A dimension-free version of the retained-energy form of the Mallat--Zeitouni conjecture is established, showing that the KL basis is within this factor of the optimal basis, and shows that the possible advantage of optimizing over all orthonormal bases vanishes as $d$ grows.
Abstract
Principal component analysis (PCA) is optimal for the linear reconstruction of Gaussian data, a foundational property underlying its central role in algorithms and signal processing. Its nonlinear analogue, however, is notoriously subtle: in 2011, Mallat and Zeitouni conjectured that the Karhunen--Lo\`eve (KL) basis remains optimal even when the retained coordinates are chosen adaptively per sample, a property that would theoretically justify the ubiquitous pipeline of PCA followed by sparse thresholding. In this paper, we establish a $1+O(1/\sqrt{d})$-approximate version of the retained-energy form of the Mallat--Zeitouni conjecture, showing that the KL basis is within this factor of the optimal basis. This dimension-free comparison depends only on the number of retained coordinates and shows that the possible advantage of optimizing over all orthonormal bases vanishes as $d$ grows. It complements the universal-constant reconstruction-error comparison of Litvak and Tikhomirov (Ann. Appl. Probab., 2018), while providing a comparison naturally suited for algorithmic analysis. Our proof rests on a clean, conceptual reduction: we relax arbitrary rotations to a deterministic threshold bound via Schur--Horn majorization, and identify the remaining loss with the correlation gap of the rank-$d$ uniform matroid over Gaussian level sets.
A universal, sample-optimal convergence theorem for the original BIHT algorithm is proved and a scalar lower bound is proved showing that any nontrivial corruption pattern, even one that involves only one flipped sign together with one clean sign, forces the iterates to oscillate indefinitely.
Exhaustive moment fitting in this constant-dimensional space produces a proper mixture and, together with the dimension-free moment characterization of Gaussian mixtures, achieves the optimal Hellinger rate in polynomial arithmetic time for every fixed $k$.
We study minimum-norm interpolation (MNI) in overparameterized linear regression with isotropic Gaussian covariates, in settings where the MNI has no closed-form formula. Whereas most prior work relied on Gaussian comparison tools such as the convex Gaussian min--max theorem (CGMT), our approach uses tools from high-dimensional geometry and probability. First, when the norm is in isotropic position, we obtain an ``offset''bound that controls the amount by which the MNI shrinks the ground truth. Second, we show that the ``intrinsic''variance of the $\ell_1$-MNI is at most $O(\tfrac{1}{n\log(d/n)^2})$, using a variant of Talagrand's $L_1$--$L_2$ inequality due to Cordero-Erausquin and Ledoux [2012], together with a classical result of Gluskin [1988]. We recover the sharp mean-squared error (MSE) bound for the $\ell_1$-MNI obtained by Wang et al. [2022], using the work of Fleury [2012] on the symmetric Gaussian polytope, which is defined via \[ P_{n,d} := \mathrm{conv}\{\pm X_i\}_{i=1}^{d} \text{ where } X_i \overset{\mathrm{i.i.d.}}{\sim} N(0,\mathrm{I}_{n \times n}), \] rather than CGMT. Our methods also imply improvements on previous results in high-dimensional geometry that may be of independent interest. First, we show that with overwhelming probability, the ratio between the isotropic constant of $P_{n,d}$ and that of the Euclidean ball in $\mathbb{R}^n$ is at most $1+O((\log(d/n))^{-2})$, improving a result of Klartag and Kozma [2009]. We also establish a refined weighted thin-shell estimate on $P_{n,d}$, and provide an elementary proof of the main theorem of Fleury [2012].
We study the squared $2$-Wasserstein distance to the standard Gaussian as a non-Gaussianity criterion and use it for linear Independent Component Analysis (ICA) and causal discovery in Linear Non-Gaussian Acyclic Models (LiNGAM). Unlike commonly used parametric contrasts and approximations of information-theoretic quantities, this criterion requires no distributional regularity beyond finite second moments, involves neither approximation nor tuning parameters, and can be computed exactly and efficiently from empirical order statistics. Our analysis relies on a strict subadditivity property of the $2$-Wasserstein distance to the Gaussian. At the population level, we prove exact identification of the ICA unmixing matrix, up to signed permutation, and give an analogous characterization of causal orders through sequential least-squares residuals. We then define empirical plug-in estimators and prove distribution-free uniform convergence under finite-moment assumptions, before detailing three practical solvers: a Picard-style orthogonal optimizer for ICA, an exhaustive dynamic program for causal order search, and a greedy order search variant. Empirically, we demonstrate competitive performance for both tasks and provide open-source implementations for source separation and causal discovery.
F'elix Laplante, C. Ambroise, Pierre Humbert· 1 citation
Wiener-Hermite cross-correlation identification represents a polynomial response in the Hermite basis. Under Gaussian excitation the basis is orthogonal and a diagonal rule recovers it exactly; under non-Gaussian excitation the same basis is kept, but its Gram matrix gains off-diagonal terms and the diagonal rule is no longer the population projection. We give the exact finite-order excess $L^2(P)$ risk of this mismatch: a moment quadratic form from two Hankel-Cholesky factorizations and one diagonal solve, at $O(s^3)$ cost from moments to order $2s$. Closed cumulant forms at orders three and four expose which non-Gaussian features drive it; symmetry protects the Gaussian basis only through order two. A bootstrap decides, from data, whether a matched basis is worth building; on a Wiener-Hammerstein benchmark it separates a near-Gaussian channel (penalty $\approx 10^{-4}$) from a skewed output (penalty $0.05$). The computation is a weighted-$L^2$ projection whose core normal-system correspondence is machine-checked in Lean 4.
We study the recovery of sparse functions from finite, noisy, and indirect observations in the framework of statistical inverse learning. The unknown is modeled as an element of $\ell^1$, and observations are generated through a possibly nonlinear forward operator $A:\ell^1\to H$, where $H$ is a vector-valued reproducing kernel Hilbert space. We propose an $\ell^1$-regularized empirical risk minimizer and develop a theoretical analysis of its statistical properties. Under mild assumptions, we establish almost-sure consistency and derive non-asymptotic high-probability convergence rates in both the prediction and $\ell^1$ reconstruction norms. The rates depend on the source smoothness parameter $r$, characterized by a variational source condition, and the effective dimension exponent $b$, describing the polynomial spectral decay of the covariance operator. We further prove matching minimax lower bounds, showing that the obtained convergence rates are optimal. To relate the theory to practical sparsity models, we consider finitely smoothing operators of the form $A=G\circ S$, where $S$ is a synthesis operator, and show that approximation-space assumptions imply the required variational source conditions. In particular, we prove that membership in the approximation space $k_t$ is equivalent to polynomial decay of the best $n$-term approximation error. Finally, we verify the assumptions for two representative inverse problems: reaction coefficient identification in elliptic PDEs and sparse computed tomography. For filtered Radon transforms, we derive explicit effective-dimension asymptotics, yielding concrete convergence rates for standard image models and sparsifying systems.
Abhishake Rastogi, T. Bubba, T. Helin et al.· 0 citations