Spectral characteristics of autoencoder parameters as a vector representation of data

This paper examines the relationship between the parameters of autoencoder models and the statistical properties of the data on which they are trained. Autoencoders are defined as models with an encoder-decoder architecture, trained to reconstruct input data through a compressed latent representation. It is proposed that the model parameters can be viewed as a dense vector representation of the corresponding sample. To test this hypothesis, a theoretical and experimental study is conducted in which a vector representation is formed based on the spectral characteristics of the autoencoder parameter matrices. Theoretical analysis shows that the singular values of the model parameter matrices are related to the eigenvalues of the covariance matrix of the training data, ensuring the transfer of information between the data space and the parameter space. Experimental results on the CIFAR-10 and FashionMNIST datasets confirm that the resulting vector representations allow for a high degree of accuracy in distinguishing between models trained on different data subsets, without resorting to complex vector generation algorithms or using the original samples. These results suggest that the parameters of trained autoencoders can be viewed as sample representations.

SMart: A Multi-source Multi-phase Time Series Representation Transfer Framework

Time series representation learning (TSRL) has attracted growing research interests in recent years. Two recent explorations in TSRL are: i) exploiting a transformer-based framework to learn time series; ii) instead of using only the targeted dataset, borrowing time series from other datasets to to facilitate representation transfer. While these two explorations are shown effective, the self-supervised time series recovery task in (i) and the single-source dataset used in (ii) are technically simple and thus can be enhanced with new ideas. In this work, we propose a new TSRL framework, namely multi-source multi-phase time series representation transfer (SMart), which has two novel mechanisms to address the aforementioned deficiencies: 1) a multi-phase recurrence plots recovery task, in three alternative modes, for guiding the encoder to embed time series dynamics into the time series representation; and 2) a source dataset selector to select multiple suitable source datasets to supplement the original target dataset for pre-training the TSRL encoder. Experimental results show that SMart outperforms several state-of-the-art models for time series representation learning, classification and regression on both uni-variate and multi-variate time series datasets, reducing mean absolute error up to 19.5% for time series regression, and increasing average accuracy up to 1.34\% for time series classification.

Superposed Latent Autoencoder

Autoencoders typically meet tight latent-memory budgets by making each latent representation smaller, sacrificing representational capacity. We ask a different question: can multiple wider latents be stored together instead? We introduce the Superposed Latent Autoencoder (SLAE), which preserves high-capacity latent representations while sharing storage through learned superposition. SLAE transforms latents into storage-friendly codes, binds them with randomized keys, superposes multiple codes into a single memory tensor, and learns to recover each latent before decoding. Under the same storage budget, SLAE replaces irreversible dimensional bottlenecks with structured interference that can be suppressed. Across CIFAR-10/100, SVHN, STL-10, Tiny ImageNet, and a wide range of memory budgets, SLAE substantially improves the reconstruction--memory tradeoff, reducing reconstruction error by up to 56% over conventional autoencoders at matched storage. Further analysis shows that SLAE's advantage comes from making wider representations usable under the same storage budget. These gains also extend beyond reconstruction: the information preserved by SLAE improves downstream classification by up to 16.79 percentage points under the same memory budget. Our results suggest a new principle for representation compression: instead of making every latent smaller, keep representations wide and let them share memory.

When Features Become Instances: Inverted Contrastive Learning for Unsupervised Feature Selection

Unsupervised feature selection seeks a compact subset of informative features without access to class labels, making feature utility difficult to define. Existing UFS methods therefore rely on indirect structural criteria, such as similarity preservation, locality, sparsity, cluster geometry, or reconstruction quality. In this paper, we instead study UFS through representation consistency and propose Inverted Contrastive Learning for Unsupervised Feature Selection (ICLFS), a feature-wise contrastive framework that reformulates UFS as a representation learning problem over features rather than samples. ICLFS first inverts the data matrix so that each feature is represented by its sample-profile vector, then constructs multiple masked positive views together with a shuffled negative view, and learns projector-space representations that remain consistent across these structured perturbations under an InfoNCE-based objective. Motivated by recent findings that cosine-based and InfoNCE-based training affect embedding norms, we use projector-space embedding magnitude as the saliency signal for ranking features. The resulting norm-based ranking is subsequently refined through Laplacian-Gated Ranking Correction, which suppresses locally redundant candidates while preserving salient ones. Extensive experiments on 12 benchmark datasets show that ICLFS achieves the best clustering accuracy on 10 datasets against both classical and neural baselines under the standard clustering-based UFS evaluation protocol, while remaining competitive on the other two. These results show that feature-wise contrastive representation consistency provides a strong and effective alternative to neighborhood, cluster, and reconstruction-based UFS formulations.

