We consider a general statistical inference model of finite-rank tensor products. For any interaction structure and order products, we identify the limit free energy in terms variational formula. Our approach consists showing first that must be viscosity solution to certain Hamilton–Jacobi equation.

Matrices with very few nonzero entries cannot have large rank. On the other hand matrices without any zero entries can have rank as low as 1. These simple observations lead us to our main question. For matrices over finite fields, what is the relationship between the rank of a matrix and the number of nonzero entries in the matrix? This question motivated a summer research project collaboration...

Factorized Hidden Layer (FHL) adaptation has been proposed for speaker adaptation of deep neural network (DNN) based acoustic models. In FHL adaptation, a speaker-dependent (SD) transformation matrix and an SD bias are included in addition to the standard affine transformation. The SD transformation is a linear combination of rank-1 matrices whereas the SD bias is a linear combination of vector...

Let A be a linear bounded operator in a Hilbert space H, N(A) and R(A) its null-space and range, and A∗ its adjoint. The operator A is called Fredholm iff dim N(A) = dim N(A∗) := n < ∞ and R(A) and R(A∗) are closed subspaces of H. A simple and short proof is given of the following known result: A is Fredholm iff A = B + F , where B is an isomorphism and F is a finite-rank operator. The proof co...