii) self-supervised representation learning. Our results show state-of-the-art performance on the Public Figures and OSR benchmarks for (i) and on the classification and segmentation tasks on the PASCAL VOC dataset for (ii).

Autoregressive models (ARMs) have become the workhorse for sequence generation tasks, since many problems can be modeled as next-token prediction. While there appears to be a natural ordering for text (i.e., left-to-right), for many data types, such as graphs, the canonical ordering is less obvious. To address this problem, we introduce a variant of ARM that generates high-dimensional data using a probabilistic ordering that is sequentially inferred from data. This model incorporates a trainable probability distribution, referred to as an order-policy, that dynamically decides the autoregressive order in a state-dependent manner. To train the model, we introduce a variational lower bound on the exact log-likelihood, which we optimize with stochastic gradient estimation. We demonstrate experimentally that our method can learn meaningful autoregressive orderings in image and graph generation. On the challenging domain of molecular graph generation, we achieve state-of-the-art results on the QM9 and ZINC250k benchmarks, evaluated using the Fr\'{e}chet ChemNet Distance (FCD).

Reordering is a preprocessing stage for Statistical Machine Translation (SMT) system where the words of the source sentence are reordered as per the syntax of the target language. We are proposing a rich set of rules for better reordering. The idea is to facilitate the training process by better alignments and parallel phrase extraction for a phrase-based SMT system. Reordering also helps the decoding process and hence improving the machine translation quality. We have observed significant improvements in the translation quality by using our approach over the baseline SMT. We have used BLEU, NIST, multi-reference word error rate, multi-reference position independent error rate for judging the improvements. We have exploited open source SMT toolkit MOSES to develop the system.

This paper discusses about a sorting algorithm which uses the concept of buckets where each bucket represents a certain number of digits. A two dimensional data structure is used where one dimension represents buckets i. e; number of digits and each bucket's corresponding dimensions represents the input numbers that belong to that bucket. Each bucket is then individually sorted. Since every preceding bucket elements will always be smaller than the succeeding buckets no comparison between them is required. By doing this we can significantly reduced the time complexity of any sorting algorithm used to sort the given set of inputs.

Macro placement is a fundamental step in modern chip physical design, playing a crucial role in determining the solution quality of high-dimensional combinatorial optimization problems. Despite recent advancements in machine learning for spatial coordinate determination, the temporal dimension of placement sequencing remains largely governed by static heuristics. In this work, we demonstrate that the placement sequence is not merely a preprocessing step but a decisive factor in optimization, where suboptimal early decisions trigger irreversible domino effects that constrain the solution space. To harness this unexplored dimension, we propose OrderPlace, a proxy-guided LLM evolution framework for automatically discovering macro placement order strategies. Instead of relying on manually crafted heuristics such as area- or connectivity-based ordering, OrderPlace explores a broader space of code-level policies, ranging from static scoring metrics to dynamic physics-inspired mechanisms. To mitigate the prohibitive cost of evaluating sequences, we introduce a lightweight proxy evaluation mechanism that efficiently filters candidates using a deterministic greedy probe. Experimental results on the standard ISPD 2005 benchmarks demonstrate that OrderPlace discovers novel ordering strategies. Compared with WireMask-EA and the state-of-the-art method EGPlace, OrderPlace reduces wirelength by 34.04\% and 14.08\%, respectively.

The higher Bruhat orders B(n,k) were introduced by Manin-Schechtman to study discriminantal hyperplane arrangements and subsequently studied by Ziegler, who connected B(n,k) to oriented matroids. In this paper, we consider the enumeration of B(n,k) and improve upon Balko's asymptotic lower and upper bounds on |B(n,k)| by a factor exponential in k. A proof of Ziegler's formula for |B(n,n-3)| is given and a bijection between a certain subset of B(n,n-4) and totally symmetric plane partitions is proved. Central to our proofs are deletion and contraction operations for the higher Bruhat orders, defined in analogy with matroids. Dual higher Bruhat orders are also introduced, and we construct isomorphisms relating the higher Bruhat orders and their duals. Additionally, weaving functions are introduced to generalize Felsner's encoding of elements in B(n,2) to all higher Bruhat orders B(n,k).

