🤖 AI Summary
Existing algorithms for high-order tensor PCA—such as Approximate Message Passing (AMP)—exhibit a statistical-computational gap relative to the Sum-of-Squares (SOS) hierarchy.
Method: This paper introduces a family of Kikuchi-based algorithms inspired by statistical physics. It is the first to incorporate Kikuchi free energy modeling into algorithm design, characterizing *t*-wise variable dependencies via the Kikuchi Hessian spectrum and integrating linearized message passing with higher-order free energy approximations.
Contribution/Results: The proposed algorithm family is the first continuous-spectrum method achieving, in subexponential time, the exact statistical-computational trade-off of SOS. For arbitrary-order tensor PCA and even-*k* XOR problems, its polynomial-time variant attains SOS-optimal statistical accuracy. Its runtime–statistics trade-off curve matches SOS identically, and its core theoretical analysis is substantially simplified compared to prior approaches.
📝 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-t algorithm can be thought of as a linearized message-passing algorithm that keeps track of t-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.
