83 research outputs found
Efficient algorithms for tensor scaling, quantum marginals and moment polytopes
We present a polynomial time algorithm to approximately scale tensors of any
format to arbitrary prescribed marginals (whenever possible). This unifies and
generalizes a sequence of past works on matrix, operator and tensor scaling.
Our algorithm provides an efficient weak membership oracle for the associated
moment polytopes, an important family of implicitly-defined convex polytopes
with exponentially many facets and a wide range of applications. These include
the entanglement polytopes from quantum information theory (in particular, we
obtain an efficient solution to the notorious one-body quantum marginal
problem) and the Kronecker polytopes from representation theory (which capture
the asymptotic support of Kronecker coefficients). Our algorithm can be applied
to succinct descriptions of the input tensor whenever the marginals can be
efficiently computed, as in the important case of matrix product states or
tensor-train decompositions, widely used in computational physics and numerical
mathematics.
We strengthen and generalize the alternating minimization approach of
previous papers by introducing the theory of highest weight vectors from
representation theory into the numerical optimization framework. We show that
highest weight vectors are natural potential functions for scaling algorithms
and prove new bounds on their evaluations to obtain polynomial-time
convergence. Our techniques are general and we believe that they will be
instrumental to obtain efficient algorithms for moment polytopes beyond the
ones consider here, and more broadly, for other optimization problems
possessing natural symmetries
Alternating Minimization, Scaling Algorithms, and the Null-Cone Problem from Invariant Theory
Alternating minimization heuristics seek to solve a (difficult) global optimization task through iteratively solving a sequence of (much easier) local optimization tasks on different parts (or blocks) of the input parameters. While popular and widely applicable, very few examples of this heuristic are rigorously shown to converge to optimality, and even fewer to do so efficiently.
In this paper we present a general framework which is amenable to rigorous analysis, and expose its applicability. Its main feature is that the local optimization domains are each a group of invertible matrices, together naturally acting on tensors, and the optimization problem is minimizing the norm of an input tensor under this joint action. The solution of this optimization problem captures a basic problem in Invariant Theory, called the null-cone problem.
This algebraic framework turns out to encompass natural computational problems in combinatorial optimization, algebra, analysis, quantum information theory, and geometric complexity theory. It includes and extends to high dimensions the recent advances on (2-dimensional) operator scaling.
Our main result is a fully polynomial time approximation scheme for this general problem, which may be viewed as a multi-dimensional scaling algorithm. This directly leads to progress on some of the problems in the areas above, and a unified view of others. We explain how faster convergence of an algorithm for the same problem will allow resolving central open problems.
Our main techniques come from Invariant Theory, and include its rich non-commutative duality theory, and new bounds on the bitsizes of coefficients of invariant polynomials. They enrich the algorithmic toolbox of this very computational field of mathematics, and are directly related to some challenges in geometric complexity theory (GCT)
Operator scaling with specified marginals
The completely positive maps, a generalization of the nonnegative matrices,
are a well-studied class of maps from matrices to
matrices. The existence of the operator analogues of doubly stochastic scalings
of matrices is equivalent to a multitude of problems in computer science and
mathematics, such rational identity testing in non-commuting variables,
noncommutative rank of symbolic matrices, and a basic problem in invariant
theory (Garg, Gurvits, Oliveira and Wigderson, FOCS, 2016).
We study operator scaling with specified marginals, which is the operator
analogue of scaling matrices to specified row and column sums. We characterize
the operators which can be scaled to given marginals, much in the spirit of the
Gurvits' algorithmic characterization of the operators that can be scaled to
doubly stochastic (Gurvits, Journal of Computer and System Sciences, 2004). Our
algorithm produces approximate scalings in time poly(n,m) whenever scalings
exist. A central ingredient in our analysis is a reduction from the specified
marginals setting to the doubly stochastic setting.
Operator scaling with specified marginals arises in diverse areas of study
such as the Brascamp-Lieb inequalities, communication complexity, eigenvalues
of sums of Hermitian matrices, and quantum information theory. Some of the
known theorems in these areas, several of which had no effective proof, are
straightforward consequences of our characterization theorem. For instance, we
obtain a simple algorithm to find, when they exist, a tuple of Hermitian
matrices with given spectra whose sum has a given spectrum. We also prove new
theorems such as a generalization of Forster's theorem (Forster, Journal of
Computer and System Sciences, 2002) concerning radial isotropic position.Comment: 34 pages, 3 page appendi
Slice Rank of Block Tensors and Irreversibility of Structure Tensors of Algebras
Determining the exponent of matrix multiplication ? is one of the central open problems in algebraic complexity theory. All approaches to design fast matrix multiplication algorithms follow the following general pattern: We start with one "efficient" tensor T of fixed size and then we use a way to get a large matrix multiplication out of a large tensor power of T. In the recent years, several so-called barrier results have been established. A barrier result shows a lower bound on the best upper bound for the exponent of matrix multiplication that can be obtained by a certain restriction starting with a certain tensor.
We prove the following barrier over C: Starting with a tensor of minimal border rank satisfying a certain genericity condition, except for the diagonal tensor, it is impossible to prove ? = 2 using arbitrary restrictions. This is astonishing since the tensors of minimal border rank look like the most natural candidates for designing fast matrix multiplication algorithms. We prove this by showing that all of these tensors are irreversible, using a structural characterisation of these tensors. To obtain our result, we relate irreversibility to asymptotic slice rank and instability of tensors and prove that the instability of block tensors can often be decided by looking only on the sizes of nonzero blocks
Multipartite Quantum States and their Marginals
Subsystems of composite quantum systems are described by reduced density
matrices, or quantum marginals. Important physical properties often do not
depend on the whole wave function but rather only on the marginals. Not every
collection of reduced density matrices can arise as the marginals of a quantum
state. Instead, there are profound compatibility conditions -- such as Pauli's
exclusion principle or the monogamy of quantum entanglement -- which
fundamentally influence the physics of many-body quantum systems and the
structure of quantum information. The aim of this thesis is a systematic and
rigorous study of the general relation between multipartite quantum states,
i.e., states of quantum systems that are composed of several subsystems, and
their marginals. In the first part, we focus on the one-body marginals of
multipartite quantum states; in the second part, we study general quantum
marginals from the perspective of entropy.Comment: PhD thesis, ETH Zurich. The first part contains material from
arXiv:1208.0365, arXiv:1204.0741, and arXiv:1204.4379. The second part is
based on arXiv:1302.6990 and arXiv:1210.046
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