从超立方体的稀疏观测中高效恢复潜在坐标结构

中文摘要 AI 辅助

从图观测中恢复潜在几何结构是统计推断中广泛研究的问题。Kapralov、Trevisan和Wrzos-Kaminska（2026）提出了从布尔超立方体的少量随机边样本中恢复其坐标结构的问题。更具体地说，存在n=2^d个顶点，每个顶点对应{±1}^d中的一个不同“特征”向量；对于特征向量汉明距离为1的每对顶点，以概率p独立观测到一条边。只要期望度数pd≳log d=log log n，我们就给出了一个多项式时间算法，该算法仅在给定观测边构成的图的情况下，能正确恢复除可忽略分数顶点外所有顶点的完整特征向量。这在多项式时间而非指数时间内达到了Kapralov等人提出的信息论保证，解决了他们工作中遗留的算法问题。我们的算法将用于平衡近最小割结构的4阶平方和（sum-of-squares）证书，与Ma、Shi和Steurer（2016）最初为张量分解开发的舍入方案相结合。分析依赖于两个新要素：布尔傅里叶分析中Friedgut-Kalai-Naor定理的平方和版本，以及超立方体观测子图的谱集中结果。最后，我们通过展示该问题基本半定规划（SDP）松弛的局限性，解释了为何可能需要更高阶的平方和。

英文摘要

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.

发表机构

• EPFL（洛桑联邦理工学院）

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