Motivated by the successful application of the theory of regular languages to formal verification of finite-state systems, there is a renewed interest in developing a theory of analyzable functions from strings to numerical values that can provide a foundation for analyzing quantitative properties of finitestate systems. In this paper, we propose a deterministic model for associating costs with strings that is parameterized by operations of interest (such as addition, scaling, and min), a notion of regularity that provides a yardstick to measure expressiveness, and study decision problems and theoretical properties of resulting classes of cost functions. Our definition of regularity relies on the theory of string-to-tree transducers, and allows associating costs with events that are conditional upon regular properties of future events. Our model of cost register automata allows computation of regular functions using multiple “write-only” registers whose values can be combined using the allowed set of operations. We show that classical shortest-path algorithms as well as algorithms designed for computing discounted costs, can be adopted for solving the min-cost problems for the more general classes of functions specified in our model. Cost register automata with min and increment give a deterministic model that is equivalent to weighted automata, an extensively studied nondeterministic model, and this connection results in new insights and new open problems.
In @cite_0 , and more recently in @cite_21 @cite_22 , the authors present the problem of deriving affine relations among variables of a program. An affine relation is a property of the form @math , where @math are program variables that range over a field such as the rationals or reals and @math are constants over the same domain. An affine program is a program with nondeterministic branching where each edge of the program is labeled with an assignment statement of the form @math , , where the RHS is an affine expression. We could define a over the cost model @math , with the cost grammar @math . An affine program is then simply obtained by ignoring the input labels of the transitions in such a . While the cost functions defined by such do not have interesting regularity properties, we remark that the equivalence of such can be checked in polynomial time by using ideas similar to the ones in @cite_21 .
@cite_0 @cite_21 @cite_22
2031373197 1592336371
This series of papers studies a geometric structure underlying Karmarkar's projective scaling algorithm for solving linear programming problems. A basic feature of the projective scaling algorithm is a vector field depending on the objective function which is defined on the interior of the polytope of feasible solutions of the linear program. The geometric structure studied is the set of trajectories obtained by integrating this vector field, which we call Ptrajectories. We also study a related vector field, the affine scaling vector field, and its associated trajectories, called A-trajectories. The affine scaling vector field is associated to another linear programming algorithm, the affine scaling algorithm. Affine and projective scaling vector fields are each defined for linear programs of a special form, called strict standard form and canonical form, respectively. This paper derives basic properties of P-trajectories and A-trajectones. It reviews the projective and affine scaling algorithms, defines the projective and affine scaling vector fields, and gives differential equations for P-trajectories and A-trajectories. It shows that projective transformations map P-trajectories into P-trajectories. It presents Karmarkar's interpretation of A-trajectories as steepest descent paths of the objective function (c, x) with respect to the Riemannian geometry ds2 Z?= dx, dx, x2 restricted to the relative interior of the polytope of feasible solutions. P-trajectories of a canonical form linear program are radial projections of A-trajectories of an associated standard form linear program. As a consequence there is a polynomial time linear programming algorithm using the affine scaling vector field of this associated linear program: This algorithm is essentially Karmarkar's algorithm. These trajectories are studied in subsequent papers by two nonlinear changes of variables called Legendre transform coordinates and projective Legendre transform coordinates, respectively. It will be shown that P-trajectories have an algebraic and a geometric interpretation. They are algebraic curves, and they are geodesics (actually distinguished chords) of a geometry isometric to a Hilbert geometry on a polytope combinatorially dual to the polytope of feasible solutions. The A-trajectories of strict standard form linear programs have similar interpretations: They are algebraic curves, and are geodesics of a geometry isometric to Euclidean geometry. Received by the editors July 28, 1986 and, in revised form, September 28, 1987 and March 21, 1988. 