Showing papers on "Product (mathematics) published in 2017"

Spectrally-normalized margin bounds for neural networks

Abstract: This paper presents a margin-based multiclass generalization bound for neural networks that scales with their margin-normalized "spectral complexity": their Lipschitz constant, meaning the product of the spectral norms of the weight matrices, times a certain correction factor. This bound is empirically investigated for a standard AlexNet network trained with SGD on the MNIST and CIFAR10 datasets, with both original and random labels; the bound, the Lipschitz constants, and the excess risks are all in direct correlation, suggesting both that SGD selects predictors whose complexity scales with the difficulty of the learning task, and secondly that the presented bound is sensitive to this complexity.

A PAC-Bayesian Approach to Spectrally-Normalized Margin Bounds for Neural Networks

Abstract: We present a generalization bound for feedforward neural networks in terms of the product of the spectral norm of the layers and the Frobenius norm of the weights. The generalization bound is derived using a PAC-Bayes analysis.

More N = 4 superconformal bootstrap.

Abstract: In this long overdue second installment, we continue to develop the conformal bootstrap program for $\mathcal{N}=4$ superconformal field theories (SCFTs) in four dimensions via an analysis of the correlation function of four stress-tensor supermultiplets. We review analytic results for this correlator and make contact with the SCFT/chiral algebra correspondence of Beem et al. [Commun. Math. Phys. 336, 1359 (2015)]. We demonstrate that the constraints of unitarity and crossing symmetry require the central charge $c$ to be greater than or equal to $3/4$ in any interacting $\mathcal{N}=4$ SCFT. We apply numerical bootstrap methods to derive upper bounds on scaling dimensions and operator product expansion coefficients for several low-lying, unprotected operators as a function of the central charge. We interpret our bounds in the context of $\mathcal{N}=4$ super Yang-Mills theories, formulating a series of conjectures regarding the embedding of the conformal manifold---parametrized by the complexified gauge coupling---into the space of scaling dimensions and operator product expansion coefficients. Our conjectures assign a distinguished role to points on the conformal manifold that are self-dual under a subgroup of the $S$-duality group. This paper contains a more detailed exposition of a number of results previously reported in Beem et al. [Phys. Rev. Lett. 111, 071601 (2013)] in addition to new results.

Global optimality in low-rank matrix optimization

Abstract: This paper considers the minimization of a general objective function $f(\boldsymbol{X})$ over the set of rectangular $n\times m$ matrices that have rank at most $r$ . To reduce the computational burden, we factorize the variable $\boldsymbol{X}$ into a product of two smaller matrices and optimize over these two matrices instead of $\boldsymbol{X}$ . Despite the resulting nonconvexity, recent studies in matrix completion and sensing have shown that the factored problem has no spurious local minima and obeys the so-called strict saddle property (the function has a directional negative curvature at all critical points but local minima). We analyze the global geometry for a general and yet well-conditioned objective function $f(\boldsymbol{X})$ whose restricted strong convexity and restricted strong smoothness constants are comparable. In particular, we show that the reformulated objective function has no spurious local minima and obeys the strict saddle property. These geometric properties imply that a number of iterative optimization algorithms (such as gradient descent) can provably solve the factored problem with global convergence.

Sample-Optimal Tomography of Quantum States

Abstract: It is a fundamental problem to decide how many copies of an unknown mixed quantum state are necessary and sufficient to determine the state. Previously, it was known only that estimating states to error $\epsilon $ in trace distance required $O(dr^{2}/\epsilon ^{2})$ copies for a $d$ -dimensional density matrix of rank $r$ . Here, we give a theoretical measurement scheme (POVM) that requires $O (dr/ \delta) \ln ~(d/\delta) $ copies to estimate $\rho $ to error $\delta $ in infidelity, and a matching lower bound up to logarithmic factors. This implies $O((dr / \epsilon ^{2}) \ln ~(d/\epsilon))$ copies suffice to achieve error $\epsilon $ in trace distance. We also prove that for independent (product) measurements, $\Omega (dr^{2}/\delta ^{2}) / \ln (1/\delta)$ copies are necessary in order to achieve error $\delta $ in infidelity. For fixed $d$ , our measurement can be implemented on a quantum computer in time polynomial in $n$ .

