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Saxena N.,Hausdorff Center for Mathematics | Seshadhri C.,IBM
SIAM Journal on Computing | Year: 2011

We study the problem of polynomial identity testing for depth-3 circuits of degree d and top fanin k. The rank of any such identity is essentially the minimum number of independent variables present. Small bounds on this quantity imply fast deterministic identity testers for these circuits. Dvir and Shpilka [SIAM J. Comput., 36 (2007), pp. 1404-1434] initiated the study of the rank and showed that any depth-3 identity (barring some uninteresting corner cases) has a rank of 2O(k2)(log d)k-2. We show that the rank of a depth-3 identity is at most O(k3 log d). This bound is almost tight, since we also provide an identity of rank ω(k log d). Our rank bound significantly improves (dependence on k exponentially reduced) the best known deterministic black-box identity tests for depth-3 circuits by Karnin and Shpilka [Z. Karnin and A. Shpilka, in Proceedings of the 23rd CCC, 2008, pp. 280-291]. Our techniques also shed light on the factorization pattern of nonzero depth-3 circuits: the rank of linear factors of a simple, minimal, and nonzero depth-3 circuit (over any field) is at most O(k3 log d). The novel feature of this work is a new notion of maps between sets of linear forms, called ideal matchings, used to study depth-3 circuits. We prove interesting structural results about depth-3 identities using these techniques. We believe that these ideas may lead to the goal of a deterministic polynomial time identity test for these circuits. © 2011 Society for Industrial and Applied Mathematics.


Agrawal M.,Indian Institute of Technology Kanpur | Saha C.,Max Planck Institute for Informatics | Saptharishi R.,Chennai Mathematical Institute | Saxena N.,Hausdorff Center for Mathematics
Proceedings of the Annual ACM Symposium on Theory of Computing | Year: 2012

We present a single common tool to strictly subsume all known cases of polynomial time blackbox polynomial identity testing (PIT), that have been hitherto solved using diverse tools and techniques, over fields of zero or large characteristic. In particular, we show that polynomial time hitting-set generators for identity testing of the two seemingly different and well studied models - depth-3 circuits with bounded top fanin, and constant-depth constant-read multilinear formulas - can be constructed using one common algebraic-geometry theme: Jacobian captures algebraic independence. By exploiting the Jacobian, we design the first efficient hitting-set generators for broad generalizations of the above-mentioned models, namely: • depth-3 (ΣΠΣ) circuits with constant transcendence degree of the polynomials computed by the product gates (no bounded top fanin restriction), and • constant-depth constant-occur formulas (no multilinear restriction). Constant-occur of a variable, as we define it, is a much more general concept than constant-read. Also, earlier work on the latter model assumed that the formula is multilinear. Thus, our work goes further beyond the related results obtained by Saxena & Seshadhri (STOC 2011), Saraf & Volkovich (STOC 2011), Anderson et al. (CCC 2011), Beecken et al. (ICALP 2011) and Grenet et al. (FSTTCS 2011), and brings them under one unifying technique. In addition, using the same Jacobian based approach, we prove exponential lower bounds for the immanant (which includes permanent and determinant) on the same depth-3 and depth-4 models for which we give efficient PIT algorithms. Our results reinforce the intimate connection between identity testing and lower bounds by exhibiting a concrete mathematical tool - the Jacobian - that is equally effective in solving both the problems on certain interesting and previously well-investigated (but not well understood) models of computation. © 2012 ACM.


Saxena N.,Hausdorff Center for Mathematics | Seshadhri C.,IBM
Proceedings - Annual IEEE Symposium on Foundations of Computer Science, FOCS | Year: 2010

We study the problem of identity testing for depth-3 circuits of top fanin k and degree d. We give a new structure theorem for such identities. A direct application of our theorem improves the known deterministic d kO(k)-time black-box identity test over rationals (Kayal & Saraf, FOCS 2009) to one that takes dO(k2)-time. Our structure theorem essentially says that the number of independent variables in a real depth-3 identity is very small. This theorem affirmatively settles the strong rank conjecture posed by Dvir & Shpilka (STOC 2005). We devise a powerful algebraic framework and develop tools to study depth-3 identities. We use these tools to show that any depth-3 identity contains a much smaller nucleus identity that contains most of the "complexity" of the main identity. The special properties of this nucleus allow us to get almost optimal rank bounds for depth-3 identities. © 2010 IEEE.