Different representation learning objectives recover distinct latent structures from the same psychometric data

Psychometric questionnaires contain rich item-level information, yet it remains unclear whether different representation learning objectives recover the same latent organization. We investigated this question using 757 matched teacher-child pairs from the baseline assessment of the Cyprus ProW preschool trial. Behavioral structure was characterized from child SDQ, ASBI, and CBRS item responses using principal component analysis and clustering, yielding four behavioral phenotypes. A contrastive objective substantially improved teacher-child retrieval relative to PCA-based representations, increasing Top-1 accuracy from 0.13% to 7.27% and Top-10 accuracy from 1.98% to 56.14%. However, contrastive representations preserved behavioral phenotype structure less effectively than PCA-based representations. A multi-task objective jointly optimizing alignment and behavioral prediction partially restored behavioral organization but reduced retrieval performance. These findings indicate that teacher-child correspondence and behavioral phenotypes represent distinct forms of latent organization and demonstrate that the latent structure recovered from linked psychometric data depends on the representation learning objective.

Learning Representations through Token Prediction: Geometry, Approximation, and Downstream Guarantees

Token prediction is a central pre-training objective for modern language models. Despite its empirical success, why token prediction learns broadly useful representations remains incompletely understood. We develop a statistical framework connecting token prediction with representation geometry, encoder approximation, and downstream performance. Under a softmax prediction head, we show that accurate token prediction organizes token embeddings according to similarities between the distributions of contexts in which different token types appear, as measured by Hellinger distance, with explicit errors governed by prediction accuracy and token frequency. Meanwhile, the contextual representation provides a low-dimensional coordinate for the conditional distribution of the target token relative to these embeddings. We further introduce a self-consistency principle showing that repeated applications of a shared representation block can progressively refine the contextual representation without introducing additional block parameters. Among representations with the same prediction accuracy, this recurrent construction favors those that can be stably reconstructed from their contexts. Finally, we establish downstream guarantees for token generation, token community recovery, and classification by a linear probe, showing how prediction accuracy and recovered geometry translate into performance beyond the pre-training objective. Together, these results explain how the simple objective of predicting tokens can recover semantic geometry and produce broadly useful representations. A controlled simulation illustrates the theoretical mechanisms.

Partially Linear Autoencoders for Manifold Learning and Dimensionality Reduction

Autoencoders are widely used for nonlinear dimensionality reduction and manifold learning. While most common implementations rely on both nonlinear encoders and decoders, we investigate the specific role of the encoder and the extent to which it can be constrained to be linear without reducing accuracy. We conduct a comparative study on four autoencoder architectures: standard fully nonlinear autoencoders (AE), linear-encoder autoencoders (Lenc-AE), linear-decoder autoencoders (Ldec-AE), and fully linear autoencoders (LAE), evaluated on synthetic manifolds, computational mechanics data sets, and real-world image data sets including MNIST. We demonstrate that imposing a linear encoder preserves most of the representational capacity of the autoencoder, provided the decoder remains nonlinear. In particular, Lenc-AE consistently outperforms both Ldec-AE and LAE, and achieves reconstruction quality comparable to fully nonlinear AE, while offering advantages in terms of parsimony and interpretability of the latent representation. These results suggest that the nonlinear decoder is the critical component for manifold learning, rather than the encoder. A geometric interpretation of this finding is developed, which identifies the precise conditions under which a linear encoder is sufficient, and the specific manifold configurations that expose its limitations.

Structured Learning on Mapper Representations

Modern machine learning (ML) methods are highly effective for prediction tasks, but many commonly used representations reduce complex data to fixed dimensional embeddings that may suppress multiscale structural organization. The Mapper algorithm from topological data analysis (TDA) provides a different perspective by decomposing data into overlapping local regions connected through a nerve construction, producing a structured representation that captures geometric organization, local statistical behavior, and relational connectivity simultaneously. In this work, we develop a framework for learning over Mapper induced structured representations. Rather than treating Mapper as a preprocessing step that produces a graph for downstream learning, we treat the full Mapper construction as part of the representation itself. We study mathematical properties of these representations, including invariance under relabeling, a distance functional on the space of Mapper representations, structural complexity of multiscale decompositions, and learning oriented stability under representation perturbations. Experiments on time series and graph classification datasets validate the proposed framework through controlled studies of representation ablation, Mapper parameter sensitivity, and the geometry of the induced representation space. Together, these results demonstrate how the proposed mathematical framework enables systematic comparison, interpretation, and analysis of Mapper representations, providing practical tools for studying representation geometry, structural complexity, and learning stability in learning tasks.

