DeepAI AI Chat
Log In Sign Up

When geometry meets optimization theory: partially orthogonal tensors

by   Ke Ye, et al.

Due to the multi-linearity of tensors, most algorithms for tensor optimization problems are designed based on the block coordinate descent method. Such algorithms are widely employed by practitioners for their implementability and effectiveness. However, these algorithms usually suffer from the lack of theoretical guarantee of global convergence and analysis of convergence rate. In this paper, we propose a block coordinate descent type algorithm for the low rank partially orthogonal tensor approximation problem and analyse its convergence behaviour. To achieve this, we carefully investigate the variety of low rank partially orthogonal tensors and its geometric properties related to the parameter space, which enable us to locate KKT points of the concerned optimization problem. With the aid of these geometric properties, we prove without any assumption that: (1) Our algorithm converges globally to a KKT point; (2) For any given tensor, the algorithm exhibits an overall sublinear convergence with an explicit rate which is sharper than the usual O(1/k) for first order methods in nonconvex optimization; (3) For a generic tensor, our algorithm converges R-linearly.


page 1

page 2

page 3

page 4


Jacobi-type algorithm for low rank orthogonal approximation of symmetric tensors and its convergence analysis

In this paper, we propose a Jacobi-type algorithm to solve the low rank ...

2D+3D facial expression recognition via embedded tensor manifold regularization

In this paper, a novel approach via embedded tensor manifold regularizat...

Tensor completion using enhanced multiple modes low-rank prior and total variation

In this paper, we propose a novel model to recover a low-rank tensor by ...

The Epsilon-Alternating Least Squares for Orthogonal Low-Rank Tensor Approximation and Its Global Convergence

The epsilon alternating least squares (ϵ-ALS) is developed and analyzed ...

Higher order Matching Pursuit for Low Rank Tensor Learning

Low rank tensor learning, such as tensor completion and multilinear mult...

Polar decomposition based algorithms on the product of Stiefel manifolds with applications in tensor approximation

In this paper, based on the matrix polar decomposition, we propose a gen...

Efficient coordinate-descent for orthogonal matrices through Givens rotations

Optimizing over the set of orthogonal matrices is a central component in...