Saxena N.,Hausdorff Center for Mathematics | Seshadhri C.,Sandia National Laboratories
Proceedings of the Annual ACM Symposium on Theory of Computing | Year: 2011

Let C be a depth-3 circuit with n variables, degree d and top fanin k (called ΣΠΣ(k,d,n) circuits) over base field FF. It is a major open problem to design a deterministic polynomial time blackbox algorithm that tests if C is identically zero. Klivans & Spielman (STOC 2001) observed that the problem is open even when k is a constant. This case has been subjected to a serious study over the past few years, starting from the work of Dvir & Shpilka (STOC 2005). We give the first polynomial time blackbox algorithm for this problem. Our algorithm runs in time poly(n)dk, regardless of the base field. The only field for which polynomial time algorithms were previously known is FF = QQ (Kayal & Saraf, FOCS 2009, and Saxena & Seshadhri, FOCS 2010). This is the first blackbox algorithm for depth-$3$ circuits that does not use the rank based approaches of Karnin & Shpilka (CCC 2008). We prove an important tool for the study of depth-3 identities. We design a blackbox polynomial time transformation that reduces the number of variables in a ΣΠΣ(k,d,n) circuit to k variables, but preserves the identity structure. © 2011 ACM.


Knupfer H.,Hausdorff Center for Mathematics | Muratov C.B.,New Jersey Institute of Technology
Journal of Nonlinear Science | Year: 2011

We investigate the ground state of a uniaxial ferromagnetic plate with perpendicular easy axis and subject to an applied magnetic field normal to the plate. Our interest is in the asymptotic behavior of the energy in macroscopically large samples near the saturation field. We establish the scaling of the critical value of the applied field strength below saturation at which the ground state changes from the uniform to a multidomain magnetization pattern and the leading order scaling behavior of the minimal energy. Furthermore, we derive a reduced sharp interface energy, giving the precise asymptotic behavior of the minimal energy in macroscopically large plates under a physically reasonable assumption of small deviations of the magnetization from the easy axis away from domain walls. On the basis of the reduced energy and by a formal asymptotic analysis near the transition, we derive the precise asymptotic values of the critical field strength at which non-trivial minimizers (either local or global) emerge. The non-trivial minimal energy scaling is achieved by magnetization patterns consisting of long slender needle-like domains of magnetization opposing the applied field. © 2011 Springer Science+Business Media, LLC.


Scheiblechner P.,Hausdorff Center for Mathematics
Foundations of Computational Mathematics | Year: 2012

We describe a parallel polynomial time algorithm for computing the topological Betti numbers of a smooth complex projective variety X. It is the first single exponential time algorithm for computing the Betti numbers of a significant class of complex varieties of arbitrary dimension. Our main theoretical result is that the Castelnuovo-Mumford regularity of the sheaf of differential p-forms on X is bounded by p(em+1)D, where e, m, and D are the maximal codimension, dimension, and degree, respectively, of all irreducible components of X. It follows that, for a union V of generic hyperplane sections in X, the algebraic de Rham cohomology of X{set minus}V is described by differential forms with poles along V of single exponential order. By covering X with sets of this type and using a Čech process, we obtain a similar description of the de Rham cohomology of X, which allows its efficient computation. Furthermore, we give a parallel polynomial time algorithm for testing whether a projective variety is smooth. © 2012 SFoCM.


