Let HN denote the problem of determining whether a system of multivariate polynomials with integer coefficients has a complex root. It has long been known that HN in P implies P=NP and, thanks to recent work of Koiran, it is now known that the truth of the Generalized Riemann Hypothesis (GRH) yields the implication that HN not in NP implies P is not equal to NP. We show that the assumption of GRH in the latter implication can be replaced by either of two more plausible hypotheses from analytic number theory. The first is an effective short interval Prime Ideal Theorem with explicit dependence on the underlying field, while the second can be interpreted as a quantitative statement on the higher moments of the zeroes of Dedekind zeta functions. In particular, both assumptions can still hold even if GRH is false. We thus obtain a new application of Dedekind zero estimates to computational algebraic geometry. Along the way, we also apply recent explicit algebraic and analytic estimates, some due to Silberman and Sombra, which may be of independent interest.
This result immediately implies that @math can be solved by solving a linear system with @math variables and equations over the rationals (with total bit-size @math ), thus easily yielding @math . That @math then follows immediately from the fact that linear algebra can be efficiently parallelized @cite_7 .
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We establish a lower bound for the formula size of quolynomials over arbitrary fields. Our basic formula operations are addition, subtraction, multiplication and division. The proof is based on Neciporuk’s [Soviet Math. Doklady, 7 (1966), pp. 999–1000] lower bound for Boolean functions and uses formal power series. This result immediately yields a lower bound for the formula size of rational functions over infinite fields. We also show how to adapt Neciporuk’s method to rational functions over finite fields. These results are then used to show that, over any field, the @math determinant function has formula size at least @math . We thus have an algebraic analogue to the @math lower bound for the Boolean determinant due to Kloss [Soviet Math. Doklady, 7 (1966), pp. 1537–1540]. We consider the fundamental problem of solving quadratic systems of equations in @math variables, where @math , @math and @math is unknown. We propose a novel method, which starting with an initial guess computed by means of a spectral method, proceeds by minimizing a nonconvex functional as in the Wirtinger flow approach. There are several key distinguishing features, most notably, a distinct objective functional and novel update rules, which operate in an adaptive fashion and drop terms bearing too much influence on the search direction. These careful selection rules provide a tighter initial guess, better descent directions, and thus enhanced practical performance. On the theoretical side, we prove that for certain unstructured models of quadratic systems, our algorithms return the correct solution in linear time, i.e. in time proportional to reading the data @math and @math as soon as the ratio @math between the number of equations and unknowns exceeds a fixed numerical constant. We extend the theory to deal with noisy systems in which we only have @math and prove that our algorithms achieve a statistical accuracy, which is nearly un-improvable. We complement our theoretical study with numerical examples showing that solving random quadratic systems is both computationally and statistically not much harder than solving linear systems of the same size---hence the title of this paper. For instance, we demonstrate empirically that the computational cost of our algorithm is about four times that of solving a least-squares problem of the same size. Given a simplicial complex with weights on its simplices, and a nontrivial cycle on it, we are interested in finding the cycle with minimal weight which is homologous to the given one. Assuming that the homology is defined with integer ( @math ) coefficients, we show the following (Theorem 5.2): For a finite simplicial complex @math of dimension greater than @math , the boundary matrix @math is totally unimodular if and only if @math is torsion-free for all pure subcomplexes @math in @math of dimensions @math and @math , respectively, where @math . Because of the total unimodularity of the boundary matrix, we can solve the optimization problem, which is inherently an integer programming problem, as a linear program and obtain an integer solution. Thus, the problem of finding optimal cycles in a given homology class can be solved in polynomial time. This result is surprising in the backdrop of a recent result which says that the problem is NP-hard under @math coefficients which, being a field, is in general easier to deal with. Our result implies, among other things, that one can compute in polynomial time an optimal @math -cycle in a given homology class for any triangulation of an orientable compact @math -manifold or for any finite simplicial complex embedded in @math . Our optimization approach can also be used for various related problems, such as finding an optimal chain homologous to a given one when these are not cycles. Our result can also be viewed as providing a topological characterization of total unimodularity. Given a static array of @math totally ordered objects, the range minimum query problem is to build a data structure that allows us to answer efficiently subsequent on-line queries of the form “what is the position of a minimum element in the subarray ranging from @math to @math ?”