Barycentric Fused Gromov-Wasserstein Balancing for Causal Inference under Multiple Treatments

Estimating heterogeneous single and interaction treatment effects from observational data under multiple simultaneous treatments is crucial for decision-making. To mitigate estimation variance, previous studies balance representation distributions between every pair of treatment patterns. However, such pairwise balancing scales quadratically with the number of treatment patterns and fails to preserve consistent local proximity structures across patterns, which degrades counterfactual estimation. To address these challenges, we propose the Causal Inference for Heterogeneous Single and Interaction Treatment Effects Network (CIHSI-Net), a deep learning framework built on a novel Barycentric Fused Gromov-Wasserstein Balancing (BFG-WB) objective. BFG-WB aligns the representation distribution of each treatment pattern with a shared Wasserstein barycenter, achieving global alignment while reducing the computational complexity from quadratic to linear, and its Fused Gromov-Wasserstein discrepancy preserves the local proximity structures essential for reliable heterogeneous effect estimation. Simulation studies show that CIHSI-Net consistently outperforms state-of-the-art baselines, and an application to real-world marketing data demonstrates its practical utility in complex multi-treatment scenarios.

Lightweight Adaptive ReduNet via Hyperspherical Manifold Learning

In recent years, a white-box neural network called ReduNet has been proposed, which employs the maximal coding rate reduction (MCR$^2$) principle to transform raw data into low-dimensional discriminative features via a forward layer-wise construction process. Unlike traditional deep networks that rely on backpropagation, ReduNet explicitly derives the parameters of each layer from the features of its preceding layer, offering a mathematically interpretable paradigm. However, this layer-wise construction often requires a large number of layers for the MCR$^2$ objective to reach a stable value, which increases the parameter storage of the unfolded module. To address this issue, we propose LA-ReduNet, a lightweight adaptive architecture that refines the layer-wise update rule and enables discriminative feature representations to be obtained with substantially fewer unfolded layers. Specifically, LA-ReduNet employs hyperspherical manifold learning and adaptive step sizes, thereby reducing by an order of magnitude the number of layers required for the MCR$^2$ objective to reach a stable value. Simulation results demonstrate that, while maintaining comparable classification accuracy, LA-ReduNet requires significantly fewer layers for the MCR$^2$ objective to reach a stable value. Remarkably, under the considered experimental settings, LA-ReduNet requires only approximately $1/29$ of the parameter storage of the unfolded ReduNet module for the MCR$^2$ objective to reach a stable value.

Conditional-Independence-Regularized Distributional Autoencoders for Mixed-Type Data

Mixed-type data containing both numerical and categorical variables arise in many scientific and real-world applications. Existing representation learning and generative modeling approaches typically focus either on reconstruction accuracy or unconditional data generation, but often fail to recover the full conditional distribution of the data while preserving interpretable structural relationships between heterogeneous variable types. In this work, we introduce Conditional-Independence-Regularized Distributional Autoencoders, a framework for learning low-dimensional representations of mixed-type data through conditional distribution matching and structural regularization. Our method combines an energy-score-based objective for numerical variables, a likelihood-based objective for categorical variables, and an auxiliary conditional independence regularization term encouraging the learned representation to capture the dependence between numerical and categorical components. We provide theoretical analysis showing that the optimal representation balances unexplained numerical variability, conditional entropy of categorical variables, and residual conditional dependence. Empirically, the proposed method achieves strong performance on both synthetic and real-world datasets, substantially improving categorical distribution recovery, achieving competitive overall conditional distribution recovery, and preserving mixed-type dependence structure. The code has been made available at GitHub.

FETERS: Few-Shot Early Time-Series Classification via Effective Ratio Selection

Early time-series classification (ETSC) aims to make accurate predictions from partially observed time series as early as possible. Although various stopping mechanisms and feature learning strategies have been developed for ETSC, most existing methods assume access to sufficient labeled training data, which may be unrealistic in applications with limited annotation. Under limited supervision, learning an additional sample-level stopping module and extracting effective classification features can both become challenging. In this paper, we propose FETERS, a few-shot ETSC framework that selects a dataset-level stopping ratio through class-wise leave-one-out (LOO) evaluation on the support set and uses a penalty-based reward function to manage the accuracy-earliness trade-off, thereby avoiding the need to train an additional stopping module. FETERS further combines Rocket-based features with frozen Chronos representations for classification. Extensive experiments on 69 public datasets spanning 14 domains show that FETERS achieves state-of-the-art (SOTA) performance in the 5-shot setting, with the highest average harmonic mean (HM) and the best HM on 38 datasets, while outperforming the current SOTA method on 44 datasets. FETERS also remains competitive in the full-shot setting, demonstrating its effectiveness in managing the accuracy-earliness trade-off.