Self-supervised learning of convolutional neural networks can harness large amounts of cheap unlabeled data to train powerful feature representations. As surrogate task, we jointly address ordering of visual data in the spatial and temporal domain. The permutations of training samples, which are at the core of self-supervision by ordering, have so far been sampled randomly from a fixed preselected set. Based on deep reinforcement learning we propose a sampling policy that adapts to the state of the network, which is being trained. Therefore, new permutations are sampled according to their expected utility for updating the convolutional feature representation. Experimental evaluation on unsupervised and transfer learning tasks demonstrates competitive performance on standard benchmarks for image and video classification and nearest neighbor retrieval.

The ability of an architecture to realize permutations is quite fundamental. For example, Large Language Models need to be able to correctly copy (and perhaps rearrange) parts of the input prompt into the output. Classical universal approximation theorems guarantee the existence of parameter configurations that solve this task but offer no insights into whether gradient-based algorithms can find them. In this paper, we address this gap by focusing on two-layer fully connected feed-forward neural networks and the task of learning permutations on nonzero binary inputs. We show that in the infinite width Neural Tangent Kernel (NTK) regime, an ensemble of such networks independently trained with gradient descent on only the k standard basis vectors out of 2^k - 1 possible inputs successfully learns any fixed permutation of length k with arbitrarily high probability. By analyzing the exact training dynamics, we prove that the network's output converges to a Gaussian process whose mean captures the ground truth permutation via sign-based features. We then demonstrate how averaging these runs (an "ensemble" method) and applying a simple rounding step yields an arbitrarily accurate prediction on any possible input unseen during training. Notably, the number of models needed to achieve exact learning with high probability (which we refer to as ensemble complexity) exhibits a linearithmic dependence on the input size k for a single test input and a quadratic dependence when considering all test inputs simultaneously.

We study the conditions under which the convex relaxation of a mixed-integer linear programming formulation for ordered optimization problems, where sorting is part of the decision process, yields integral optimal solutions. Thereby solving the problem exactly in polynomial time. Our analysis identifies structural properties of the input data that influence the integrality of the relaxation. We show that incorporating ordered components introduces additional layers of combinatorial complexity that invalidate the exactness observed in classical (non-ordered) settings. In particular, for certain ordered problems such as the min--max case, the linear relaxation never recovers the integral solution. These results clarify the intrinsic hardness introduced by sorting and reveal that the strength of the relaxation depends critically on the ``proximity'' of the ordered problem to its classical counterpart: problems closer to the non-ordered case tend to admit tighter relaxations, while those further away exhibit substantially weaker behavior. Computational experiments on benchmark instances confirm the predictive value of the integrality conditions and demonstrate the practical implications of exact relaxations for ordered location problems.

We study the problem of designing models for machine learning tasks defined on sets. In contrast to traditional approach of operating on fixed dimensional vectors, we consider objective functions defined on sets that are invariant to permutations. Such problems are widespread, ranging from estimation of population statistics poczos13aistats, to anomaly detection in piezometer data of embankment dams Jung15Exploration, to cosmology Ntampaka16Dynamical,Ravanbakhsh16ICML1. Our main theorem characterizes the permutation invariant functions and provides a family of functions to which any permutation invariant objective function must belong. This family of functions has a special structure which enables us to design a deep network architecture that can operate on sets and which can be deployed on a variety of scenarios including both unsupervised and supervised learning tasks. We also derive the necessary and sufficient conditions for permutation equivariance in deep models. We demonstrate the applicability of our method on population statistic estimation, point cloud classification, set expansion, and outlier detection.

Recently, Ainsworth et al. showed that using weight matching (WM) to minimize the L_2 distance in a permutation search of model parameters effectively identifies permutations that satisfy linear mode connectivity (LMC), in which the loss along a linear path between two independently trained models with different seeds remains nearly constant. This paper provides a theoretical analysis of LMC using WM, which is crucial for understanding stochastic gradient descent's effectiveness and its application in areas like model merging. We first experimentally and theoretically show that permutations found by WM do not significantly reduce the L_2 distance between two models and the occurrence of LMC is not merely due to distance reduction by WM in itself. We then provide theoretical insights showing that permutations can change the directions of the singular vectors, but not the singular values, of the weight matrices in each layer. This finding shows that permutations found by WM mainly align the directions of singular vectors associated with large singular values across models. This alignment brings the singular vectors with large singular values, which determine the model functionality, closer between pre-merged and post-merged models, so that the post-merged model retains functionality similar to the pre-merged models, making it easy to satisfy LMC. Finally, we analyze the difference between WM and straight-through estimator (STE), a dataset-dependent permutation search method, and show that WM outperforms STE, especially when merging three or more models.

