- University of Colorado at Boulder
- http://www.cs.colorado.edu/~jgrochow/
- Joined 8 Dec 2016

- cond-mat.dis-nn
- cond-mat.stat-mech
- cs.AI
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- q-bio.GN
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- quant-ph

Showing all papers from arXiv author **grochow_j_1**

- In 1969, Strassen shocked the world by showing that two n x n matrices could be multiplied in time asymptotically less than $O(n^3)$. While the recursive construction in his algorithm is very clear, the key gain was made by showing that 2 x 2 matrix multiplication could be performed with only 7 multiplications instead of 8. The latter construction was arrived at by a process of elimination and appears to come out of thin air. Here, we give the simplest and most transparent proof of Strassen's algorithm that we are aware of, using only a simple unitary 2-design and a few easy lines of calculation. Moreover, using basic facts from the representation theory of finite groups, we use 2-designs coming from group orbits to generalize our construction to all n (although the resulting algorithms aren't optimal for n at least 3).
- In the past three decades, many theoretical measures of complexity have been proposed to help understand complex systems. In this work, for the first time, we place these measures on a level playing field, to explore the qualitative similarities and differences between them, and their shortcomings. Specifically, using the Boltzmann machine architecture (a fully connected recurrent neural network) with uniformly distributed weights as our model of study, we numerically measure how complexity changes as a function of network dynamics and network parameters. We apply an extension of one such information-theoretic measure of complexity to understand incremental Hebbian learning in Hopfield networks, a fully recurrent architecture model of autoassociative memory. In the course of Hebbian learning, the total information flow reflects a natural upward trend in complexity as the network attempts to learn more and more patterns.
- Andrew Berdahl, Uttam Bhat, Vanessa Ferdinand, Joshua Garland, Keyan Ghazi-Zahedi, Justin Grana, Joshua A. Grochow, Elizabeth Hobson, Yoav Kallus, Christopher P. Kempes, Artemy Kolchinsky, Daniel B. Larremore, Eric Libby, Eleanor A. Power, Brendan D. TraceyMay 15 2017 physics.soc-ph arXiv:1705.04353v2World record setting has long attracted public interest and scientific investigation. Extremal records summarize the limits of the space explored by a process, and the historical progression of a record sheds light on the underlying dynamics of the process. Existing analyses of prediction, statistical properties, and ultimate limits of record progressions have focused on particular domains. However, a broad perspective on how record progressions vary across different spheres of activity needs further development. Here we employ cross-cutting metrics to compare records across a variety of domains, including sports, games, biological evolution, and technological development. We find that these domains exhibit characteristic statistical signatures in terms of rates of improvement, "burstiness" of record-breaking time series, and the acceleration of the record breaking process. Specifically, sports and games exhibit the slowest rate of improvement and a wide range of rates of "burstiness." Technology improves at a much faster rate and, unlike other domains, tends to show acceleration in records. Many biological and technological processes are characterized by constant rates of improvement, showing less burstiness than sports and games. It is important to understand how these statistical properties of record progression emerge from the underlying dynamics. Towards this end, we conduct a detailed analysis of a particular record-setting event: elite marathon running. In this domain, we find that studying record-setting data alone can obscure many of the structural properties of the underlying process. The marathon study also illustrates how some of the standard statistical assumptions underlying record progression models may be inappropriate or commonly violated in real-world datasets.
- We observe that a certain kind of algebraic proof - which covers essentially all known algebraic circuit lower bounds to date - cannot be used to prove lower bounds against VP if and only if what we call succinct hitting sets exist for VP. This is analogous to the Razborov-Rudich natural proofs barrier in Boolean circuit complexity, in that we rule out a large class of lower bound techniques under a derandomization assumption. We also discuss connections between this algebraic natural proofs barrier, geometric complexity theory, and (algebraic) proof complexity.