1980 Mathematics Subject Classification (1985 Revision). Primary 90C05; Secondary 52A40, 34A34. Research of the first author partially supported by ONR contract N00014-87-K0214. (D 1989 American Mathematical Society 0002-9947 89 @math .25 per page In our LICS 2004 paper we introduced an approach to the study of the local structure of finite algebras and relational structures that aims at applications in the Constraint Satisfaction Problem (CSP). This approach involves a graph associated with an algebra @math or a relational structure A, whose vertices are the elements of @math (or A), the edges represent subsets of @math such that the restriction of some term operation of @math is ‘good’ on the subset, that is, act as an operation of one of the 3 types: semilattice, majority, or affine. In this paper we significantly refine and advance this approach. In particular, we prove certain connectivity and rectangularity properties of relations over algebras related to components of the graph connected by semilattice and affine edges. We also prove a result similar to 2-decomposition of relations invariant under a majority operation, only here we do not impose any restrictions on the relation. These results allow us to give a new, somewhat more intuitive proof of the bounded width theorem: the CSP over algebra @math has bounded width if and only if @math does not contain affine edges. Actually, this result shows that bounded width implies width (2,3). We also consider algebras with edges from a restricted set of types. In particular, it can be proved that type restrictions are preserved under the standard algebraic constructions. Finally, we prove that algebras without semilattice edges have few subalgebras of powers, that is, the CSP over such algebras is also polynomial time. We investigate the minimization of Newton's functional for the problem of the body of minimal resistance with maximal height @math cite butt in the class of convex developable functions defined in a disc. This class is a natural candidate to find a (non-radial) minimizer in accordance with the results of cite lrp2. We prove that the minimizer in this class has a minimal set in the form of a regular polygon with @math sides centered in the disc, and numerical experiments indicate that the natural number @math is a non-decreasing function of @math . The corresponding functions all achieve a lower value of the functional than the optimal radially symmetric function with the same height @math . We give a simple formulation of Karr’s algorithm for computing all affine relationships in affine programs. This simplified algorithm runs in time ( O (nk^ 3 ) ) where n is the program size and k is the number of program variables assuming unit cost for arithmetic operations. This improves upon the original formulation by a factor of k. Moreover, our re-formulation avoids exponential growth of the lengths of intermediately occurring numbers (in binary representation) and uses less complicated elementary operations. We also describe a generalization that determines all polynomial relations up to degree d in time ( O (nk^ 3d ) ). We exhibit an algorithm to compute the strongest polynomial (or algebraic) invariants that hold at each location of a given affine program (i.e., a program having only non-deterministic (as opposed to conditional) branching and all of whose assignments are given by affine expressions). Our main tool is an algebraic result of independent interest: given a finite set of rational square matrices of the same dimension, we show how to compute the Zariski closure of the semigroup that they generate. We prove the existence of a polynomial time computable pseudorandom generator that @math -fools constant width regular read-once branching programs of length @math using a seed of length @math . The previous best pseudorandom generator for regular branching programs used a seed of length @math , and was due to Braverman et. al. and Brody and Verbin (FOCS 2010). Our pseudorandom generator is the INW generator due to Impagliazzo et. al. (STOC 1994). Our work shares some similarity with the recent work of Kouck ' y et. al. (STOC 2011)who get similar seed length for permutation branching programs. However, our work proceeds by analyzing the eigenvalues of the stochastic matrices that arise in the transitions of the branching program which arguably makes the technique more robust. As a corollary of our techniques, we present new results on the small biased spaces for group products'' problem by Meka and Zuckerman (RANDOM 2009). We get a pseudorandom generator with seed length @math . Previously, using the result of Kouck ' y et. al., it was possible to get a seed length of @math for this problem. 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. State-of-the-art subspace clustering methods are based on expressing each data point as a linear combination of other data points while regularizing the matrix of coefficients with @math , @math or nuclear norms. @math regularization is guaranteed to give a subspace-preserving affinity (i.e., there are no connections between points from different subspaces) under broad theoretical conditions, but the clusters may not be connected. @math and nuclear norm regularization often improve connectivity, but give a subspace-preserving affinity only for independent subspaces. Mixed @math , @math and nuclear norm regularizations offer a balance between the subspace-preserving and connectedness properties, but this comes at the cost of increased computational complexity. This paper studies the geometry of the elastic net regularizer (a mixture of the @math and @math norms) and uses it to derive a provably correct and scalable active set method for finding the optimal coefficients. Our