Task-Anchored Representation Shaping for Pre-Trained Model-Based Continual Learning

Pre-trained models (PTMs) provide a strong foundation for continual learning by offering stable representations that facilitate lightweight adaptation to new tasks. However, adapting well to each task does not ensure reliable inference over all learned tasks. Since task boundaries are often artificial and semantically entangled, an input from an unknown task can remain ambiguous even with strong PTM features, making cross-task prediction a key bottleneck. We propose Task-Anchored Inference Latent Shaping (TAILS), a lightweight post-PTM module that can be integrated into diverse continual learners and optimized through a decoupled step. TAILS uses fixed task anchors as persistent references to accumulated knowledge. It interprets each sample's feature representation relative to these references, then composes relevant evidence across tasks into latent recall. Rather than selecting a task-specific path or adjusting classifier outputs, TAILS uses latent recall to directly correct the feature representation before prediction. It therefore resolves cross-task ambiguity at the representation level, while leaving the original PTM, method-specific modules, and classifier unchanged. Extensive experiments across multiple PTM-based continual learning paradigms show that TAILS can improve classification and task-inference performance with modest parameter overhead and negligible inference cost.

The Trade-off Between Covariate Dependence and Latent Structure in Representation Learning

Disentangled representation learning seeks latent representations whose indicidual dimensions each align with a distinct covariate. Unsupervised approaches typically target latent dimension independence, yet this gives no guarantee that the resulting dimensions align with semantically meaningful covariates. Supervised approaches structure the latent space using observed covariates, but under correlated covariates they cannot simultaneously control one-to-one latent-covariate alignment and latent independence. We introduce a unified, supervised framework that couples latent dimension-covariate dependence with constraints on the latent structure. Within this framework, we show an inherent trade-off, where enforcing latent independence or exclusive one-to-one latent-covariate dependence comes at a provable cost in latent-covariate alignment. We prove that the resulting disentanglement regimes are ordered by the strength of that alignment. Each regime admits a closed-form transformation of the latent space. We apply these transformations post-hoc to realign the representations of pretrained models such as CLIP, DINOv2, and ViT, and we fold them into the inference of informed factor analysis (iFA), a probabilistic model with covariate-informed factors. On simulated and real multi-omics data, we show that both post-hoc alignment and iFA enable controllability of structured latent representations.

Geometry of Forgetting: Representation Flux in Continual Learning

Catastrophic forgetting remains a fundamental obstacle to continual learning, where neural networks lose previously acquired knowledge while learning new tasks. Existing methods primarily mitigate forgetting through parameter regularization or experience replay, while the representation-space dynamics associated with forgetting remain less understood. We investigate latent representation evolution during sequential learning and introduce representation flux, a geometric measure of sample-level representation displacement across training. We show that representation flux is strongly associated with catastrophic forgetting across multiple benchmarks, with temporal analyses indicating that elevated flux can precede subsequent performance degradation. Representation displacement is also associated with confidence degradation, while complementary geometric properties provide additional information about sample-level forgetting. Motivated by these observations, we propose FlowLess-R, a representation-space regularization method that constrains replay representations relative to stored references while allowing continued learning. FlowLess-R is architecture-agnostic and integrates into replay-based methods through a representation-matching term. Experiments on SplitMNIST, SplitFashionMNIST, SplitCIFAR10, and SplitTinyImageNet show improved final average accuracy and reduced forgetting with ER, DER++, and ER-ACE. Our results identify representation flux as an informative geometric marker of forgetting and show that stabilizing latent representations provides a simple strategy for mitigating catastrophic forgetting.

