# The Kikuchi Hierarchy and Tensor PCA

## TL;DR

This paper introduces a new hierarchy of algorithms inspired by statistical physics for tensor PCA, matching sum-of-squares performance with simpler proofs and extending to related problems like refuting random XOR formulas.

## Contribution

It proposes a Kikuchi Hessian-based hierarchy that generalizes belief propagation, achieving SOS-level performance for tensor PCA and related problems with simpler proofs.

## Key findings

- Hierarchy matches SOS performance for tensor PCA
- Provides polynomial-time algorithms with optimal runtime-statistical tradeoff
- Simplifies proofs and extends to refuting random XOR formulas

## Abstract

For the tensor PCA (principal component analysis) problem, we propose a new hierarchy of increasingly powerful algorithms with increasing runtime. Our hierarchy is analogous to the sum-of-squares (SOS) hierarchy but is instead inspired by statistical physics and related algorithms such as belief propagation and AMP (approximate message passing). Our level-$\ell$ algorithm can be thought of as a linearized message-passing algorithm that keeps track of $\ell$-wise dependencies among the hidden variables. Specifically, our algorithms are spectral methods based on the Kikuchi Hessian, which generalizes the well-studied Bethe Hessian to the higher-order Kikuchi free energies. It is known that AMP, the flagship algorithm of statistical physics, has substantially worse performance than SOS for tensor PCA. In this work we 'redeem' the statistical physics approach by showing that our hierarchy gives a polynomial-time algorithm matching the performance of SOS. Our hierarchy also yields a continuum of subexponential-time algorithms, and we prove that these achieve the same (conjecturally optimal) tradeoff between runtime and statistical power as SOS. Our proofs are much simpler than prior work, and also apply to the related problem of refuting random $k$-XOR formulas. The results we present here apply to tensor PCA for tensors of all orders, and to $k$-XOR when $k$ is even. Our methods suggest a new avenue for systematically obtaining optimal algorithms for Bayesian inference problems, and our results constitute a step toward unifying the statistical physics and sum-of-squares approaches to algorithm design.