geometric analysis also provides a theoretical justification and a geometric interpretation for the balance between the connectedness (due to @math regularization) and subspace-preserving (due to @math regularization) properties for elastic net subspace clustering. Our experiments show that the proposed active set method not only achieves state-of-the-art clustering performance, but also efficiently handles large-scale datasets. We prove a homological mirror symmetry equivalence between an @math -brane category for the pair of pants, computed as a wrapped microlocal sheaf category, and a @math -brane category for a mirror LG model, understood as a category of matrix factorizations. The equivalence improves upon prior results in two ways: it intertwines evident affine Weyl group symmetries on both sides, and it exhibits the relation of wrapped microlocal sheaves along different types of Lagrangian skeleta for the same hypersurface. The equivalence proceeds through the construction of a combinatorial realization of the @math -model via arboreal singularities. The constructions here represent the start of a program to generalize to higher dimensions many of the structures which have appeared in topological approaches to Fukaya categories of surfaces. We characterize the set of properties of Boolean-valued functions on a finite domain @math that are testable with a constant number of samples. Specifically, we show that a property @math is testable with a constant number of samples if and only if it is (essentially) a @math -part symmetric property for some constant @math , where a property is @math -part symmetric if there is a partition @math of @math such that whether @math satisfies the property is determined solely by the densities of @math on @math . We use this characterization to obtain a number of corollaries, namely: (i) A graph property @math is testable with a constant number of samples if and only if whether a graph @math satisfies @math is (essentially) determined by the edge density of @math . (ii) An affine-invariant property @math of functions @math is testable with a constant number of samples if and only if whether @math satisfies @math is (essentially) determined by the density of @math . (iii) For every constant @math , monotonicity of functions @math on the @math -dimensional hypergrid is testable with a constant number of samples. We propose a novel framework for the deterministic construction of linear, near-isometric embeddings of a finite set of data points. Given a set of training points @math , we consider the secant set @math that consists of all pairwise difference vectors of @math , normalized to lie on the unit sphere. We formulate an affine rank minimization problem to construct a matrix @math that preserves the norms of all the vectors in @math up to a distortion parameter @math . While affine rank minimization is NP-hard, we show that this problem can be relaxed to a convex formulation that can be solved using a tractable semidefinite program (SDP). In order to enable scalability of our proposed SDP to very large-scale problems, we adopt a two-stage approach. First, in order to reduce compute time, we develop a novel algorithm based on the Alternating Direction Method of Multipliers (ADMM) that we call Nuclear norm minimization with Max-norm constraints (NuMax) to solve the SDP. Second, we develop a greedy, approximate version of NuMax based on the column generation method commonly used to solve large-scale linear programs. We demonstrate that our framework is useful for a number of signal processing applications via a range of experiments on large-scale synthetic and real datasets. We present two automatic program analyses. The first analysis checks if a given polynomial relation holds among the program variables whenever control reaches a given program point. It fully interprets assignment statements with polynomial expressions on the right-hand side and polynomial disequality guards. Other assignments are treated as non-deterministically assigning any value and guards that are not polynomial disequalities are ignored. The second analysis extends this checking procedure. It computes the set of all polynomial relations of an arbitrary given form that are valid at a given target program point. It is also complete up to the abstraction described above. It is shown that, over an arbitrary ring, the functions computed by polynomial-size algebraic formulas are also computed by polynomial-length algebraic straight-line programs that use only three registers. This was previously known for Boolean formulas [D. A. Barrington, J. Comput. System Sci., 38 (1989), pp. 150–164], which are equivalent to algebraic formulas over the ring @math . For formulas over arbitrary rings, the result is an improvement over previous methods that require the number of registers to be logarithmic in the size of the formulas in order to obtain polynomial-length straight-line programs. Moreover, the straight-line programs that arise in these constructions have the property that they consist of statements whose actions on the registers are linear and bijective. A consequence of this is that the problem of determining the iterated product of @math matrices is complete (under P-projections) for algebraic @math . Also, when the ring is @math , the programs that arise in the c... We study the convex hull