arXiv research

Study explores relationship between Hölder and FDPD divergences.

New ONMF model minimizes KL divergence for better sparse data modeling.

New probabilistic model for semi-nonnegative matrix factorization using Skellam distribution.

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

The paper develops new algorithms for KL-divergence NMF, proving convergence and performance.

New algorithm speeds up NMF with $β$-divergence.

Develops deep NMF models using β-divergences for feature extraction.

NNEinFact fits any nonnegative tensor factorization quickly and accurately.

The study proves inequalities for complex operators on curved spaces.

New integral estimates on substatic manifolds improve Alexandrov Theorem.

The multiplicative update (MU) algorithm has been extensively used to estimate the basis and coefficient matrices in nonnegative matrix factorization (NMF) problems under a wide range of divergences and regularizers. However, theoretical convergence guarantees have only been derived for a few special divergences withou…

We present a novel nonnegative tensor decomposition method, called Legendre decomposition, which factorizes an input tensor into a multiplicative combination of parameters. Thanks to the well-developed theory of information geometry, the reconstructed tensor is unique and always minimizes the KL divergence from an inpu…

This paper addresses the estimation of the latent dimensionality in nonnegative matrix factorization (NMF) with the β-divergence. The β-divergence is a family of cost functions that includes the squared Euclidean distance, Kullback-Leibler and Itakura-Saito divergences as special cases. Learning the model order is impo…

This paper introduces a robust mixing model to describe hyperspectral data resulting from the mixture of several pure spectral signatures. This new model not only generalizes the commonly used linear mixing model, but also allows for possible nonlinear effects to be easily handled, relying on mild assumptions regarding…

We develop a unified and systematic framework for performing online nonnegative matrix factorization under a wide variety of important divergences. The online nature of our algorithm makes it particularly amenable to large-scale data. We prove that the sequence of learned dictionaries converges almost surely to the set…

We study the problem of nonnegative rank-one approximation of a nonnegative tensor, and show that the globally optimal solution that minimizes the generalized Kullback-Leibler divergence can be efficiently obtained, i.e., it is not NP-hard. This result works for arbitrary nonnegative tensors with an arbitrary number of…

Nonnegative matrix factorization (NMF) is now a common tool for audio source separation. When learning NMF on large audio databases, one major drawback is that the complexity in time is O(FKN) when updating the dictionary (where (F;N) is the dimension of the input power spectrograms, and K the number of basis spectra),…

New divergence identity for scalar curvature helps prove rigidity of tensors.

The paper proves rigidity results for manifolds with special holonomy.

Paper proves convergence of warped product manifolds to a nonnegative scalar curvature limit.

In this paper, we extend the $β$-CNMF to two dimensions and derive exact multiplicative updates for its factors. The new updates generalize and correct the nonnegative matrix factor deconvolution previously proposed by Schmidt and Mørup. We show by simulation that the updates lead to a monotonically decreasing $β$-dive…

Proposes a new neural head for asymmetric representation learning.

We consider complete asymptotically flat Riemannian manifolds that are the graphs of smooth functions over $\mathbb R^n$. By recognizing the scalar curvature of such manifolds as a divergence, we express the ADM mass as an integral of the product of the scalar curvature and a nonnegative potential function, thus provin…

We investigate the $m$-relative entropy, which stems from the Bregman divergence, on weighted Riemannian and Finsler manifolds. We prove that the displacement $K$-convexity of the $m$-relative entropy is equivalent to the combination of the nonnegativity of the weighted Ricci curvature and the $K$-convexity of the weig…

The aim of this paper is to study complete (noncompact) steady $m$-quasi-Einstein manifolds satisfying a fourth-order vanishing condition on the Weyl tensor. In this case, we are able to prove that a steady $m$-quasi-Einstein manifold ($m>1$) on a simply connected $n$-dimensional manifold $(M^{n},g)$, $(n\geq4),$ with …

The study proves uniqueness of large isoperimetric sets in specific noncompact manifolds.

Non-negative matrix factorization (NMF) is a knowledge discovery method that is used in many fields. Variational inference and Gibbs sampling methods for it are also wellknown. However, the variational approximation error has not been clarified yet, because NMF is not statistically regular and the prior distribution us…

Considering a mixed signal composed of various audio sources and recorded with a single microphone, we consider on this paper the blind audio source separation problem which consists in isolating and extracting each of the sources. To perform this task, nonnegative matrix factorization (NMF) based on the Kullback-Leibl…

New tensors reveal full curvature structure from Riemann tensor.

Study optimal transport costs with zero MTW tensor, finding new families of costs and divergence functions.

We introduce a class of generalized relative entropies (inspired by the Bregman divergence in information theory) on the Wasserstein space over a weighted Riemannian or Finsler manifold. We prove that the convexity of all the entropies in this class is equivalent to the combination of the nonnegative weighted Ricci cur…

Study well-posedness of fast diffusion equation on noncompact manifolds.

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

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

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…

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

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

Sharp inequalities for manifolds with nonnegative curvature.

Study on Kähler manifolds with nonnegative Ricci curvature, focusing on rigidity.

We provide techniques for studying the nonnegatively curved left-invariant metrics on a compact Lie group. For "straight" paths of left-invariant metrics starting at bi-invariant metrics and ending at nonnegatively curved metrics, we deduce a nonnegativity property of the initial derivative of curvature. We apply this …

Survey on open manifolds with nonnegative Ricci curvature and open questions.

EGAB algorithms improve online portfolio selection.

We discuss the cobordism type of spin manifolds with nonnegative sectional curvature. We show that in each dimension $4k \geq 12$, there are infinitely many cobordism types of simply connected and nonnegatively curved spin manifolds. Moreover, we raise and analyze a question about possible cobordism obstructions to non…

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…

The exact nonnegative matrix factorization (exact NMF) problem is the following: given an $m$-by-$n$ nonnegative matrix $X$ and a factorization rank $r$, find, if possible, an $m$-by-$r$ nonnegative matrix $W$ and an $r$-by-$n$ nonnegative matrix $H$ such that $X = WH$. In this paper, we propose two heuristics for exac…

Given compact Lie groups H\subset G, we study the space of G-invariant metrics on G/H with nonnegative sectional curvature. For an intermediate subgroup K between H and G, we derive conditions under which enlarging the Lie algebra of K maintains nonnegative curvature on G/H. Such an enlarging is possible if (K,H) is a …

Huisken's isoperimetric mass is always nonnegative.

We study manifolds with almost nonnegative curvature operator (ANCO) and provide first examples of closed simply connected ANCO mannifolds that do not admit nonnegative curvature operator.