We characterize the first-order sensitivity of approximately recovering a low-rank matrix from linear measurements, standard problem in compressed sensing. A special case covered by our analysis is approximating an incomplete matrix. This one customary approach to build recommender systems. give algorithm for computing associated condition number and demonstrate experimentally how measurements ...

Properties of nonlinear perfect binary codes are investigated and several new constructions of perfect codes are derived from these properties. An upper bound on the cardinality of the intersection of two perfect codes of length n is presented, and perfect codes whose intersection attains the upper bound are constructed for all n. As an immediate consequence of the proof of the upper bound we o...

ui = Mvi ‖Mvi‖ = Mvi √ λi = Mvi σi Note that singular values σi are equal to √ λi; since M M is PSD, λi ≥ 0 and σi is well defined. In particular, observe that if M is a symmetric matrix, σi is the absolute value of the i-th eigenvalue of M . Now, we want to show that these vis and uis meet SVD conditions. Recall that vi’s are orthonormal because they are eigenvectors of MM , and uis are orthon...

Matrix Factorization (MF) is among the most widely used techniques for collaborative filtering based recommendation. Along this line, a critical demand is to incrementally refine the MF models when new ratings come in an online scenario. However, most of existing incremental MF algorithms are limited by specific MF models or strict use restrictions. In this paper, we propose a general increment...

We shall show that the Picard number of the generic fiber of an abelian fibered hyperkähler manifold over the projective space is always one. We then give a few applications for the Mordell-Weil group. In particular, by deforming O’Grady’s 10dimensional manifold, we construct an abelian fibered hyperkähler manifold of MordellWeil rank 20, which is the maximum possible among all known ones.

Earlier work has shown that no extension of the Eckart–Young SVD approximation theorem can be made to the strong orthogonal rank tensor decomposition. Here, we present a counterexample to the extension of the Eckart–Young SVD approximation theorem to the orthogonal rank tensor decomposition, answering an open question previously posed by Kolda [SIAM J. Matrix Anal. Appl., 23 (2001), pp. 243–355].

The purpose of this paper is to present a largely self-contained proof of the singular value decomposition (SVD), and to explore its application to the low rank approximation problem. We begin by proving background concepts used throughout the paper. We then develop the SVD by way of the polar decomposition. Finally, we show that the SVD can be used to achieve the best low rank approximation of...