Permutation-Invariant Learning

Decision models often score a variable-sized set of candidate actions encoded in a single sequence. This setting is increasingly relevant for System 1 components inside generative systems, where candidates may be proposed or ordered differently across runs. Standard causal cross-encoding is expressive, but it can make a candidate's score depend on serialization order rather than on the underlying decision problem. We introduce candidate-independent block-causal attention, which preserves causal computation within the shared context and each candidate while blocking cross-candidate information flow and resetting candidate positions. We compare this architecture with standard causal attention and complementary invariant baselines across Gemma 3 1B, Qwen3 1.7B, and Qwen3 4B backbones. Candidate-independent attention consistently reduces permutation sensitivity while retaining competitive decision quality; ablations indicate that candidate isolation is the primary source of the effect, with position resetting completing the intended symmetry. A larger Qwen3-4B study further examines the behavior of the proposed architecture with substantially more training data. Code is available at the \href{https://github.com/guyAmit/ci-decision-models}{\textcolor{blue}{project repository}}, and the \href{https://huggingface.co/Guy-Amit/qwen3-4b-ci-decision-4096-poc}{\textcolor{blue}{Qwen3-4B model artifact}} is available on Hugging Face.

Audio decisions often depend on evidence that a transcript does not preserve, while their probability estimates can depend on how answer options are ordered. AudioJev maps a waveform, question and supplied alternatives directly to a candidate distribution through one shared full-parameter model. We define order calibration as preserving an answer's probability under meaning-preserving option permutations. Random-derangement SKL training pairs each question with a reordered view in which every alternative changes position, supervises both answers, and aligns the two distributions before applying a symmetric KL penalty. Inference retains a single candidate-scoring forward, with no calibration head or order ensemble. Across three training seeds, AudioJev reaches 68.88%/55.33% mean accuracy on complete MMAU/MMAR and reduces random-order SKL by 43.8%/60.5% relative to the single-view removal ablation. The same model handles intent, environmental sound, note properties, speech activity and conversational transitions. Paired removal ablations and multi-order evaluation measure predictive accuracy and probability stability together, establishing a direct audio interface whose calibration objective acts on candidate meaning rather than presentation position. Inference code and model weights are available at https://github.com/SihanLv/AudioJev-Inference and https://huggingface.co/shlv/AudioJev.

Physiological recordings often contain nonstationary oscillatory components whose number and dynamics vary across signals. Amplitude- and frequency-modulated (AM-FM) representations are well suited to characterizing such dynamics and have shown broad utility in biomedical signal analysis. Recent approaches have incorporated neural networks to learn mode decomposition patterns from data, but component cardinality is often predefined or determined through separate stopping or selection mechanisms. We propose a permutation-invariant neural framework for variable-cardinality AM-FM mode decomposition (PI-AMFM). PI-AMFM combines a multiscale temporal encoder, Mamba backbone, and component-presence estimation, with permutation-invariant Hungarian matching during training. On synthetic AM-FM signals, PI-AMFM achieved lower decomposition, instantaneous-frequency, reconstruction, and mode-count errors than the compared methods while preserving the overall trajectory pattern in a crossing-chirp example. On photoplethysmographic recordings, recovered modes captured cardiac and respiratory dynamics despite training only on synthetic signals. These results support the feasibility of PI-AMFM for variable-cardinality decomposition of nonstationary biomedical signals.

Feature transformation improves predictive performance on tabular data by constructing informative abstractions from raw features. Recent generative approaches encode transformation knowledge into continuous embedding spaces for efficient exploration of candidate strategies, but face three key limitations: (1) overlooking hierarchical relationships between low-level features, operations, and high-level abstractions; (2) enforcing order-sensitive embeddings on inherently permutation-invariant transformation sequences, thereby introducing systematic bias; and (3) relying on gradient-based search, which is ill-suited to non-convex transformation spaces. We propose a framework with two complementary components. First, a permutation-invariant hierarchical module captures interactions across features, operations, and abstraction levels, with a self-attention pooling mechanism that maps semantically equivalent structures to consistent embeddings aligned with downstream performance. Second, a policy-guided multi-objective reinforcement learning strategy initializes the search from empirically strong seeds and jointly optimizes predictive accuracy and transformation efficiency. Extensive experiments on diverse tabular benchmarks demonstrate the effectiveness and robustness of our framework against strong baselines. Our code and data are publicly available at: https://github.com/RayLiu1103/PHER.

