Local learning methods offer an alternative to end-to-end backpropagation, but their unstructured local objectives can produce weak feature learning in deep networks. We study whether the order of hidden-state optimization can address this limitation. We propose a boundary-first inference schedule that partitions a model into chunks, first coordinates hidden states at chunk boundaries, and then refines representations within each chunk. We instantiate this schedule in predictive coding networks (PCNs), a local-learning framework in which hidden activities and prediction errors are explicitly exposed during inference. On CIFAR-10, the resulting boundary-first predictive-coding instantiation improves accuracy over standard predictive coding by $9.77\%$ under a standard parametrization and by $5.51\%$ under a $μ$-parametrization. Diagnostic analyses further show more non-trivial early-layer updates, lower initial-to-final CKA, and more diverse layerwise gradients, consistent with stronger feature learning. These results support boundary-first, chunk-based inference as a practical design principle for predictive-coding training and motivate its study in broader local-learning systems.
Over the course of the last decade, neural networks have grown from an academic curiosity to moving the markets of nations. Despite this explosion in both research and deployment, relatively little is understood about how they achieve the solutions they do. This is both scientifically relevant, and pressing for society. When neural networks make decisions across self-driving, construction, law, hiring and health, there have been and will continue to be unintended consequences. However, attempting to generalize the failures of the largest and most important production systems makes for a very difficult task. Yet signs of these failures exist at all scales of neural networks, so we should be able to study a much more tractable setting. All neural networks must undergo an optimization process, called training, to be useful. To a great degree, understanding neural networks is understanding their optimization: through what process and exposure to which data did they arrive at their results. Yet our knowledge on this topic as a field is quite imprecise. In particular, a curious phenomenon called mode connectivity, the ability to connect neural networks in the loss surface, defies explanation entirely. This dissertation elucidates, explains and exploits this special structure in the loss landscape...
Understanding how neural networks learn and organize features is central to understanding their behavior. Much existing theory of feature learning has focused on the emergence of a global low-dimensional predictive geometry. We show that this picture is incomplete. In regression problems with clustered data, we demonstrate that multilayer perceptrons (MLPs) naturally develop monosemantic specialized neurons: individual neurons become strongly aligned with a specific predictive feature relevant to a particular region of the input space. Rather than learning a single global low-dimensional representation, MLPs learn a collection of local low-dimensional representations that can collectively span a high-dimensional space. This specialization provably gives MLPs a data-efficiency advantage over feature-learning methods based on a global low-dimensional representation.
In this work we present a method to accelerate the optimization of learning high dimensional functions using deep neural network (DNN). This optimization procedure introduces contextual features into the first layer of a DNN. The parameters of DNN are optimized via standard gradient descent while keeping the input-feature basis fixed. After optimization of the DNN parameters, the feature layer is provided a chance to update and change before DNN optimization resumes. The feature layer has two types of functions: those that can be evaluated quickly in a matrix-free way on the domain (i.e. rank-1 features) and more complex features that must first be decomposed using tensor network (TN) decomposition strategies (tensor features). In particular, we study the effect of adding features which distill pretrained DNN into TNs using a discretize and decompose strategy. To efficiently decompose high-dimensional functions constructed from discretized DNN, we leverage a randomized tensor decomposition strategy. Using randomization, we are able to reduce the storage cost of decomposing high dimensional functions by at least 8 orders of magnitude. Using this approach, we are able to efficiently train models between 5 and 40 dimensions.
Training changes a network's predictions while allocating task-relevant structure across its internal units. In an overparameterized ReLU network, several neurons can begin with exactly the same functional role, yet one may acquire a teacher feature while the others become redundant. We call the identity of that neuron feature ownership and ask whether it can be controlled by a parameter choice invisible to the initial predictor. In a tractable Gaussian teacher--student model, we fix the complete initial function and vary only a positive-homogeneous scaling gauge. Opposite gauges produce distinct feature trajectories and a sharp $Θ(D^2)$ separation in specialization time that no global change of clock can explain. Among any fixed number of initially duplicate students, assigning the favorable gauge to one neuron deterministically selects it as the owner and drives the remaining functional contribution to zero. An exact reaction--transport decomposition attributes the effect to different mobilities for changing a feature's coefficient and direction. We prove global selection and functional pruning, extend finite-time selection to visible perturbations and small-step full-batch gradient descent, and verify the predicted loss, alignment, pruning, and dissipation trajectories in population and finite-sample training. The initial predictor therefore determines neither when the feature is learned nor which neuron learns it.
