Related papers: Spectrum Estimation from a Few Entries
We consider the twin problems of estimating the effective rank and the Schatten norms $\|{\bf A}\|_{s}$ of a rectangular $p\times q$ matrix ${\bf A}$ from noisy observations. When $s$ is an even integer, we introduce a polynomial-time…
Finding eigenvalue distributions for a number of sparse random matrix ensembles can be reduced to solving nonlinear integral equations of the Hammerstein type. While a systematic mathematical theory of such equations exists, it has not been…
On the heels of compressed sensing, a remarkable new field has very recently emerged. This field addresses a broad range of problems of significant practical interest, namely, the recovery of a data matrix from what appears to be…
The paper studies the problem of recovering a spectrally sparse object from a small number of time domain samples. Specifically, the object of interest with ambient dimension $n$ is assumed to be a mixture of $r$ complex multi-dimensional…
Spectral clustering is one of the most widely used techniques for extracting the underlying global structure of a data set. Compressed sensing and matrix completion have emerged as prevailing methods for efficiently recovering sparse and…
We present a new sublinear time algorithm for approximating the spectral density (eigenvalue distribution) of an $n\times n$ normalized graph adjacency or Laplacian matrix. The algorithm recovers the spectrum up to $\epsilon$ accuracy in…
The matrix recovery (completion) problem, a central problem in data science and theoretical computer science, is to recover a matrix $A$ from a relatively small sample of entries. While such a task is impossible in general, it has been…
We study how well one can recover sparse principal components of a data matrix using a sketch formed from a few of its elements. We show that for a wide class of optimization problems, if the sketch is close (in the spectral norm) to the…
In this paper, we consider matrix completion from non-uniformly sampled entries including fully observed and partially observed columns. Specifically, we assume that a small number of columns are randomly selected and fully observed, and…
We show how spectral functions for quantum impurity models can be calculated very accurately using a complete set of ``discarded'' numerical renormalization group eigenstates, recently introduced by Anders and Schiller. The only…
Compressed Sensing suggests that the required number of samples for reconstructing a signal can be greatly reduced if it is sparse in a known discrete basis, yet many real-world signals are sparse in a continuous dictionary. One example is…
Spectral estimators are fundamental in lowrank matrix models and arise throughout machine learning and statistics, with applications including network analysis, matrix completion and PCA. These estimators aim to recover the leading…
While Spectral Methods have long been used for Principal Component Analysis, this survey focusses on work over the last 15 years with three salient features: (i) Spectral methods are useful not only for numerical problems, but also discrete…
This is a survey about spectral sets, to appear in the second edition of Handbook of Linear Algebra (L. Hogben, ed.). Spectral sets and K-spectral sets, introduced by John von Neumann, offer a possibility to estimate the norm of functions…
Matrix completion algorithms recover a low rank matrix from a small fraction of the entries, each entry contaminated with additive errors. In practice, the singular vectors and singular values of the low rank matrix play a pivotal role for…
In many machine learning and data related applications, it is required to have the knowledge of approximate ranks of large data matrices at hand. In this paper, we present two computationally inexpensive techniques to estimate the…
This paper introduces a novel algorithm to approximate the matrix with minimum nuclear norm among all matrices obeying a set of convex constraints. This problem may be understood as the convex relaxation of a rank minimization problem, and…
We explore the impact of coarse quantization on matrix completion in the extreme scenario of dithered one-bit sensing, where the matrix entries are compared with time-varying threshold levels. In particular, instead of observing a subset of…
Many important problems are characterized by the eigenvalues of a large matrix. For example, the difficulty of many optimization problems, such as those arising from the fitting of large models in statistics and machine learning, can be…
Networked sensing, where the goal is to perform complex inference using a large number of inexpensive and decentralized sensors, has become an increasingly attractive research topic due to its applications in wireless sensor networks and…