Title: Reconstructing Shredded Random Matrices
| Speaker: | Massimo Vicenzo |
| Affiliation: | University of Waterloo |
| Location: | MC 5479 |
Abstract: The Graph Reconstruction Conjecture states that if we are given the set of vertex-deleted subgraphs of some graph, then there is a unique graph G that can be reconstructed from them. This conjecture has been open since the 60s, and has only been solved for certain classes of graphs with not much progress towards the general case. We instead study adjacent reconstruction problems, for example, studying matrices instead of graphs: Given some binary matrix M, suppose we are presented with the collection of its rows and columns in independent arbitrary orderings. From this information, are we able to recover the original matrix and will it be unique? We present an algorithm that identifies whether there is a unique ordering associated with a set of rows and columns, and outputs either the unique correct orderings for the rows and columns or the full collection of all valid orderings and valid matrices. We show that for matrices with entries that are i.i.d. Bernoulli(p), that for p >2log(n)/n that the matrix is indeed unique with high probability. This is a joint work with Caelan Atamanchuk and Luc Devroye.