Related papers: Dequantization and Hardness of Spectral Sum Estima…
Quantum error correction is indispensable to achieving reliable quantum computation. When quantum information is encoded redundantly, a larger Hilbert space is constructed using multiple physical qubits, and the computation is performed…
Recently, several works have shown that natural modifications of the classical conditional gradient method (aka Frank-Wolfe algorithm) for constrained convex optimization, provably converge with a linear rate when: i) the feasible set is a…
We study planted problems---finding hidden structures in random noisy inputs---through the lens of the sum-of-squares semidefinite programming hierarchy (SoS). This family of powerful semidefinite programs has recently yielded many new…
In the quest for quantum advantage, a central question is under what conditions can classical algorithms achieve a performance comparable to quantum algorithms--a concept known as dequantization. Random Fourier features (RFFs) have…
We present a general approach to rounding semidefinite programming relaxations obtained by the Sum-of-Squares method (Lasserre hierarchy). Our approach is based on using the connection between these relaxations and the Sum-of-Squares proof…
We study the decomposition of multivariate polynomials as sums of powers of linear forms. As one of our main results we give an algorithm for the following problem: given a homogeneous polynomial of degree 3, decide whether it can be…
Solving eigenproblem of the Laplacian matrix of a fully connected weighted graph has wide applications in data science, machine learning, and image processing, etc. However, this is very challenging because it involves expensive matrix…
In this paper, a fast algorithm for overcomplete sparse decomposition, called SL0, is proposed. The algorithm is essentially a method for obtaining sparse solutions of underdetermined systems of linear equations, and its applications…
Matrices with the displacement structures of circulant, Toeplitz, and Hankel types as well as matrices with structures generalizing these types are omnipresent in computations of sciences and engineering. In this paper, we present efficient…
Quantum computing holds the promise of solving computational mechanics problems in polylogarithmic time, meaning computational time scales as $\mathscr{O}((\log N)^c)$, where $N$ is the problem size and $c$ a constant. We propose a quantum…
The local Hamiltonian (LH) problem, the quantum analog of the classical constraint satisfaction problem, is a cornerstone of quantum computation and complexity theory. It is known to be QMA-complete, indicating that it is challenging even…
We prove a quantitative deterministic equivalence theorem for the logarithmic potentials of deterministic complex $N\times N$ matrices subject to small random perturbations. We show that with probability close to $1$ this log-potential is,…
We consider sparse matrix estimation where the goal is to estimate an $n\times n$ matrix from noisy observations of a small subset of its entries. We analyze the estimation error of the popularly utilized collaborative filtering algorithm…
Block encoding is a successful technique used in several powerful quantum algorithms. In this work we provide an explicit quantum circuit for block encoding a sparse matrix with a periodic diagonal structure. The proposed methodology is…
The ultimate goal of any sparse coding method is to accurately recover from a few noisy linear measurements, an unknown sparse vector. Unfortunately, this estimation problem is NP-hard in general, and it is therefore always approached with…
In this paper, we analyse classical variants of the Spectral Clustering (SC) algorithm in the Dynamic Stochastic Block Model (DSBM). Existing results show that, in the relatively sparse case where the expected degree grows logarithmically…
We present novel algorithmic techniques to efficiently verify the Kruskal rank of matrices that arise in sparse linear regression, tensor decomposition, and latent variable models. Our unified framework combines randomized hashing…
In this study, the problem of computing a sparse representation of multi-dimensional visual data is considered. In general, such data e.g., hyperspectral images, color images or video data consists of signals that exhibit strong local…
We present an algorithm computing the determinant of an integer matrix A. The algorithm is introspective in the sense that it uses several distinct algorithms that run in a concurrent manner. During the course of the algorithm partial…
This paper establishes a statistical versus computational trade-off for solving a basic high-dimensional machine learning problem via a basic convex relaxation method. Specifically, we consider the {\em Sparse Principal Component Analysis}…