Shape complexity and process energy consumption in electron beam melting: a case of something for nothing in additive manufacturing?

Abstract: Additive manufacturing (AM) technology is capable of building up component geometry in a layer-by-layer process, entirely without tools, molds, or dies. One advantage of the approach is that it is capable of efficiently creating complex product geometry. Using experimental data collected during the manufacture of a titanium test part on a variant of AM technology, electron beam melting (EBM), this research studies the effect of a variation in product shape complexity on process energy consumption. This is done by applying a computationally quantifiable convexity-based characteristic associated with shape complexity to the test part and correlating this quantity with per-layer process energy consumption on the EBM system. Only a weak correlation is found between the complexity metric and energy consumption (ρ = .35), suggesting that process energy consumption is indeed not driven by shape complexity. This result is discussed in the context of the energy consumption of computer-controlled machining technology, which forms an important substitute to EBM. This article further discusses the impact of available additional shape complexity at the manufacturing process level on the incentives toward minimization of energy inputs, additional benefits arising later within the product’s life cycle, and its implications for value creation possibilities.

Double ramification cycles on the moduli spaces of curves

Abstract: Curves of genus $g$ which admit a map to $\mathbf {P}^{1}$ with specified ramification profile $\mu$ over $0\in \mathbf {P}^{1}$ and $ u$ over $\infty\in \mathbf {P}^{1}$ define a double ramification cycle $\mathsf{DR}_{g}(\mu, u)$ on the moduli space of curves. The study of the restrictions of these cycles to the moduli of nonsingular curves is a classical topic. In 2003, Hain calculated the cycles for curves of compact type. We study here double ramification cycles on the moduli space of Deligne-Mumford stable curves. The cycle $\mathsf{DR}_{g}(\mu, u)$ for stable curves is defined via the virtual fundamental class of the moduli of stable maps to rubber. Our main result is the proof of an explicit formula for $\mathsf{DR}_{g}(\mu, u)$ in the tautological ring conjectured by Pixton in 2014. The formula expresses the double ramification cycle as a sum over stable graphs (corresponding to strata classes) with summand equal to a product over markings and edges. The result answers a question of Eliashberg from 2001 and specializes to Hain’s formula in the compact type case. When $\mu= u=\emptyset$ , the formula for double ramification cycles expresses the top Chern class $\lambda_{g}$ of the Hodge bundle of $\overline {\mathcal{M}}_{g}$ as a push-forward of tautological classes supported on the divisor of non-separating nodes. Applications to Hodge integral calculations are given.

Warped Product Bi-slant Immersions in Kaehler Manifolds

Abstract: A submanifold M of an almost Hermitian manifold $$(\widetilde{M},g,J)$$ is called slant, if for each point $$p\in M$$ and $$0 e X\in T_p M$$ , the angle between JX and $$T_p M$$ is constant (see Chen in Bull Aust Math Soc 41:135–147, 1990). Later, Carriazo (in: Proceedings of the ICRAMS 2000, Kharagpur, 2000) defined the notion of bi-slant immersions as an extension of slant immersions. In this paper, we study warped product bi-slant submanifolds in Kaehler manifolds and provide some examples of warped product bi-slant submanifolds in some complex Euclidean spaces. Our main theorem states that every warped product bi-slant submanifold in a Kaehler manifold is either a Riemannian product or a warped product hemi-slant submanifold.

Distributed PCP Theorems for Hardness of Approximation in P

Abstract: We present a new distributed} model of probabilistically checkable proofs (PCP). A satisfying assignment x ∊ \{0,1\}^n to a CNF formula \phi is shared between two parties, where Alice knows x_1, \dots, x_{n/2, Bob knows x_{n/2+1},\dots,x_n, and both parties know \phi. The goal is to have Alice and Bob jointly write a PCP that x satisfies \phi, while exchanging little or no information. Unfortunately, this model as-is does not allow for nontrivial query complexity. Instead, we focus on a non-deterministic} variant, where the players are helped by Merlin, a third party who knows all of x.Using our framework, we obtain, for the first time, PCP-like reductions from the Strong Exponential Time Hypothesis (SETH) to approximation problems in \P. In particular, under SETH we show that %(assuming SETH) there are no truly-subquadratic approximation algorithms for %the following problems: Maximum Inner Product over \{0,1\}-vectors, LCS Closest Pair over permutations, Approximate Partial Match, Approximate Regular Expression Matching, and Diameter in Product Metric. All our inapproximability factors are nearly-tight. In particular, for the first three problems we obtain nearly-polynomial factors of 2^{(log n)^{1-o(1)}};only (1+o(1))-factor lower bounds (under SETH) were known before.As an additional feature of our reduction, we obtain new SETH lower bounds for the exact} monochromatic Closest Pair problem in the Euclidean, Manhattan, and Hamming metrics.