Multi-head attention combines an input-dependent softmax routing with an input-independent linear value projection, so the per-sample operator mapping aggregated values to outputs is the same for every input set. We study the consequences of this asymmetry for permutation-invariant set targets. We introduce the Transformation Degrees of Freedom (TDOF) of a target operator, a complexity measure counting the input-dependent directions an exact representation requires, and present a depth-separation analysis showing that context-rigid attention needs depth proportional to the target's TDOF, whereas a single layer with a context-adaptive value family can represent the same target. Building on this analysis, we propose Matrix Zonotopic Attention (MZAttn), which replaces the fixed value projection with a context-adaptive matrix-zonotope family: a centre matrix plus a sum of generator matrices weighted by input-dependent gates. The construction reduces to standard multi-head attention at initialisation, preserves permutation equivariance, and admits a data-driven reachability interpretation. Experiments on a range of set-prediction tasks are consistent with the TDOF prediction that the architectural advantage is selective: it appears on targets that depend on the input set in a high-rank, sparsely combinatorial way, and is small on aggregate-statistic targets where parameter-matched standard attention is already competitive.

Tensorial multi-view clustering (TMC) has achieved strong performance due to its ability to capture high-order correlations across multiple views. Most existing t-SVD-based TMC frameworks apply the Fast Fourier Transform (FFT) along the sample mode to impose frequency-domain low-rank constraints. However, we reveal that this widely adopted design critically relies on an implicit ``periodicity assumption'' induced by the sample arrangement. When samples are ordered by class, neighboring indices tend to be semantically similar, creating artificial local continuity along the sample mode and a favorable spectral structure for FFT-based low-rank regularization. Once this ordering is removed by random permutation, existing t-SVD-based TMC methods suffer severe performance degradation. This strong sensitivity to class ordering conflicts with the permutation-invariant nature of clustering and indicates that part of the reported performance may be attributed to a privileged sample arrangement rather than genuine high-order structure modeling. In this paper, we systematically investigate this phenomenon and its underlying algebraic and spectral mechanisms. To address this fundamental flaw, we further propose a graph-spectral low-rank tensor learning framework based on the Graph Fourier Transform (GFT), which replaces the fixed Fourier basis along the sample mode with a data-driven graph spectral basis, thereby capturing the intrinsic manifold structure without relying on a particular sample ordering. Moreover, we develop an anchor-based variant to address large-scale datasets efficiently. Extensive experiments on various benchmarks validate our findings and demonstrate the competitive or superior performance of the proposed methods compared with state-of-the-art TMC approaches.

Chest radiograph datasets often combine multiple images with Clinical Indication, Findings, and Impression, although these inputs are produced at different stages of care. We evaluated 15,000 ReXGradient-160K studies with two readable images and five CheXbert-derived report observations. Frozen DenseNet-121 and Bio+ClinicalBERT encoders were used to compare image-only, Indication-only, fixed-order multimodal, random-swap, DeepSets, and SectionGuard-MI models. Findings and Impression were evaluated only as post-hoc leakage controls. Models were trained with five seeds, and public-test uncertainty was estimated with 2,000 patient-cluster bootstrap replicates. Under U-Ones, macro AUROC was 0.643 for the primary image, 0.694 for two images, 0.749 for Indication, and 0.780 for ordinary two-image-plus-Indication fusion. SectionGuard-MI achieved AUROC 0.783 and AUPRC 0.260. Relative to ordinary fusion, its paired AUROC difference was 0.0031 (95% CI, -0.0042 to 0.0104; adjusted p=0.374), while its AUPRC difference was 0.0289 (95% CI, 0.0095 to 0.0413; adjusted p=0.004). DeepSets had the highest prospective AUROC point estimate (0.787), and random-swap fusion had the highest prospective AUPRC point estimate (0.265) with better calibration than SectionGuard-MI. Full report text alone reached AUROC 0.979 and AUPRC 0.836; AUROC remained above 0.973 after exact or expanded masking. These results show that prospective Indication is strongly associated with report-derived targets, permutation-aware fusion is competitive, and post-hoc report text creates substantial report-label circularity.

