When a group acts on a set, it naturally partitions it into orbits, giving rise to orbit problems. These are natural algorithmic problems, as symmetries are central in numerous questions and structures in physics, mathematics, computer science, optimization, and more. Accordingly, it is of high interest to understand their computational complexity. Recently, Bürgisser et al. gave the first polynomial-time algorithms for orbit problems of torus actions, that is, actions of commutative continuous groups on Euclidean space. In this work, motivated by theoretical and practical applications, we study the computational complexity of robust generalizations of these orbit problems, which amount to approximating the distance of orbits in $\mathbb{C}^n$ up to a factor $γ>1$. In particular, this allows deciding whether two inputs are approximately in the same orbit or far from being so. On the one hand, we prove the NP-hardness of this problem for $γ= n^{Ω(1/\log\log n)}$ by reducing the closest vector problem for lattices to it. On the other hand, we describe algorithms for solving this problem for an approximation factor $γ= \exp(\mathrm{poly}(n))$. Our algorithms combine tools from invariant theory and algorithmic lattice theory, and they also provide group elements witnessing the proximity of the given orbits (in contrast to the algebraic algorithms of prior work). We prove that they run in polynomial time if and only if a version of the famous number-theoretic $abc$-conjecture holds -- establishing a new and surprising connection between computational complexity and number theory.

The problem PosSLP involves determining whether an integer computed by a given straight-line program is positive. This problem has attracted considerable attention within the field of computational complexity as it provides a complete characterization of the complexity associated with numerical computation. However, non-trivial lower bounds for PosSLP remain unknown. In this paper, we demonstrate that PosSLP ∈ BPP would imply that NP ⊆ BPP, under the assumption of a conjecture concerning the complexity of the radical of a polynomial proposed by Dutta, Saxena, and Sinhababu (STOC’2018). Our proof builds upon the established NP-hardness of determining if a univariate polynomial computed by an SLP has a real root, as demonstrated by Perrucci and Sabia (JDA’2005).

Consider a system f1(x) = 0, …, fn(x) = 0 of n random real polynomials in n variables, where each fi has a prescribed set of exponent vectors in a set of cardinality ti, whose convex hull is denoted Pi. Assuming that the coefficients of the fi are independent standard Gaussian, we prove that the expected number of zeros of the random system in the positive orthant is at most . Here V0 denotes the number of vertices of the Minkowski sum P1 + … + Pn. However, this bound does not improve over the bound in [8] for the unmixed case, where all supports Ai are equal. All arguments equally work for real exponent vectors.

An action of a group on a vector space partitions the latter into a set of orbits. We consider three natural and useful algorithmic "isomorphism" or "classification" problems, namely, orbit equality, orbit closure intersection, and orbit closure containment. These capture and relate to a variety of problems within mathematics, physics and computer science, optimization and statistics. These orbit problems extend the more basic null cone problem, whose algorithmic complexity has seen significant progress in recent years. In this paper, we initiate a study of these problems by focusing on the actions of commutative groups (namely, tori). We explain how this setting is motivated from questions in algebraic complexity, and is still rich enough to capture interesting combinatorial algorithmic problems. While the structural theory of commutative actions is well understood, no general efficient algorithms were known for the aforementioned problems. Our main results are polynomial time algorithms for all three problems. We also show how to efficiently find separating invariants for orbits, and how to compute systems of generating rational invariants for these actions (in contrast, for polynomial invariants the latter is known to be hard). Our techniques are based on a combination of fundamental results in invariant theory, linear programming, and algorithmic lattice theory.

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 noncommutative 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 apriori 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 noncommutative 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.

We describe and analyze an algorithm for computing the homology (Betti numbers and torsion coefficients) of basic semialgebraic sets that works in weak exponential time. That is, of a set of exponentially small measure in the space of data, the cost of the algorithm is exponential in the size of the data. All algorithms previously proposed for this problem have a complexity that is doubly exponential (and this is so for almost all data).

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.

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).

