arXiv subjects

Kevin Han Huang

Publications and source records attributed to Kevin Han Huang.

9 recordsLinked to original sources

arXiv2026

The Zero Pattern of a Design Matrix Drives Multiple Descent in Over-parameterized Regression

Over-parameterized linear regression has been widely studied over the last decade. However, most existing works assume that the covariates are independent and that their covariance matrices are non-degenerate. In this paper, we relax both assumptions and derive deterministic equivalents for the prediction risk in a vanishing-ridge regime. We show that degeneracy of the covariance matrices and dependence can lead to multiple descent, and characterize where the corresponding peaks can occur. Our proofs use a novel graph representation of the variance profile. We show that maximum matchings and the Dulmage--Mendelsohn decomposition of the associated bipartite graph identify the configurations at which the variance becomes singular.

math.ST↗

arXiv2026

Top Singular Value in Sum-Products of Random Matrices

We study the top singular value for a sum of $m$ independent $n \times n$ random matrices, each of which is a product of $N$ i.i.d. $n\times n$ Gaussian matrices. Our main conceptual observation is that when $m,n,N\rightarrow \infty$, the top singular value coincides with the partition function in a random energy model at the inverse temperature $\beta=\sqrt{2(N-1)/(n\log m)}$, with energies depending on the ratio $N/n$. We provide several non-asymptotic results making this approximation precise.

math.PR↗

arXiv2026

Data augmented bootstrap: Unifying confidence interval construction by approximate invariance

We propose the data augmented bootstrap (DAB), a framework for constructing confidence intervals from approximately invariant transformations of the data. As special cases, DAB recovers popular methods that rely on exact group symmetries, such as conformal prediction, wild bootstrap for Maximum Mean Discrepancy U-statistics and the recently proposed SymmPI. Meanwhile, DAB also recovers the classical bootstrap method, which exploits the dataset's approximate invariance under uniform sampling of data indices as the dataset size grows. For all DAB methods, we establish theoretical coverage results that interpolate between finite-sample and asymptotic guarantees according to the strength of the invariance, and without assuming a group structure. The approximate invariance is measured in the Kolmogorov distance and, for statistics that satisfy Gaussian universality, reduces to conditional mean and variance matching. This allows us to incorporate data augmentation (DA), a widely used machine learning heuristic based on approximate invariances, into known statistical methods. We empirically test the performance of incorporating DA into bootstrap, wild bootstrap and conformal prediction for simulated settings as well as for image, language and scientific data.

stat.ME↗

arXiv2025

Near-Optimality of Contrastive Divergence Algorithms

We perform a non-asymptotic analysis of the contrastive divergence (CD) algorithm, a training method for unnormalized models. While prior work has established that (for exponential family distributions) the CD iterates asymptotically converge at an $O(n^{-1 / 3})$ rate to the true parameter of the data distribution, we show, under some regularity assumptions, that CD can achieve the parametric rate $O(n^{-1 / 2})$. Our analysis provides results for various data batching schemes, including the fully online and minibatch ones. We additionally show that CD can be near-optimal, in the sense that its asymptotic variance is close to the Cramér-Rao lower bound.

stat.ML↗

arXiv2025

Diagonal Symmetrization of Neural Network Solvers for the Many-Electron Schrödinger Equation

Incorporating group symmetries into neural networks has been a cornerstone of success in many AI-for-science applications. Diagonal groups of isometries, which describe the invariance under a simultaneous movement of multiple objects, arise naturally in many-body quantum problems. Despite their importance, diagonal groups have received relatively little attention, as they lack a natural choice of invariant maps except in special cases. We study different ways of incorporating diagonal invariance in neural network ansätze trained via variational Monte Carlo methods, and consider specifically data augmentation, group averaging and canonicalization. We show that, contrary to standard ML setups, in-training symmetrization destabilizes training and can lead to worse performance. Our theoretical and numerical results indicate that this unexpected behavior may arise from a unique computational-statistical tradeoff not found in standard ML analyses of symmetrization. Meanwhile, we demonstrate that post hoc averaging is less sensitive to such tradeoffs and emerges as a simple, flexible and effective method for improving neural network solvers.