Accurately representing atmospheric aerosol populations is essential for simulating aerosol-cloud interactions, radiative forcing, and ice nucleation, yet existing reduced schemes impose structural assumptions that limit their ability to capture composition diversity and mixing state. Machine-learning approaches offer more flexible representations, but standard autoencoders do not preserve the mathematical structure of aerosol populations and therefore cannot support physically meaningful process operators. We introduce AeroMELD (Aerosol Measure Embedding for Latent Dynamics), a mathematically grounded framework for constructing low-dimensional latent variables that retain this structure. We show that any permutation-invariant linear encoder must take a scale-shape decomposition, with total number concentration represented explicitly and latent shape given by a barycentric combination of per-particle embeddings. This aggregated latent state retains the diagnostic expressiveness of a Deep Sets model by moving the nonlinear post-aggregation stage into the learned diagnostic map while preserving latent linearity. Using particle-resolved data as ground truth, we encode weighted particle populations directly rather than binned aerosol states; size-resolved mass and number distributions serve only as diagnostic targets and visual summaries. The latent space accurately reconstructs these distributions, CCN spectra, optical coefficients, and immersion-freezing behavior while preserving the linear population structure needed for hybrid ML-physics models. Although the experiments focus on diagnostic reconstruction, the embedding is designed so that emissions and mixing can be represented exactly and nonlinear microphysical processes learned in a controlled latent space. This work establishes a foundation for learning aerosol-process evolution directly in latent space.

Modern LLM workflows move coordinate-indexed objects across checkpoints: steering vectors, sparse autoencoders, top-$k$ neuron sets, attribution lists, and merge alignments. This is only well posed after fixing the model's residual-stream gauge, which we show is architecture-dependent: LayerNorm residual charts have permutation gauge $S_d$ (up to a global sign flip), while RMSNorm charts with generic per-channel gain have signed-permutation gauge $B_d = S_d \ltimes \{\pm 1\}^d$. Permutation-only alignment is therefore symmetry-incomplete for RMSNorm models. We introduce sign-marginalized Hungarian matching and prove a sharp failure mode: with decorrelated coordinates, raw signed-correlation matching has a structural permutation-accuracy ceiling at the positive-sign fraction of the true gauge, which sign-marginalization removes. We then make coordinate-preserving transport, not function-level merging, the primary object: composing saved-checkpoint local $B_d$ gauges along same-base fine-tuning trajectories recovers 91.1% of cross-run coordinates at 1500 steps versus 60.3% for endpoint matching, and the gain is not explained by merely routing through the base. The recovered gauge transfers tools that permutation-only alignment breaks: TinyLlama SAE reconstruction has NMSE 0.004 under $B_d$ versus 1.08 under $S_d$; Qwen sentiment steering preserves 95.8% of its effect versus 17.2%; refusal steering reverses sign under $S_d$; coordinate-preserving merges behave the same way. The same covariance governs stateful training: signed transport of AdamW state preserves the resumed trajectory, while permutation-only state follows a different one from a functionally identical checkpoint. Finally, gauge-sweep audits show index-level interpretability claims are reproducible only relative to an explicit gauge.

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 ($\textbf{PI-FT}$), 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. $\textbf{DevDataBench}$ 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 $\texttt{text-embedding-3-large}$ (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.

