In collaboration with Robert Krone, we use ideas from rigidity theory to study uniqueness of nonnegative matrix factorizations in the case when nonnegative rank of a matrix is equal to its rank. We characterize infinitesimally rigid nonnegative factorizations, prove that a nonnegative factorization is infinitesimally rigid if and only if it is locally rigid and a certain matrix achieves its maximal possible Kruskal rank, and show that locally rigid nonnegative factorizations can be extended to globally rigid nonnegative factorizations. We also explore connections between rigidity of nonnegative factorizations and boundaries of the set of matrices of fixed nonnegative rank. Finally we extend these results from nonnegative factorizations to completely positive factorizations. These results appear in a preprint on arXiv and are submitted for publication.
One direction in the study of nonnegative rank is motivated by linear relaxations for hard combinatorial optimization problems. Nonnegative rank of the slack matrix of the feasible region of the relaxation measures the complexity of the linear relaxation and hence provides lower bounds for the complexity of the original optimization problem. In joint article with Per Austrin and Petteri Kaski, we study another restricted model of computation which is given by tensor networks for evaluating multilinear maps. We show that this model of computation captures the best algorithms for several problems, such as matrix multiplication, discrete Fourier transform, (3t)-clique counting and computing the permanent of a matrix. For counting homomorphisms of a general pattern graph P into a host graph on n vertices we obtain an upper bound that essentially matches the bound for counting cliques and yields small improvements over previous algorithms for many choices of P. There results are published in Proceedings of 10th Innovations in Theoretical Computer Science Conference.
In joint article with Elizabeth Allman, Hector Banos Cervantes, Robin Evans, Serkan Hosten, Daniel Lemke, John Rhodes and Piotr Zwiernik, we study binary tensors of nonnegative rank at most two and three. We give the boundary stratification of binary tensors of nonnegative rank at most two. For small tensors, we show that this stratification can be used for exact computation of maximum likelihood estimates (MLEs). This method guarantees finding the MLE whereas the EM algorithm does not. We show how the EM fixed point ideal provides an alternative method for obtaining the boundary decomposition and for computing MLEs. These results are published in Journal of Algebraic Statistics.
In the joint paper with Anastasiya Belyaeva, Lawrence J. Sun and Caroline Uhler, we study the problem of reconstructing the 3D organization of the genome from such whole-genome contact frequencies. We prove that the 3D organization of the DNA is not identifiable from pairwise distance measurements derived from Hi-C for diploid organisms. In fact, there are infinitely many solutions even in the noise-free setting. We then discuss various additional biologically relevant constraints and prove identifiability under these conditions. Finally, we provide SDP formulations for computing the 3D embedding of the DNA with these additional constraints and show that we can recover the true 3D embedding with high accuracy also from both noiseless and noisy measurements. These formulations minimize the trace of a Gram matrix as an approximation of rank minimization and finally use eigendecomposition to get a rank three approximation of the Gram matrix. These results are in the last stages of preparation and will be submitted in the near future.
In joint article with Carlos Amendola and Dimitra Kosta, we study the maximum likelihood estimation problem for several classes of toric Fano models. We start by exploring the maximum likelihood degree for all 2-dimensional Gorenstein toric Fano varieties. We then explore the reasons for the ML degree drop using A-discriminants and intersection theory. Finally, we show that toric Fano varieties associated to 3-valent phylogenetic trees have ML degree one and provide a formula for the maximum likelihood estimate. We prove it as a corollary to a more general result about the multiplicativity of ML degrees of codimension zero toric fiber products, and it also follows from a connection to a recent result about staged trees. These results appear in a preprint on arXiv and are submitted for publication.
These results have been presented at international seminars, workshops and conferences.