Saxena N.,Hausdorff Center for Mathematics | Seshadhri C.,Sandia National Laboratories
SIAM Journal on Computing | Year: 2012

Let C be a depth-3 circuit with n variables, degree d, and top-fanin k (called ∑π∑ (k, d, n) circuits) over base field F. It is a major open problem to design a deterministic polynomial time blackbox algorithm that tests whether C is identically zero. Klivans and Spielman [Proceedings of the 33rd Annual Symposium on Theory of Computing (STOC), 2001, pp. 216-223] observed that the problem is open even when k is a constant. This case has been subjected to serious scrutiny over the past few years, starting from the work of Dvir and Shpilka [SIAM J. Comput., 36 (2007), pp. 1404-1434]. We give the first polynomial time blackbox algorithm for this problem. Our algorithm runs in time poly(n)d k, regardless of the base field. The only field for which polynomial time algorithms were previously known is F = Q [N. Kayal and S. Saraf, Proceedings of the 50th Annual Symposium on Foundations of Computer Science (FOCS), 2009, pp. 198-207; N. Saxena and C. Seshadhri, Proceedings of the 51st Annual Symposium on Foundations of Computer Science (FOCS), 2010, pp. 21-29]. This is the first blackbox algorithm for depth-3 circuits that does not use the rank-based approaches of Karnin and Shpilka [Proceedings of the 24th Annual Conference on Computational Complexity (CCC), 2009, pp. 274-285]. We prove an important tool for the study of depth-3 identities. We design a blackbox polynomial time transformation that reduces the number of variables in a ∑π∑ (k, d, n) circuit to k variables but preserves the identity structure. © 2012 Society for Industrial and Applied Mathematics.


de Philippis G.,Hausdorff Center for Mathematics | Figalli A.,University of Texas at Austin
Archive for Rational Mechanics and Analysis | Year: 2014

We prove higher integrability for the gradient of local minimizers of the Mumford-Shah energy functional, providing a positive answer to a conjecture of De Giorgi (Free discontinuity problems in calculus of variations. Frontiers in pure and applied mathematics, North-Holland, Amsterdam, pp 55-62, 1991). © 2014 Springer-Verlag Berlin Heidelberg.


Alim M.,Arnold Sommerfeld Center for Theoretical Physics | Alim M.,Hausdorff Center for Mathematics | Hecht M.,Arnold Sommerfeld Center for Theoretical Physics | Jockers H.,Stanford University | And 3 more authors.
Nuclear Physics B | Year: 2010

We perform a Hodge theoretic study of parameter dependent families of D-branes on compact Calabi-Yau manifolds in type II and F-theory compactifications. Starting from a geometric Gauss-Manin connection for B-type branes we study the integrability and flatness conditions. The B-model geometry defines an interesting ring structure of operators. For the mirror A-model this indicates the existence of an open-string extension of the so-called A-model connection, whereas the discovered ring structure should be part of the open-string A-model quantum cohomology. We obtain predictions for genuine Ooguri-Vafa invariants for Lagrangian branes on the quintic in P4 that pass some non-trivial consistency checks. We discuss the lift of the brane compactifications to F-theory on Calabi-Yau four-folds and the effective couplings in the effective supergravity action as determined by the N=1 special geometry of the open-closed deformation space. © 2010 Elsevier B.V.


Saxena N.,Hausdorff Center for Mathematics | Seshadhri C.,Sandia National Laboratories | Seshadhri C.,IBM
Journal of the ACM | Year: 2013

We study the problem of identity testing for depth-3 circuits of top fanin k and degree d. We give a newstructure theorem for such identities that improves the known deterministic dkO(k) -time blackbox identitytest over rationals [Kayal and Saraf, 2009] to one that takes dO(k2)-time. Our structure theorem essentiallysays that the number of independentvariables in a real depth-3 identity is very small. This theoremaffirmatively settles the strong rankconjecture posed by Dvir and Shpilka [2006].We devise various algebraic tools to study depth-3 identities, and use these tools to show that any depth-3identity contains a much smaller nucleus identity that contains most of the "complexity" of the main identity.The special properties of this nucleus allow us to get near optimal rank bounds for depth-3 identities. Themost important aspect of this work is relating a field-dependent quantity, the Sylvester-Gallai rank bound,to the rankof depth-3 identities. We also prove a high-dimensional Sylvester-Gallai theorem for all fields,and get a general depth-3 identity rank bound (slightly improving previous © 2013 ACM.

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