Jan Draisma

The Euclidean Distance Degree of an Algebraic Variety

The nearest point map of a real algebraic variety with respect to Euclidean distance is an algebraic function. For instance, for varieties of low-rank matrices, the Eckart–Young Theorem states that this map is given by the singular value decomposition. This article develops a theory of such nearest point maps from the perspective of computational algebraic geometry. The Euclidean distance degree of a variety is the number of critical points of the squared distance to a general point outside the variety. Focusing on varieties seen in applications, we present numerous tools for exact computations.

Trek separation for Gaussian graphical models

Gaussian graphical models are semi-algebraic subsets of the cone of positive definite covariance matrices. Submatrices with low rank correspond to generalizations of conditional independence constraints on collections of random variables. We give a precise graph-theoretic characterization of when submatrices of the covariance matrix have small rank for a general class of mixed graphs that includes directed acyclic and undirected graphs as special cases. Our new trek separation criterion generalizes the familiar d-separation criterion. Proofs are based on the trek rule, the resulting matrix factorizations and classical theorems of algebraic combinatorics on the expansions of determinants of path polynomials.

Bounded-rank tensors are defined in bounded degree

Matrices of rank at most k are defined by the vanishing of polynomials of degree k+1 in their entries (namely, their ((k+1)×(k+1))-subdeterminants), regardless of the size of the matrix. We prove a qualitative analogue of this statement for tensors of arbitrary dimension, where matrices correspond to two-dimensional tensors. More specifically, we prove that for each k there exists an upper bound d=d(k) such that tensors of border rank at most k are defined by the vanishing of polynomials of degree at most d, regardless of the dimension of the tensor and regardless of its size in each dimension. Our proof involves passing to an infinite-dimensional limit of tensor powers of a vector space, whose elements we dub infinite-dimensional tensors, and exploiting the symmetries of this limit in crucial ways.

Polarization of Separating Invariants

We prove a characteristic free version of Weyls theorem on polarization. Our result is an exact analogue of Weyls theorem, the difference being that our statement is about separating invariants rather than generating invariants. For the special case of finite group actions we introduce the concept of cheap polarization, and show that it is enough to take cheap polarizations of invariants of just one copy of a representation to obtain separating vector invariants for any number of copies. This leads to upper bounds on the number and degrees of separating vector invariants of finite groups.

Equivariant Gröbner bases and the Gaussian two-factor model

Exploiting symmetry in Grobner basis computations is difficult when the symmetry takes the form of a group acting by automorphisms on monomials in finitely many variables. This is largely due to the fact that the group elements, being invertible, cannot preserve a term order. By contrast, inspired by work of Aschenbrenner and Hillar, we introduce the concept of equivariant Grobner basis in a setting where a monoid acts by homomorphisms on monomials in potentially infinitely many variables. We require that the action be compatible with a term order, and under some further assumptions derive a Buchberger-type algorithm for computing equivariant Grobner bases. Using this algorithm and the monoid of strictly increasing functions ℕ → ℕ we prove that the kernel of the ring homomorphism ℝ[y ij |i, j ∈ ℕ, i>j] → ℝ[s i , t i | i ∈ ℕ], y ij ↦ s i s j + t i t j is generated by two types of polynomials: off-diagonal 3 x 3-minors and pentads. This confirms a conjecture by Drton, Sturmfels, and Sullivant on the Gaussian two-factor model from algebraic statistics.

Secant dimensions of minimal orbits: computations and conjectures

We present an algorithm for computing the dimensions of higher secant varieties of minimal orbits. Experiments with this algorithm lead to many conjectures on secant dimensions, especially of Grassmannians and Segre products. For these two classes of minimal orbits we give a short proof of the relation—known from the work of Ehrenborg, Catalisano–Geramita–Gimigliano, and Sturmfels–Sullivant—between the existence of certain codes and nondefectiveness of certain higher secant varieties.

Finiteness results for Abelian tree models

Equivariant tree models are statistical models used in the reconstruction of phylogenetic trees from genetic data. Here equivariant§ refers to a symmetry group imposed on the root distribution and on the transition matrices in the model. We prove that if that symmetry group is Abelian, then the Zariski closures of these models are defined by polynomial equations of bounded degree, independent of the tree. Moreover, we show that there exists a polynomial-time membership test for that Zariski closure. This generalises earlier results on tensors of bounded rank, which correspond to the case where the group is trivial and the tree is a star, and implies a qualitative variant of a quantitative conjecture by Sturmfels and Sullivant in the case where the group and the alphabet coincide. Our proofs exploit the symmetries of an infinite-dimensional projective limit of Abelian star models. Keywords: Phylogenetic tree models, tensor rank, noetherianity up to symmetry, applied algebraic geometry

Small Maximal Spaces of Non-Invertible Matrices

The rank of a vector space A of n x n-matrices is by definition the maximal rank of an element of A. The space A is called rank-critical if any matrix space that properly contains A has a strictly higher rank. This paper exhibits a sufficient condition for rank-criticality, which is then used to prove that the images of certain Lie algebra representations are rank-critical. A rather counter-intuitive consequence, and the main novelty in this paper, is that for infinitely many n, there exists an eight-dimensional rank-critical space of n x n-matrices having generic rank n – 1, or, in other words: an eight-dimensional maximal space of non-invertible matrices. This settles the question, posed by Fillmore, Laurie, and Radjavi in 1985, of whether such a maximal space can have dimension smaller than n. Another consequence is that the image of the adjoint representation of any semisimple Lie algebra is rank-critical; in both results, the ground field is assumed to have characteristic zero.

Noetherianity up to Symmetry

These lecture notes for the 2013 CIME/CIRM summer school Combinatorial Algebraic Geometry deal with manifestly infinite-dimensional algebraic varieties with large symmetry groups. So large, in fact, that subvarieties stable under those symmetry groups are defined by finitely many orbits of equations—whence the title Noetherianity up to symmetry. It is not the purpose of these notes to give a systematic, exhaustive treatment of such varieties, but rather to discuss a few “personal favourites”: exciting examples drawn from applications in algebraic statistics and multilinear algebra. My hope is that these notes will attract other mathematicians to this vibrant area at the crossroads of combinatorics, commutative algebra, algebraic geometry, statistics, and other applications.

Orthogonal and unitary tensor decomposition from an algebraic perspective

While every matrix admits a singular value decomposition, in which the terms are pairwise orthogonal in a strong sense, higher-order tensors typically do not admit such an orthogonal decomposition. Those that do have attracted attention from theoretical computer science and scientific computing. We complement this existing body of literature with an algebro-geometric analysis of the set of orthogonally decomposable tensors.More specifically, we prove that they form a real-algebraic variety defined by polynomials of degree at most four. The exact degrees, and the corresponding polynomials, are different in each of three times two scenarios: ordinary, symmetric, or alternating tensors; and real-orthogonal versus complex-unitary. A key feature of our approach is a surprising connection between orthogonally decomposable tensors and semisimple algebras—associative in the ordinary and symmetric settings and of compact Lie type in the alternating setting.