We give a short combinatorial proof that the nonnegative matrix factorization is an NP-hard problem. Moreover, we prove that NMF remains NP-hard when restricted to 01-matrices, answering a recent question of Moitra. The (exact) nonnegative matrix factorization is the following problem. Given an integer k and a matrix A with nonnegative entries, do there exist k nonnegative rank-one matrices tha...

Nonnegative matrix factorization has been offered as a fast and effective method for analyzing nonnegative two-mode proximity data. The goal is to structurally represent a nonnegative proximity matrix as the product of two lower-dimensional nonnegative matrices. Goodness of fit is typically measured as the sum of the squared deviations between the observed matrix elements and the estimated elem...

Where N is a nite set of the cardinality n and P the family of all its subsets, we study real functions on P having nonnegative diierences of orders n?2, n?1 and n. Nonnegative diierences of zeroth order, rst order, and second order may be interpreted as nonnegativity, nonincreasingness and convexity, respectively. If all diierences up to order n of a function are nonnegative, the set function ...

We say that a square real matrix $M$ is \emph{off-diagonal nonnegative} if and only all entries outside its diagonal are nonnegative numbers. In this note we show for any off-diagonal symmetric $M$, there exists $\widehat{M}$ which sparse close in spectrum to $M$.

We show that there exist real numbers λ1, λ2, . . . , λn that occur as the eigenvalues of an entry-wise nonnegative n-by-n matrix but do not occur as the eigenvalues of a symmetric nonnegative n-by-n matrix. This solves a problem posed by Boyle and Handelman, Hershkowitz, and others. In the process, recent work by Boyle and Handelman that solves the nonnegative inverse eigenvalue problem by app...

This paper approximates simulation models by B-splines with a penalty on high-order finite differences of the coefficients of adjacent B-splines. The penalty prevents overfitting. The simulation output is assumed to be nonnegative. The nonnegative spline simulation metamodel is casted as a second-order cone programming problem, which can be solved efficiently by modern optimization techniques. ...

R d with a Lévy density given by c|x| 1{|x|<1} for some constant c. In this paper we study the potential theory of truncated symmetric stable processes in detail. We prove a Harnack inequality for nonnegative harmonic nonnegative functions of these processes. We also establish a boundary Harnack principle for nonnegative functions which are harmonic with respect to these processes in bounded co...

We describe the convex set of the eigenvalues of hermitian matrices which are majorized by sum of m hermitian matrices with prescribed eigenvalues. We extend our characterization to selfadjoint nonnegative (definite) compact operators on a separable Hilbert space. We give necessary and sufficient conditions on the eigenvalue sequence of a selfadjoint nonnegative compact operator of trace class ...

In this paper we consider the study of the servomechanism problem for multi-input multi-output (MIMO) linear time-invariant (LTI) positive systems. In particular, we present results on the tracking problem of nonnegative constant reference signals for unknown stable MIMO positive LTI systems with nonnegative constant disturbances. Our results exploit the structure of tuning regulators and show ...

Nonnegative Matrix Factorization (NMF) can be used to approximate a large nonnegative matrix as a product of two smaller nonnegative matrices. This paper shows in detail how an NMF algorithm based on Newton iteration can be derived utilizing the general Karush-KuhnTucker (KKT) conditions for first-order optimality. This algorithm is suited for parallel execution on shared-memory systems. It was...