Effect of electron-hole asymmetry on optical conductivity in 8 -P m m n borophene

Abstract: We present a detailed theoretical study of the Drude weight and optical conductivity of 8-$Pmmn$ borophene having tilted anisotropic Dirac cones. We provide exact analytical expressions of $xx$ and $yy$ components of the Drude weight as well as maximum optical conductivity. We also obtain exact analytical expressions of the minimum energy (${\ensuremath{\epsilon}}_{1}$) required to trigger the optical transitions and energy (${\ensuremath{\epsilon}}_{2}$) needed to attain maximum optical conductivity. We find that the Drude weight and optical conductivity are highly anisotropic as a consequence of the anisotropic Dirac cone. The optical conductivities have a nonmonotonic behavior with photon energy in the regime between ${\ensuremath{\epsilon}}_{1}$ and ${\ensuremath{\epsilon}}_{2}$, as a result of the tilted parameter ${v}_{t}$. The tilted parameter can be extracted by knowing ${\ensuremath{\epsilon}}_{1}$ and ${\ensuremath{\epsilon}}_{2}$ from optical measurements. The maximum values of the components of the optical conductivity do not depend on the carrier density and the tilted parameter. The product of the maximum values of the anisotropic conductivities has the universal value ${({e}^{2}/4\ensuremath{\hbar})}^{2}$. The tilted anisotropic Dirac cones in 8-$Pmmn$ borophene can be realized by the optical conductivity measurement.

Bounding the Costs of Quantum Simulation of Many-Body Physics in Real Space

Abstract: We present a quantum algorithm for simulating the dynamics of a first-quantized Hamiltonian in real space based on the truncated Taylor series algorithm. We avoid the possibility of singularities by applying various cutoffs to the system and using a high-order finite difference approximation to the kinetic energy operator. We find that our algorithm can simulate $\eta$ interacting particles using a number of calculations of the pairwise interactions that scales, for a fixed spatial grid spacing, as $\tilde{O}(\eta^2)$, versus the $\tilde{O}(\eta^5)$ time required by previous methods (assuming the number of orbitals is proportional to $\eta$), and scales super-polynomially better with the error tolerance than algorithms based on the Lie-Trotter-Suzuki product formula. Finally, we analyze discretization errors that arise from the spatial grid and show that under some circumstances these errors can remove the exponential speedups typically afforded by quantum simulation.

The $K$-theoretic bulk-edge correspondence for topological insulators

Abstract: We study the application of Kasparov theory to topological insulator systems and the bulk–edge correspondence. We consider observable algebras as modelled by crossed products, where bulk and edge systems may be linked by a short exact sequence. We construct unbounded Kasparov modules encoding the dynamics of the crossed product. We then link bulk and edge Kasparov modules using the Kasparov product. Because of the anti-linear symmetries that occur in topological insulator models, real \(C^*\)-algebras and KKO-theory must be used.

The local indistinguishability of multipartite product states

Abstract: We study the perfectly local indistinguishability of multipartite product states. Firstly, we follow the method of Zhang et al. (Phys Rev A 93:012314, 2016) to give another more concise set of $$2n-1$$2n-1 orthogonal product states in $${\mathbb {C}}^m\otimes {\mathbb {C}}^n$$Cm?Cn$$(4\le m\le n)$$(4≤m≤n) which can not be distinguished by local operations and classical communication. Then we use the three-dimensional cubes to present some product states which give us an intuitive view on how to construct locally indistinguishable product states in tripartite quantum systems. At last, we give an explicit construction of locally indistinguishable orthogonal product states for general multipartite systems.

