Complete quadrics: Schubert calculus for Gaussian models and semidefinite programming

Abstract

We establish connections between: the maximum likelihood degree (MLdegree) for linear concentration models, the algebraic degree of semidefinite programming (SDP), and Schubert calculus for complete quadrics. We prove a conjecture by Sturmfels and Uhler on the polynomiality of the ML-degree. We also prove a conjecture by Nie, Ranestad and Sturmfels providing an explicit formula for the degree of SDP. The interactions between the three fields shed new light on the asymptotic behaviour of enumerative invariants for the variety of complete quadrics. We also extend these results to spaces of general matrices and of skew-symmetric matrices.

Characteristic numbers and chromatic polynomial of a tensor

“…Next, we provide relations to algebraic statistics, based on the results from [20,16,15]. This explains Example 2.3.…”

“…Thus special subspaces L give us smaller (or equal) charactistic numbers than general ones. For general L (thus for general tensors) we have explicit methods to compute the characteristic numbers [15]. This means that if we take a general tensor of high enough rank r, we know b a,n,r =: b a,n .…”

“…Here our inspirations are drawn from the maximum likelihood for linear concentration models as described in [20]. In the recent papers [16,15] connections to the cohomology of complete quadrics were made. This topic may be seen as the 'symmetric case' of our construction.…”

The enumerative geometry of cubic hypersurfaces: point and line conditions

“…This is a compactification of the space of smooth quadric hypersurfaces in P(W ) = P n , where W is an (n + 1)-dimensional vector space over an algebraically closed field k. To construct this space, one starts with V 0 = P(Sym 2 (W )) and considers the sequence of n blow-ups obtained by iteratively blowing up the proper transforms of the loci of matrices with rank at most i. For more details, we refer to [Man+20] and the references therein. This variety has been used to answer the degree 2 case of questions like: How many smooth degree d hypersurfaces in P n are tangent to n+d n − 1 general linear spaces of various dimensions?…”

“…What might sound like a rather basic question was later translated into a problem about the Chow ring of the space of complete quadrics, where beautiful results were achieved [DP85;Sem48;Vai82]. More recently, the space of complete quadrics has proved useful to study some classical problems in algebraic statistics related to maximum likelihood estimation [Man+20;MMW21]. For quadrics, we know the characteristic numbers and also a space where to translate the question above into a cohomological problem.…”