We study the set of networks, which consist of sources, sinks and neutral points, bijective to the permutations. The set of directed edges, which characterizes a network, is constructed from a polyomino or a Rothe diagram of a permutation through a Dyck tiling on a ribbon. We introduce a new combinatorial object similar to a tree-like tableau, which we call a forest. A forest is shown to give a permutation, and be bijective to a network corresponding to the inverse of the permutation. We show that the poset of networks is a finite graded lattice and admits an EL-labeling. By use of this EL-labeling, we show the lattice is supersolvable and compute the M\"obius function of an interval of the poset.

Large language models (LLMs) have accomplished remarkable reasoning performance in various domains. However, in the domain of reasoning tasks, we discover a frailty: LLMs are surprisingly brittle to the ordering of the premises, despite the fact that such ordering does not alter the underlying task. In particular, we observe that LLMs achieve the best performance when the premise order aligns with the context required in intermediate reasoning steps. For example, in deductive reasoning tasks, presenting the premises in the same order as the ground truth proof in the prompt (as opposed to random ordering) drastically increases the model's accuracy. We first examine the effect of premise ordering on deductive reasoning on a variety of LLMs, and our evaluation shows that permuting the premise order can cause a performance drop of over 30%. In addition, we release the benchmark R-GSM, based on GSM8K, to examine the ordering effect for mathematical problem-solving, and we again observe a significant drop in accuracy, relative to the original GSM8K benchmark.

Autoregressive models, such as the GPT family, use a fixed order, usually left-to-right, to generate sequences. However, this is not a necessity. In this paper, we challenge this assumption and show that by simply adding a positional encoding for the output, this order can be modulated on-the-fly per-sample which offers key advantageous properties. It allows for the sampling of and conditioning on arbitrary subsets of tokens, and it also allows sampling in one shot multiple tokens dynamically according to a rejection strategy, leading to a sub-linear number of model evaluations. We evaluate our method across various domains, including language modeling, path-solving, and aircraft vertical rate prediction, decreasing the number of steps required for generation by an order of magnitude.

We introduce a framework for automatically defining and learning deep generative models with problem-specific structure. We tackle problem domains that are more traditionally solved by algorithms such as sorting, constraint satisfaction for Sudoku, and matrix factorization. Concretely, we train diffusion models with an architecture tailored to the problem specification. This problem specification should contain a graphical model describing relationships between variables, and often benefits from explicit representation of subcomputations. Permutation invariances can also be exploited. Across a diverse set of experiments we improve the scaling relationship between problem dimension and our model's performance, in terms of both training time and final accuracy. Our code can be found at https://github.com/plai-group/gsdm.

For a sequence of random variables (X_1, X_2, ldots, X_n), n geq 1, that are independent and identically distributed with a regularly varying tail with index -alpha, alpha geq 0, we show that the contribution of the maximum term M_n triangleq max(X_1,ldots,X_n) in the sum S_n triangleq X_1 + cdots +X_n, as n to infty, decreases monotonically with alpha in stochastic ordering sense.

We propose Monte Carlo Permutation Search (MCPS), a general-purpose Monte Carlo Tree Search (MCTS) algorithm that improves upon the GRAVE algorithm. MCPS is relevant when deep reinforcement learning is not an option, or when the computing power available before play is not substantial, such as in General Game Playing, for example. The principle of MCPS is to include in the exploration term of a node the statistics on all the playouts that contain all the moves on the path from the root to the node. We extensively test MCPS on a variety of games: board games, wargame, investment game, video game and multi-player games. MCPS has better results than GRAVE in all the two-player games. It has equivalent results for multi-player games because these games are inherently balanced even when players have different strengths. We also show that using abstract codes for moves instead of exact codes can be beneficial to both MCPS and GRAVE, as they improve the permutation statistics and the AMAF statistics. We also provide a mathematical derivation of the formulas used for weighting the three sources of statistics. These formulas are an improvement on the GRAVE formula since they no longer use the bias hyperparameter of GRAVE. Moreover, MCPS is not sensitive to the ref hyperparameter.