Learning reshapes power-law anisotropy in internal representations

Power-law anisotropy in internal representations has been observed across a wide range of biological and artificial neural systems, from state-of-the-art language models to the mouse cerebral cortex. This anisotropy is a key geometric property of high-dimensional information processing and underlies a variety of theoretical analyses. However, the mechanism by which it emerges from input structure and task-driven learning has remained unclear. Here, we characterize this formation process by exactly solving the learning dynamics of a wide two-layer linear neural network in a teacher--student setting with power-law input and teacher structures. We show that, in the feature-learning regime, the local power-law exponent of the internal-representation spectrum evolves nonmonotonically over the course of training and exhibits up to four distinct asymptotic regimes across modes and training times. By contrast, in the lazy regime, the exponent remains essentially unchanged. We further demonstrate numerically that similar exponent dynamics arise in more realistic nonlinear networks. Together, these results suggest a general mechanism by which the dynamic interaction between input statistics and task structure gives rise to power-law internal representations.

Training Fair Tabular Foundation Models

Tabular Foundation Models (TFMs) have emerged as leading methods for tabular predictive tasks, leveraging in-context learning to predict on new data without task-specific training. Despite the increased use of TFMs in high-stakes decision-making, their fairness properties remain largely unexplored. In this work, we incorporate fairness constraints directly into TFM training, enabling fair predictions in a single forward pass. Our approach addresses two key challenges: limited access to sensitive attributes in training data, and the incompatibility of existing fairness techniques with the in-context learning paradigm. We propose FairTFM, a scalable training strategy based on synthetic fairness tasks and a fairness-aware architecture using a gradient reversal layer, which encourages the model to learn representations invariant to sensitive attributes. Experiments on 132 fairness tasks show consistent improvements in fairness while maintaining competitive accuracy.

TabSOM: A tabular-to-image encoding method based on self-organizing maps

Tabular-to-image methods have emerged as novel approaches to leverage the high predictive performance of convolutional neural networks and vision transformers. They convert tabular data into image representations, mapping each feature at a fixed pixel location derived from a dimensionality-reduction method (e.g., t-SNE, UMAP, PCA). However, they encode only the marginal value of each feature and discard information about feature relationships. We propose TabSOM, a tabular-to-image encoding built on the Self-Organizing Map (SOM), which provides: (i) a spatial layout in which every input feature occupies a fixed canvas position derived from its component plane via collision-free Hungarian assignment; and (ii) a graph that captures pairwise feature relationships derived from the SOM component planes. The resulting image stacks two multi-scale node channels: one encodes feature values at fixed scales, while the other encodes pairwise feature interactions as spatial connections between related features. Two SOM-derived interpretability approaches are introduced: a prototype-inspired partial dependence plot and a class--separation importance score. Benchmarked against twelve existing tabular-to-image methods across public binary-classification datasets, TabSOM ranks first or second on every dataset and achieves the lowest variance of any method evaluated. Interpretability obtained with TabSOM was validated against Random Forest, XGBoost, and SHAP, the class-separation score shows reasonable agreement with established baselines on the top-ranked features while capturing complementary structural information from input data. These results demonstrate that TabSOM provides an effective and interpretable approach for applying deep learning architectures to tabular data, bridging the performance--interpretability gap in this domain.

Your Probabilistic JEPA Is Secretly a Hidden Markov Model: A State-Space Interpretation of Joint-Embedding Predictive Learning

A hidden Markov model (HMM) combines three roles: inference of a hidden-state belief from observations, propagation through a Markov transition, and emission back to observation space. We show that full, time-indexed Predictive Information Bottleneck VJEPA (PIB-VJEPA) exposes the same computational structure: a stochastic context encoder plays the role of an amortized filtering distribution, a probabilistic predictor defines latent-state dynamics, and a decoder, inverse target encoder, or induced implicit conditional supplies the emission direction. We distinguish 4 progressively stronger levels of correspondence and give sufficient conditions for exact sequence-level HMM equivalence. To make the connection concrete, we introduce Markov-Chain JEPA (MCJEPA), which replaces the latent predictor by a learned transition matrix; in the finite time-homogeneous case, matrix powers guarantee exact multi-horizon Chapman--Kolmogorov consistency. Conditioned discrete-state transitions, continuous-state Markov kernels, and continuous-time dynamics extend this construction, while deterministic temporal JEPA appears as a degenerate Dirac-kernel special case. We further interpret predictive information-bottleneck learning as seeking a compact predictive state: compression promotes minimality, while residual predictability tests sufficiency. Controlled experiments support transition composition, the filtering interpretation, predictive Markovization in a known synthetic process, and the distinction between JEPA latent prediction and HMM-style sequence learning. Together, these results give temporal JEPA a principled state-space interpretation.

