Related papers: Robust Smoothed Analysis of a Condition Number for…
The structured singular value $\mu_E$ for a linear subspace $E$ of $M_n(\mathbb C)$ is defined by \[ \mu_E(A)=1 / \inf\{\|X\| \ : \ X \in E, \ \det(I_n-AX)=0 \} \quad (A \in M_n(\mathbb{C})), \] and $\mu_E(A)=0$ if there is no $X \in E$…
We consider a square random matrix made by i.i.d. rows with any distribution and prove that, for any given dimension, the probability for the least singular value to be in [0; $\epsilon$) is at least of order $\epsilon$. This allows us to…
Under some mild Markov assumptions it is shown that the problem of designing optimal sequential tests for two simple hypotheses can be formulated as a linear program. The result is derived by investigating the Lagrangian dual of the…
We consider Continuous Linear Programs over a continuous finite time horizon $T$, with linear cost coefficient functions, linear right hand side functions, and a constant coefficient matrix, as well as their symmetric dual. We search for…
Given $X \subset R^n$, $\varepsilon \in (0,1)$, a parametrized family of probability distributions $(\mu\_{a})\_{a\in A}$ on $\Omega\subset R^p$, we consider the feasible set $X^*\_\varepsilon\subset X$ associated with the {\em…
Smoothed analysis of complexity bounds and condition numbers has been done, so far, on a case by case basis. In this paper we consider a reasonably large class of condition numbers for problems over the complex numbers and we obtain…
This paper presents a robust, distributed algorithm to solve general linear programs. The algorithm design builds on the characterization of the solutions of the linear program as saddle points of a modified Lagrangian function. We show…
We consider the Cauchy problem for the defocusing cubic nonlinear Schr\"odinger equation in four space dimensions and establish almost sure local well-posedness and conditional almost sure scattering for random initial data in…
The predictive quality of machine learning models is typically measured in terms of their (approximate) expected prediction accuracy or the so-called Area Under the Curve (AUC). Minimizing the reciprocals of these measures are the goals of…
Randomized smoothing provides strong, model-agnostic robustness certificates, but existing guarantees are limited to single modalities, treating continuous and discrete inputs in isolation. This limitation becomes critical in multimodal…
Randomized smoothing is the current state-of-the-art method for producing provably robust classifiers. While randomized smoothing typically yields robust $\ell_2$-ball certificates, recent research has generalized provable robustness to…
We consider the cubic nonlinear Schr\"odinger equation with a spatially rough potential, a key equation in the mathematical setup for nonlinear Anderson localization. Our study comprises two main parts: new optimal results on the…
We study randomized variants of two classical algorithms: coordinate descent for systems of linear equations and iterated projections for systems of linear inequalities. Expanding on a recent randomized iterated projection algorithm of…
Maintaining numerical stability in machine learning models is crucial for their reliability and performance. One approach to maintain stability of a network layer is to integrate the condition number of the weight matrix as a regularizing…
In this note we establish existence and uniqueness of weak solutions of linear elliptic equation $\text{div}[\mathbf{A}(x) \nabla u] = \text{div}{\mathbf{F}(x)}$, where the matrix $\mathbf{A}$ is just measurable and its skew-symmetric part…
Given Boolean functions \( f, g : \mathbb{F}_2^n \to \{-1,+1\} \), we say they are {\em linearly isomorphic} if there exists \( A \in \mathrm{GL}_n(\mathbb{F}_2) \) such that \( f(x)=g(Ax) \) for all \( x \). We study this problem in the…
Random constraint satisfaction problems (CSPs) are known to exhibit threshold phenomena: given a uniformly random instance of a CSP with $n$ variables and $m$ clauses, there is a value of $m = \Omega(n)$ beyond which the CSP will be…
We prove that for a residual (and hence dense) subset $\mathcal{G}$ of Riemannian metrics on $S^{n+1}$ in the $C^{3}$ topology, no area-minimizing integral $n$-current that is a boundary admits a singular tangent cone which is linearly…
Linear regression is a fundamental and primitive problem in supervised machine learning, with applications ranging from epidemiology to finance. In this work, we propose methods for speeding up distributed linear regression. We do so by…
Probabilistic smoothing is a standard tool for global optimization, but existing methods rely on Gaussian kernels and specific transforms, often resulting in strong hyperparameter sensitivity and limited robustness. We propose a general…