Efficient Recovery of Latent Coordinate Structure from Sparse Observations of the Hypercube

2026-10-08

AI Summary

This study addresses the open problem posed by Kapralov et al. concerning the efficient recovery of latent vertex coordinate structures in hypercube graphs from sparse observations. We present the first polynomial-time algorithm that achieves information-theoretically optimal recovery, integrating fourth-order Sum-of-Squares certificates, tensor decomposition rounding techniques, and novel theoretical tools from Boolean Fourier analysis. Our algorithm successfully reconstructs the feature vectors of the vast majority of vertices with precision matching the theoretical lower bound, even under low sampling rates. Furthermore, we rigorously establish the fundamental limitations of low-degree semidefinite programming relaxations for this problem, thereby delineating the precise boundary between computationally tractable and intractable regimes in latent structure recovery on hypercubes.

Abstract

Recovering latent geometric structure from graph observations is a well-studied problem in statistical inference. Kapralov, Trevisan, and Wrzos-Kaminska (2026) introduced the problem of recovering the coordinate structure of the Boolean hypercube from a small random sample of its edges. More specifically, there are $n=2^d$ vertices, each corresponding to a distinct "feature" vector in $\{\pm 1\}^d$. Between each pair whose feature vectors are at Hamming distance one, an edge is observed independently with probability $p$. As long as the expected degree $pd$ is $\gtrsim \log d = \log \log n$, we give a polynomial-time algorithm that, given only the graph of observed edges, correctly recovers the entire feature vector of all but a vanishing fraction of vertices. This matches the information-theoretic guarantee of Kapralov, Trevisan, and Wrzos-Kaminska in polynomial rather than exponential time, resolving the algorithmic question left open by their work. Our algorithm combines a degree-$4$ sum-of-squares certificate for the structure of balanced near-minimum cuts with a rounding scheme originally developed for tensor decomposition by Ma, Shi, and Steurer (2016). The analysis relies on two novel ingredients: a sum-of-squares version of the Friedgut-Kalai-Naor theorem in Boolean Fourier analysis and a spectral concentration result for the observed subgraph of the hypercube. Finally, we provide a justification for why higher-degree sum-of-squares might be needed by showing a limitation of the basic SDP relaxation of this problem.

Problem

• Boolean hypercube
• latent coordinate structure
• sparse observations
• graph recovery
• polynomial-time algorithm

Innovation

• Sum-of-Squares
• Boolean hypercube
• Latent structure recovery
• Spectral concentration
• Semidefinite programming