On the complexity of evaluating highest weight vectors

Geometric complexity theory (GCT) is an approach towards separating algebraic complexity classes through algebraic geometry and representation theory. Originally Mulmuley and Sohoni proposed (SIAM J Comput 2001, 2008) to use occurrence obstructions to prove Valiant's determinant vs permanent conjecture, but recently B\"urgisser, Ikenmeyer, and Panova (Journal of the AMS 2019) proved this impossible. However, fundamental theorems of algebraic geometry and representation theory grant that every lower bound in GCT can be proved by the use of so-called highest weight vectors (HWVs). In the setting of interest in GCT (namely in the setting of polynomials) we prove the NP-hardness of the evaluation of HWVs in general, and we give efficient algorithms if the treewidth of the corresponding Young-diagram is small, where the point of evaluation is concisely encoded as a noncommutative algebraic branching program! In particular, this gives a large new class of separating functions that can be efficiently evaluated at points with low (border) Waring rank

The border support rank of two-by-two matrix multiplication is seven

We show that the border support rank of the tensor corresponding to two-by-two matrix multiplication is seven over the complex numbers. We do this by constructing two polynomials that vanish on all complex tensors with format four-by-four-by-four and border rank at most six, but that do not vanish simultaneously on any tensor with the same support as the two-by-two matrix multiplication tensor. This extends the work of Hauenstein, Ikenmeyer, and Landsberg. We also give two proofs that the support rank of the two-by-two matrix multiplication tensor is seven over any field: one proof using a result of De Groote saying that the decomposition of this tensor is unique up to sandwiching, and another proof via the substitution method. These results answer a question asked by Cohn and Umans. Studying the border support rank of the matrix multiplication tensor is relevant for the design of matrix multiplication algorithms, because upper bounds on the border support rank of the matrix multiplication tensor lead to upper bounds on the computational complexity of matrix multiplication, via a construction of Cohn and Umans. Moreover, support rank has applications in quantum communication complexity

An Agent-Based Algorithm exploiting Multiple Local Dissimilarities for Clusters Mining and Knowledge Discovery

We propose a multi-agent algorithm able to automatically discover relevant regularities in a given dataset, determining at the same time the set of configurations of the adopted parametric dissimilarity measure yielding compact and separated clusters. Each agent operates independently by performing a Markovian random walk on a suitable weighted graph representation of the input dataset. Such a weighted graph representation is induced by the specific parameter configuration of the dissimilarity measure adopted by the agent, which searches and takes decisions autonomously for one cluster at a time. Results show that the algorithm is able to discover parameter configurations that yield a consistent and interpretable collection of clusters. Moreover, we demonstrate that our algorithm shows comparable performances with other similar state-of-the-art algorithms when facing specific clustering problems

On the Quantum Computational Complexity of the Ising Spin Glass Partition Function and of Knot Invariants

It is shown that the canonical problem of classical statistical thermodynamics, the computation of the partition function, is in the case of +/-J Ising spin glasses a particular instance of certain simple sums known as quadratically signed weight enumerators (QWGTs). On the other hand it is known that quantum computing is polynomially equivalent to classical probabilistic computing with an oracle for estimating QWGTs. This suggests a connection between the partition function estimation problem for spin glasses and quantum computation. This connection extends to knots and graph theory via the equivalence of the Kauffman polynomial and the partition function for the Potts model.Comment: 8 pages, incl. 2 figures. v2: Substantially rewritte

Membership in moment polytopes is in NP and coNP

We show that the problem of deciding membership in the moment polytope associated with a finite-dimensional unitary representation of a compact, connected Lie group is in NP and coNP. This is the first non-trivial result on the computational complexity of this problem, which naively amounts to a quadratically-constrained program. Our result applies in particular to the Kronecker polytopes, and therefore to the problem of deciding positivity of the stretched Kronecker coefficients. In contrast, it has recently been shown that deciding positivity of a single Kronecker coefficient is NP-hard in general [Ikenmeyer, Mulmuley and Walter, arXiv:1507.02955]. We discuss the consequences of our work in the context of complexity theory and the quantum marginal problem.Comment: 20 page