HYDRA: Hyperbolic Dynamic Representation Architecture for Kolmogorov-Arnold Networks

Kolmogorov-Arnold Networks (KANs) enhance nonlinear function approximation by replacing scalar weights with learnable univariate functions. However, assigning an independent function to every connection results in substantial parameter redundancy, limiting their scalability and efficiency. To reduce this redundancy, we introduce \textbf{HY}perbolic \textbf{D}ynamic \textbf{R}epresentation \textbf{A}rchitecture (HYDRA), a parameter-efficient hyperbolic extension of KAN that combines spline-based functional learning with representations in the Poincaré ball. HYDRA maps vector-valued inputs into a bounded hyperbolic latent space, performs KAN-style updates in tangent space, and employs a low-rank prototype block to share functional transformations across hidden dimensions. The resulting hyperbolic representations provide a structured radial coordinate for interpretation, while radius control improves training stability by preventing boundary saturation. Extensive experiments across eight benchmark datasets demonstrate that HYDRA consistently achieves competitive or superior predictive performance while improving parameter efficiency and representation interpretability.

Low-Interaction-Rank Learning: Unifying Multiplicative Dual-Encoder Heads

A multiplicative dual-encoder network computes a real-valued output for a pair of inputs as the inner product of their separate encodings. This architecture has been developed independently in operator learning, bipartite matching, contrastive vision-language models, retrieval, and other areas, yet no unified theory guides the basic design decisions: how many interaction modes to represent, how to normalize the encoders, and when the architecture should be avoided. We provide such a foundation by introducing the class of functions of low interaction rank, a class whose intrinsic complexity is measured by its interaction spectrum. Within this framework, approximation error decomposes into a spectral truncation term and an encoder-realization term; sample complexity is governed by the sum of the two encoder complexities rather than their product; and a usability criterion based on spectral decay determines when the architecture can succeed. The same framework exposes a central identifiability problem: the encoders are defined only up to a linear gauge symmetry that leaves the learned coordinates arbitrary. We show that normalization is gauge fixing and that whitening pins the interaction modes up to permutation and sign, thereby explaining the uninterpretability of contrastive dimensions and providing a constructive remedy. Experiments on synthetic kernels, operator learning, and CLIP models validate the theoretical predictions: spectral decay rates match the predicted scaling, whitening recovers the true modes, and independently trained CLIP models are related by a single rotation which, after removal by whitening, exposes interpretable concept axes. The code of this paper is provided at https://github.com/RS2002/Mul-Net .

FiGuRO: Intrinsic Dimension Estimation for Multi-Modal Data

Determining the complexity, or Intrinsic Dimension (ID), of data is fundamental to efficient and interpretable representation learning. This is particularly challenging in multi-modal settings when trying to learn disentangled representations for shared and private information. Existing techniques leave a critical gap: they are often static, uni-modal, or in the case of contrastive methods, adapt only to the shared ID implicitly. We introduce Fidelity-Guided Rank Optimization (FiGuRO), a framework for approximating the ID of uni- and multi-modal data under constraints of model capacity and hyperparameters. FiGuRO learns the dimensions of low-rank projections using truncated singular value decomposition and an algorithm that determines when to reduce or increase dimension and in which latent space. Disentanglement of shared and private information arises as an emergent property of this optimization, eliminating the need for complex auxiliary loss functions. We demonstrate that FiGuRO outperforms existing ID estimation techniques and is more robust to hyperparameter changes. Across simulations and real-world data, FiGuRO captures distinct ID scales and varying subspace ratios, and decomposes shared and private information successfully. Furthermore, we show that FiGuRO can be applied to modern uni-modal pretrained models, enabling efficient, post-hoc disentanglement of multi-modal representations.

From Objectives to What Models Learn: A Landau Theory of Invariant Learning

Invariant learning seeks representations that remain predictive across environments, yet the behavior of its objectives along the regularization path is often opaque. We address this objective-behavior gap by viewing representation learning as multimode magnetization and deriving, from concrete invariant-learning objectives, a Landau-type effective free energy whose low-order coefficients form objective signatures and induce distinct regularization phenotypes. Effective quadratic corrections move the phase boundary and enable finite-strength mode elimination; quartic corrections regulate post-onset amplitude and typically leave residual loading at finite strength; higher-order structure governs non-monotone tails, instability, and collapse at large regularization. In a canonical bilinear model, the theory yields closed-form phase boundaries and steady-state loadings, as well as distinct critical strengths for shortcut and stable modes that define a selective-retention window. Controlled experiments confirm the predicted phase boundaries, loadings, and regularization phenotypes. In one- and two-hidden-layer ReLU networks, the same signatures remain predictive of qualitative regularization-path behavior despite depth-dependent shifts in scale. A matrix extension generalizes the framework to coupled collective modes and yields a spectral phase-boundary criterion. Together, the framework turns low-order objective signatures into predictions of regularization phenotypes and, ultimately, of what models learn as regularization varies.

