Related papers: SoS Certificates for Sparse Singular Values and Th…
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,…
Methods to certify the robustness of neural networks in the presence of input uncertainty are vital in safety-critical settings. Most certification methods in the literature are designed for adversarial or worst-case inputs, but researchers…
We present a new computational approach to approximating a large, noisy data table by a low-rank matrix with sparse singular vectors. The approximation is obtained from thresholded subspace iterations that produce the singular vectors…
A low rank matrix X has been contaminated by uniformly distributed noise, missing values, outliers and corrupt entries. Reconstruction of X from the singular values and singular vectors of the contaminated matrix Y is a key problem in…
We study the algorithmic problem of sparse mean estimation in the presence of adversarial outliers. Specifically, the algorithm observes a \emph{corrupted} set of samples from $\mathcal{N}(\mu,\mathbf{I}_d)$, where the unknown mean $\mu \in…
We propose an approximation method for thresholding of singular values using Chebyshev polynomial approximation (CPA). Many signal processing problems require iterative application of singular value decomposition (SVD) for minimizing the…
Modern state estimation is often formulated as an optimization problem and solved using efficient local search methods. These methods at best guarantee convergence to local minima, but, in some cases, global optimality can also be…
In this paper, we derive nearly tight probabilistic norm bounds for a class of random matrices we call graph matrices. While the classical case of symmetric matrices with independent random entries (Wigner's matrices) is a special case, in…
Many differentially private and classical non-private graph algorithms rely crucially on determining whether some property of each vertex meets a threshold. For example, for the $k$-core decomposition problem, the classic peeling algorithm…
We study certificates in static data structures. In the cell-probe model, certificates are the cell probes which can uniquely identify the answer to the query. As a natural notion of nondeterministic cell probes, lower bounds for…
The recently introduced notions of ranking functions and closure certificates utilize well-foundedness arguments to facilitate the verification of dynamical systems against $\omega$-regular properties. A ranking function and a closure…
We consider certificates of positivity for univariate polynomials with rational coefficients that are positive over (an interval of)~$\mathbb{R}$. Such certificates take the form of weighted sums of squares (SOS) of polynomials with…
A challenging problem in computational mathematics is to compute roots of a high-degree univariate random polynomial. We combine an efficient multiprecision implementation for solving high-degree random polynomials with two certification…
Sparse coding (SC) is attracting more and more attention due to its comprehensive theoretical studies and its excellent performance in many signal processing applications. However, most existing sparse coding algorithms are nonconvex and…
Models for image segmentation, node classification and many other tasks map a single input to multiple labels. By perturbing this single shared input (e.g. the image) an adversary can manipulate several predictions (e.g. misclassify several…
Motivated by a sampling problem basic to computational statistical inference, we develop a nearly optimal algorithm for a fundamental problem in spectral graph theory and numerical analysis. Given an $n\times n$ SDDM matrix ${\bf…
We study the rank of the Sum of Squares (SoS) hierarchy over the Boolean hypercube for Symmetric Quadratic Functions (SQFs) in $n$ variables with roots placed in points $k-1$ and $k$. Functions of this type have played a central role in…
We consider the matrix completion problem where the aim is to esti-mate a large data matrix for which only a relatively small random subset of its entries is observed. Quite popular approaches to matrix completion problem are iterative…
We discuss optimization problems over convex cones in which membership is difficult to verify directly. In the standard theory of duality, vectors in the dual cone $K^*$ are associated with separating hyperplanes and interpreted as…
In this work, we show that for all statistical estimation problems, a natural MMSE instability (discontinuity) condition implies the failure of stable algorithms, serving as a version of OGP for estimation tasks. Using this criterion, we…