We consider the challenging task of training models for image-to-video deblurring, which aims to recover a sequence of sharp images corresponding to a given blurry image input. A critical issue disturbing the training of an image-to-video model is the ambiguity of the frame ordering since both the forward and backward sequences are plausible solutions. This paper proposes an effective self-supervised ordering scheme that allows training high-quality image-to-video deblurring models. Unlike previous methods that rely on order-invariant losses, we assign an explicit order for each video sequence, thus avoiding the order-ambiguity issue. Specifically, we map each video sequence to a vector in a latent high-dimensional space so that there exists a hyperplane such that for every video sequence, the vectors extracted from it and its reversed sequence are on different sides of the hyperplane. The side of the vectors will be used to define the order of the corresponding sequence. Last but not least, we propose a real-image dataset for the image-to-video deblurring problem that covers a variety of popular domains, including face, hand, and street. Extensive experimental results confirm the effectiveness of our method. Code and data are available at https://github.com/VinAIResearch/HyperCUT.git

We study the problem of detecting the correlation between two Gaussian databases XinR^{ntimes d} and Y^{ntimes d}, each composed of n users with d features. This problem is relevant in the analysis of social media, computational biology, etc. We formulate this as a hypothesis testing problem: under the null hypothesis, these two databases are statistically independent. Under the alternative, however, there exists an unknown permutation sigma over the set of n users (or, row permutation), such that X is rho-correlated with Y^sigma, a permuted version of Y. We determine sharp thresholds at which optimal testing exhibits a phase transition, depending on the asymptotic regime of n and d. Specifically, we prove that if rho^2dto0, as dtoinfty, then weak detection (performing slightly better than random guessing) is statistically impossible, irrespectively of the value of n. This compliments the performance of a simple test that thresholds the sum all entries of X^TY. Furthermore, when d is fixed, we prove that strong detection (vanishing error probability) is impossible for any rho<rho^star, where rho^star is an explicit function of d, while weak detection is again impossible as long as rho^2dto0. These results close significant gaps in current recent related studies.

In this technical note, we study the problem of inverse permutation learning in decoder-only transformers. Given a permutation and a string to which that permutation has been applied, the model is tasked with producing the original (``canonical'') string. We argue that this task models a natural robustness property across a variety of reasoning tasks, including long-context retrieval, multiple choice QA and in-context learning. Our primary contribution is an impossibility result: we show that an arbitrary depth, decoder-only transformer cannot learn this task. This result concerns the expressive capacity of decoder-only transformer models and is agnostic to training dynamics or sample complexity. We give a pair of alternative constructions under which inverse permutation learning is feasible. The first of these highlights the fundamental role of the causal attention mask, and reveals a gap between the expressivity of encoder-decoder transformers and the more popular decoder-only architecture. The latter result is more surprising: we show that simply padding the input with ``scratch tokens" yields a construction under which inverse permutation learning is possible. We conjecture that this may suggest an alternative mechanism by which chain-of-thought prompting or, more generally, intermediate ``thinking'' tokens can enable reasoning in large language models, even when these tokens encode no meaningful semantic information (e.g., the results of intermediate computations).

Let L be a finite lattice. An Ungar move sends an element xin L to the meet of {x}cup T, where T is a subset of the set of elements covered by x. We introduce the following Ungar game. Starting at the top element of L, two players -- Atniss and Eeta -- take turns making nontrivial Ungar moves; the first player who cannot do so loses the game. Atniss plays first. We say L is an Atniss win (respectively, Eeta win) if Atniss (respectively, Eeta) has a winning strategy in the Ungar game on L. We first prove that the number of principal order ideals in the weak order on S_n that are Eeta wins is O(0.95586^nn!). We then consider a broad class of intervals in Young's lattice that includes all principal order ideals, and we characterize the Eeta wins in this class; we deduce precise enumerative results concerning order ideals in rectangles and type-A root posets. We also characterize and enumerate principal order ideals in Tamari lattices that are Eeta wins. Finally, we conclude with some open problems and a short discussion of the computational complexity of Ungar games.