Residual Algebra for Representation-Preserving Learning

Learning from heterogeneous representations is often reduced to feature concatenation, erasing which representation produced each error. We propose residual algebra, in which each representation retains its coordinate system and owns its unresolved residual until an explicit aggregation boundary. Fold instantiates representations as point-in-time conditional-mean fields on 10x10 rank grids, and FPRC-PQ composes them through relax-aggregate-close: each field first fits a correction to its own residual, corrected fields then meet at a fixed mean, and a shared learner closes only the aggregate's fresh residual. We formalize aggregation as a quotient by the zero-sum redistribution kernel, characterizing legal post-aggregation operators as those constant on its cosets. The resulting composition separates representation, local residual estimation, and residual-of-residual estimation, with population variance reduction and first-order coupled-path mean orthogonality. Rumination-B and Rumination-H extend the algebra with quotient-legal finite correction and feedback. On 3.67M Chinese A-share stock-day observations (2023-2026) under a frozen point-in-time protocol, FPRC-PQ raises net-of-cost return from 13.52% to 19.10% and Sharpe from 1.42 to 2.09, outperforming matched-capacity, unified-residual, identity-free two-stage, and pairwise-only controls. The gain is thus attributable to explicit residual ownership and composition rather than additional features or trees.

PRISM: Powerful Time Series to Image (TS2I) Representations for Multivariate Anomaly Detection

Time series anomaly detection (TSAD) underpins applications in predictive maintenance, finance, and cloud computing, however performance remains sensitive to representation choices, especially in multivariate settings. While transforming time series into images has shown success in forecasting and classification, it remains unclear how multivariate, high-dimensional series should be mapped to multi-channel images and whether vision backbones can match time-domain baselines in TSAD. We introduce PRISM, a plug-and-play meta-workflow enabling systematic construction and evaluation of image-based representations for multivariate TSAD. Our evaluation spanning over 7,000 experiments shows that well-designed PRISM configurations are competitive with 24 time-domain baselines, achieving the best VUS-PR on 10 of 14 datasets, with an average improvement of 41% over the best competing method on those datasets. Further, we identify channelization - how the channel dimension of multi-channel images is constructed - as a critical and previously understudied design dimension, and introduce MSM, a novel statistics-based scheme achieving 11-27% gains over PCA-based alternatives. Finally, ImageNet-pretrained encoders transfer effectively to TSAD, with frozen encoders retaining 92% of fine-tuned performance while training 1.8 times faster. Our code is available at: https://github.com/Smendowski/PRISM.

Topological Simplification in Predictive Coding Networks

We study the topology of learned representations in predictive coding networks (PCNs), a neuro-inspired bidirectional architecture, using a quantitative layer-wise persistent homology analysis. We train well-performing PCNs on a synthetic classification dataset ($\geq 99.9\%$ test accuracy) and on MNIST ($\geq 95\%$ test accuracy), and measure how topological features change across layers for different architectures and activation functions. We find that smaller PCNs collapse connected components across layers earlier than larger models (Spearman $\unicode{x1D70C} \in [0.72, 0.79]$ across activations), with model size measured as the sum of hidden-layer widths. We also observe a strong negative correlation ($\unicode{x1D70C} = -0.58$) between the depth at which simplification occurs and reconstruction error; i.e., architectures that simplify later reconstruct better. Finally, a seed-level bootstrap comparison across architectures and activations shows that PCNs consistently collapse connected components later than matched MLPs, with an average difference of $3.6$ layers. These results suggest that persistent homology offers a useful quantitative lens on the compression--reconstruction tradeoff in PCNs, and that both model capacity and the recurrent, bidirectional dynamics of predictive coding inference shape when this tradeoff is resolved across layers.