Multi-task table recognition jointly addresses table structure prediction, cell localization, and cell content recognition within a unified framework. Existing approaches often rely on autoregressive decoders to generate table structures and reuse their hidden states for cell localization and content recognition. This autoregressive generation process can make cell representations order-dependent, degrading global consistency across cells. This paper proposes a structural refinement module that produces order-independent cell features through non-causal attention. This design enables parallel inference of cell contents while conditioning each cell on global context encoded in the refined features. Experiments on two large datasets demonstrate consistent gains in cell localization and end-to-end recognition, while reducing overall inference time by around threefold.

Multi-pitch estimation (MPE) typically predicts which pitches are active in a mixture, but not which instrument or source produced them. This paper investigates a lightweight slot-attention framework for multi-instrument MPE (MI-MPE), where a mixture CQT is mapped to an unordered set of source-like pitch maps. The model uses permutation-invariant Hungarian matching to avoid fixed output semantics and treats the number of slots as an upper bound on the number of active sources. We further study two modular extensions: a self-supervised timbre encoder that provides training-time targets for slot-level timbre embeddings, and a polyphony branch that regularizes the pitch density of mixture- and slot-level predictions. Experiments show that Hungarian matching substantially improves instrument family decomposition on URMP. Stem-level prediction remains more challenging: timbre and polyphony supervision improve selected configurations, but do not consistently resolve source assignment. The results suggest that slot-based architectures are a promising direction for source-aware MPE, while highlighting the need to couple auxiliary musical cues to slot identity more carefully.

Radio frequency spectrum awareness requires the ability to detect, localize, and characterize emitters in dense and contested wireless environments. In this work, we propose a task-oriented distributed compression framework for joint multi-emitter localization and characterization using spatially distributed receivers. Each receiver observes a short window of complex IQ samples, converts the observation to a time--frequency representation, and encodes it into a compact latent vector. A central fusion decoder combines the receiver latents to estimate an unordered set of active emitters, including their locations, center-frequency offsets, occupied bandwidths, and waveform families. A permutation-invariant training objective is used to handle the arbitrary ordering of emitters and predictions. Experiments on synthetic multi-emitter scenes with spectral overlap show that even extremely compact receiver-side representations can preserve useful information for emitter counting and waveform-family estimation. However, accurate localization and spectral-parameter regression require larger latent dimensions. Increasing the receiver latent dimension from $d_{\mathrm{rx}}=1$ to $d_{\mathrm{rx}}=16$ provides the largest improvement, while further increasing to $d_{\mathrm{rx}}=64$ gives smaller gains. These results demonstrate the potential of learned task-oriented compression for communication-efficient distributed spectrum awareness.

Accurately modeling the macroscopic dynamics of high-dimensional microscopic systems is of broad interest across the sciences. Many data-driven approaches learn a low-dimensional latent state through an autoencoder trained for pointwise input reconstruction. These methods typically assume a fixed ordering of microscopic degrees of freedom in the input. However, in many settings, such as particle systems, the microscopic state is inherently unordered. This motivates an autoencoder framework that learns permutation-invariant latent representations. To this end, we adopt a permutation-invariant encoder and design the decoder to reconstruct the mass distribution centered at the observed points rather than per-sample reconstruction. We then jointly learn the macroscopic dynamics of the observables together with the latent states. We demonstrate the effectiveness and robustness of the proposed method across a range of microscopic settings, including learning the energy dynamics in interacting particle systems, predicting mixing dynamics in Lennard-Jones fluids, and modeling the stretching dynamics from video data of polymers moving in an elongational force field.

We present Orbit-Space Geometric Probability Paths (OGPP), a particle-native flow-matching framework for generative modeling of particle systems. OGPP is motivated by two insights: (i) particles are defined up to permutation symmetries, so anonymous indexing inflates per-index target variance and yields curved, hard-to-learn flows; and (ii) particles live in physical space, so the flow terminal velocity has physical meaning and can encode geometric attributes, e.g., surface normals. OGPP instantiates three key components: (1) orbit-space canonicalization of the probability-path terminal endpoint, (2) particle index embeddings for role specialization, and (3) geometric probability paths with arc-length-aware terminal velocities that generate normals as a byproduct of the flow. We evaluate OGPP on minimal-surface benchmarks, where it reduces metric error by up to two orders of magnitude in a single inference step; on ShapeNet, where it matches the state of the art with 5x fewer steps and reaches airplane EMD comparable to DiT-3D with 26x fewer parameters and 5x fewer steps; and on single-shape encoding, where it produces normals and reconstructions competitive with 6D generators while operating entirely in 3D.