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 nontrivial 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 [C. Ikenmeyer, K. D. Mulmuley, and M. Walter, preprint, arXiv:1507.02955, 2015]. We discuss the consequences of our work in the context of complexity theory and the quantum marginal problem.

We prove the lower bound R Mm) ≥ 3/2 m2-2 on the border rank of m x m matrix multiplication by exhibiting explicit representation theoretic (occurence) obstructions in the sense of Mulmuley and Sohoni's geometric complexity theory (GCT) program. While this bound is weaker than the one recently obtained by Landsberg and Ottaviani, these are the first significant lower bounds obtained within the GCT program. Behind the proof is an explicit description of the highest weight vectors in Symd⊗3 (Cn)* in terms of combinatorial objects, called obstruction designs. This description results from analyzing the process of polarization and Schur-Weyl duality.

Starting with Knutson and Tao's hive model [J. Amer. Math. Soc., 12 (1999), pp. 1055--1090] we characterize the Littlewood--Richardson coefficient ${c_{\lambda,\mu}^{\nu}}$ of given partitions $\lambda,\mu,\nu\in\mathbb{N}^n$ as the number of capacity achieving hive flows on the honeycomb graph. Based on this, we design a polynomial time algorithm for deciding ${c_{\lambda,\mu}^{\nu}} >0$. This algorithm is easy to state and takes $\mathcal{O}(n^3\log\nu_1)$ arithmetic operations and comparisons. We further show that the capacity achieving hive flows can be seen as the vertices of a connected graph, which leads to new structural insights into Littlewood--Richardson coefficients.

It is a remarkable fact that two prominent problems of algebraic complexity theory, the permanent versus determinant problem and the tensor rank problem (matrix multiplication), can be restated as explicit orbit closure problems. This offers the potential to prove lower complexity bounds by relying on methods from algebraic geometry and representation theory. While this basic idea for the tensor rank problem goes back to work by Volker Strassen from the mid eighties, the geometric complexity program has gained visibility and momentum in the past years. Some modest lower bounds for border rank have recently been proven by the construction of explicit obstructions. For further progress, a better understanding of irreducible representions of symmetric groups (tensor products and plethysms) is required. Interestingly, asymptotic versions of the the latter questions are of relevance in quantum information theory.

Condition numbers are well known in numerical linear algebra. It is less known that this concept also plays a crucial part in understanding the efficiency of algorithms in linear programming, convex optimization, and for solving systems of polynomial equations. Indeed, the running time of such algorithms may be often effectively bounded in terms of the condition underlying the problem.

Mulmuley and Sohoni [GCT1, SICOMP 2001; GCT2, SICOMP 2008] proposed to view the permanent versus determinant problem as a specific orbit closure problem and to attack it by methods from geometric invariant and representation theory. We adopt these ideas towards the goal of showing lower bounds on the border rank of specific tensors, in particular for matrix multiplication. We thus study specific orbit closure problems for the group G =GL(W1) x GL(W2) x GL(W3) acting on the tensor product W=W1 ⊗ W2 ⊗ W3 of complex finite dimensional vector spaces. Let Gs =SL(W1) x SL(W2) x SL(W3). A key idea from [GCT2] is that the irreducible Gs-representations occurring in the coordinate ring of the G-orbit closure of a stable tensor w ∈ W are exactly those having a nonzero invariant with respect to the stabilizer group of w.

We discuss the geometry of orbit closures and the asymptotic behavior of Kronecker coefficients in the context of the geometric complexity theory program to prove a variant of Valiant's algebraic analogue of the $\mathbf{P}\neq\mathbf{NP}$ conjecture. We also describe the precise separation of complexity classes that their program proposes to demonstrate.

