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 alwaysminimizes the KL divergence from an input tensor. We empirically show that Legendre decomposition can more accurately ...

How can we decompose a data tensor if the indices are partially missing? Tensor decomposition is a fundamental tool to analyze the tensor data. Suppose, for example, we have a 3rd-order tensor X where each element Xijk takes 1 if user i posts word j at location k on Twitter. Standard tensor decomposition expects all the indices are observed. However, in some tweets, location k can be missing. I...

Tensor decomposition is a powerful tool for multiway data analysis. Many popular tensor decomposition approaches—such as the Tucker decomposition and CANDECOMP/PARAFAC (CP)—conduct multi-linear factorization. They are insufficient to model (i) complex interactions between data entities, (ii) various data types (e.g. missing data and binary data), and (iii) noisy observations and outliers. To ad...

In cosmological perturbation theory a first major step consists in the decomposition of the various perturbation amplitudes into scalar, vector and tensor perturbations, which mutually decouple. In performing this decomposition one uses – beside the Hodge decomposition for one-forms – an analogous decomposition of symmetric tensor fields of second rank on Riemannian manifolds with constant curv...

As it is known, groups of correlated 2D images of various kind could be represented as 3D images, which are mathematically described as 3 rd order tensors. Various generalizations of the Singular Value Decomposition (SVD) exist, aimed at the tensor description reduction. In this work, new approach is presented for 3 rd order tensor decomposition, where unlike the famous methods for decompositio...

We discuss a multilinear generalization of the singular value decomposition. There is a strong analogy between several properties of the matrix and the higher-order tensor decomposition; uniqueness, link with the matrix eigenvalue decomposition, first-order perturbation effects, etc., are analyzed. We investigate how tensor symmetries affect the decomposition and propose a multilinear generaliz...