Sequences have become first class citizens in supervised learning thanks to the resurgence of recurrent neural networks. Many complex tasks that require mapping from or to a sequence of observations can now be formulated with the sequence-to-sequence (seq2seq) framework which employs the chain rule to efficiently represent the joint probability of sequences. In many cases, however, variable sized inputs and/or outputs might not be naturally expressed as sequences. For instance, it is not clear how to input a set of numbers into a model where the task is to sort them; similarly, we do not know how to organize outputs when they correspond to random variables and the task is to model their unknown joint probability. In this paper, we first show using various examples that the order in which we organize input and/or output data matters significantly when learning an underlying model. We then discuss an extension of the seq2seq framework that goes beyond sequences and handles input sets in a principled way. In addition, we propose a loss which, by searching over possible orders during training, deals with the lack of structure of output sets. We show empirical evidence of our claims regarding ordering, and on the modifications to the seq2seq framework on benchmark language modeling and parsing tasks, as well as two artificial tasks -- sorting numbers and estimating the joint probability of unknown graphical models.

Reinforcement learning from verifiable rewards (e.g. GRPO) is the engine behind today's reasoning models, yet it grades only the final answer. On hard problems this trains models to write more rather than to think better, since the trace itself is never graded and no label for good thinking exists. We introduce Agon, which makes two competing models each other's graders. Both attempt the same problem; in alternating roles, one drafts a solution and the other reads it while solving, and each is rewarded for out-solving the other. To win, a model must out-reason a rival that has seen its work, so reasoning is judged implicitly during training, with no process labels and no reward model. Because both models are optimized, each faces a progressively stronger rival, which single-model RL cannot provide. The two need only be comparably strong and behaviorally different. At inference the pair deploys as it trains, a two-stage cascade in which one model drafts and the other answers after reading the draft. On the hard split of DeepMath with Qwen3, this doubles GRPO's pass@1, roughly eight times the gain of an untrained Mixture-of-Agents pass over the same base. The ordering replicates on competitive-programming code and across model families (Qwen3.5, Gemma 4). For now the models talk in text; the next step is to let them reason together in latent space.

We introduce refutationally complete superposition calculi for intentional and extensional clausal lambda-free higher-order logic, two formalisms that allow partial application and applied variables. The calculi are parameterized by a term order that need not be fully monotonic, making it possible to employ the lambda-free higher-order lexicographic path and Knuth-Bendix orders. We implemented the calculi in the Zipperposition prover and evaluated them on Isabelle/HOL and TPTP benchmarks. They appear promising as a stepping stone towards complete, highly efficient automatic theorem provers for full higher-order logic.

Revolutionizing the field of deep learning, Transformer-based models have achieved remarkable performance in many tasks. Recent research has recognized these models are robust to shuffling but are limited to inter-token permutation in the forward propagation. In this work, we propose our definition of permutation equivariance, a broader concept covering both inter- and intra- token permutation in the forward and backward propagation of neural networks. We rigorously proved that such permutation equivariance property can be satisfied on most vanilla Transformer-based models with almost no adaptation. We examine the property over a range of state-of-the-art models including ViT, Bert, GPT, and others, with experimental validations. Further, as a proof-of-concept, we explore how real-world applications including privacy-enhancing split learning, and model authorization, could exploit the permutation equivariance property, which implicates wider, intriguing application scenarios.