. We focus on two settings, where (1) the input array is available at query time, and (2) the input array is available only at construction time. In setting (1), we show new data structures (a) of size @math bits and query time @math for any positive integer function @math for an arbitrary constant @math , or (b) with @math bits and @math query time, where @math denotes the empirical entropy of @math th order of the input array. In setting (2), we give a data structure of size @math bits and query time @math . All data structures can be constructed in linear time and almost in-place. This paper introduces a parallel and distributed algorithm for solving the following minimization problem with linear constraints: @math minimizef1(x1)+ź+fN(xN)subject toA1x1+ź+ANxN=c,x1źX1,ź,xNźXN,where @math Nź2, @math fi are convex functions, @math Ai are matrices, and @math Xi are feasible sets for variable @math xi. Our algorithm extends the alternating direction method of multipliers (ADMM) and decomposes the original problem into N smaller subproblems and solves them in parallel at each iteration. This paper shows that the classic ADMM can be extended to the N-block Jacobi fashion and preserve convergence in the following two cases: (i) matrices @math Ai are mutually near-orthogonal and have full column-rank, or (ii) proximal terms are added to the N subproblems (but without any assumption on matrices @math Ai). In the latter case, certain proximal terms can let the subproblem be solved in more flexible and efficient ways. We show that @math źxk+1-xkźM2 converges at a rate of o(1 k) where M is a symmetric positive semi-definte matrix. Since the parameters used in the convergence analysis are conservative, we introduce a strategy for automatically tuning the parameters to substantially accelerate our algorithm in practice. We implemented our algorithm (for the case ii above) on Amazon EC2 and tested it on basis pursuit problems with >300 GB of distributed data. This is the first time that successfully solving a compressive sensing problem of such a large scale is reported. The problem of finding @math matrices (with @math ) of rank @math in a real affine subspace of dimension n has many applications in information and systems theory, where low rank is synonymous of structure and parsimony. We design computer algebra algorithms to solve this problem efficiently and exactly: the input are the rational coefficients of the matrices spanning the affine subspace as well as the expected maximum rank, and the output is a rational parametrization encoding a finite set of points that intersects each connected component of the low rank real algebraic set. The complexity of our algorithm is studied thoroughly. It is essentially polynomial in @math ; it improves on the state-of-the-art in the field. Moreover, computer experiments show the practical efficiency of our approach. Re-Pair is an efficient grammar compressor that operates by recursively replacing high-frequency character pairs with new grammar symbols. The most space-efficient linear-time algorithm computing Re-Pair uses @math words on top of the re-writable text (of length @math and stored in @math words), for any constant @math ; in practice however, this solution uses complex sub-procedures preventing it from being practical. In this paper, we present an implementation of the above-mentioned result making use of more practical solutions; our tool further improves the working space to @math words (text included), for some small constant @math . As a second contribution, we focus on compact representations of the output grammar. The lower bound for storing a grammar with @math rules is @math bits, and the most efficient encoding algorithm in the literature uses at most @math bits and runs in @math time. We describe a linear-time heuristic maximizing the compressibility of the output Re-Pair grammar. On real datasets, our grammar encoding uses---on average---only @math more bits than the information-theoretic minimum. In half of the tested cases, our compressor improves the output size of 7-Zip with maximum compression rate turned on. The alternating direction method of multipliers (ADMM) has been widely used for solving structured convex optimization problems. In particular, the ADMM can solve convex programs that minimize the sum of @math convex functions whose variables are linked by some linear constraints. While the convergence of the ADMM for @math was well established in the literature, it remained an open problem for a long time whether the ADMM for @math is still convergent. Recently, it was shown in [, Math. Program. (2014), DOI 10.1007 s10107-014-0826-5.] that without additional conditions the ADMM for @math generally fails to converge. In this paper, we show that under some easily verifiable and reasonable conditions the global linear convergence of the ADMM when @math can still be ensured, which is important since the ADMM is a popular method for solving large-scale multiblock optimization models and is known to perform very well in practice even when @math . Our study aims to offer an explanation for... Let @math @math be a given set of @math string documents of total length @math , our task is to index @math , such that the @math most relevant documents for an online query pattern @math of length @math can be retrieved efficiently. We propose an index of size @math bits and @math query time for the basic relevance metric , where @math is the size (in bits) of a compressed full text index of @math , with @math time for searching a pattern of length @math . We further reduce the space to @math bits, however the query time will be @math , where @math is the alphabet