- We show how to construct highly symmetric algorithms for matrix multiplication. In particular, we consider algorithms which decompose the matrix multiplication tensor into a sum of rank-1 tensors, where the decomposition itself consists of orbits under some finite group action. We show how to use the representation theory of the corresponding group to derive simple constraints on the decomposition, which we solve by hand for n=2,3,4,5, recovering Strassen's algorithm (in a particularly symmetric form) and new algorithms for larger n. While these new algorithms do not improve the known upper bounds on tensor rank or the matrix multiplication exponent, they are beautiful in their own right, and we point out modifications of this idea that could plausibly lead to further improvements. Our constructions also suggest further patterns that could be mined for new algorithms, including a tantalizing connection with lattices. In particular, using lattices we give the most transparent proof to date of Strassen's algorithm; the same proof works for all n, to yield a decomposition with $n^3 - n + 1$ terms.
- Dec 02 2016 q-bio.OT arXiv:1612.00036v1The organism is a fundamental concept in biology. However there is no universally accepted, formal, and yet broadly applicable definition of what an organism is. Here we introduce a candidate definition. We adopt the view that the "organism" is a functional concept, used by scientists to address particular questions concerning the future state of a biological system, rather than something wholly defined by that system. In this approach organisms are a coarse-graining of a fine-grained dynamical model of a biological system. Crucially, the coarse-graining of the system into organisms is chosen so that their dynamics can be used by scientists to make accurate predictions of those features of the biological system that interests them, and do so with minimal computational burden. To illustrate our framework we apply it to a dynamic model of lichen symbiosis---a system where either the lichen or its constituent fungi and algae could reasonably be considered "organisms." We find that the best choice for what organisms are in this scenario are complex mixtures of many entities that do not resemble standard notions of organisms. When we restrict our allowed coarse-grainings to more traditional types of organisms, we find that ecological conditions, such as niche competition and predation pressure, play a significant role in determining the best choice for organisms.
- Mahaney's Theorem states that, assuming $\mathsf{P} \neq \mathsf{NP}$, no NP-hard set can have a polynomially bounded number of yes-instances at each input length. We give an exposition of a very simple unpublished proof of Manindra Agrawal whose ideas appear in Agrawal-Arvind ("Geometric sets of low information content," Theoret. Comp. Sci., 1996). This proof is so simple that it can easily be taught to undergraduates or a general graduate CS audience - not just theorists! - in about 10 minutes, which the author has done successfully several times. We also include applications of Mahaney's Theorem to fundamental questions that bright undergraduates would ask which could be used to fill the remaining hour of a lecture, as well as an application (due to Ikenmeyer, Mulmuley, and Walter, arXiv:1507.02955) to the representation theory of the symmetric group and the Geometric Complexity Theory Program. To this author, the fact that sparsity results on NP-complete sets have an application to classical questions in representation theory says that they are not only a gem of classical theoretical computer science, but indeed a gem of mathematics.
- In 2003, Cohn and Umans described a framework for proving upper bounds on the exponent $\omega$ of matrix multiplication by reducing matrix multiplication to group algebra multiplication, and in 2005 Cohn, Kleinberg, Szegedy, and Umans proposed specific conjectures for how to obtain $\omega=2$. In this paper we rule out obtaining $\omega=2$ in this framework from abelian groups of bounded exponent. To do this we bound the size of tricolored sum-free sets in such groups, extending the breakthrough results of Croot, Lev, Pach, Ellenberg, and Gijswijt on cap sets. As a byproduct of our proof, we show that a variant of tensor rank due to Tao gives a quantitative understanding of the notion of unstable tensor from geometric invariant theory.
