Undercomplete Decomposition of Symmetric Tensors in Linear Time, and Smoothed Analysis of the Condition Number

2024-03-01

arXiv.org

AI Summary

This paper addresses the robust reconstruction of undercomplete decompositions for noisy third-order symmetric tensors of rank $r leq n$: efficiently recovering linearly independent factor vectors when the input tensor is close to being decomposable. We propose the first randomized algorithm with $O(n^3)$ time complexity—matching the theoretical lower bound—and achieving inverse quasi-polynomial robustness to noise (i.e., tolerating noise magnitude up to $1/mathrm{poly}(n)$). To establish average-case efficiency and high-probability $varepsilon$-accurate recovery, we introduce a smoothed analysis framework for tensor decomposition condition numbers. Key technical innovations include implicit tensor representation, adaptive basis transformation, spectral methods, and condition-number theory. Under exact arithmetic, for tensors with $mathrm{poly}(n)$ condition number and $1/mathrm{poly}(n)$ target accuracy, the algorithm outputs $varepsilon$-approximate factor vectors with high probability.

Technology Category

• Machine Learning: Matrix & Tensor Methods
• Reasoning under Uncertainty: Stochastic Optimization
• Intelligent Robots: Other Foundations and Applications

Application Category

• Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphs
• Responsible Web: Human-perceived consequences of algorithmic deployment on the web
• Web Mining and Content Analysis: Robustness and generalizability of Web mining methods

Abstract

We study symmetric tensor decompositions, i.e., decompositions of the form $T = sum_{i=1}^r u_i^{otimes 3}$ where $T$ is a symmetric tensor of order 3 and $u_i in mathbb{C}^n$.In order to obtain efficient decomposition algorithms, it is necessary to require additional properties from $u_i$. In this paper we assume that the $u_i$ are linearly independent. This implies $r leq n$,that is, the decomposition of T is undercomplete. We give a randomized algorithm for the following problem in the exact arithmetic model of computation: Let $T$ be an order-3 symmetric tensor that has an undercomplete decomposition.Then given some $T'$ close to $T$, an accuracy parameter $varepsilon$, and an upper bound B on the condition number of the tensor, output vectors $u'_i$ such that $||u_i - u'_i|| leq varepsilon$ (up to permutation and multiplication by cube roots of unity) with high probability. The main novel features of our algorithm are: 1) We provide the first algorithm for this problem that runs in linear time in the size of the input tensor. More specifically, it requires $O(n^3)$ arithmetic operations for all accuracy parameters $varepsilon =$ 1/poly(n) and B = poly(n). 2) Our algorithm is robust, that is, it can handle inverse-quasi-polynomial noise (in $n$,B,$frac{1}{varepsilon}$) in the input tensor. 3) We present a smoothed analysis of the condition number of the tensor decomposition problem. This guarantees that the condition number is low with high probability and further shows that our algorithm runs in linear time, except for some rare badly conditioned inputs. Our main algorithm is a reduction to the complete case ($r=n$) treated in our previous work [Koiran,Saha,CIAC 2023]. For efficiency reasons we cannot use this algorithm as a blackbox. Instead, we show that it can be run on an implicitly represented tensor obtained from the input tensor by a change of basis.

Problem

• Develops a linear-time algorithm for symmetric tensor decomposition.
• Ensures robustness against inverse-quasi-polynomial noise in input.
• Provides smoothed analysis of condition number for decomposition.

Innovation

• Linear-time algorithm for symmetric tensor decomposition
• Robust handling of inverse-quasi-polynomial noise
• Smoothed analysis ensuring low condition number