arXiv research

The key condition A3w of Ma, Trudinger and Wang for regularity of optimal transportation maps is implied by the nonnegativity of a pseudo-Riemannian curvature -- which we call cross-curvature -- induced by the transportation cost. For the Riemannian distance squared cost, it is shown that (1) cross-curvature nonnegativ…

Paper proposes ONTD for nonnegative tensor data.

Sharp inequalities in nonnegative Ricci curvature spaces using mass transport.

An algorithm simplifies optimization with nonnegative and orthogonal constraints.

New algorithm for nonnegative tensor completion with linear convergence rate.

Paper uses ABP method to prove logarithmic Sobolev inequalities on curved spaces.

Nonnegative matrix factorization (NMF) is a powerful tool for data mining. However, the emergence of `big data' has severely challenged our ability to compute this fundamental decomposition using deterministic algorithms. This paper presents a randomized hierarchical alternating least squares (HALS) algorithm to comput…

New algorithm improves clustering accuracy without sacrificing scalability.

Proves optimal volume growth for certain nonnegative Ricci curvature manifolds.

Normalized nonnegative models assign probability distributions to users and random variables to items; see [Stark, 2015]. Rating an item is regarded as sampling the random variable assigned to the item with respect to the distribution assigned to the user who rates the item. Models of that kind are highly expressive. F…

New method reduces computational cost for nonnegative low rank matrix approximation.

New NMF method tackles nonnegative data with separability relaxed.

The paper proves rigidity results for manifolds with nonnegative scalar curvature.

A Bayesian approach termed BAyesian Least Squares Optimization with Nonnegative L1-norm constraint (BALSON) is proposed. The error distribution of data fitting is described by Gaussian likelihood. The parameter distribution is assumed to be a Dirichlet distribution. With the Bayes rule, searching for the optimal parame…

The paper proves a Minkowski inequality on specific Riemannian manifolds.

The nonnegative matrix factorization is a widely used, flexible matrix decomposition, finding applications in biology, image and signal processing and information retrieval, among other areas. Here we present a related matrix factorization. A multi-objective optimization problem finds conical combinations of templates …

Nonnegative CANDECOMP/PARAFAC (NCP) decomposition is an important tool to process nonnegative tensor. Sometimes, additional sparse regularization is needed to extract meaningful nonnegative and sparse components. Thus, an optimization method for NCP that can impose sparsity efficiently is required. In this paper, we co…

Unified approach to analysis on Ricci nonnegative manifolds and flows using optimal transport.

Given a matrix $M$ (not necessarily nonnegative) and a factorization rank $r$, semi-nonnegative matrix factorization (semi-NMF) looks for a matrix $U$ with $r$ columns and a nonnegative matrix $V$ with $r$ rows such that $UV$ is the best possible approximation of $M$ according to some metric. In this paper, we study th…

Nonnegative low-rank matrix recovery can have spurious local minima.

Proposes a variational NNCC formulation for infinite dimensions.

We show that one-dimensional circle is the only case for closed smooth metric measure spaces with nonnegative Bakry-Émery Ricci curvature whose spectrum of the weighted Laplacian has an optimal positive upper bound. This result extends the work of Hang-Wang in the manifold case (Int. Math. Res. Not. 18 (2007), Art. ID …

Nonnegative Matrix Factorization (NMF) has been a popular representation method for pattern classification problem. It tries to decompose a nonnegative matrix of data samples as the product of a nonnegative basic matrix and a nonnegative coefficient matrix, and the coefficient matrix is used as the new representation. …

In this paper, we propose a general framework to accelerate significantly the algorithms for nonnegative matrix factorization (NMF). This framework is inspired from the extrapolation scheme used to accelerate gradient methods in convex optimization and from the method of parallel tangents. However, the use of extrapola…

Nonnegative Matrix Factorization (NMF) is a widely used technique in many applications such as face recognition, motion segmentation, etc. It approximates the nonnegative data in an original high dimensional space with a linear representation in a low dimensional space by using the product of two nonnegative matrices. …