- One fundamental question in the context of the geometric complexity theory approach to the VP vs. VNP conjecture is whether VP = $\overline{\textrm{VP}}$, where VP is the class of families of polynomials that are of polynomial degree and can be computed by arithmetic circuits of polynomial size, and $\overline{\textrm{VP}}$ is the class of families of polynomials that are of polynomial degree and can be approximated infinitesimally closely by arithmetic circuits of polynomial size. The goal of this article is to study the conjecture in (Mulmuley, FOCS 2012) that $\overline{\textrm{VP}}$ is not contained in VP. Towards that end, we introduce three degenerations of VP (i.e., sets of points in $\overline{\textrm{VP}}$), namely the stable degeneration Stable-VP, the Newton degeneration Newton-VP, and the p-definable one-parameter degeneration VP*. We also introduce analogous degenerations of VNP. We show that Stable-VP $\subseteq$ Newton-VP $\subseteq$ VP* $\subseteq$ VNP, and Stable-VNP = Newton-VNP = VNP* = VNP. The three notions of degenerations and the proof of this result shed light on the problem of separating $\overline{\textrm{VP}}$ from VP. Although we do not yet construct explicit candidates for the polynomial families in $\overline{\textrm{VP}}\setminus$VP, we prove results which tell us where not to look for such families. Specifically, we demonstrate that the families in Newton-VP $\setminus$ VP based on semi-invariants of quivers would have to be non-generic by showing that, for many finite quivers (including some wild ones), any Newton degeneration of a generic semi-invariant can be computed by a circuit of polynomial size. We also show that the Newton degenerations of perfect matching Pfaffians, monotone arithmetic circuits over the reals, and Schur polynomials have polynomial-size circuits.
- Pathogens can spread epidemically through populations. Beneficial contagions, such as viruses that enhance host survival or technological innovations that improve quality of life, also have the potential to spread epidemically. How do the dynamics of beneficial biological and social epidemics differ from those of detrimental epidemics? We investigate this question using three theoretical approaches. First, in the context of population genetics, we show that a horizontally-transmissible element that increases fitness, such as viral DNA, spreads superexponentially through a population, more quickly than a beneficial mutation. Second, in the context of behavioral epidemiology, we show that infections that cause increased connectivity lead to superexponential fixation in the population. Third, in the context of dynamic social networks, we find that preferences for increased global infection accelerate spread and produce superexponential fixation, but preferences for local assortativity halt epidemics by disconnecting the infected from the susceptible. We conclude that the dynamics of beneficial biological and social epidemics are characterized by the rapid spread of beneficial elements, which is facilitated in biological systems by horizontal transmission and in social systems by active spreading behavior of infected individuals.
- Nov 26 2015 cs.CC arXiv:1511.08189v1We show that the Graph Automorphism problem is ZPP-reducible to MKTP, the problem of minimizing time-bounded Kolmogorov complexity. MKTP has previously been studied in connection with the Minimum Circuit Size Problem (MCSP) and is often viewed as essentially a different encoding of MCSP. All prior reductions to MCSP have applied equally well to MKTP, and vice-versa, and all such reductions have relied on the fact that functions computable in polynomial time can be inverted with high probability relative to MCSP and MKTP. Our reduction uses a different approach, and consequently yields the first example of a problem -- other than MKTP itself -- that is in ZPP^MKTP but that is not known to lie in NP intersect coNP. We also show that this approach can be used to provide a reduction of the Graph Isomorphism problem to MKTP.
- We introduce a new network statistic that measures diverse structural properties at the micro-, meso-, and macroscopic scales, while still being easy to compute and easy to interpret at a glance. Our statistic, the onion spectrum, is based on the onion decomposition, which refines the k-core decomposition, a standard network fingerprinting method. The onion spectrum is exactly as easy to compute as the k-cores: It is based on the stages at which each vertex gets removed from a graph in the standard algorithm for computing the k-cores. But the onion spectrum reveals much more information about a network, and at multiple scales; for example, it can be used to quantify node heterogeneity, degree correlations, centrality, and tree- or lattice-likeness of the whole network as well as of each k-core. Furthermore, unlike the k-core decomposition, the combined degree-onion spectrum immediately gives a clear local picture of the network around each node which allows the detection of interesting subgraphs whose topological structure differs from the global network organization. This local description can also be leveraged to easily generate samples from the ensemble of networks with a given joint degree-onion distribution. We demonstrate the utility of the onion spectrum for understanding both static and dynamic properties on several standard graph models and on many real-world networks.