of @math , the set of @math orthogonal matrices with unit determinant, from the point of view of semidefinite programming. We show that the convex hull of @math is doubly spectrahedral, i.e., both it and its polar have a description as the intersection of a cone of positive semidefinite matrices with an affine subspace. Our spectrahedral representations are explicit and are of minimum size, in the sense that there are no smaller spectrahedral representations of these convex bodies. We consider integer arithmetic modulo a power of 2 as providedby mainstream programming languages like Java or standardimplementations of C. The difficulty here is that, for w> 1, the ring Zm of integers modulom = 2w has zero divisors and thus cannotbe embedded into a field. Not withstanding that, we present intra-and interprocedural algorithms for inferring for every programpoint u affine relations between program variables valid atu. If conditional branching is replaced withnondeterministic branching, our algorithms are not only sound butalso complete in that they detect all valid affinerelations in a natural class of programs. Moreover, they run intime linear in the program size and polynomial in the number ofprogram variables and can be implemented by using the same modularinteger arithmetic as the target language to be analyzed. We alsoindicate how our analysis can be extended to deal with equalityguards, even in an interprocedural setting. We consider mirror symmetry for (essentially arbitrary) hypersurfaces in (possibly noncompact) toric varieties from the perspective of the Strominger-Yau-Zaslow (SYZ) conjecture. Given a hypersurface @math in a toric variety @math we construct a Landau-Ginzburg model which is SYZ mirror to the blowup of @math along @math , under a positivity assumption. This construction also yields SYZ mirrors to affine conic bundles, as well as a Landau-Ginzburg model which can be naturally viewed as a mirror to @math . The main applications concern affine hypersurfaces of general type, for which our results provide a geometric basis for various mirror symmetry statements that appear in the recent literature. We also obtain analogous results for complete intersections. NNScore is a neural-network-based scoring function designed to aid the computational identification of small-molecule ligands. While the test cases included in the original NNScore article demonstrated the utility of the program, the application examples were limited. The purpose of the current work is to further confirm that neural-network scoring functions are effective, even when compared to the scoring functions of state-of-the-art docking programs, such as AutoDock, the most commonly cited program, and AutoDock Vina, thought to be two orders of magnitude faster. Aside from providing additional validation of the original NNScore function, we here present a second neural-network scoring function, NNScore 2.0. NNScore 2.0 considers many more binding characteristics when predicting affinity than does the original NNScore. The network output of NNScore 2.0 also differs from that of NNScore 1.0; rather than a binary classification of ligand potency, NNScore 2.0 provides a single estimate of the pKd. To fac... We classify fully commutative elements in the affine Coxeter group of type @math . We give a normal form for such elements, then we propose an application of this normal form: we lift these fully commutative elements to the affine braid group of type @math and we get a form for "fully commutative braids". We develop the theory of regular cost functions over finite trees: aquantitative extension to the notion of regular languages of trees: Cost functions map each input (tree) to a value in @math , and are considered modulo an equivalence relation which forgets about specific values, but preserves boundedness of functions on all subsets of the domain. We introduce nondeterministic and alternating finite tree cost automata for describing cost functions. We show that all these forms of automata are effectively equivalent. We also provide decision procedures for them. Finally, following B "uchi's seminal idea, we use cost automata for providing decision procedures for cost monadic logic, a quantitative extension of monadic second order logic. Consider the recovery of an unknown signal @math from quantized linear measurements. In the one-bit compressive sensing setting, one typically assumes that @math is sparse, and that the measurements are of the form @math . Since such measurements give no information on the norm of @math , recovery methods typically assume that @math . We show that if one allows more generally for quantized affine measurements of the form @math , and if the vectors @math are random, an appropriate choice of the affine shifts @math allows norm recovery to be easily incorporated into existing methods for one-bit compressive sensing. In addition, we show that for arbitrary fixed @math in the annulus @math , one may estimate the norm @math up to additive error @math from @math such binary measurements through a single evaluation of the inverse Gaussian error function. Finally, all of our recovery guarantees can be made universal over sparse vectors in the sense that with high probability, one set of measurements and thresholds can successfully estimate all sparse vectors @math in a Euclidean ball of known radius.