The 17th of the problems proposed by Steve Smale for the 21st century asks for the existence of a deterministic algorithm computing an approximate solution of a system of n complex polynomials in $n$ unknowns in time polynomial, on the average, in the size N of the input system. A partial solution to this problem was given by Carlos Beltran and Luis Miguel Pardo who exhibited a randomized algorithm, call it LV, doing so. In this paper we further extend this result in several directions. Firstly, we perform a smoothed analysis (in the sense of Spielman and Teng) of algorithm LV and prove that its smoothed complexity is polynomial in the input size and σ-1, where σ controls the size of the random perturbation of the input systems. Secondly, we perform a condition-based analysis of LV. That is, we give a bound, for each system f, of the expected running time of LV with input f. In addition to its dependence on N this bound also depends on the condition of f. Thirdly, and to conclude, we return to Smale's 17th problem as originally formulated for deterministic algorithms. We exhibit such an algorithm and show that its average complexity is NO(log log N). This is nearly a solution to Smale's 17th problem.

We study two quite different approaches to understanding the complexity of fundamental problems in numerical analysis: (a) the Blum–Shub–Smale model of computation over the reals; and (b) a problem we call the “generic task of numerical computation,” which captures an aspect of doing numerical computation in floating point, similar to the “long exponent model” that has been studied in the numerical computing community. We show that both of these approaches hinge on the question of understanding the complexity of the following problem, which we call PosSLP: Given a division-free straight-line program producing an integer N, decide whether $N>0$. In the Blum–Shub–Smale model, polynomial-time computation over the reals (on discrete inputs) is polynomial-time equivalent to PosSLP when there are only algebraic constants. We conjecture that using transcendental constants provides no additional power, beyond nonuniform reductions to PosSLP, and we present some preliminary results supporting this conjecture. The generic task of numerical computation is also polynomial-time equivalent to PosSLP. We prove that PosSLP lies in the counting hierarchy. Combining this with work of Tiwari, we obtain that the Euclidean traveling salesman problem lies in the counting hierarchy—the previous best upper bound for this important problem (in terms of classical complexity classes) being PSPACE. In the course of developing the context for our results on arithmetic circuits, we present some new observations on the complexity of the arithmetic circuit identity testing (ACIT) problem. In particular, we show that if $n!$ is not ultimately easy, then ACIT has subexponential complexity.

We give a uniform method for the two problems #CCC and #ICC of counting connected and irreducible components of complex algebraic varieties, respectively. Our algorithms are purely algebraic, i.e., they use only the field structure of C. They work efficiently in parallel and can be implemented by algebraic circuits of polynomial depth, i.e., in parallel polynomial time. The design of our algorithms relies on the concept of algebraic differential forms. A further important building block is an algorithm of Szántó [40] computing a variant of characteristic sets. The crucial complexity parameter for #ICC turns out to be the number of equations. We describe a randomised algorithm solving #ICC for a fixed number of rational equations given by straight-line programs (slps), which runs in parallel polylogarithmic time in the length and the degree of the slps.

We study two quite different approaches to understanding the complexity of fundamental problems in numerical analysis. We show that both hinge on the question of understanding the complexity of the following problem, which we call PosSLP; given a division-free straight-line program producing an integer N, decide whether N > 0. We show that PosSLP lies in the counting hierarchy, and combining our results with work of Tiwari, we show that the Euclidean traveling salesman problem lies in the counting hierarchy - the previous best upper bound for this important problem (in terms of classical complexity classes) being PSPACE.

We define counting classes #PR and #PC in the Blum-Shub-Smale setting of computations over the real or complex numbers, respectively. The problems of counting the number of solutions of systems of polynomial inequalities over R, or of systems of polynomial equalities over C, respectively, turn out to be natural complete problems in these classes. We investigate to what extent the new counting classes capture the complexity of computing basic topological invariants of semialgebraic sets (over R) and algebraic sets (over C). We prove that the problem to compute the (modified) Euler characteristic of semialgebraic sets is FPR#P RR-complete, and that the problem to compute the geometric degree of complex algebraic sets is FPR#PCC-complete. We also define new counting complexity classes GCR and GCC in the classical Turing model via taking Boolean parts of the classes above, and show that the problems to compute the Euler characteristic and the geometric degree of (semi)algebraic sets given by integer polynomials are complete in these classes. We complement the results in the Turing model by proving, for all k ∈ N, the FPSPACE-hardness of the problem of computing the kth Betti number of the set of real zeros of a given integer polynomial. This holds with respect to the singular homology as well as for the Borel-Moore homology.