- Oct 29 2015 cs.CC arXiv:1510.08417v3In this short note, we show that the Hamiltonian Cycle polynomial is not a monotone sub-exponential-size projection of the permanent; this both rules out a natural attempt at a monotone lower bound on the Boolean permanent, and shows that the permanent is not complete for non-negative polynomials in $VNP_{\mathbb{R}}$ under monotone p-projections. We also show that the cut polynomials and the perfect matching polynomial (or "unsigned Pfaffian") are not monotone p-projections of the permanent. The latter can be interpreted as saying that there is no monotone projection reduction from counting perfect matchings in general graphs to counting perfect matchings in bipartite graphs, putting at least one theorem behind the well-established intuition. As the permanent is universal for monotone formulas, these results also imply exponential lower bounds on the monotone formula size and monotone circuit size of these polynomials. To prove these results we introduce a new connection between monotone projections of polynomials and extended formulations of linear programs that may have further applications.
- We give new polynomial-time algorithms for testing isomorphism of a class of groups given by multiplication tables (GpI). Two results (Cannon & Holt, J. Symb. Comput. 2003; Babai, Codenotti & Qiao, ICALP 2012) imply that GpI reduces to the following: given groups G, H with characteristic subgroups of the same type and isomorphic to $\mathbb{Z}_p^d$, and given the coset of isomorphisms $Iso(G/\mathbb{Z}_p^d, H/\mathbb{Z}_p^d)$, compute Iso(G, H) in time poly(|G|). Babai & Qiao (STACS 2012) solved this problem when a Sylow p-subgroup of $G/\mathbb{Z}_p^d$ is trivial. In this paper, we solve the preceding problem in the so-called "tame" case, i.e., when a Sylow p-subgroup of $G/\mathbb{Z}_p^d$ is cyclic, dihedral, semi-dihedral, or generalized quaternion. These cases correspond exactly to the group algebra $\overline{\mathbb{F}}_p[G/\mathbb{Z}_p^d]$ being of tame type, as in the celebrated tame-wild dichotomy in representation theory. We then solve new cases of GpI in polynomial time. Our result relies crucially on the divide-and-conquer strategy proposed earlier by the authors (CCC 2014), which splits GpI into two problems, one on group actions (representations), and one on group cohomology. Based on this strategy, we combine permutation group and representation algorithms with new mathematical results, including bounds on the number of indecomposable representations of groups in the tame case, and on the size of their cohomology groups. Finally, we note that when a group extension is not tame, the preceding bounds do not hold. This suggests a precise sense in which the tame-wild dichotomy from representation theory may also be a dividing line between the (currently) easy and hard instances of GpI.
- To analyze high-dimensional systems, many fields in science and engineering rely on high-level descriptions, sometimes called "macrostates," "coarse-grainings," or "effective theories". Examples of such descriptions include the thermodynamic properties of a large collection of point particles undergoing reversible dynamics, the variables in a macroeconomic model describing the individuals that participate in an economy, and the summary state of a cell composed of a large set of biochemical networks. Often these high-level descriptions are constructed without considering the ultimate reason for needing them in the first place. Here, we formalize and quantify one such purpose: the need to predict observables of interest concerning the high-dimensional system with as high accuracy as possible, while minimizing the computational cost of doing so. The resulting State Space Compression (SSC) framework provides a guide for how to solve for the optimal high-level description of a given dynamical system, rather than constructing it based on human intuition alone. In this preliminary report, we introduce SSC, and illustrate it with several information-theoretic quantifications of "accuracy", all with different implications for the optimal compression. We also discuss some other possible applications of SSC beyond the goal of accurate prediction. These include SSC as a measure of the complexity of a dynamical system, and as a way to quantify information flow between the scales of a system.
