arXiv subjects

Hung-Hsu Chou

Publications and source records attributed to Hung-Hsu Chou.

13 recordsLinked to original sources

arXiv2026

Compressive Sensing - Introduction and Relations to Deep Learning

Compressive sensing predicts that sparse vectors (signals) can be recovered from a small number of linear measurements via efficient algorithms. This finding, which dates back two decades, has triggered a paradigm shift in signal processing and initiated many developments both in practical signal-processing applications, such as medical imaging, radar, and astronomy, and on the theoretical side. More recently, seminal connections to the field of deep learning have led to further advances in the field, such as the use of unrolled neural networks for sparse recovery and the discovery that common training algorithms (variants of gradient descent) favor sparsity in overparameterized scenarios -- the so-called implicit bias phenomenon. This article gives an introduction to compressive sensing and outlines connections to deep learning. In particular, we will discuss generalization for neural networks generated by unrolling sparse recovery algorithms and implicit regularization for gradient descent applied to learning simplified linear neural networks.

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arXiv2026

Stability of Low-Rank Implicit Regularization in Perturbed Deep Matrix Factorization

This paper studies the stability of low-rank implicit regularization in deep matrix factorization, a tractable model for understanding how gradient-based training can favor low-complexity structure. We first revisit the noiseless setting and derive sufficient spectral conditions under which gradient descent exhibits a nonempty low-rank interval. These conditions clarify how the target spectrum, initialization, and step size jointly determine when a low-rank phase is observable along the optimization trajectory. We then analyze the perturbed problem, where the target matrix is subject to an additive perturbation. By studying the perturbed gradient descent dynamics at the eigenvalue level, we prove convergence guarantees and quantify how the perturbation size affects iteration complexity and eigenvalue recovery. Finally, we establish stability of the low-rank phase under perturbation: the effective rank of the iterates remains close to that of the rank-L approximation of the noiseless target over a perturbed low-rank interval, with explicit dependence on the perturbation size. Numerical illustrations support the theoretical predictions and illustrate the role of spectral structure in determining when this stability is observed.

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arXiv2026

How to induce regularization in linear models: A guide to reparametrizing gradient flow

In this work, we analyze the relation between reparametrizations of gradient flow and the induced implicit bias in linear models, which encompass various basic regression tasks. In particular, we study how reparametrization, loss function, and link function influence the convergence behavior and implicit bias of gradient flow. Our results provide conditions under which the implicit bias can be well-described and convergence of the flow is guaranteed. We furthermore show how to use these insights for designing reparametrization functions that lead to specific implicit biases which are closely connected to $\ell_p$- or trigonometric regularizers.

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arXiv2026

Conflicting Biases at the Edge of Stability: Norm versus Sharpness Regularization

The remarkable generalization properties of overparameterized networks are often attributed to implicit biases, such as norm minimization at small learning rates and low sharpness in the Edge-of-Stability regime. In this work, we argue that a comprehensive understanding of the generalization performance of gradient descent requires analyzing the interaction between these various forms of implicit regularization. We empirically demonstrate that the learning rate interpolates between low parameter norm and low sharpness of the trained model. We furthermore prove that neither implicit bias alone minimizes the generalization error for diagonal linear networks trained on a simple regression task. These findings demonstrate that focusing on a single implicit bias is insufficient to explain good generalization, and they motivate a broader view of implicit regularization that captures the dynamic trade-off between norm and sharpness induced by non-negligible learning rates.

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arXiv2026

Fast One-Pass Sparse Approximation of the Top Eigenvectors of Huge Approximately Low-Rank Matrices? Yes, $MAM^*$!

Motivated by applications such as sparse PCA, in this paper we present provably-accurate one-pass algorithms for the sparse approximation of the top eigenvectors of extremely massive matrices based on a single compact linear sketch. The resulting compressive-sensing-based approaches can approximate the leading eigenvectors of huge approximately low-rank matrices that are too large to store in memory based on a single pass over its entries while utilizing a total memory footprint on the order of the much smaller desired sparse eigenvector approximations. Finally, the compressive sensing recovery algorithm itself (which takes the gathered compressive matrix measurements as input, and then outputs sparse approximations of its top eigenvectors) can also be formulated to run in a time which principally depends on the size of the sought sparse approximations, making its runtime sublinear in the size of the large matrix whose eigenvectors one aims to approximate. Preliminary experiments on huge matrices having $\sim 10^{16}$ entries illustrate the developed theory and demonstrate the practical potential of the proposed approach.

