Konstantin Tikhomirov

Dylan J. Altschuler, Konstantin Tikhomirov

Determining the capacity $α_c$ of the Binary Perceptron is a long-standing problem. Krauth and Mezard (1989) conjectured an explicit value of $α_c$, approximately equal to .833, and a rigorous lower bound matching this prediction was recently established by Ding and Sun (2019). Regarding the upper bound, Kim and Roche (1998) and Talagrand (1999) independently showed that $α_c$ < .996, while Krauth and Mezard outlined an argument which can be used to show that $α_c$ < .847. The purpose of this expository note is to record a complete proof of the bound $α_c$ < .847. The proof is a conditional first moment method combined with known results on the spherical perceptron

Cheng Mao, Mark Rudelson, Konstantin Tikhomirov

This paper deals with the problem of graph matching or network alignment for Erdős--Rényi graphs, which can be viewed as a noisy average-case version of the graph isomorphism problem. Let $G$ and $G'$ be $G(n, p)$ Erdős--Rényi graphs marginally, identified with their adjacency matrices. Assume that $G$ and $G'$ are correlated such that $\mathbb{E}[G_{ij} G'_{ij}] = p(1-α)$. For a permutation $π$ representing a latent matching between the vertices of $G$ and $G'$, denote by $G^π$ the graph obtained from permuting the vertices of $G$ by $π$. Observing $G^π$ and $G'$, we aim to recover the matching $π$. In this work, we show that for every $\varepsilon \in (0,1]$, there is $n_0>0$ depending on $\varepsilon$ and absolute constants $α_0, R > 0$ with the following property. Let $n \ge n_0$, $(1+\varepsilon) \log n \le np \le n^{\frac{1}{R \log \log n}}$, and $0 < α< \min(α_0,\varepsilon/4)$. There is a polynomial-time algorithm $F$ such that $\mathbb{P}\{F(G^π,G')=π\}=1-o(1)$. This is the first polynomial-time algorithm that recovers the exact matching between vertices of correlated Erdős--Rényi graphs with constant correlation with high probability. The algorithm is based on comparison of partition trees associated with the graph vertices.

Cheng Mao, Mark Rudelson, Konstantin Tikhomirov

Graph matching, also known as network alignment, refers to finding a bijection between the vertex sets of two given graphs so as to maximally align their edges. This fundamental computational problem arises frequently in multiple fields such as computer vision and biology. Recently, there has been a plethora of work studying efficient algorithms for graph matching under probabilistic models. In this work, we propose a new algorithm for graph matching: Our algorithm associates each vertex with a signature vector using a multistage procedure and then matches a pair of vertices from the two graphs if their signature vectors are close to each other. We show that, for two Erdős--Rényi graphs with edge correlation $1-α$, our algorithm recovers the underlying matching exactly with high probability when $α\le 1 / (\log \log n)^C$, where $n$ is the number of vertices in each graph and $C$ denotes a positive universal constant. This improves the condition $α\le 1 / (\log n)^C$ achieved in previous work.