Deep learning has been widely adopted for WiFi CSI-based human activity recognition (HAR) due to its ability to learn spatio-temporal features in a privacy-preserving and cost-effective manner. However, DL-based models generalize poorly across environments, a challenge amplified in multi-user settings where overlapping activities cause CSI entanglement and domain shifts. Practical deployments often limit access to labeled source data due to privacy constraints, motivating source-free adaptation using only unlabeled target-domain CSI and a pre-trained source model. In this paper, we propose MU-SHOT-Fi, a source-free unsupervised domain adaptation framework for single- and multi-user Wi-Fi sensing. MU-SHOT-Fi employs permutation-invariant set prediction with Hungarian matching during source training, followed by frozen-classifier backbone adaptation in the target domain. To enable stable adaptation without labels, we introduce occupancy-weighted information maximization that prevents model collapse by focusing diversity regularization on likely-occupied slots while excluding the dominant class from marginal entropy. Additionally, we employ binary rotation prediction as spatial self-supervision that exploits CSI frequency-time structure to learn domain-invariant features. For single-user scenarios, we introduce SU-SHOT-Fi by replacing occupancy weighting with standard information maximization and incorporating contrastive predictive coding to exploit temporal consistency. Extensive experiments on the WiMANS and Widar 3.0 datasets across cross-environment, cross-frequency, cross-orientation, and combined domain shifts demonstrate that MU-SHOT-Fi effectively recovers multi-user exact-activity classification performance under large domain shifts while maintaining accurate occupancy estimation and preventing collapse toward dominant classes.

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 $\textit{invariant}$ 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/

Many learning problems are organized by group symmetries. While invariance is often imposed through architectures or group averaging, we ask when it can emerge from training on a finite random subset of an orbit. We study this question in classical Hopfield networks, where strict memorization can be expressed as a linear margin problem. Reparameterizing minimization of energy flow (MEF) as an exponential loss connects gradient descent to the corresponding minimum-norm hard-margin memorizer. Our main result shows that, for independent uniform samples from any finite permutation orbit, the exact sample hard-margin support vector machine (HSVM) concentrates exponentially around the invariant full-orbit HSVM. Consequently, an orbit-size-independent polynomial number of samples suffices both for approximate parameter invariance and for simultaneous memorization of every orbit element; directional convergence transfers this conclusion asymptotically to MEF gradient descent. For graph-isomorphism orbits, we characterize the invariant parameters as a three-dimensional subspace and show that every such orbit is memorizable. For cliques of fixed linear density, additional symmetry sharpens the uniform memorization bound to $O(v^4\log(1/δ))$, exponentially smaller than the orbit size. Together with experiments across several learning rules, these results give a finite-sample account of how optimization bias can recover symmetry from partial group-structured data.

Tabular prediction has long been dominated by gradient-boosted decision trees and specialized deep tabular models, while large language models (LLMs) remain difficult to make competitive despite their cross-task adaptability and transparent reasoning traces. We address this gap by incorporating tabular structural priors into LLM post-training. Specifically, we propose Permutation Relative Policy Optimization (PRPO), which operationalizes column-permutation invariance through label-preserving column permutations and two-level advantage estimation. This design converts sparse outcome rewards into denser and more stable optimization signals. Extensive experiments on 139 OpenML datasets show that our 8B model reaches a genuinely competitive regime against strong specialized tabular baselines. It achieves strong fully supervised performance, dominates zero-shot settings, and performs on par with 32-shot strong baselines. Moreover, it substantially outperforms much larger general-purpose and reasoning LLMs, including up to a 53.17% improvement over DeepSeek-R1 (685B). These results show that structural-prior RL post-training is an effective route for making LLMs competitive in tabular prediction.