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arXiv2026

GradPCA: Leveraging NTK Alignment for Reliable Out-of-Distribution Detection

We introduce GradPCA, an Out-of-Distribution (OOD) detection method that exploits the low-rank structure of neural network gradients induced by Neural Tangent Kernel (NTK) alignment. GradPCA applies Principal Component Analysis (PCA) to gradient class-means, achieving more consistent performance than existing methods across standard image classification benchmarks. We provide a theoretical perspective on spectral OOD detection in neural networks to support GradPCA, highlighting feature-space properties that enable effective detection and naturally emerge from NTK alignment. Our analysis further reveals that feature quality -- particularly the use of pretrained versus non-pretrained representations -- plays a crucial role in determining which detectors will succeed. Extensive experiments validate the strong performance of GradPCA, and our theoretical framework offers guidance for designing more principled spectral OOD detectors.

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arXiv2026

Latent Structure Emergence in Diffusion Models via Confidence-Based Filtering

Diffusion models rely on a high-dimensional latent space of initial noise seeds, yet it remains unclear whether this space contains sufficient structure to predict properties of the generated samples, such as their classes. In this work, we investigate the emergence of latent structure through the lens of confidence scores assigned by a pre-trained classifier to generated samples. We show that while the latent space appears largely unstructured when considering all noise realizations, restricting attention to initial noise seeds that produce high-confidence samples reveals pronounced class separability. By comparing class predictability across noise subsets of varying confidence and examining the class separability of the latent space, we find evidence of class-relevant latent structure that becomes observable only under confidence-based filtering. As a practical implication, we discuss how confidence-based filtering enables conditional generation as an alternative to guidance-based methods.

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arXiv2025

Get rid of your constraints and reparametrize: A study in NNLS and implicit bias

Over the past years, there has been significant interest in understanding the implicit bias of gradient descent optimization and its connection to the generalization properties of overparametrized neural networks. Several works observed that when training linear diagonal networks on the square loss for regression tasks (which corresponds to overparametrized linear regression) gradient descent converges to special solutions, e.g., non-negative ones. We connect this observation to Riemannian optimization and view overparametrized GD with identical initialization as a Riemannian GD. We use this fact for solving non-negative least squares (NNLS), an important problem behind many techniques, e.g., non-negative matrix factorization. We show that gradient flow on the reparametrized objective converges globally to NNLS solutions, providing convergence rates also for its discretized counterpart. Unlike previous methods, we do not rely on the calculation of exponential maps or geodesics. We further show accelerated convergence using a second-order ODE, lending itself to accelerated descent methods. Finally, we establish the stability against negative perturbations and discuss generalization to other constrained optimization problems.

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arXiv2025

More is Less: Inducing Sparsity via Overparameterization

In deep learning it is common to overparameterize neural networks, that is, to use more parameters than training samples. Quite surprisingly training the neural network via (stochastic) gradient descent leads to models that generalize very well, while classical statistics would suggest overfitting. In order to gain understanding of this implicit bias phenomenon we study the special case of sparse recovery (compressed sensing) which is of interest on its own. More precisely, in order to reconstruct a vector from underdetermined linear measurements, we introduce a corresponding overparameterized square loss functional, where the vector to be reconstructed is deeply factorized into several vectors. We show that, if there exists an exact solution, vanilla gradient flow for the overparameterized loss functional converges to a good approximation of the solution of minimal $\ell_1$-norm. The latter is well-known to promote sparse solutions. As a by-product, our results significantly improve the sample complexity for compressed sensing via gradient flow/descent on overparameterized models derived in previous works. The theory accurately predicts the recovery rate in numerical experiments. Our proof relies on analyzing a certain Bregman divergence of the flow. This bypasses the obstacles caused by non-convexity and should be of independent interest.

