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Henry Cohn

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4 papers

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JMLR Journal 2018 Journal Article

Sparse Exchangeable Graphs and Their Limits via Graphon Processes

Christian Borgs
Jennifer T. Chayes
Henry Cohn
Nina Holden

In a recent paper, Caron and Fox suggest a probabilistic model for sparse graphs which are exchangeable when associating each vertex with a time parameter in $\mathbb{R}_+$. Here we show that by generalizing the classical definition of graphons as functions over probability spaces to functions over $\sigma$-finite measure spaces, we can model a large family of exchangeable graphs, including the Caron-Fox graphs and the traditional exchangeable dense graphs as special cases. Explicitly, modelling the underlying space of features by a $\sigma$-finite measure space $(S,\mathcal{S},\mu)$ and the connection probabilities by an integrable function $W\colon S\times S\to [0,1]$, we construct a random family $(G_t)_{t\geq 0}$ of growing graphs such that the vertices of $G_t$ are given by a Poisson point process on $S$ with intensity $t\mu$, with two points $x,y$ of the point process connected with probability $W(x,y)$. We call such a random family a graphon process. We prove that a graphon process has convergent subgraph frequencies (with possibly infinite limits) and that, in the natural extension of the cut metric to our setting, the sequence converges to the generating graphon. We also show that the underlying graphon is identifiable only as an equivalence class over graphons with cut distance zero. More generally, we study metric convergence for arbitrary (not necessarily random) sequences of graphs, and show that a sequence of graphs has a convergent subsequence if and only if it has a subsequence satisfying a property we call uniform regularity of tails. Finally, we prove that every graphon is equivalent to a graphon on $\mathbb{R}_+$ equipped with Lebesgue measure. [abs] [ pdf ][ bib ] &copy JMLR 2018. ( edit, beta )

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SODA Conference 2013 Conference Paper

Fast matrix multiplication using coherent configurations

Henry Cohn
Chris Umans

We introduce a relaxation of the notion of tensor rank, called s-rank, and show that upper bounds on the s -rank of the matrix multiplication tensor imply upper bounds on the ordinary rank. In particular, if the “ s -rank exponent of matrix multiplication” equals 2, then ω = 2. This connection between the s -rank exponent and the ordinary exponent enables us to significantly generalize the group-theoretic approach of Cohn and Umans, from group algebras to general algebras. Embedding matrix multiplication into general algebra multiplication yields bounds on s -rank (not ordinary rank) and, prior to this paper, that had been a barrier to working with general algebras. We identify adjacency algebras of coherent configurations as a promising family of algebras in the generalized framework. Coherent configurations are combinatorial objects that generalize groups and group actions; adjacency algebras are the analogue of group algebras and retain many of their important features. As with groups, coherent configurations support matrix multiplication when a natural combinatorial condition is satisfied, involving triangles of points in their underlying geometry. Finally, we prove a closure property involving symmetric powers of adjacency algebras, which enables us to prove nontrivial bounds on ω using commutative coherent configurations and suggests that commutative coherent configurations may be sufficient to prove ω = 2. Altogether, our results show that bounds on ω can be established by embedding large matrix multiplication instances into small commutative coherent configurations.

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FOCS Conference 2005 Conference Paper

Group-theoretic Algorithms for Matrix Multiplication

Henry Cohn
Robert Kleinberg
Balázs Szegedy
Chris Umans

We further develop the group-theoretic approach to fast matrix multiplication introduced by Cohn and Umans, and for the first time use it to derive algorithms asymptotically faster than the standard algorithm. We describe several families of wreath product groups that achieve matrix multiplication exponent less than 3, the asymptotically fastest of which achieves exponent 2. 41. We present two conjectures regarding specific improvements, one combinatorial and the other algebraic. Either one would imply that the exponent of matrix multiplication is 2.

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FOCS Conference 2003 Conference Paper

A Group-Theoretic Approach to Fast Matrix Multiplication

Henry Cohn
Chris Umans

We develop a new, group-theoretic approach to bounding the exponent of matrix multiplication. There are two components to this approach: (1) identifying groups G that admit a certain type of embedding of matrix multiplication into the group algebra /spl Copf/[G], and (2) controlling the dimensions of the irreducible representations of such groups. We present machinery and examples to support (1), including a proof that certain families of groups of order n/sup 2+o(1)/ support n /spl times/ n matrix multiplication, a necessary condition for the approach to yield exponent 2. Although we cannot yet completely achieve both (1) and (2), we hope that it may be possible, and we suggest potential routes to that result using the constructions in this paper.

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