Dropout and Random Gradient Masking (RaM) are two training techniques used to improve performance in deep learning. Both techniques inject randomness into the training dynamics, but in significantly different ways: dropout applies random masks to the activations in the forward pass, whereas RaM leaves the forward pass unchanged and instead masks the gradients. In particular, the noise induced by RaM in the parameter updates is unbiased, so standard explanations for the effectiveness of dropout, such as the penalization effect or the prevention of co-adaptation between neurons, do not apply to RaM. In this work, we show that the difference between the two methods disappears for ResNets in the large depth and width asymptotics: in the complete feature learning regime, they both converge to the same large-scale limiting dynamics. This asymptotic equivalence holds for several variants of dropout and RaM, including layerwise dropout as used in stochastic-depth ResNets, albeit at slower quantitative rates. In fact, we also show that several of these variants collapse to the same limit asymptotically.
Saad Ejaz, Miguel Fernandez-Cortizas, Javier Civera +2cs.CV cs.RO
CAD-to-image alignment aims to estimate an object's 9D pose (rotation, translation, and anisotropic scale) from a single RGB image, enabling applications in robotics and augmented reality. Recent zero-shot methods use visual foundation models to match image regions to CAD models, yet typically their correspondences are appearance-driven and degrade under occlusion or sim-to-real domain shift. To address these limitations, we introduce SUFLECA (Scaling Up Feature LEarning for CAD Alignment), a weakly-supervised framework for zero-shot CAD alignment with two key contributions. First, SUFLECA scales up geometry-grounded feature learning from pretrained visual representations through Normalized Object Coordinates (NOCs) supervision on 674K images spanning 12 real and synthetic datasets, learning compact geometry-aware features that generalize across domains. Second, we propose a geometrically consistent matching algorithm that establishes reliable one-to-one CAD-to-image correspondences. Together, these contributions enable accurate, sub-second alignment per object instance without iterative pose refinement. On ScanNet25k, SUFLECA achieves 33.4%/42.3% category/instance accuracy, outperforming, with a smaller computational footprint, the strongest zero-shot baseline by 10.3/12.2 percentage points and, for the first time on this benchmark, even surpassing fully supervised methods. Code is available at: https://github.com/snt-arg/SUFLECA
This paper investigates feature learning within the framework of the deep Ritz method for solving the stationary Schrödinger equation with Neumann boundary conditions. We first analyze the convergence of Riemannian gradient descent in an agnostic setting, where the hypothesis function is restricted to a single-index model while the PDE solution is arbitrary. We prove that gradient descent reaches an approximate global minimum: after T = O(log(1/ε)) iterations, the loss is within εof a constant multiple of the optimal loss. We then examine the loss landscape when the source term of the PDE itself follows a single-index model, considering hypothesis functions given by either a single-index model or a two-neuron multi-index model. In the single-index case, we show that the minimum Ritz energy is attained at the feature vector aligned with that of the source term. In the two-neuron case, we study the landscape of regularized Ritz losses and characterize how a second feature emerges, given that the first feature is aligned with the source, as the regularization parameter varies. Finally, numerical experiments are presented to validate the feature emergence theory in the two-neuron setting.
We consider the Multiscale Single-Index Model (MSIM), first introduced in \cite{oymak2021learning}, as a stylized model for hierarchical learning with \emph{scale separation}. Each layer extracts a shared single-index feature at one physical scale and passes it to the next, thus defining a tractable setting in which to study how deep architectures learn multiscale representations. Under non-degeneracy and delocalization assumptions on the link function and planted features respectively, for fixed depth $K$ and local scale $d$, the first Wiener chaos of the target behaves as a perturbed spiked tensor, where the perturbation of order $d^{-1/2}$ comes from the non-linearity -- revealing the MSIM as a natural non-linear analogue of the Tensor PCA model \cite{montanari2014statistical}. While this perturbative picture is sufficient to enable efficient spectral recovery based on Tensor unfolding (as already observed in \cite{oymak2021learning}), it is not precise enough for the analysis of backpropagation gradient-based methods. In this work, we address this limitation by performing a fine-grained analysis of the Wiener chaos using Edgeworth expansions. In the first chaos, this gives a finite-rank hierarchy at scales $d^{-q/2}$. In higher chaoses, balanced flattenings exhibit staircase singular-value plateaus of size $d^{-ρ/2}$ and multiplicity $d^ρ$ under a natural higher-chaos non-cancellation condition. Using this higher-chaos structure, and under an additional slow Hermite-energy tail condition, we first establish shallow-network approximation lower bounds, quantifying the benefit of depth in this model. Next, and most importantly, we prove that online SGD on the correlation objective, where all layers evolve in the same timescale, achieves $1 - o_d(1)$ recovery with $n = \widetilde{O}( d^{K-1})$ samples, recovering the same sample complexity as in the linear counterpart.