Autoregressive patch-based image generation has recently shown competitive results in terms of image quality and scalability. It can also be easily integrated and scaled within Vision-Language models. Nevertheless, autoregressive models require a defined order for patch generation. While a natural order based on the dictation of the words makes sense for text generation, there is no inherent generation order that exists for image generation. Traditionally, a raster-scan order (from top-left to bottom-right) guides autoregressive image generation models. In this paper, we argue that this order is suboptimal, as it fails to respect the causality of the image content: for instance, when conditioned on a visual description of a sunset, an autoregressive model may generate clouds before the sun, even though the color of clouds should depend on the color of the sun and not the inverse. In this work, we show that first by training a model to generate patches in any-given-order, we can infer both the content and the location (order) of each patch during generation. Secondly, we use these extracted orders to finetune the any-given-order model to produce better-quality images. Through our experiments, we show on two datasets that this new generation method produces better images than the traditional raster-scan approach, with similar training costs and no extra annotations.

In statistics, independent, identically distributed random samples do not carry a natural ordering, and their statistics are typically invariant with respect to permutations of their order. Thus, an n-sample in a space M can be considered as an element of the quotient space of M^n modulo the permutation group. The present paper takes this definition of sample space and the related concept of orbit types as a starting point for developing a geometric perspective on statistics. We aim at deriving a general mathematical setting for studying the behavior of empirical and population means in spaces ranging from smooth Riemannian manifolds to general stratified spaces. We fully describe the orbifold and path-metric structure of the sample space when M is a manifold or path-metric space, respectively. These results are non-trivial even when M is Euclidean. We show that the infinite sample space exists in a Gromov-Hausdorff type sense and coincides with the Wasserstein space of probability distributions on M. We exhibit Fr\'echet means and k-means as metric projections onto 1-skeleta or k-skeleta in Wasserstein space, and we define a new and more general notion of polymeans. This geometric characterization via metric projections applies equally to sample and population means, and we use it to establish asymptotic properties of polymeans such as consistency and asymptotic normality.

A key to knowledge graph embedding (KGE) is to choose a proper representation space, e.g., point-wise Euclidean space and complex vector space. In this paper, we propose a unified perspective of embedding and introduce uncertainty into KGE from the view of group theory. Our model can incorporate existing models (i.e., generality), ensure the computation is tractable (i.e., efficiency) and enjoy the expressive power of complex random variables (i.e., expressiveness). The core idea is that we embed entities/relations as elements of a symmetric group, i.e., permutations of a set. Permutations of different sets can reflect different properties of embedding. And the group operation of symmetric groups is easy to compute. In specific, we show that the embedding of many existing models, point vectors, can be seen as elements of a symmetric group. To reflect uncertainty, we first embed entities/relations as permutations of a set of random variables. A permutation can transform a simple random variable into a complex random variable for greater expressiveness, called a normalizing flow. We then define scoring functions by measuring the similarity of two normalizing flows, namely NFE. We construct several instantiating models and prove that they are able to learn logical rules. Experimental results demonstrate the effectiveness of introducing uncertainty and our model. The code is available at https://github.com/changyi7231/NFE.

Multi-hop reasoning, which requires multi-step reasoning based on the supporting documents within a given context, remains challenging for large language models (LLMs). LLMs often struggle to filter out irrelevant documents within the context, and their performance is sensitive to the absolute position of supporting documents within that context. In this paper, we identify an additional challenge: LLMs' performance is also sensitive to the order, relative position, in which the supporting documents are presented. We refer to this as the misordered context problem. To address this issue, based on the theoretical approach, we propose a simple yet effective method called context repetition (CoRe), which involves prompting the model by repeatedly presenting the context. This ensures that certain contiguous reasoning segments within supporting documents are presented in the optimal order, effectively guiding the model's reasoning in the appropriate direction. Applying CoRe, we improve the F1 score by up to 30%p on multi-hop QA tasks and increase accuracy by up to 70%p on a synthetic task. Additionally, CoRe helps mitigate the well-known "lost-in-the-middle" problem in LLMs and can be effectively combined with retrieval-based approaches utilizing Chain-of-Thought (CoT) reasoning.

Neural network parameterizations exhibit inherent symmetries that yield multiple equivalent minima within the loss landscape. Scale Graph Metanetworks (ScaleGMNs) explicitly leverage these symmetries by proposing an architecture equivariant to both permutation and parameter scaling transformations. Previous work by Ainsworth et al. (2023) addressed permutation symmetries through a computationally intensive combinatorial assignment problem, demonstrating that leveraging permutation symmetries alone can map networks into a shared loss basin. In this work, we extend their approach by also incorporating scaling symmetries, presenting an autoencoder framework utilizing ScaleGMNs as invariant encoders. Experimental results demonstrate that our method aligns Implicit Neural Representations (INRs) and Convolutional Neural Networks (CNNs) under both permutation and scaling symmetries without explicitly solving the assignment problem. This approach ensures that similar networks naturally converge within the same basin, facilitating model merging, i.e., smooth linear interpolation while avoiding regions of high loss. The code is publicly available on our GitHub repository.

