Related papers: Von Neumann Entropy Penalization and Low Rank Matr…
We propose a general framework for reconstructing and denoising single entries of incomplete and noisy entries. We describe: effective algorithms for deciding if and entry can be reconstructed and, if so, for reconstructing and denoising…
We analyze quantum state tomography in scenarios where measurements and states are both constrained. States are assumed to live in a semi-algebraic subset of state space and measurements are supposed to be rank-one POVMs, possibly with…
Rank estimation is a classical model order selection problem that arises in a variety of important statistical signal and array processing systems, yet is addressed relatively infrequently in the extant literature. Here we present sample…
The entropy accumulation theorem states that the smooth min-entropy of an $n$-partite system $A = (A_1, \ldots, A_n)$ is lower-bounded by the sum of the von Neumann entropies of suitably chosen conditional states up to corrections that are…
A general many quantiles + noise model is studied in the robust formulation (allowing non-normal, non-independent observations), where the identifiability requirement for the noise is formulated in terms of quantiles rather than the…
We study modeling and identification of processes with a spectral density matrix of low rank. Equivalently, we consider processes having an innovation of reduced dimension for which Prediction Error Methods (PEM) algorithms are not directly…
Quantum information-theoretic approach has been identified as a way to understand the foundations of quantum mechanics as early as 1950 due to Shannon. However there hasn't been enough advancement or rigorous development of the subject. In…
The problem of estimating the smallest singular value of random square matrices is important in connection with matrix computations and analysis of the spectral distribution. In this survey, we consider recent developments in the study of…
Intuitively, if a density operator has small rank, then it should be easier to estimate from experimental data, since in this case only a few eigenvectors need to be learned. We prove two complementary results that confirm this intuition.…
We provide new general kernel selection rules thanks to penalized least-squares criteria. We derive optimal oracle inequalities using adequate concentration tools. We also investigate the problem of minimal penalty as described in [BM07].
We propose a quantum algorithm for finding eigenvalues of non-unitary matrices. We show how to construct, through interactions in a quantum system and projective measurements, a non-Hermitian or non-unitary matrix and obtain its eigenvalues…
We study the Riemannian optimization methods on the embedded manifold of low rank matrices for the problem of matrix completion, which is about recovering a low rank matrix from its partial entries. Assume $m$ entries of an $n\times n$ rank…
Several quantities of interest in quantum information, including entanglement and purity, are nonlinear functions of the density matrix and cannot, even in principle, correspond to proper quantum observables. Any method aimed to determine…
An algorithm to classify a general Hermitian matrix according to its signature (positive semi-definite, negative or indefinite) is presented. It builds on the Quantum Phase Estimation algorithm, which stores the sign of the eigenvalues of a…
This paper deals with the trace regression model where $n$ entries or linear combinations of entries of an unknown $m_1\times m_2$ matrix $A_0$ corrupted by noise are observed. We propose a new nuclear norm penalized estimator of $A_0$ and…
We propose a unified framework for estimating low-rank matrices through nonconvex optimization based on gradient descent algorithm. Our framework is quite general and can be applied to both noisy and noiseless observations. In the general…
The problem of estimating ARMA models is computationally interesting due to the nonconcavity of the log-likelihood function. Recent results were based on the convex minimization. Joint model selection using penalization by a convex norm,…
Classification is a machine learning method used in many practical applications: text mining, handwritten character recognition, face recognition, pattern classification, scene labeling, computer vision, natural langage processing. A…
We consider the problem of noisy matrix completion, in which the goal is to reconstruct a structured matrix whose entries are partially observed in noise. Standard approaches to this underdetermined inverse problem are based on assuming…
In classical physics, entropy quantifies the randomness of large systems, where the complete specification of the state, though possible in theory, is not possible in practice. In quantum physics, despite its inherently probabilistic…