Understanding Submodular Information Measure Based Objectives for Representation Learning: A Variance and Separation Perspective

Submodular Information Measures (SIMs) have recently emerged as a powerful framework for representation learning and multimodal learning. In particular, the SCORE framework~\cite{majee2024score} demonstrated that SIMs can serve as effective objectives for supervised contrastive learning. Despite their empirical success, however, the geometric and statistical properties induced by different submodular information measures remain poorly understood. In this work, we develop a unified theoretical framework connecting SIMs to classical concepts in representation learning and statistical pattern recognition. We show that Total Information (TI) objectives characterize intra-class structure: Graph Cut TI recovers within-class variance, LogDet TI recovers generalized variance and covariance volume, and Facility Location TI induces imbalance-aware separation that emphasizes rare and confusable classes. We further show that Mutual Information (MI) objectives capture complementary notions of inter-class structure: Graph Cut MI is closely related to centroid separation and Fisher-style discrimination, LogDet MI captures covariance-aware separation through Mahalanobis distance, and Facility Location MI measures nearest-mode representational overlap. We validate these theoretical characterizations using controlled synthetic experiments that independently vary variance, covariance, class imbalance, class separation, and multimodal overlap. Across all settings, the empirical behavior closely matches the proposed theory. Our results provide the first unified geometric and statistical understanding of submodular information measures and offer principled guidance for selecting and designing SIM-based objectives for representation learning.

Uncertainty-Guided LLM Semantic Augmentation for Heterogeneous Treatment Effect Estimation

Estimating heterogeneous treatment effects is central to targeted interventions, such as personalized promotions and precision medicine. We focus on the conditional average treatment effect (CATE), a standard estimand for characterizing such heterogeneity. Even under standard identification conditions, finite-sample CATE estimation requires learning the nuisance structure for covariate adjustment and treatment-effect heterogeneity, often together with an effective representation of X. Raw numerical and categorical encodings can leave semantic relations and higher-order interactions implicit, making this joint task locally unstable. A motivating study further shows that this instability appears through partially separable assignment- and heterogeneity-side channels. Building on this observation, we propose CURL (Causal Uncertainty-guided Representation Learning), a plug-in adapter that uses estimator uncertainty to allocate pretrained semantic capacity to locally unstable units. CURL queries a frozen LLM through two role-conditioned prompts, constructs assignment- and heterogeneity-oriented representations from the observed covariates, and routes them through separated pathways. On four benchmarks, CURL improves ten host learners in most settings, while ablation, refinement-dynamics, route-reassignment, and probe analyses support the intended design and roles of the two channels.

Phasor Attention: Mean Root Square Normalization for Phase Manifold Preservation

While Root Mean Square Normalization has become the de facto standard for accelerating modern sequence models, its reliance on the quadratic accumulation of independent scalars ($\sum x^2$) inherently triggers outlier-induced numerical instability, gradient starvation, and anisotropic phase distortion. We introduce Mean Root Square Normalization (MRSNorm). By structurally pairing channels into 2D phasors, MRSNorm mathematically inverts the traditional scaling paradigm: it computes the localized $L_2$ magnitudes (Root Square) before aggregating them via a global $L_1$ average (Mean). This operational inversion strictly constrains activations to a phasor manifold, preserving conformal invariance. By sharing a single affine weight across phasor components, MRSNorm halves the total number of learnable parameters, proving that unconstrained spatial scaling in standard norms is a harmful redundancy. We analytically demonstrate that this geometric constraint yields a built-in, trigonometric gradient clipper governed by the Pythagorean identity, unconditionally equalizing the local gradient norm to ensure Gradient Homogeneity. Empirical evaluations on a ResNet with CIFAR-100 show that despite halved parameters, MRSNorm provides critical structural stability under rigorous stress tests. Under extreme hyperparameter settings where standard normalizations suffer from gradient divergence, MRSNorm successfully prevents numerical explosion and secures stable optimization trajectories. Our findings propose a fundamental paradigm shift toward phasor-based deep representation learning. The implementation of MRSNorm is available at Appendix C.

Contrastive-Collapsed Loss for Flexible and Geometrically Optimal Embeddings and Faster Convergence

In this work, we introduce CoCo, a loss function aimed at learning normalized and well-structured representations. The proposed loss encourages intra-class collapse and inter-class contrast while preserving sufficient flexibility for neural networks to approximate geometrically optimal embeddings with large angular separation between classes. We provide a theoretical analysis positioning CoCo with respect to related objectives such as dot regression and cross-entropy, showing that the new proposed loss benefits from closer initialization to the optimal configuration, more informative gradients, and stronger incentives for class-wise representation collapse. Extensive experiments on diverse tabular datasets from the OpenML-CC18 benchmark show that CoCo achieves competitive performance with state-of-the-art methods, including kernel SVM, Random Forest, dot regression, and cross-entropy-based neural networks. In addition, both theoretical arguments and empirical analyses demonstrate that the proposal promotes tighter class clustering and faster convergence. These results highlight CoCo loss as an effective objective for learning discriminative representations while maintaining competitive predictive performance.