This paper presents RevOrder, a novel technique aimed at improving arithmetic operations in large language models (LLMs) by reversing the output digits in addition, subtraction, and n-digit by 1-digit (nD by 1D) multiplication tasks. Our method significantly reduces the Count of Sequential Intermediate Digits (CSID) to O(1), a new metric we introduce to assess equation complexity. Through comprehensive testing, RevOrder not only achieves perfect accuracy in basic arithmetic operations but also substantially boosts LLM performance in division tasks, particularly with large numbers where traditional models struggle. Implementation of RevOrder is cost-effective for both training and inference phases. Moreover, applying RevOrder to fine-tune the LLaMA2-7B model on the GSM8K math task results in a considerable improvement, reducing equation calculation errors by 46% and increasing overall scores from 41.6 to 44.4.

While plan-and-infill decoding in Masked Diffusion Models (MDMs) shows promise for mathematical and code reasoning, performance remains highly sensitive to slot infilling order, often yielding substantial output variance. We introduce McDiffuSE, a framework that formulates slot selection as decision making and optimises infilling orders through Monte Carlo Tree Search (MCTS). McDiffuSE uses look-ahead simulations to evaluate partial completions before commitment, systematically exploring the combinatorial space of generation orders. Experiments show an average improvement of 3.2% over autoregressive baselines and 8.0% over baseline plan-and-infill, with notable gains of 19.5% on MBPP and 4.9% on MATH500. Our analysis reveals that while McDiffuSE predominantly follows sequential ordering, incorporating non-sequential generation is essential for maximising performance. We observe that larger exploration constants, rather than increased simulations, are necessary to overcome model confidence biases and discover effective orderings. These findings establish MCTS-based planning as an effective approach for enhancing generation quality in MDMs.

We discuss the numerical solution of initial value problems for varepsilon^2,varphi''+a(x),varphi=0 in the highly oscillatory regime, i.e., with a(x)>0 and 0<varepsilonll 1. We analyze and implement an approximate solution based on the well-known WKB-ansatz. The resulting approximation error is of magnitude O(varepsilon^{N}) where N refers to the truncation order of the underlying asymptotic series. When the optimal truncation order N_{opt} is chosen, the error behaves like O(varepsilon^{-2}exp(-cvarepsilon^{-1})) with some c>0.

The quality of training data impacts the performance of pre-trained large language models (LMs). Given a fixed budget of tokens, we study how to best select data that leads to good downstream model performance across tasks. We develop a new framework based on a simple hypothesis: just as humans acquire interdependent skills in a deliberate order, language models also follow a natural order when learning a set of skills from their training data. If such an order exists, it can be utilized for improved understanding of LMs and for data-efficient training. Using this intuition, our framework formalizes the notion of a skill and of an ordered set of skills in terms of the associated data. First, using both synthetic and real data, we demonstrate that these ordered skill sets exist, and that their existence enables more advanced skills to be learned with less data when we train on their prerequisite skills. Second, using our proposed framework, we introduce an online data sampling algorithm, Skill-It, over mixtures of skills for both continual pre-training and fine-tuning regimes, where the objective is to efficiently learn multiple skills in the former and an individual skill in the latter. On the LEGO synthetic in the continual pre-training setting, Skill-It obtains 36.5 points higher accuracy than random sampling. On the Natural Instructions dataset in the fine-tuning setting, Skill-It reduces the validation loss on the target skill by 13.6% versus training on data associated with the target skill itself. We apply our skills framework on the recent RedPajama dataset to continually pre-train a 3B-parameter LM, achieving higher accuracy on the LM Evaluation Harness with 1B tokens than the baseline approach of sampling uniformly over data sources with 3B tokens.

