[Paper Review] Deterministic and Probabilistic Conditions for Finite Completability of Low-Tucker-Rank Tensor

Morteza Ashraphijuo, Vaneet Aggarwal+1 more|arXiv (Cornell University)|Dec 6, 2016

Tensor decomposition and applications60 references3 citations

TL;DR

This paper proposes deterministic and probabilistic conditions for finite and unique completability of low-Tucker-rank tensors using algebraic geometry on the Tucker manifold. By modeling sampled entries as polynomials in Tucker decomposition components, it establishes algebraic independence conditions that guarantee finite or unique completions, extending prior Grassmannian-based methods to handle multiple rank components in tensors.

ABSTRACT

We investigate the fundamental conditions on the sampling pattern, i.e., locations of the sampled entries, for finite completability of a low-rank tensor given some components of its Tucker rank. In order to find the deterministic necessary and sufficient conditions, we propose an algebraic geometric analysis on the Tucker manifold, which allows us to incorporate multiple rank components in the proposed analysis in contrast with the conventional geometric approaches on the Grassmannian manifold. This analysis characterizes the algebraic independence of a set of polynomials defined based on the sampling pattern, which is closely related to finite completion. Probabilistic conditions are then studied and a lower bound on the sampling probability is given, which guarantees that the proposed deterministic conditions on the sampling patterns for finite completability hold with high probability. Furthermore, using the proposed geometric approach for finite completability, we propose a sufficient condition on the sampling pattern that ensures there exists exactly one completion for the sampled tensor.

Motivation & Objective

• To identify deterministic necessary and sufficient conditions on sampling patterns for finite completability of low-Tucker-rank tensors.
• To extend algebraic geometric analysis from the Grassmannian manifold to the Tucker manifold to incorporate multiple rank components.
• To derive probabilistic sampling conditions ensuring finite completability with high probability.
• To propose a sufficient condition on sampling patterns that guarantees unique completion of the tensor.
• To overcome limitations of prior Grassmannian-based methods that cannot handle multiple rank constraints simultaneously.

Proposed method

• Model the low-Tucker-rank tensor completion problem by parameterizing the Tucker decomposition and expressing observed entries as polynomial equations in the decomposition components.
• Apply algebraic geometry to analyze the algebraic independence of these polynomials, which directly relates to finite completability.
• Use the concept of genericity to ensure that the polynomial system's structure reflects the underlying manifold geometry of the Tucker decomposition.
• Derive a lower bound on sampling probability that ensures the deterministic finite completability conditions hold with high probability.
• Introduce a canonical basis representation (Definition 12) to simplify the decomposition and facilitate algebraic analysis.
• Apply an extended Hall’s theorem framework to analyze rank constraints across multiple matricizations of the tensor.

Experimental results

Research questions

• RQ1 What deterministic sampling pattern ensures finite completability of a low-Tucker-rank tensor when multiple Tucker rank components are known?
• RQ2 How can algebraic geometry on the Tucker manifold be used to analyze finite completability beyond the Grassmannian manifold framework?
• RQ3 What is the minimal sampling probability that guarantees finite completability with high probability for low-Tucker-rank tensors?
• RQ4 Under what sampling conditions does a low-Tucker-rank tensor have a unique completion?
• RQ5 Can the proposed method handle multiple rank constraints simultaneously, unlike prior Grassmannian-based approaches?

Key findings

• The paper establishes that finite completability is equivalent to the algebraic independence of a set of polynomials derived from the sampling pattern and Tucker decomposition components.
• A sampling pattern ensures finite completability if and only if the corresponding polynomials are algebraically independent, providing a necessary and sufficient condition.
• The required number of samples per dimension is on the order of O(max{log(n), r}) for finite completability, significantly lower than optimization-based methods requiring O(log(n)r^{2.5}) samples.
• For the specific rank-2 matrix in Example 5, the analysis confirms exactly two possible completions, validating the method’s precision in counting solutions.
• The method enables unique completion when the sampling pattern satisfies a sufficient condition derived from the extended Hall’s theorem and algebraic independence.
• The probabilistic analysis provides a lower bound on sampling probability such that the deterministic finite completability conditions hold with high probability, enabling robust recovery under random sampling.