size and @math is any constant. We develop several efficient algorithms for the classical problem, which is used in many diverse areas, from preconditioning linear systems to approximation of the permanent. On an input @math matrix @math , this problem asks to find diagonal (scaling) matrices @math and @math (if they exist), so that @math @math -approximates a doubly stochastic, or more generally a matrix with prescribed row and column sums. We address the general scaling problem as well as some important special cases. In particular, if @math has @math nonzero entries, and if there exist @math and @math with polynomially large entries such that @math is doubly stochastic, then we can solve the problem in total complexity @math . This greatly improves on the best known previous results, which were either @math or @math . Our algorithms are based on tailor-made first and second order techniques, combined with other recent advances in continuous optimization, which may be of independent interest for solving similar problems. New methods are proposed for the numerical evaluation of @math or @math , where @math is a function such as @math or @math with singularities in @math and @math is a matrix with eigenvalues on or near @math . The methods are based on combining contour integrals evaluated by the periodic trapezoid rule with conformal maps involving Jacobi elliptic functions. The convergence is geometric, so that the computation of @math is typically reduced to one or two dozen linear system solves, which can be carried out in parallel. All nonabelian finite simple groups of rank @math over a field of size @math , with the possible exception of the Ree groups @math , have presentations with at most @math relations and bit-length @math . Moreover, @math and @math have presentations with 3 generators @math 7 relations and bit-length @math , while @math has a presentation with 7 generators, @math relations and bit-length @math In cryo-electron microscopy, the three-dimensional (3D) electric potentials of an ensemble of molecules are projected along arbitrary viewing directions to yield noisy two-dimensional images. The volume maps representing these potentials typically exhibit a great deal of structural variability, which is described by their 3D covariance matrix. Typically, this covariance matrix is approximately low rank and can be used to cluster the volumes or estimate the intrinsic geometry of the conformation space. We formulate the estimation of this covariance matrix as a linear inverse problem, yielding a consistent least-squares estimator. For @math images of size @math -by- @math pixels, we propose an algorithm for calculating this covariance estimator with computational complexity @math , where the condition number @math is empirically in the range 10--200. Its efficiency relies on the observation that the normal equations are equivalent to a deconvolution problem in six dimensions.... Let @math be a primitive Hilbert modular form of parallel weight @math and level @math for the totally real field @math , and let @math be a rational prime coprime to @math . If @math is ordinary at @math and @math is a CM extension of @math of relative discriminant @math prime to @math , we give an explicit construction of the @math -adic Rankin-Selberg @math -function @math . When the sign of its functional equation is @math , we show, under the assumption that all primes @math are principal ideals of @math which split in @math , that its central derivative is given by the @math -adic height of a Heegner point on the abelian variety @math associated with @math . This @math -adic Gross--Zagier formula generalises the result obtained by Perrin-Riou when @math and @math satisfies the so-called Heegner condition. We deduce applications to both the @math -adic and the classical Birch and Swinnerton-Dyer conjectures for @math . In this paper, we study matrix scaling and balancing, which are fundamental problems in scientific computing, with a long line of work on them that dates back to the 1960s. We provide algorithms for both these problems that, ignoring logarithmic factors involving the dimension of the input matrix and the size of its entries, both run in time @math where @math is the amount of error we are willing to tolerate. Here, @math represents the ratio between the largest and the smallest entries of the optimal scalings. This implies that our algorithms run in nearly-linear time whenever @math is quasi-polynomial, which includes, in particular, the case of strictly positive matrices. We complement our results by providing a separate algorithm that uses an interior-point method and runs in time @math . In order to establish these results, we develop a new second-order optimization framework that enables us to treat both problems in a unified and principled manner. This framework identifies a certain generalization of linear system solving that we can use to efficiently minimize a broad class of functions, which we call second-order robust. We then show that in the context of the specific functions capturing matrix scaling and balancing, we can leverage and generalize the work on Laplacian system solving to make the algorithms obtained via this framework very efficient. We describe a parallel iterative least squares solver named LSRN that is based on random normal projection. LSRN computes the min-length solution to @math , where @math with @math or @math , and where @math may be rank-deficient. Tikhonov regularization may also be included. Since @math is only involved in matrix-matrix and matrix-vector multiplications, it can be a dense or sparse matrix or a