cs.LG↗

arXiv2025

Universality of High-Dimensional Logistic Regression and a Novel CGMT under Dependence with Applications to Data Augmentation

Over the last decade, a wave of research has characterized the exact asymptotic risk of many high-dimensional models in the proportional regime. Two foundational results have driven this progress: Gaussian universality, which shows that the asymptotic risk of estimators trained on non-Gaussian and Gaussian data is equivalent, and the convex Gaussian min-max theorem (CGMT), which characterizes the risk under Gaussian settings. However, these results rely on the assumption that the data consists of independent random vectors--an assumption that significantly limits its applicability to many practical setups. In this paper, we address this limitation by generalizing both results to the dependent setting. More precisely, we prove that Gaussian universality still holds for high-dimensional logistic regression under block dependence, $m$-dependence and special cases of mixing, and establish a novel CGMT framework that accommodates for correlation across both the covariates and observations. Using these results, we establish the impact of data augmentation, a widespread practice in deep learning, on the asymptotic risk.

math.ST↗

arXiv2024

Slow rates of approximation of U-statistics and V-statistics by quadratic forms of Gaussians

We construct examples of degree-two U- and V-statistics of $n$ i.i.d.~heavy-tailed random vectors in $\mathbb{R}^{d(n)}$, whose $ν$-th moments exist for ${ν> 2}$, and provide tight bounds on the error of approximating both statistics by a quadratic form of Gaussians. In the case ${ν=3}$, the error of approximation is $Θ(n^{-1/12})$. The proof adapts a result of Huang, Austern and Orbanz [12] to U- and V-statistics. The lower bound for U-statistics is a simple example of the concept of variance domination used in [12].

math.ST↗

arXiv2024

Gaussian universality for approximately polynomial functions of high-dimensional data

Gaussian universality results assert that the properties of many estimators remain unchanged when the input data are replaced by Gaussians. Such results have gained popularity in high-dimensional statistics and machine learning, as Gaussianity often substantially simplifies downstream analyses. Yet, an open question remains on when universality may cease to hold. To address this, we establish nearly optimal upper and lower bounds for Gaussian universality approximation, measured in Kolmogorov distance, over the class of approximately polynomial functions of high-dimensional random vectors. The upper bounds adapt the invariance principle of Mossel, O'Donnell and Oleszkiewicz (2010) for high-dimensional vectors and functions beyond multilinear forms. As applications, we obtain a delta method for high-dimensional data with non-Gaussian limits, a necessary and sufficient condition for asymptotic normality, and simple estimators that are asymptotically normal but for which bootstrap fails to be consistent. We also extend recent results on the high-dimensional degeneracy of non-degenerate U-statistics, phase transition of MMD in two-sample tests with imbalanced data, and confidence spheres for high-dimensional averages. Our lower bound is constructive and shows that, for polynomials of even degree $m$, universality holds up to $m=o(\log n)$. As a corollary, the Gaussian polynomial approximation error of $\Omega(n^{-1/6m})$ is not improvable for even-degree U-statistics and V-statistics. Our results also explain how universality results for U-statistics and V-statistics differ significantly in their dependence on dimensions.

math.PR↗

arXiv2022

Gaussian and Non-Gaussian Universality of Data Augmentation

We provide universality results that quantify how data augmentation affects the variance and limiting distribution of estimates through simple surrogates, and analyze several specific models in detail. The results confirm some observations made in machine learning practice, but also lead to unexpected findings: Data augmentation may increase rather than decrease the uncertainty of estimates, such as the empirical prediction risk. It can act as a regularizer, but fails to do so in certain high-dimensional problems, and it may shift the double-descent peak of an empirical risk. Overall, the analysis shows that several properties data augmentation has been attributed with are not either true or false, but rather depend on a combination of factors -- notably the data distribution, the properties of the estimator, and the interplay of sample size, number of augmentations, and dimension. As our main theoretical tool, we develop an adaptation of Lindeberg's technique for block dependence. The resulting universality regime may be Gaussian or non-Gaussian.

cs.LG↗