Related papers: Functions preserving positive definiteness for spa…
We introduce a new consistency-based approach for defining and solving nonnegative/positive matrix and tensor completion problems. The novelty of the framework is that instead of artificially making the problem well-posed in the form of an…
Many problems in nonlinear analysis and optimization, among them variational inequalities and minimization of convex functions, can be reduced to finding zeros (namely, roots) of set-valued operators. Hence numerous algorithms have been…
The low-rank matrix completion problem can be succinctly stated as follows: given a subset of the entries of a matrix, find a low-rank matrix consistent with the observations. While several low-complexity algorithms for matrix completion…
Many problems of theoretical and practical interest involve finding an optimum over a family of convex functions. For instance, finding the projection on the convex functions in $H^k(\Omega)$, and optimizing functionals arising from some…
In this paper, we focus on approximating a natural class of functions that are compositions of smooth functions. Unlike the low-dimensional support assumption on the covariate, we demonstrate that composition functions have an intrinsic…
Let $\mathbf{A}_{n,m;k}$ be a random $n \times m$ matrix with entries from some field $\mathbb{F}$ where there are exactly $k$ non-zero entries in each column, whose locations are chosen independently and uniformly at random from the set of…
In this paper, we develop verifiable and computable performance analysis of sparsity recovery. We define a family of goodness measures for arbitrary sensing matrices as a set of optimization problems, and design algorithms with a…
We investigate positive definiteness, Hurwitz stability and Schur stability of parametric interval matrices. We give a verifiable sufficient condition for positive definiteness of parametric interval matrices with non-linear dependencies.…
In this paper, we investigate the relationship between positive definite functions on the unit sphere $\sph$ and on the Euclidean space $\RR^d$. For the dimension $d$ to be odd, a new technique is developed to establish the inheritance of…
The relation between representations and positive definite functions is a key concept in harmonic analysis on topological groups. Recently this relation has been studied on topological groupoids. This is the second in a series of papers in…
Let $\mathcal{P}$ be a property of function $\mathbb{F}_p^n \to \{0,1\}$ for a fixed prime $p$. An algorithm is called a tester for $\mathcal{P}$ if, given a query access to the input function $f$, with high probability, it accepts when $f$…
Submodularity is a discrete domain functional property that can be interpreted as mimicking the role of the well-known convexity/concavity properties in the continuous domain. Submodular functions exhibit strong structure that lead to…
Incomplete factorizations have long been popular general-purpose algebraic preconditioners for solving large sparse linear systems of equations. Guaranteeing the factorization is breakdown free while computing a high quality preconditioner…
We resolve a conjecture of Cooper-Fenner-Purewal that a certain sequence of combinatorial matrices which can be used to bound small product-Ramsey numbers is positive semidefinite. Because the connection to Ramsey Theory involves solving…
This article is centered around generalizing a previous implicit function theorem of the author to be applicable for maps f:E sqcap F to F which can be lifted to Keller C^k_pi maps f_i:E sqcap F_i to F_i with F_i Banach and F=projlim F_i .…
The approximation properties of the finite element method can often be substantially improved by choosing smooth high-order basis functions. It is extremely difficult to devise such basis functions for partitions consisting of arbitrarily…
The paper studies coincidence points of parameterized set-valued mappings (multifunctions), which provide an extended framework to cover several important topics in variational analysis and optimization that include the existence of…
Sparse covariance matrices play crucial roles by encoding the interdependencies between variables in numerous fields such as genetics and neuroscience. Despite substantial studies on sparse covariance matrices, existing methods face several…
We give new characterizations for matrix monotonicity and convexity of fixed order which connects previous characterizations by Loewner, Dobsch, Donoghue, Kraus and Bendat--Sherman. The ideas introduced are then used to characterize matrix…
Copositive and completely positive matrices play an increasingly important role in Applied Mathematics, namely as a key concept for approximating NP-hard optimization problems. The cone of copositive matrices of a given order and the cone…