Constructive non-commutative rank computation is in deterministic polynomial time

Proceedings of the 50th Annual ACM SIGACT Symposium on Theory of Computing

et al. 2018

We propose a new second-order method for geodesically convex optimization on the natural hyperbolic metric over positive denite matrices. We apply it to solve the operator scaling problem in time polynomial in the input size and logarithmic in the error. This is an exponential improvement over previous algorithms which were analyzed in the usual Euclidean, commutative metric (for which the above problem is not convex). Our method is general and applicable to other settings.As a consequence, we solve the equivalence problem for the left-right group action underlying the operator scaling problem. This yields a deterministic polynomial-time algorithm for a new class of Polynomial Identity Testing (PIT) problems, which was the original motivation for studying operator scaling. * Microsoft Research Redmond, the downstream applications (such as orbit-closure intersection). Remark 1.1. A special case of operator scaling is the matrix scaling problem (cf. [7,19] and references therein). In matrix scaling, we are given a real matrix with non-negative entries, and asked to re-scale its rows and columns to make it doubly stochastic. In this very special case, one can make a change of variables in the appropriate capacity, and make it convex in the Euclidean metric. This aords standard convex optimization techniques, and for this special case, algorithms running in time poly(n, log M, log 1/ε) are known [7,19,52,60].It is known that for every positive operator T , log(det(T (X))) is geodesically convex in X [74]. Also, it is simple to verify that log(det(X)) is geodesically linear (i.e., both convex and concave) 3 . Hence, if we dene the following alternative objective (removing the hard constraint on det(X)) logcap(X) = log det (T (X)) − log det X (1.1) 1 It is also known as a completely positive operator.2 One can assume Ai's are integral or complex integral without loss of generality. 3 This should be contrasted with the fact that log(det(X)) is a concave function in the Euclidean geometry.

Section: Introductionsupporting

confidence: 62%

Section: Orbit-closure Intersection For Left-right Actionmentioning

confidence: 99%

Section: Algorithm For Checking Unitary Equivalencementioning

confidence: 99%

See 1 more Smart Citation

Operator scaling via geodesically convex optimization, invariant theory and polynomial identity testing

Allen-Zhu

Proceedings of the 50th Annual ACM SIGACT Symposium on Theory of Computing

et al. 2018

“…The underlying algebraic problem associated with operator scaling, namely non-commutative singularity and rank of symbolic matrices found a different, algebraic algorithm in the works of [IQS17,DM15] Corollary 1.8. There is a randomized algorithm running in time polypN, 1{εq, that takes as input X P Tenpn 0 ; n 1 , .…”

Section: Summary Of Resultsmentioning

confidence: 99%

Efficient Algorithms for Tensor Scaling, Quantum Marginals, and Moment Polytopes

Bürgisser

Franks

2018 IEEE 59th Annual Symposium on Foundations of Computer Science (FOCS)

et al. 2018

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.Beyond these applications, the algorithm enriches the arsenal of "numerical" methods for classical problems in invariant theory that are significantly faster than "symbolic" methods which explicitly compute invariants or covariants of the relevant action. We stress that (like almost all past algorithms) our convergence rate is polynomial in the approximation parameter; it is an intriguing question to achieve exponential convergence rate, beating symbolic algorithms exponentially, and providing strong membership and separation oracles for the problems above.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.

“…The null cone for this action captures non-commutative singularity (see, e.g., [IQS17b,GGOW16,DM17,IQS17a]) and Problem 1 in Section 1.2. The left-right action has been crucial in getting deterministic polynomial time algorithms for the non-commutative rational identity testing problem [IQS17b,GGOW16,DM17,IQS17a]. The commutative analogue is the famous polynomial identity testing (PIT) problem, for which designing a deterministic polynomial time algorithm remains a major open question in derandomization and complexity theory.…”

Section: Null Conementioning

confidence: 99%

Towards a Theory of Non-Commutative Optimization: Geodesic 1st and 2nd Order Methods for Moment Maps and Polytopes

Bürgisser

Franks

2019 IEEE 60th Annual Symposium on Foundations of Computer Science (FOCS)

et al. 2019

This paper initiates a systematic development of a theory of non-commutative optimization, a setting which greatly extends ordinary (Euclidean) convex optimization. It aims to unify and generalize a growing body of work from the past few years which developed and analyzed algorithms for natural geodesically convex optimization problems on Riemannian manifolds that arise from the symmetries of non-commutative groups. More specifically, these are algorithms to minimize the moment map (a non-commutative notion of the usual gradient), and to test membership in moment polytopes (a vast class of polytopes, typically of exponential vertex and facet complexity, which quite magically arise from this a-priori non-convex, non-linear setting).The importance of understanding this very general setting of geodesic optimization, as these works unveiled and powerfully demonstrate, is that it captures a diverse set of problems, many non-convex, in different areas of CS, math, and physics. Several of them were solved efficiently for the first time using non-commutative methods; the corresponding algorithms also lead to solutions of purely structural problems and to many new connections between disparate fields.In the spirit of standard convex optimization, we develop two general methods in the geodesic setting, a first order and a second order method, which respectively receive first and second order information on the "derivatives" of the function to be optimized. These in particular subsume all past results. The main technical work, again unifying and extending much of the previous work, goes into identifying the key parameters of the underlying group actions which control convergence to the optimum in each of these methods. These non-commutative analogues of "smoothness" in the commutative case are far more complex, and require significant algebraic and analytic machinery (much existing and some newly developed here). Despite this complexity, the way in which these parameters control convergence in both methods is quite simple and elegant. We also bound these parameters in several general cases.Our work points to intriguing open problems and suggests further research directions. We believe that extending this theory, namely understanding geodesic optimization better, is both mathematically and computationally fascinating; it provides a great meeting place for ideas and techniques from several very different research areas, and promises better algorithms for existing and yet unforeseen applications.