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In this article we study the problem of recovering the unknown solution of a linear ill-posed problem, via iterative regularization methods. We review the problem of projection-regularization from a statistical point of view. A basic…
LSQR and its mathematically equivalent CGLS have been popularly used over the decades for large-scale linear discrete ill-posed problems, where the iteration number $k$ plays the role of the regularization parameter. It has been long known…
We introduce a new method for sparse principal component analysis, based on the aggregation of eigenvector information from carefully-selected axis-aligned random projections of the sample covariance matrix. Unlike most alternative…
This paper considers the sparse generalized eigenvalue problem (SGEP), which aims to find the leading eigenvector with at most $k$ nonzero entries. SGEP naturally arises in many applications in machine learning, statistics, and scientific…
Recently, a class of algorithms combining classical fixed point iterations with repeated random sparsification of approximate solution vectors has been successfully applied to eigenproblems with matrices as large as $10^{108} \times…
Given an approximate eigenvector, its (standard) Rayleigh quotient and harmonic Rayleigh quotient are two well-known approximations of the corresponding eigenvalue. We propose a new type of Rayleigh quotient, the homogeneous Rayleigh…
Generalized eigenvalue problems (GEPs) find applications in various fields of science and engineering. For example, principal component analysis, Fisher's discriminant analysis, and canonical correlation analysis are specific instances of…
The entropy-based moment method is a well-known discretization for the velocity variable in kinetic equations which has many desirable theoretical properties but is difficult to implement with high-order numerical methods. The regularized…
This note considers the unstructured sparse recovery problems in a general form. Examples include rational approximation, spectral function estimation, Fourier inversion, Laplace inversion, and sparse deconvolution. The main challenges are…
This paper describes a set of rational filtering algorithms to compute a few eigenvalues (and associated eigenvectors) of non-Hermitian matrix pencils. Our interest lies in computing eigenvalues located inside a given disk, and the proposed…
In this article, we study eigenvalue problems associated to self-adjoint operators and their approximation obtained by subspace projection, as used in the reduced basis method for instance. We provide error bounds between the exact…
We develop an iterative refinement method that improves the accuracy of a user-chosen subset of $k$ eigenvectors ($k\ll n$) of an $n\times n$ real symmetric matrix. Using an orthogonal matrix represented in compact WY form, the method…
Many problems in physics, chemistry and other fields are perturbative in nature, i.e. differ only slightly from related problems with known solutions. Prominent among these is the eigenvalue perturbation problem, wherein one seeks the…
Allocation of samples in stratified and/or multistage sampling is one of the central issues of sampling theory. In a survey of a population often the constraints for precision of estimators of subpopulations parameters have to be taken care…
It is well known that a family of $n\times n$ commuting matrices can be simultaneously triangularized by a unitary similarity transformation. The diagonal entries of the triangular matrices define the $n$ joint eigenvalues of the family. In…
The recently developed data-driven eigenmatrix method shows very promising reconstruction accuracy in sparse recovery for a wide range of kernel functions and random sample locations. However, its current implementation can lead to…
The goal of this work is to obtain optimal rates for the convergence problem in mean field control. Our analysis covers cases where the solutions to the limiting problem may not be unique nor stable. Equivalently the value function of the…
We introduce a novel eigenvalue algorithm for near-diagonal matrices inspired by Rayleigh-Schr\"odinger perturbation theory and termed Iterative Perturbative Theory (IPT). Contrary to standard eigenvalue algorithms, which are either…
Recent work has shown that the (block) Lanczos algorithm can be used to extract approximate energy spectra and matrix elements from (matrices of) correlation functions in quantum field theory, and identified exact coincidences between…
The sparse portfolio selection problem is one of the most famous and frequently-studied problems in the optimization and financial economics literatures. In a universe of risky assets, the goal is to construct a portfolio with maximal…