Preston Pitzer, Anish Pradhan, Harpreet S. Dhilloneess.SP cs.LG
Mutual information (MI)-inspired feature learning techniques are capable of generating low-dimensional embeddings that retain nonlinear dependence structures, but direct estimations of MI suffer from noisy probability distribution estimates in the low-data regime. The H-Score objective, computed from second-order statistics, provides a practical proxy metric for training feature extraction networks. We prove that H-Score is invariant to invertible transformations in the unrestricted functional setting, but becomes sensitive to input basis rotations under constrained approximation classes. Consequently, we study unitary preconditioning for H-Score networks and show that selecting an appropriate basis rotation reduces finite-width truncation error by concentrating predictive dependence into fewer dominant modes. We identify the fast Fourier transform (FFT) as an effective data-independent, low-cost preconditioner for approximately stationary processes, where spectral structure induces concentration of the cross-covariance singular value spectrum. We introduce training-free metrics based on spectral entropy and cumulative dependence energy to quantify basis suitability and predict downstream inference gains prior to network training. Experiments across eight multivariate datasets demonstrate that FFT preconditioning is particularly useful in resource-constrained regimes, achieving up to 50% normalized mean squared error (NMSE) reduction, while the proposed metrics correlate with observed performance gains and correctly identify cases where spectral preconditioning is detrimental.
Recursive Feature Machines (RFMs) are a class of kernel machines that utilize the Average Gradient Outer Product (AGOP) as a mechanism for feature learning. They have been shown to effectively replicate the learning dynamics and feature representations of Feedforward Neural Networks (FNNs) across various settings. However, despite comparable capacity for feature learning and the similarities in the features they acquire, RFMs exhibit significantly lower performance than neural networks in certain data-corrupted scenarios. In this work, we investigate these limitations in mathematical problems. As a solution, we introduce a remarkably effective transformation applied to the training labels which promotes learning in noisy, complexly represented, and class-imbalanced data. This simple yet powerful adjustment enables RFMs to close the performance gap with FNNs and, in some cases, even surpass them.
Sehwan Kim, Yan Sun, Faming Liangstat.ML cs.LG math.ST
Over the past decade, deep neural networks (DNNs) have achieved remarkable success on complex machine-learning tasks, yet the theoretical foundations of their performance remain incomplete. From a statistical viewpoint, a natural question is: can DNNs attain feature-learning and prediction consistency comparable to that of classical models? While a full characterization is open, we provide positive results for a broad subclass. We establish feature-learning consistency guarantees for sublinearly structured DNNs-architectures whose input/output dimensions and number of hidden neurons grow sublinearly with the sample size-when learning hierarchically compositional target functions. Importantly, this consistency still holds even in the conventional "over-parameterized" regime where the total number of parameters exceeds the number of training samples. Empirically, sublinearly structured DNNs match or surpass wide DNNs in prediction. A structural audit further indicates that widely used convolutional neural networks (CNNs), including AlexNet, VGGNet, ResNet, GoogLeNet, are sublinearly structured on their image classification benchmarks. We further prove that the sublinearly structured DNNs achieve universal approximation for hierarchically compositional functions in the large-sample limit. Moreover, images exhibit an inherent hierarchical, compositional structure. Taken together, these results explain, through a statistical lens, why many large-scale deep learning models succeed after adequate training on massive image datasets.
Sparse autoencoders (SAEs) detect features via inner product, so a feature's activation scales with both its directional alignment and the input's norm. Features that fire on token norm therefore claim dictionary slots regardless of content alignment. This matters because sublayer normalization has already discarded the magnitude the score measures, so the encoder detects a quantity the model does not read. We replace the score with a learned blend of cosine similarity and input magnitude, letting the optimizer choose how much norm to use; a per-feature extension lets each feature decide independently. In both regimes, training is free to recover inner product but never does, with no feature ever choosing more than half-magnitude dependence. At matched reconstruction, the cosine encoder learns features that align with human-recognizable concepts far more often than standard, filling dictionary slots that inner product wastes on norm detectors. Loss reweighting that equalizes gradients barely closes the gap, confirming forward-pass score geometry as the lever. The advantage is not universal across tasks or depths, but we believe cosine scoring should be the default for dictionary learning on normalized representations.
Tianyu Ruan, Fengzhuo Zhang, Shuche Wang +1cs.LG cs.AI
Muon has recently emerged as a state-of-the-art optimizer for pretraining Large Language Models (LLMs) and vision classifiers. Despite its efficiency advantage over Adam and SGD, the feature-learning advantage of Muon remains unclear. This paper investigates Muon's feature-learning advantage through the lens of robustness and transferability. First, by evaluating pretrained models on corrupted images and texts, we show that features learned by Muon are consistently more robust than those learned by Adam and SGD across different architectures, including transformers and Convolutional Neural Networks (CNNs). Using trained layer-wise probes, we further show that this robustness advantage is reflected in larger logit margins across layers. Second, by training linear classifiers or fine-tuning full models from pretrained parameters on downstream tasks, we demonstrate that Muon-learned features transfer more effectively than those learned by Adam and SGD. This transferability advantage is further supported by the diversity of hidden states across layers, as measured by effective rank. Finally, in a representative classification problem with multi-component features, we prove that Muon attains larger margins and higher effective rank than Adam and SGD, providing theoretical support for our empirical findings.
