Tensor decomposition is an important tool for big data analysis. In this paper,we resolve many of the key algorithmic questions regarding robustness, memoryefficiency, and differential privacy of tensor decomposition. We propose simplevariants of the tensor power method which enjoy these strong properties. We presentthe first guarantees for online tensor power method which has a...

This survey provides an overview of higher-order tensor decompositions, their applications, and available software. A tensor is a multidimensional or N-way array. Decompositions of higher-order tensors (i.e., N-way arrays with N ≥ 3) have applications in psychometrics, chemometrics, signal processing, numerical linear algebra, computer vision, numerical analysis, data mining, neuroscience, grap...

Developing efficient and guaranteed nonconvex algorithms has been an important challenge in modern machine learning. Algorithms with good empirical performance such as stochastic gradient descent often lack theoretical guarantees. In this paper, we analyze the class of homotopy or continuation methods for global optimization of nonconvex functions. These methods start from an objective function...

We introduce a Bayesian extension of the tensor factorization problem to multiple coupled tensors. For a single tensor it reduces to standard PARAFAC-type Bayesian factorization, and for two tensors it is the first Bayesian Tensor Canonical Correlation Analysis method. It can also be seen to solve a tensorial extension of the recent Group Factor Analysis problem. The method decomposes the set o...

In this paper, we propose a Riemannian framework for statistical analysis of tensor fields. Existing approaches to this problem have been mainly voxel-based that overlook the correlation between tensors at different voxels. In our approach, the tensor fields are considered as points in a high-dimensional Riemannian product space and accordingly, we extend Principal Geodesic Analysis (PGA) to th...

Matrix factorizations and their extensions to tensor factorizations and decompositions have become prominent techniques for linear and multilinear blind source separation (BSS), especially multiway Independent Component Analysis (ICA), Nonnegative Matrix and Tensor Factorization (NMF/NTF), Smooth Component Analysis (SmoCA) and Sparse Component Analysis (SCA). Moreover, tensor decompositions hav...

Tensors, or multi dimensional arrays, are receiving significant attentions due to the various types of data that can be modeled by them; examples include call graphs (sender, receiver, time), knowledge bases (subject, verb, object), 3-dimensional web graphs augmented with anchor texts, to name a few. Scalable tensor mining aims to extract important patterns and anomalies from a large amount of ...

Background The use of a medical simulator should enhance the goals of minimally invasive surgery: patient safety, cosmesis, shortening the length of hospital admissions, and reducing cost. Using an innovative approach to the handling of ultrasound images in virtual reality simulation, this article describes a process that employs a hybrid model of deformable models that can be applied in the te...