- We introduce a new algebraic proof system, which has tight connections to (algebraic) circuit complexity. In particular, we show that any super-polynomial lower bound on any Boolean tautology in our proof system implies that the permanent does not have polynomial-size algebraic circuits (VNP is not equal to VP). As a corollary to the proof, we also show that super-polynomial lower bounds on the number of lines in Polynomial Calculus proofs (as opposed to the usual measure of number of monomials) imply the Permanent versus Determinant Conjecture. Note that, prior to our work, there was no proof system for which lower bounds on an arbitrary tautology implied any computational lower bound. Our proof system helps clarify the relationships between previous algebraic proof systems, and begins to shed light on why proof complexity lower bounds for various proof systems have been so much harder than lower bounds on the corresponding circuit classes. In doing so, we highlight the importance of polynomial identity testing (PIT) for understanding proof complexity. More specifically, we introduce certain propositional axioms satisfied by any Boolean circuit computing PIT. We use these PIT axioms to shed light on AC^0[p]-Frege lower bounds, which have been open for nearly 30 years, with no satisfactory explanation as to their apparent difficulty. We show that either: a) Proving super-polynomial lower bounds on AC^0[p]-Frege implies VNP does not have polynomial-size circuits of depth d - a notoriously open question for d at least 4 - thus explaining the difficulty of lower bounds on AC^0[p]-Frege, or b) AC^0[p]-Frege cannot efficiently prove the depth d PIT axioms, and hence we have a lower bound on AC^0[p]-Frege. Using the algebraic structure of our proof system, we propose a novel way to extend techniques from algebraic circuit complexity to prove lower bounds in proof complexity.
- The isomorphism problem for finite groups of order n (GpI) has long been known to be solvable in $n^{\log n+O(1)}$ time, but only recently were polynomial-time algorithms designed for several interesting group classes. Inspired by recent progress, we revisit the strategy for GpI via the extension theory of groups. The extension theory describes how a normal subgroup N is related to G/N via G, and this naturally leads to a divide-and-conquer strategy that splits GpI into two subproblems: one regarding group actions on other groups, and one regarding group cohomology. When the normal subgroup N is abelian, this strategy is well-known. Our first contribution is to extend this strategy to handle the case when N is not necessarily abelian. This allows us to provide a unified explanation of all recent polynomial-time algorithms for special group classes. Guided by this strategy, to make further progress on GpI, we consider central-radical groups, proposed in Babai et al. (SODA 2011): the class of groups such that G mod its center has no abelian normal subgroups. This class is a natural extension of the group class considered by Babai et al. (ICALP 2012), namely those groups with no abelian normal subgroups. Following the above strategy, we solve GpI in $n^{O(\log \log n)}$ time for central-radical groups, and in polynomial time for several prominent subclasses of central-radical groups. We also solve GpI in $n^{O(\log\log n)}$ time for groups whose solvable normal subgroups are elementary abelian but not necessarily central. As far as we are aware, this is the first time there have been worst-case guarantees on a $n^{o(\log n)}$-time algorithm that tackles both aspects of GpI---actions and cohomology---simultaneously.
- Aug 13 2013 math.CO arXiv:1308.2677v2The sandpile group Pic^0(G) of a finite graph G is a discrete analogue of the Jacobian of a Riemann surface which was rediscovered several times in the contexts of arithmetic geometry, self-organized criticality, random walks, and algorithms. Given a ribbon graph G, Holroyd et al. used the "rotor-routing" model to define a free and transitive action of Pic^0(G) on the set of spanning trees of G. However, their construction depends a priori on a choice of basepoint vertex. Ellenberg asked whether this action does in fact depend on the choice of basepoint. We answer this question by proving that the action of Pic^0(G) is independent of the basepoint if and only if G is a planar ribbon graph.
