Trading off 1-norm and sparsity against rank for linear models using mathematical optimization: 1-norm minimizing partially reflexive ah-symmetric generalized inverses

The M-P (Moore-Penrose) pseudoinverse has as a key application the computation of least-squares solutions of inconsistent systems of linear equations. Irrespective of whether a given input matrix is sparse, its M-P pseudoinverse can be dense, potentially leading to high computational burden, especially when we are dealing with high-dimensional matrices. The M-P pseudoinverse is uniquely characterized by four properties, but only two of them need to be satisfied for the computation of least-squares solutions. Fampa and Lee (2018) and Xu, Fampa, Lee, and Ponte (2019) propose local-search procedures to construct sparse block-structured generalized inverses that satisfy the two key M-P properties, plus one more (the so-called reflexive property). That additional M-P property is equivalent to imposing a minimum-rank condition on the generalized inverse. (Vector) 1-norm minimization is used to induce sparsity and, importantly, to keep the magnitudes of entries under control for the generalized-inverses constructed. Here, we investigate the trade-off between low 1-norm and low rank for generalized inverses that can be used in the computation of least-squares solutions. We propose several algorithmic approaches that start from a $1$-norm minimizing generalized inverse that satisfies the two key M-P properties, and gradually decrease its rank, by iteratively imposing the reflexive property. The algorithms iterate until the generalized inverse has the least possible rank. During the iterations, we produce intermediate solutions, trading off low 1-norm (and typically high sparsity) against low rank

The Residual Method for Regularizing Ill-Posed Problems

Although the \emph{residual method}, or \emph{constrained regularization}, is frequently used in applications, a detailed study of its properties is still missing. This sharply contrasts the progress of the theory of Tikhonov regularization, where a series of new results for regularization in Banach spaces has been published in the recent years. The present paper intends to bridge the gap between the existing theories as far as possible. We develop a stability and convergence theory for the residual method in general topological spaces. In addition, we prove convergence rates in terms of (generalized) Bregman distances, which can also be applied to non-convex regularization functionals. We provide three examples that show the applicability of our theory. The first example is the regularized solution of linear operator equations on $L^p$-spaces, where we show that the results of Tikhonov regularization generalize unchanged to the residual method. As a second example, we consider the problem of density estimation from a finite number of sampling points, using the Wasserstein distance as a fidelity term and an entropy measure as regularization term. It is shown that the densities obtained in this way depend continuously on the location of the sampled points and that the underlying density can be recovered as the number of sampling points tends to infinity. Finally, we apply our theory to compressed sensing. Here, we show the well-posedness of the method and derive convergence rates both for convex and non-convex regularization under rather weak conditions.Comment: 29 pages, one figur

Nearest Neighbor Methods for Testing Reflexivity and Species-Correspondence

Nearest neighbor (NN) methods are employed for drawing inferences about spatial patterns of points from two or more classes. We consider Pielou's test of niche specificity which is defined using a contingency table based on the NN relationships between the data points. We demonstrate that Pielou's contingency table for niche specificity is actually more appropriate for testing reflexivity in NN structure, hence we call this table as NN reflexivity contingency table (NN-RCT) henceforth. We also derive an asymptotic approximation for the distribution of the entries of the NN-RCT and consider variants of Fisher's exact test on it. Moreover, we introduce a new test of class- or species-correspondence inspired by spatial niche/habitat specificity and the associated contingency table called species-correspondence contingency table (SCCT). We also determine the appropriate null hypotheses and the underlying conditions appropriate for these tests. We investigate the finite sample performance of the tests in terms of empirical size and power by extensive Monte Carlo simulations and the methods are illustrated on a real-life ecological data set.Comment: 23 pages, 1 figur