Few-shot class-incremental learning (FSCIL) in synthetic aperture radar imagery presents unique challenges due to severe data scarcity and SAR-specific variability. In particular, strong azimuth sensitivity in SAR induces large intra-class variation and inter-class confusion, and FSCIL sequential updates further lead to catastrophic forgetting of previously learned classes. Inspired by neural collapse, we propose an optical-guided SAR FSCIL framework, which derives orthogonal feature subspaces from a data-rich optical ATR dataset and uses them as geometric priors to guide SAR feature learning. SAR features are projected onto these orthogonal subspaces via principal angle constraints, effectively transferring discriminative structure from the optical to the SAR domain. Specifically, our projection loss and the classifier loss optimized with a frozen simplex-ETF geometry jointly induce neural collapse by concentrating features around class means while maintaining large inter-class angles. We evaluate the approach on a benchmark comprising an optical ATR dataset and a SAR ATR dataset with 24 target classes, organized into a base training session and seven incremental sessions. Compared with recent FSCIL methods including NCFSCIL and so on, our method achieves the highest final accuracy and a favorable trade-off between final performance and performance degradation. Moreover, neural collapse metrics show improved intra-class compactness and inter-class separability, indicating that the learned features more closely approximate the ideal simplex-ETF geometry.
Understanding how structured internal structure emerges during neural network training is central to the study of deep learning. We investigate this phenomenon through the group composition task, where a two-layer neural network is trained to predict $g_1 \star g_2$ for elements of a finite group $G$. By lifting the projected gradient flow to the Fourier domain, we demonstrate that the training dynamics are governed by a Riemannian gradient ascent on a representation-theoretic energy functional. We prove that, under random initialization, this flow drives each neuron to converge almost surely toward a single irreducible representation, while the cross-layer Fourier coefficients achieve a rotational rank-one alignment. This framework provides a representation-theoretic account of feature learning and characterizes a novel low-rank compression phenomenon for matrix-valued group representations. Moreover, for Abelian groups, we provide a complete population-level description: random initialization promotes uniform diversification across nontrivial representations and induces Haar-uniform phases, jointly approximating the indicator via a majority-vote mechanism. We further prove that both phase alignment and representation competition emerge with exponential convergence rates.
Time series classification is an important analytical task across diverse domains. However, its practical application is often hindered by the scarcity of labeled data and the requirement for substantial computational resources. To address these challenges, this paper proposes EvoTSC, a novel genetic programming approach designed to automatically evolve lightweight feature learning models for time series classification. The core of EvoTSC is a carefully designed multi-layer program structure that strategically embeds diverse forms of prior expert knowledge into the evolutionary process, effectively guiding the search toward operations known to be highly effective for time series analysis. To mitigate the common overfitting problem in time series classification, a tailored Pareto tournament selection strategy is proposed to favor models that perform consistently well across varying training data subsets, promoting the discovery of highly generalizable models. Extensive experiments conducted on univariate time series classification datasets demonstrate that EvoTSC significantly outperforms eleven benchmark methods in most comparisons. Further analyses verify the contribution of each component and the resource efficiency of the evolved models.
We show that replacing the rolling SVD of AdamW updates with a rolling SVD of loss gradients changes the diagnostic by 1-2 orders of magnitude. Performing SVD on the loss gradient instead of the AdamW update increases the measured perturbative coupling between SED directions and Linear Centroid Hypothesis (LCH) features from $ \bar{R}_k \approx 3 $--$9\times$ to $100$--$330\times$ across four single-task modular arithmetic operations, eliminating the apparent operation dependence in the original measurement. On a multitask transformer with a shared encoder, update-based SED gives $ \bar{R}_k \leq 1 $ -- an apparent failure of the diagnostic -- while per-operation gradient-based SED recovers $ \bar{R}_k = 20 $--$45\times$ across all four operations. Gradient aggregation across competing tasks is the main obstruction; performing SVD on per-task gradients resolves it. A causal intervention shows that constraining attention updates to any rank-3 subspace (whether SED-derived or random) accelerates grokking by approximately $2.3\times$ across random seeds and operations, while removing the rank-3 component has negligible effect under proper gradient-projection methodology. The SED-LCH coupling is therefore a strong diagnostic of where feature formation concentrates in parameter space, but it is not a unique causal pathway: the natural full-rank AdamW attention update is highly rank-redundant under our hyperparameters.