Related papers: Rate-Optimal Perturbation Bounds for Singular Subs…
Witsenhausen's problem asks for the maximum fraction $\alpha_n$ of the $n$-dimensional unit sphere that can be covered by a measurable set containing no pairs of orthogonal points. The best upper bounds for $\alpha_n$ are given by…
Constructing an efficient parameterization of a large, noisy data set of points lying close to a smooth manifold in high dimension remains a fundamental problem. One approach consists in recovering a local parameterization using the local…
In this paper, we study high-dimensional sparse Quadratic Discriminant Analysis (QDA) and aim to establish the optimal convergence rates for the classification error. Minimax lower bounds are established to demonstrate the necessity of…
We consider mappings satisfying an upper bound for the distortion of families of curves. We establish lower bounds for the distortion of distances under such mappings. As applications, we obtain theorems on the discreteness of the limit…
We perturb a real matrix $A$ of full column rank, and derive lower bounds for the smallest singular values of the perturbed matrix, in terms of normwise absolute perturbations. Our bounds, which extend existing lower-order expressions,…
During recent years the interest of optimization and machine learning communities in high-probability convergence of stochastic optimization methods has been growing. One of the main reasons for this is that high-probability complexity…
The eigenproblem of low-rank updated matrices are of crucial importance in many applications. Recently, an upper bound on the number of distinct eigenvalues of a perturbed matrix was established. The result can be applied to estimate the…
Our aim in the current article is to extend the developments in Kruger, Ngai & Th\'era, SIAM J. Optim. 20(6), 3280-3296 (2010) and, more precisely, to characterize, in the Banach space setting, the stability of the local and global error…
This paper revisits the rate-distortion theory from the perspective of optimal weak transport, as recently introduced by Gozlan et al. While the conditions for optimality and the existence of solutions are well-understood in the case of…
Combined perturbation bounds are presented for eigenvalues and eigenspaces of Hermitian matrices or singular values and singular subspaces of general matrices. The bounds are derived based on the smooth decompositions and elementary…
Understanding how the optimal value of an optimisation problem changes when its input data is modified is an old question in mathematical optimisation. This paper investigates the computation of the optimal values of a family of (possibly…
The use of Variational Autoencoders in different Machine Learning tasks has drastically increased in the last years. They have been developed as denoising, clustering and generative tools, highlighting a large potential in a wide range of…
In this paper, we study the relative perturbation bounds for joint eigenvalues of commuting tuples of normal $n \times n$ matrices. Some Hoffman-Wielandt type relative perturbation bounds are proved using the Clifford algebra technique. A…
We establish the optimal nonergodic sublinear convergence rate of the proximal point algorithm for maximal monotone inclusion problems. First, the optimal bound is formulated by the performance estimation framework, resulting in an infinite…
This paper derives a perturbation bound on the optimal subspace estimator obtained from a subset of its canonical projections contaminated by noise. This fundamental result has important implications in matrix completion, subspace…
We develop a new technique for constructing sparse graphs that allow us to prove near-linear lower bounds on the round complexity of computing distances in the CONGEST model. Specifically, we show an $\widetilde{\Omega}(n)$ lower bound for…
This paper is concerned with the analysis of the randomized subspace iteration for the computation of low-rank approximations. We present three different kinds of bounds. First, we derive both bounds for the canonical angles between the…
Optimization under structural constraints is typically analyzed through projection or penalty methods, obscuring the geometric mechanism by which constraints shape admissible dynamics. We propose an operator-theoretic formulation in which…
In this article we derive an (almost) optimal scaling law for a singular perturbation problem associated with the Tartar square. As in \cite{W97, C99}, our upper bound quantifies the well-known construction which is used in the literature…
This paper develops a spatially resolved perturbation theory for singular vectors under high-dimensional separable noise and applies it to data-driven matrix recovery. In the asymptotic regime where the matrix dimensions are proportional…