Automatic Differentiation (AD) has become a dominant technique in ML. AD frameworks have first been implemented for imperative languages using tapes. Meanwhile, functional implementations of AD have been developed, often based on dual numbers, which are close to the formal specification of differentiation and hence easier to prove correct. But these papers have focussed on correctness not efficiency. Recently, it was shown how an approach using dual numbers could be made efficient through the right optimizations. Optimizations are highly dependent on order, as one optimization can enable another. It can therefore be useful to have fine-grained control over the scheduling of optimizations. One method expresses compiler optimizations as rewrite rules, whose application can be combined and controlled using strategy languages. Previous work describes the use of term rewriting and strategies to generate high-performance code in a compiler for a functional language. In this work, we implement dual numbers AD in a functional array programming language using rewrite rules and strategy combinators for optimization. We aim to combine the elegance of differentiation using dual numbers with a succinct expression of the optimization schedule using a strategy language. We give preliminary evidence suggesting the viability of the approach on a micro-benchmark.

A challenging problem in many modern machine learning tasks is to process weight-space features, i.e., to transform or extract information from the weights and gradients of a neural network. Recent works have developed promising weight-space models that are equivariant to the permutation symmetries of simple feedforward networks. However, they are not applicable to general architectures, since the permutation symmetries of a weight space can be complicated by recurrence or residual connections. This work proposes an algorithm that automatically constructs permutation equivariant models, which we refer to as universal neural functionals (UNFs), for any weight space. Among other applications, we demonstrate how UNFs can be substituted into existing learned optimizer designs, and find promising improvements over prior methods when optimizing small image classifiers and language models. Our results suggest that learned optimizers can benefit from considering the (symmetry) structure of the weight space they optimize. We open-source our library for constructing UNFs at https://github.com/AllanYangZhou/universal_neural_functional.

Pairwise Ranking Prompting (PRP) elicits pairwise preference judgments from an LLM, which are then aggregated into a ranking, usually via classical sorting algorithms. However, judgments are noisy, order-sensitive, and sometimes intransitive, so sorting assumptions do not match the setting. Because sorting aims to recover a full permutation, truncating it to meet a call budget does not produce a dependable top-K. We thus reframe PRP reranking as active learning from noisy pairwise comparisons and show that active rankers are drop-in replacements that improve NDCG@10 per call in the call-constrained regime. Our noise-robust framework also introduces a randomized-direction oracle that uses a single LLM call per pair. This approach converts systematic position bias into zero-mean noise, enabling unbiased aggregate ranking without the cost of bidirectional calls.

We investigate how to make small tabular foundation models effective for High-Dimensional, Low-Sample Size (HDLSS) tabular prediction without retraining large backbones. We introduce Graph-guided Ordering with Local Refinement (GO-LR), show its equivalence to weighted Minimum Linear Arrangement, and interpret the practical solver as a TSP-path-style surrogate. We propose GOTabPFN,which builds on GO-LR, and a Neuro-Inspired Subunit Compression (NSC) unit to pool locally adjacent ordered features into meta-features, yielding a compact representation that makes TabPFN-style prediction practical in HDLSS regimes. Across tabular benchmarks, GOTabPFN improves stability and accuracy under tight token budgets.

The last few years have seen significant experimental progress in characterizing the copper-based hole-doped high temperature superconductors in the regime of low hole density, p. Quantum oscillations, NMR, X-ray, and STM experiments have shed much light on the nature of the ordering at low temperatures. We review evidence that the order parameter in the non-Lanthanum-based cuprates is a d-form factor density-wave. This novel order acts as an unexpected window into the electronic structure of the pseudogap phase at higher temperatures in zero field: we argue in favor of a `fractionalized Fermi liquid' (FL*) with 4 pockets of spin S=1/2, charge +e fermions enclosing an area specified by p.

Recent investigations on the set of commutators between the elements of a finite group having relatively prime orders have prompt us to propose a variant of the Ore conjecture: For every finite non-abelian simple group and for every gin G, there exist x,yin G with g=[y,x] and with the order of x relatively prime to the order of y. In this note we present some evidence towards the veracity of this conjecture by proving it for alternating groups and some sporadic simple groups.