Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Probabilistic bounds on best rank-one approximation ratio

Khazhgali Kozhasov, Josué Tonelli-Cueto

Abstract

We provide new upper and lower bounds on the minimum possible ratio of the spectral and Frobenius norms of a (partially) symmetric tensor. In the particular case of general tensors our result recovers a known upper bound. For symmetric tensors our upper bound unveils that the ratio of norms has the same order of magnitude as the trivial lower bound $1/\sqrt{n^{d-1}}$, when the order of a tensor $d$ is fixed and the dimension of the underlying vector space $n$ tends to infinity. However, when $n$ is fixed and $d$ tends to infinity, our lower bound is better than $1/\sqrt{n^{d-1}}$.

Probabilistic bounds on best rank-one approximation ratio

Abstract

We provide new upper and lower bounds on the minimum possible ratio of the spectral and Frobenius norms of a (partially) symmetric tensor. In the particular case of general tensors our result recovers a known upper bound. For symmetric tensors our upper bound unveils that the ratio of norms has the same order of magnitude as the trivial lower bound , when the order of a tensor is fixed and the dimension of the underlying vector space tends to infinity. However, when is fixed and tends to infinity, our lower bound is better than .
Paper Structure (19 sections, 18 theorems, 145 equations)

This paper contains 19 sections, 18 theorems, 145 equations.

Table of Contents

  1. Introduction
  2. General tensors
  3. Symmetric tensors
  4. Partially symmetric tensors
  5. Preliminaries
  6. Symmetric tensors and homogeneous polynomials
  7. Harmonic polynomials
  8. Probabilistic models
  9. Upper bounds
  10. Subgaussian estimates for the projection of a Gaussian vector
  11. Estimates for a random Lipschitz map on a product of spheres
  12. Upper bound for general tensors
  13. Upper bounds for symmetric tensors
  14. Upper bounds for partially symmetric tensors
  15. Lower bounds for (partially) symmetric tensors
  16. ...and 4 more sections

Key Result

Theorem 1.1

For any $d\geq 3$ and $\boldsymbol{n}=(n_1,\dots, n_d)$ with $n_1,\dots, n_d\geq 2$ we have

Theorems & Definitions (40)

  • Theorem 1.1
  • Theorem 1.2
  • Remark 1.3
  • Theorem 1.4
  • Lemma 2.1
  • proof
  • Proposition 3.1
  • Remark 3.2
  • Remark 3.3
  • Proposition 3.4
  • ...and 30 more