We study the problem of maximum likelihood estimation of densities that are log-concave and lie in the graphical model corresponding to a given undirected graph G. More precisely, we assume that each density in our family factorizes according to the graph G and all factors are log-concave. We show that the maximum likelihood estimate (MLE) is the product of the exponentials of several tent functions, one for each maximal clique of G. While the set of log-concave densities in a graphical model is infinite-dimensional, our results imply that the MLE can be found by solving a finite-dimensional convex optimization problem. We provide an implementation and a few examples. Furthermore, we show that the MLE exists and is unique with probability 1 as long as the number of sample points is larger than the size of the largest clique of G when G is chordal. We show that the MLE is consistent when the graph G is a disjoint union of cliques. Finally, we discuss the conditions under which a log-concave density in the graphical model of G has a log-concave factorization according to G.

We study the algebraic complexity of Euclidean Distance (ED) minimization from a generic tensor to a variety of rank-one tensors. The ED degree of the Segre-Veronese variety counts the number of complex critical points of this optimization problem. We regard this invariant as a function of inner products. We prove that Frobenius inner product is a local minimum of the ED degree, and conjecture that it is a global minimum. We prove our conjecture in the case of matrices and symmetric binary and 3 x 3 x 3 tensors. We discuss the above optimization problem for other algebraic varieties, classifying all possible values of the ED degree. Our approach combines tools from Singularity Theory, Morse Theory, and Algebraic Geometry.

The factor analysis model is a statistical model where a certain number of hidden random variables, called factors, affect linearly the behavior of another set of observed random variables, with additional random noise. The main assumption of the model is that the factors and the noise are Gaussian random variables. This implies that the feasible set lies in the cone of positive semidefinite matrices. In this paper, we do not assume that the factors and the noise are Gaussian, hence the higher order moment and cumulant tensors of the observed variables are generally nonzero. This motivates the notion of a k-th order factor analysis model, which is the family of all random vectors in a factor analysis model where the factors and the noise have finite and possibly nonzero moment and cumulant tensors up to order k. This subset may be described as the image of a polynomial map onto a Cartesian product of symmetric tensor spaces. Our goal is to compute its dimension and we provide conditions under which the image has positive codimension.

We introduce a theory of relative tangency for projective algebraic varieties. The dual variety X^∨_Z of a variety X relative to a subvariety Z is the set of hyperplanes tangent to X at a point of Z. We also introduce the concept of polar classes of X relative to Z. We explore the duality of varieties of low rank matrices relative to special linear sections. In this framework, we study the critical points of the Euclidean Distance function from a data point to X, lying on Z. The locus where the number of such conditional critical points is positive is called the ED data locus of X given Z. The generic number of such critical points defines the conditional ED degree of X given Z.We show the irreducibility of ED data loci, and we compute their dimensions and degrees in terms of relative characteristic classes