We prove lower bounds of order n log n for both the problem of multiplying polynomials of degree n , and of dividing polynomials with remainder, in the model of bounded coefficient arithmetic circuits over the complex numbers. These lower bounds are optimal up to order of magnitude. The proof uses a recent idea of R. Raz [ Proc. 34th STOC 2002 ] proposed for matrix multiplication. It reduces the linear problem of multiplying a random circulant matrix with a vector to the bilinear problem of cyclic convolution. We treat the arising linear problem by extending J. Morgenstern's bound [ J. ACM 20, pp. 305--306, 1973 ] in a unitarily invariant way. This establishes a new lower bound on the bounded coefficient complexity of linear forms in terms of the singular values of the corresponding matrix. In addition, we extend these lower bounds for linear and bilinear maps to a model of circuits that allows a restricted number of unbounded scalar multiplications.

We define a counting class ${\rm #P}_\add$ in the Blum--Shub--Smale setting of additive computations over the reals. Structural properties of this class are studied, including a characterization in terms of the classical counting class $#{\sf P}$ introduced by Valiant. We also establish transfer theorems for both directions between the real additive and the discrete setting. Then we characterize in terms of completeness results the complexity of computing basic topological invariants of semilinear sets given by additive circuits. It turns out that the computation of the Euler characteristic is ${\rm FP}_{\rm add}^{{\rm #P}_{\rm add}}$-complete, while for fixed k the computation of the kth Betti number is ${\rm FPAR}_{\rm add}$-complete. Thus the latter is more difficult under standard complexity theoretic assumptions. We use all of the above to prove some analogous completeness results in the classical setting.

We prove lower bounds of order n log n for both the problem to multiply polynomials of degree n, and to divide polynomials with remainder, in the model of bounded coefficient arithmetic circuits over the complex numbers. These lower bounds are optimal up to order of magnitude. The proof uses a recent idea of R. Raz [Proc. 34th STOC 2002] proposed for matrix multiplication. It reduces the linear problem to multiply a random circulant matrix with a vector to the bilinear problem of cyclic convolution. We treat the arising linear problem by extending J. Morgenstern's bound [J. ACM 20, pp. 305-306, 1973] in a unitarily invariant way. This establishes a new lower bound on the bounded coefficient complexity of linear forms in terms of the singular values of the corresponding matrix.

The existence of string functions, which are not polynomial time computable, but whose graph is checkable in polynomial time, is a basic assumption in cryptography. We prove that in the framework of algebraic complexity, there are no such families of polynomial functions of-bounded degree over fields of characteristic zero. The proof relies on a polynomial upper bound on the approximative complexity of a factor of a polynomial in terms of the (approximative) complexity of and the degree of the factor. This extends a result by Kaltofen (STOC 1986). The concept of approximative complexity allows to cope with the case that a factor has an exponential multiplicity, by using a perturbation argument. Our result extends to randomized (twosided error) decision complexity. 1

We describe a fast algorithm to evaluate irreducible matrix representations of complex general linear groups ${\rm GL}_{m}$ with respect to a symmetry adapted basis (Gelfand--Tsetlin basis). This is complemented by a lower bound, which shows that our algorithm is optimal up to a factor $m^2$ with regard to nonscalar complexity. Our algorithm can be used for the fast evaluation of special functions: for instance, we obtain an $O(\ell\log\ell)$ algorithm to evaluate all associated Legendre functions of degree $\ell$. As a further application we obtain an algorithm to evaluate immanants, which is faster than previous algorithms due to Hartmann and Barvinok.

Permanents and determinants are special cases of immanants. The latter are polynomial matrix functions defined in terms of characters of symmetric groups and corresponding to Young diagrams. Valiant has proved that the evaluation of permanents is a complete problem in both the Turing machine model (#P-completeness) as well as in his algebraic model (VNP-completeness). We show that the evaluation of immanants corresponding to hook diagrams or rectangular diagrams of polynomially growing width is both #P-complete and VNP-complete.