Speaker: Hanbaek Lyu (UW Madison)
Title: Large random matrices with given margins
< p dir="ltr" style="line-height: 1.38; margin-top: 0pt; margin-bottom: 0pt;" >Abstract: We study large random matrices with i.i.d. entries conditioned to have prescribed row and column sums (mar gin). This problem has rich connections to relative entropy minimization,&n bsp; Schr"{o}dinger bridge, the enumeration of contingency tables, and rand om graphs with given degree sequences. We show that such margin-constrained random matrix is sharply concentrated around a certain deterministic matri x, which we call the typical table. Typical tables have dual characterizati ons: (1) the expectation of the random matrix ensemble with minimum relativ e entropy from the base model constrained to have the expected target margi n, and (2) the expectation of the maximum likelihood model obtained by rank -one exponential tilting of the base model. The structure of the typical ta ble is dictated by two dual variables, which give the maximum likelihood es timates of the tilting parameters. Based on these results, for a sequence o f "tame" margins that converges in $L^{1}$ to a limiting continuum margin a s the size of the matrix diverges, we show that the sequence of margin-cons trained random matrices converges in cut norm to a limiting kernel, which i s the $L^{2}$-limit of the corresponding rescaled typical tables. The rate of convergence is controlled by how fast the margins converge in $L^{1}$.&n bsp; We also propose a Sinkhorn-type 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.
Based on a joint work with Sumit Mukherjee.
DTSTAMP:20260828T150904 DTSTART;TZID=America/New_York:20241111T140000 DTEND;TZID=America/New_York:20241111T150000 SEQUENCE:0 TRANSP:OPAQUE END:VEVENT END:VCALENDAR