Abstract. We survey results in algebraic complexity theory, focusing on matrix multiplication. Our goals are (i) to show how open questions in algebraic complexity theory are naturally posed as questions in geometry and representation theory, (ii) to motivate researchers to work on these questions, and (iii) to point out relations with more general problems in geometry. The key geometric objects for our study are the secant varieties of Segre varieties. We explain how these varieties are also useful for algebraic statistics, the study of phylogenetic invariants, and quantum computing.

This work was originally motivated by a classification of tensors proposed by Richard Harshman. In particular, we focus on simple and multiple “bottlenecks”, and on “swamps”. Existing theoretical results are surveyed, some numerical algorithms are described in details, and their numerical complexity is calculated. In particular, the interest in using the ELS enhancement in these algorithms is discussed. Computer simulations feed this discussion.

While latent class models of various types arise in many statistical applications, it is often difficult to establish their identifiability. Focusing on models in which there is some structure of independence of some of the observed variables conditioned on hidden ones, we demonstrate a general approach for establishing identifiability, utilizing algebraic arguments. A theorem of J. Kruskal for a simple latent class model with finite state space lies at the core of our results, though we apply it to a diverse set of models. These include mixtures of both finite and non-parametric product distributions, hidden Markov models, and random graph models, and lead to a number of new results and improvements to old ones. In the parametric setting we argue that the classical definition of identifiability is too strong, and should be replaced by the concept of generic identifiability. Generic identifiability implies that the set of non-identifiable parameters has zero measure, so that the model remains useful for inference. In particular, this sheds light on the properties of finite mixtures of Bernoulli products, which have been used for decades despite being known to be non-identifiable models. In the non-parametric setting, we again obtain identifiability only when certain restrictions are placed on the distributions that are mixed, but we explicitly describe the conditions.

The ideal of a Segre variety P n1

This paper illustrates how methods such as homotopy continuation and monodromy, when combined with a numerical version of Terracini’s lemma, can be used to produce a high probability algorithm for computing the dimensions of secant and join varieties. The use of numerical methods allows applications to problems that are difficult to handle by purely symbolic algorithms.

Abstract. We investigate differences between X-rank and X-border rank, focusing on the cases of tensors and partially symmetric tensors. As an aid to our study, and as an object of interest in its own right, we define notions of X-rank and border rank for a linear subspace. Results include determining and bounding the maximum X-rank of points in several cases of interest. 1.

Abstract. Let Gr(k, n) be the Plücker embedding of the Grassmann variety of projective k-planes in P n. For a projective variety X, let σs(X) denote the variety of its s − 1 secant planes. More precisely, σs(X) denotes the Zariski closure of the union of linear spans of s-tuples of points lying on X. We exhibit two functions s0(n) ≤ s1(n) such that σs(Gr(2, n)) has the expected dimension whenever n ≥ 9 and either s ≤ s0(n) or s1(n) ≤ s. Both s0(n) and s1(n) are asymptotic to n2. This yields, asymptotically, the typical rank of 18 an element of ∧3 Cn+1. Finally, we classify all defective σs(Gr(k, n)) for s ≤ 6 and provide geometric arguments underlying each defective case. 1.

The concept of tensor rank was introduced in the twenties. In the seventies, when methods of Component Analysis on arrays with more than two indices became popular, tensor rank became a much studied topic. The generic rank may be seen as an upper bound to the number of factors that are needed to construct a random tensor. We explain in this paper how to obtain numerically in the complex field the generic rank of tensors of arbitrary dimensions, based on Terracini’s lemma, and compare it with the algebraic results already known in the real or complex fields. In particular, we examine the cases of symmetric tensors, tensors with symmetric matrix slices, complex tensors enjoying Hermitian symmetries, or merely tensors with free entries.

Abstract. For any irreducible non-degenerate variety X ⊂ P r, we relate the dimension of the s-th secant varieties of the Segre embedding of P k × X to the dimension of the (k, s)-Grassmann secant variety GSX(k, s) of X. We also give a criterion for the s-identifiability of X.

This work was originally motivated by a classification of tensors proposed by Richard Harshman. In particular, we focus on simple and multiple “bottlenecks”, and on “swamps”. Existing theoretical results are surveyed, some numerical algorithms are described in details, and their numerical complexity is calculated. In particular, the interest in using the ELS enhancement in these algorithms is discussed. Computer simulations feed this discussion. 1