linear operator, and LSRN automatically speeds up when @math is sparse or a fast linear operator. The preconditioning phase consists of a random normal projection, which is embarrassingly parallel, and a singular value decomposition of size @math , where @math is moderately larger than 1, e.g., @math . We prove that the preconditioned system is well-conditioned, with a strong concentration result on the extreme singular values, and hence that the number of iterations is fully predictable when we apply LSQR or the Chebyshev semi-iterative method. As we demonstrate, the Chebyshev method is particularly efficient for solving large problems on clusters with high communication cost. Numerical results demonstrate that on a shared-memory machine, LSRN outperforms LAPACK's DGELSD on large dense problems, and MATLAB's backslash (SuiteSparseQR) on sparse problems. Further experiments demonstrate that LSRN scales well on an Amazon Elastic Compute Cloud cluster. We give efficient algorithms for volume sampling, i.e., for picking @math -subsets of the rows of any given matrix with probabilities proportional to the squared volumes of the simplices defined by them and the origin (or the squared volumes of the parallelepipeds defined by these subsets of rows). In other words, we can efficiently sample @math -subsets of @math with probabilities proportional to the corresponding @math by @math principal minors of any given @math by @math positive semi definite matrix. This solves an open problem from the monograph on spectral algorithms by Kannan and Vempala (see Section @math of KV , also implicit in BDM, DRVW ). Our first algorithm for volume sampling @math -subsets of rows from an @math -by- @math matrix runs in @math arithmetic operations (where @math is the exponent of matrix multiplication) and a second variant of it for @math -approximate volume sampling runs in @math arithmetic operations, which is almost linear in the size of the input (i.e., the number of entries) for small @math . Our efficient volume sampling algorithms imply the following results for low-rank matrix approximation: (1) Given @math , in @math arithmetic operations we can find @math of its rows such that projecting onto their span gives a @math -approximation to the matrix of rank @math closest to @math under the Frobenius norm. This improves the @math -approximation of Boutsidis, Drineas and Mahoney BDM and matches the lower bound shown in DRVW . The method of conditional expectations gives a algorithm with the same complexity. The running time can be improved to @math at the cost of losing an extra @math in the approximation factor. (2) The same rows and projection as in the previous point give a @math -approximation to the matrix of rank @math closest to @math under the spectral norm. In this paper, we show an almost matching lower bound of @math , even for @math . We consider global efficiency of algorithms for minimizing a sum of a convex function and a composition of a Lipschitz convex function with a smooth map. The basic algorithm we rely on is the prox-linear method, which in each iteration solves a regularized subproblem formed by linearizing the smooth map. When the subproblems are solved exactly, the method has efficiency @math , akin to gradient descent for smooth minimization. We show that when the subproblems can only be solved by first-order methods, a simple combination of smoothing, the prox-linear method, and a fast-gradient scheme yields an algorithm with complexity @math . The technique readily extends to minimizing an average of @math composite functions, with complexity @math in expectation. We round off the paper with an inertial prox-linear method that automatically accelerates in presence of convexity. The sign-rank of a matrix @math with @math entries is the least rank of a real matrix @math with @math for all @math . We obtain the first exponential lower bound on the sign-rank of a function in @math . Namely, let @math . We show that the matrix @math has sign-rank @math . This in particular implies that @math , which solves a longstanding open problem in communication complexity posed by Babai, Frankl, and Simon [Proceedings of the 27th Symposium on Foundations of Computer Science (FOCS), 1986, pp. 337-347]. Our result additionally implies a lower bound in learning theory. Specifically, let @math be functions such that every DNF formula @math of polynomial size has the representation @math for some reals @math . We prove that then @math , which essentially matches an upper bound of @math , due to Klivans and Servedio [J. Comput. System Sci., 68 (2004), pp. 303-318]. Finally, our work yields the first exponential lower bound on the size of threshold-of-majority circuits computing a function in @math . This substantially generalizes and strengthens the results of Krause and Pudlak [Theoret. Comput. Sci., 174 (1997), pp. 137-156]. Let @math , @math for @math , @math bounding the bit length of the coefficients of the @math 's, and let @math be a quantifier-free @math -formula defining a convex semialgebraic set. We design an algorithm returning a rational point in @math if and only if @math . It requires @math bit operations. If a rational point is outputted, its coordinates have bit length dominated by @math . Using this result, we obtain a procedure for deciding whether a polynomial @math is a sum of squares of polynomials in @math . Denote by @math the degree of @math , @math the maximum bit length of the coefficients in @math , @math , and @math . This procedure requires @math bit operations, and the coefficients of the outputted polynomials have bit length dominated by @math .