- Apr 24 2013 cs.CC arXiv:1304.6333v1We show that most arithmetic circuit lower bounds and relations between lower bounds naturally fit into the representation-theoretic framework suggested by geometric complexity theory (GCT), including: the partial derivatives technique (Nisan-Wigderson), the results of Razborov and Smolensky on $AC^0[p]$, multilinear formula and circuit size lower bounds (Raz et al.), the degree bound (Strassen, Baur-Strassen), the connected components technique (Ben-Or), depth 3 arithmetic circuit lower bounds over finite fields (Grigoriev-Karpinski), lower bounds on permanent versus determinant (Mignon-Ressayre, Landsberg-Manivel-Ressayre), lower bounds on matrix multiplication (Bürgisser-Ikenmeyer) (these last two were already known to fit into GCT), the chasms at depth 3 and 4 (Gupta-Kayal-Kamath-Saptharishi; Agrawal-Vinay; Koiran), matrix rigidity (Valiant) and others. That is, the original proofs, with what is often just a little extra work, already provide representation-theoretic obstructions in the sense of GCT for their respective lower bounds. This enables us to expose a new viewpoint on GCT, whereby it is a natural unification and broad generalization of known results. It also shows that the framework of GCT is at least as powerful as known methods, and gives many new proofs-of-concept that GCT can indeed provide significant asymptotic lower bounds. This new viewpoint also opens up the possibility of fruitful two-way interactions between previous results and the new methods of GCT; we provide several concrete suggestions of such interactions. For example, the representation-theoretic viewpoint of GCT naturally provides new properties to consider in the search for new lower bounds.
- This is a report on a workshop held August 1 to August 5, 2011 at the Institute for Computational and Experimental Research in Mathematics (ICERM) at Brown University, Providence, Rhode Island, organized by Saugata Basu, Joseph M. Landsberg, and J. Maurice Rojas. We provide overviews of the more recent results presented at the workshop, including some works-in-progress as well as tentative and intriguing ideas for new directions. The main themes we discuss are representation theory and geometry in the Mulmuley-Sohoni Geometric Complexity Theory Program, and number theory and other ideas in the Blum-Shub-Smale model.
- We study the problem of matrix Lie algebra conjugacy. Lie algebras arise centrally in areas as diverse as differential equations, particle physics, group theory, and the Mulmuley--Sohoni Geometric Complexity Theory program. A matrix Lie algebra is a set L of matrices such that $A, B\in L$ implies $AB - BA \in L$. Two matrix Lie algebras are conjugate if there is an invertible matrix $M$ such that $L_1 = M L_2 M^{-1}$. We show that certain cases of Lie algebra conjugacy are equivalent to graph isomorphism. On the other hand, we give polynomial-time algorithms for other cases of Lie algebra conjugacy, which allow us to essentially derandomize a recent result of Kayal on affine equivalence of polynomials. Affine equivalence is related to many complexity problems such as factoring integers, graph isomorphism, matrix multiplication, and permanent versus determinant. Specifically, we show: Abelian Lie algebra conjugacy is equivalent to the code equivalence problem, and hence is as hard as graph isomorphism. Abelian Lie algebra conjugacy of $n \times n$ matrices can be solved in poly(n) time when the Lie algebras have dimension O(1). Semisimple Lie algebra conjugacy is equivalent to graph isomorphism. A Lie algebra is semisimple if it is a direct sum of simple Lie algebras. Semisimple Lie algebra conjugacy of $n \times n$ matrices can be solved in polynomial time when the Lie algebras consist of only $O(\log n)$ simple direct summands. Conjugacy of completely reducible Lie algebras---that is, a direct sum of an abelian and a semisimple Lie algebra---can be solved in polynomial time when the abelian part has dimension O(1) and the semisimple part has $O(\log n)$ simple direct summands.
- Jul 29 2009 cs.CC arXiv:0907.4775v2To determine if two lists of numbers are the same set, we sort both lists and see if we get the same result. The sorted list is a canonical form for the equivalence relation of set equality. Other canonical forms arise in graph isomorphism algorithms, and the equality of permutation groups given by generators. To determine if two graphs are cospectral (have the same eigenvalues), however, we compute their characteristic polynomials and see if they are the same; the characteristic polynomial is a complete invariant for the equivalence relation of cospectrality. This is weaker than a canonical form, and it is not known whether a polynomial-time canonical form for cospectrality exists. Note that it is a priori possible for an equivalence relation to be decidable in polynomial time without either a complete invariant or canonical form. Blass and Gurevich (SIAM J. Comput., 1984) ask whether these conditions on equivalence relations -- having an FP canonical form, having an FP complete invariant, and simply being in P -- are in fact different. They showed that this question requires non-relativizing techniques to resolve. Here we extend their results, and give new connections to probabilistic and quantum computation.