We study large random matrices with i.i.d. entries conditioned to have prescribed row and column sums (margin). This problem has rich connections to relative entropy minimization, Schrodinger bridge, the enumeration of contingency tables, and random graphs with given degree sequences. Such margin-constrained random matrix turns out to be sharply concentrated around a certain deterministic matrix, which we call the "typical table". Typical tables have dual characterizations: (1) the expectation of the random matrix ensemble with minimum relative entropy from the base model constrained to have the expected target margin, and (2) the expectation of the maximum likelihood model obtained by rank-one exponential tilting of the base model. The structure of the typical table is dictated by two dual variables, which give the maximum likelihood estimates of the tilting parameters. Based on these results, for a sequence of "tame" margins that converges in $L^{1}$ to a limiting continuum margin as the size of the matrix diverges, we show that the sequence of margin-constrained random matrices converges in cut norm to a limiting kernel, which is the $L^{2}$-limit of the corresponding rescaled typical tables. The rate of convergence is controlled by how fast the margins converge in $L^{1}$. We also propose a Sinkhorn-type alternating minimization algorithm for computing typical tables, which speicalizes to the classical Sinkhorn algorithm for the Poisson base measure. We derive several new results for random contingency tables from our general framework.
This talk is based on a Joint work with Sumit Mukherjee (Columbia).
|
