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

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.