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arXiv2024

Robust Implicit Regularization via Weight Normalization

Overparameterized models may have many interpolating solutions; implicit regularization refers to the hidden preference of a particular optimization method towards a certain interpolating solution among the many. A by now established line of work has shown that (stochastic) gradient descent tends to have an implicit bias towards low rank and/or sparse solutions when used to train deep linear networks, explaining to some extent why overparameterized neural network models trained by gradient descent tend to have good generalization performance in practice. However, existing theory for square-loss objectives often requires very small initialization of the trainable weights, which is at odds with the larger scale at which weights are initialized in practice for faster convergence and better generalization performance. In this paper, we aim to close this gap by incorporating and analyzing gradient flow (continuous-time version of gradient descent) with weight normalization, where the weight vector is reparameterized in terms of polar coordinates, and gradient flow is applied to the polar coordinates. By analyzing key invariants of the gradient flow and using Lojasiewicz Theorem, we show that weight normalization also has an implicit bias towards sparse solutions in the diagonal linear model, but that in contrast to plain gradient flow, weight normalization enables a robust bias that persists even if the weights are initialized at practically large scale. Experiments suggest that the gains in both convergence speed and robustness of the implicit bias are improved dramatically by using weight normalization in overparameterized diagonal linear network models.

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arXiv2023

Neural (Tangent Kernel) Collapse

This work bridges two important concepts: the Neural Tangent Kernel (NTK), which captures the evolution of deep neural networks (DNNs) during training, and the Neural Collapse (NC) phenomenon, which refers to the emergence of symmetry and structure in the last-layer features of well-trained classification DNNs. We adopt the natural assumption that the empirical NTK develops a block structure aligned with the class labels, i.e., samples within the same class have stronger correlations than samples from different classes. Under this assumption, we derive the dynamics of DNNs trained with mean squared (MSE) loss and break them into interpretable phases. Moreover, we identify an invariant that captures the essence of the dynamics, and use it to prove the emergence of NC in DNNs with block-structured NTK. We provide large-scale numerical experiments on three common DNN architectures and three benchmark datasets to support our theory.

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arXiv2023

Gradient Descent for Deep Matrix Factorization: Dynamics and Implicit Bias towards Low Rank

In deep learning, it is common to use more network parameters than training points. In such scenarioof over-parameterization, there are usually multiple networks that achieve zero training error so that thetraining algorithm induces an implicit bias on the computed solution. In practice, (stochastic) gradientdescent tends to prefer solutions which generalize well, which provides a possible explanation of thesuccess of deep learning. In this paper we analyze the dynamics of gradient descent in the simplifiedsetting of linear networks and of an estimation problem. Although we are not in an overparameterizedscenario, our analysis nevertheless provides insights into the phenomenon of implicit bias. In fact, wederive a rigorous analysis of the dynamics of vanilla gradient descent, and characterize the dynamicalconvergence of the spectrum. We are able to accurately locate time intervals where the effective rankof the iterates is close to the effective rank of a low-rank projection of the ground-truth matrix. Inpractice, those intervals can be used as criteria for early stopping if a certain regularity is desired. Wealso provide empirical evidence for implicit bias in more general scenarios, such as matrix sensing andrandom initialization. This suggests that deep learning prefers trajectories whose complexity (measuredin terms of effective rank) is monotonically increasing, which we believe is a fundamental concept for thetheoretical understanding of deep learning.

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arXiv2021

Overparameterization and generalization error: weighted trigonometric interpolation

Motivated by surprisingly good generalization properties of learned deep neural networks in overparameterized scenarios and by the related double descent phenomenon, this paper analyzes the relation between smoothness and low generalization error in an overparameterized linear learning problem. We study a random Fourier series model, where the task is to estimate the unknown Fourier coefficients from equidistant samples. We derive exact expressions for the generalization error of both plain and weighted least squares estimators. We show precisely how a bias towards smooth interpolants, in the form of weighted trigonometric interpolation, can lead to smaller generalization error in the overparameterized regime compared to the underparameterized regime. This provides insight into the power of overparameterization, which is common in modern machine learning.

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