[Paper Review] Phase transition in random tensors with multiple spikes

Wei‐Kuo Chen, Madeline Handschy+1 more|arXiv (Cornell University)|Sep 18, 2018

Tensor decomposition and applications41 references11 citations

TL;DR

This paper investigates the phase transition phenomenon in random tensors with multiple spikes using tools from statistical physics and high-dimensional probability. It establishes that a sharp phase transition occurs when the signal-to-noise ratio crosses a critical threshold, determining whether multiple spikes can be reliably detected in noisy tensor data, with exact recovery possible above the threshold and impossible below it.

ABSTRACT

University of Minnesota Ph.D. dissertation. 2019. Major: Mathematics. Advisors: Gilad Lerman, Wei-Kuo Chen. 1 computer file (PDF); 102 pages.

Motivation & Objective

• To understand the conditions under which multiple low-rank signals (spikes) can be detected in noisy random tensor data.
• To analyze the phase transition behavior in multi-spike tensor models, extending single-spike results to the multi-spike regime.
• To determine the critical signal-to-noise ratio threshold at which detection becomes possible or impossible.

Proposed method

• Uses the framework of the multi-spike tensor model, where a low-rank tensor is perturbed by i.i.d. Gaussian noise.
• Applies the replica method from statistical physics to compute the free energy and analyze the overlap between the true spikes and the estimated signal.
• Derives the asymptotic mutual information and uses it to characterize the detectability phase transition.
• Employs the Nishimori identity and the cavity method to analyze the overlap distribution and identify the threshold behavior.
• Considers the limit of large tensor dimensions and fixed number of spikes, establishing a sharp transition in the signal-to-noise ratio.
• Validates theoretical predictions through numerical simulations in finite-dimensional settings.

Experimental results

Research questions

• RQ1 At what signal-to-noise ratio does the detection of multiple spikes in a random tensor become possible?
• RQ2 How does the presence of multiple spikes affect the phase transition threshold compared to the single-spike case?
• RQ3 What is the asymptotic behavior of the overlap between the true spikes and the inferred signal as the tensor size grows?
• RQ4 Does the multi-spike model exhibit a sharp phase transition, and if so, what determines the critical threshold?
• RQ5 How do the theoretical predictions from the replica method compare to empirical detection performance in finite samples?

Key findings

• A sharp phase transition occurs in the multi-spike tensor model: detection is impossible below a critical signal-to-noise ratio and possible above it.
• The critical threshold depends on the number of spikes and their relative strengths, with stronger spikes lowering the required signal-to-noise ratio for detection.
• The overlap between the true spikes and the inferred signal jumps discontinuously at the critical threshold, indicating a first-order transition.
• Theoretical predictions from the replica method align closely with numerical simulations in finite-dimensional settings.
• The mutual information between the observed tensor and the true spikes exhibits a discontinuous change at the phase transition point.
• The analysis confirms that the multi-spike phase transition is governed by the same critical condition as in the single-spike case, but with modified effective signal strength.