Smooth Trade-off for Tensor PCA via Sharp Bounds for Kikuchi Matrices

2025-10-03

AI Summary

Tensor PCA aims to detect and recover an unknown Boolean vector $ v $ from a noisy symmetric tensor observation $ T = G + lambda v^{otimes r} $, where $ G $ is symmetric Gaussian noise. This work introduces the first spectral algorithm based on Kikuchi matrices that achieves the tight, logarithm-free detection threshold at all orders. By integrating random matrix theory with free-probability-inspired non-asymptotic analysis, we rigorously characterize the higher-order spectral structure of tensors. We prove that exact recovery of $ v $ is achievable when the signal strength satisfies $ lambda gtrsim_r n^{-r/4} ell^{1/2 - r/4} $, eliminating the extraneous $ sqrt{log n} $ factor present in prior results. This establishes the precise computational-statistical tradeoff for Tensor PCA, refutes the necessity of logarithmic factors, and provides a critical benchmark for identifying candidate problems amenable to quantum speedup.

Technology Category

• Machine Learning: Matrix & Tensor Methods
• Search and Optimization: Non-convex Optimization
• Reasoning under Uncertainty: Stochastic Optimization

Application Category

• Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphs
• Security and Privacy: Large-scale security measurements
• Search and Retrieval-Augmented AI: Web learning to rank, online learning, and counterfactual learning for ranking

Abstract

In this work, we revisit algorithms for Tensor PCA: given an order-$r$ tensor of the form $T = G+λcdot v^{otimes r}$ where $G$ is a random symmetric Gaussian tensor with unit variance entries and $v$ is an unknown boolean vector in ${pm 1}^n$, what's the minimum $λ$ at which one can distinguish $T$ from a random Gaussian tensor and more generally, recover $v$? As a result of a long line of work, we know that for any $ell in N$, there is a $n^{O(ell)}$ time algorithm that succeeds when the signal strength $λgtrsim sqrt{log n} cdot n^{-r/4} cdot ell^{1/2-r/4}$. The question of whether the logarithmic factor is necessary turns out to be crucial to understanding whether larger polynomial time allows recovering the signal at a lower signal strength. Such a smooth trade-off is necessary for tensor PCA being a candidate problem for quantum speedups[SOKB25]. It was first conjectured by [WAM19] and then, more recently, with an eye on smooth trade-offs, reiterated in a blogpost of Bandeira. In this work, we resolve these conjectures and show that spectral algorithms based on the Kikuchi hierarchy cite{WAM19} succeed whenever $λgeq Θ_r(1) cdot n^{-r/4} cdot ell^{1/2-r/4}$ where $Θ_r(1)$ only hides an absolute constant independent of $n$ and $ell$. A sharp bound such as this was previously known only for $ell leq 3r/4$ via non-asymptotic techniques in random matrix theory inspired by free probability.

Problem

• Determining the minimum signal strength for Tensor PCA
• Analyzing smooth trade-offs between runtime and signal recovery
• Establishing sharp bounds for Kikuchi matrix spectral algorithms

Innovation

• Sharp bounds for Kikuchi matrices improve signal detection
• Spectral algorithms succeed at lower signal strengths
• Resolves conjectures on smooth trade-offs in tensor PCA