In this paper, we study global properties of continuous plurisubharmonic functions on complete noncompact Kähler manifolds with nonnegative bisectional curvature and their applications to the structure of such manifolds. We prove that continuous plurisubharmonic functions with reasonable growth rate on such manifolds c…

Symmetric nonnegative matrix factorization has found abundant applications in various domains by providing a symmetric low-rank decomposition of nonnegative matrices. In this paper we propose a Frank-Wolfe (FW) solver to optimize the symmetric nonnegative matrix factorization problem under a simplicial constraint, whic…

The study finds a limit on the volume growth of certain 3-manifolds.

New method for NMF without tuning parameter.

Method uses NMF for clustering with partial distance measurements.

In this paper we consider complete noncompact Riemannian manifolds $(M, g)$ with nonnegative Ricci curvature and Euclidean volume growth, of dimension $n \geq 3$. We prove a sharp Willmore-type inequality for closed hypersurfaces $\partial Ω$ in $M$, with equality holding true if and only if $(M{\setminus}Ω, g)$ is iso…

On a closed weighted Riemannian manifold with nonnegative Bakry-Émery Ricci curvature, it is shown that the ratio of the $k$-th to first eigenvalues of the weighted Laplacian is dominated by $641k^2$, using an argument via the Cheeger constant. While improving the previous exponential upper bound, the order of $k$ here…

We consider dimension reduction for solutions of the Kähler-Ricci flow with nonegative bisectional curvature. When the complex dimension $n=2$, we prove an optimal dimension reduction theorem for complete translating Kähler-Ricci solitons with nonnegative bisectional curvature. We also prove a general dimension reducti…

State spaces of multifactor approximations of nonnegative Volterra processes are linear transformations of the nonnegative orthant.

The paper studies nonlinear mass concepts in 3-manifolds with nonnegative scalar curvature.

Fixed points of nonnegative neural networks are analyzed using fixed point theory.

The SCMU algorithm computes cone factorizations for symmetric cones, improving upon existing methods.

In this paper, we propose a new fast and robust recursive algorithm for near-separable nonnegative matrix factorization, a particular nonnegative blind source separation problem. This algorithm, which we refer to as the successive nonnegative projection algorithm (SNPA), is closely related to the popular successive pro…

SON-NMF estimates nonnegative rank on-the-fly for NMF.

We prove short time existence for the Ricci flow on open manifolds of nonnegative complex sectional curvature. We do not require upper curvature bounds. By considering the doubling of convex sets contained in a Cheeger-Gromoll convex exhaustion and solving the singular initial value problem for the Ricci flow on these …

This paper develops a low-nonnegative-rank approximation method to identify the state aggregation structure of a finite-state Markov chain under an assumption that the state space can be mapped into a handful of meta-states. The number of meta-states is characterized by the nonnegative rank of the Markov transition mat…

Neural NMF discovers hierarchical topics in multilayer data.

Nonnegative matrix factorization (NMF) has become a very popular technique in machine learning because it automatically extracts meaningful features through a sparse and part-based representation. However, NMF has the drawback of being highly ill-posed, that is, there typically exist many different but equivalent facto…

Sharp rigidity theorem for quasilinear Liouville equation on manifolds with nonnegative Ricci curvature.

In this article, we will use inverse mean curvature flow to establish an optimal Sobolev-type inequality for hypersurfaces $Σ$ with nonnegative sectional curvature in $\mathbb{H}^n$. As an application, we prove the hyperbolic Alexandrov-Fenchel inequalities for hypersurfaces with nonnegative sectional curvature in $\ma…

Study Euler characteristic of manifolds with almost nonnegative curvature operator, showing nonnegativity under certain conditions.

The paper introduces MU for NMF with $β$-divergences and disjoint constraints.

Study open Alexandrov spaces with nonnegative curvature, proving structural results.