Construction of nonlocal multipartite quantum states

Abstract: For general bipartite quantum systems, many sets of locally indistinguishable orthogonal product states have been constructed so far. Here, we first present a general method to construct multipartite orthogonal product states in ${d}_{1}\ensuremath{\bigotimes}{d}_{2}\ensuremath{\bigotimes}\ensuremath{\cdots}\ensuremath{\bigotimes}{d}_{n}({d}_{1,2,\ensuremath{\cdots},n}\ensuremath{\ge}3,n\ensuremath{\ge}4)$ by using some locally indistinguishable bipartite orthogonal product states. And we prove that these multipartite orthogonal quantum states cannot be distinguished by local operations and classical communication. Furthermore, in ${d}_{1}\ensuremath{\bigotimes}{d}_{2}\ensuremath{\bigotimes}\ensuremath{\cdots}\ensuremath{\bigotimes}{d}_{n}({d}_{1,2,\ensuremath{\cdots},n}\ensuremath{\ge}3,n\ensuremath{\ge}5)$, we give a general method to construct a much smaller number of locally indistinguishable multipartite orthogonal product states for even and odd $n$ separately. In addition, we also present a general method to construct complete orthogonal product bases for the multipartite quantum systems. Our results demonstrate the phenomenon of nonlocality without entanglement for the multipartite quantum systems.

Import competition and vertical integration: Evidence from India

Abstract: Recent theoretical contributions provide conflicting predictions about the effects of product market competition on firms' organizational choices. This paper uses a rich firm-product-level panel data set of Indian manufacturing firms to analyze the relationship between import competition and vertical integration. Exploiting exogenous variation from changes in India's trade policy, we find that foreign competition, induced by falling output tariffs, increases backward vertical integration by domestic firms. The effects are concentrated in rather homogenous product categories, among firms that mainly operate on the domestic market, and in relatively large firms. Our results are robust towards different sub-samples and hold with or without conditioning on various firm- and product-level characteristics including input tariffs and firm-year fixed effects. We also provide evidence that vertical integration is associated with higher physical productivity, lower marginal costs and rising markups.

$M$-fuzzifying interval spaces

Abstract: In this paper, we introduce the notion of $M$-fuzzifying interval spaces, and discuss the relationship between $M$-fuzzifying interval spaces and $M$-fuzzifying convex structures.It is proved that the category {bf MYCSA2} can be embedded in the category {bf MYIS} as a reflective subcategory, where {bf MYCSA2} and {bf MYIS} denote the category of $M$-fuzzifying convex structures of $M$-fuzzifying arity $leq 2$ and the category of $M$-fuzzifying interval spaces, respectively. Under the framework of $M$-fuzzifying interval spaces, subspaces and product spaces are presented and some of their fundamental properties are obtained.

Variational calculation of the ground state of closed-shell nuclei up to A =40

Abstract: Variational calculations of ground-state properties of $^{4}\mathrm{He},^{16}\mathrm{O}$, and $^{40}\mathrm{Ca}$ are carried out employing realistic phenomenological two- and three-nucleon potentials. The trial wave function includes two- and three-body correlations acting on a product of single-particle determinants. Expectation values are evaluated with a cluster expansion for the spin-isospin dependent correlations considering up to five-body cluster terms. The optimal wave function is obtained by minimizing the energy expectation value over a set of up to 20 parameters by means of a nonlinear optimization library. We present results for the binding energy, charge radius, one- and two-body densities, single-nucleon momentum distribution, charge form factor, and Coulomb sum rule. We find that the employed three-nucleon interaction becomes repulsive for $A\ensuremath{\ge}16$. In $^{16}\mathrm{O}$ the inclusion of such a force provides a better description of the properties of the nucleus. In $^{40}\mathrm{Ca}$ instead, the repulsive behavior of the three-body interaction fails to reproduce experimental data for the charge radius and the charge form factor. We find that the high-momentum region of the momentum distributions, determined by the short-range terms of nuclear correlations, exhibits a universal behavior independent of the particular nucleus. The comparison of the Coulomb sum rules for $^{4}\mathrm{He},^{16}\mathrm{O}$, and $^{40}\mathrm{Ca}$ reported in this work will help elucidate in-medium modifications of the nucleon form factors.

Semi-tensor product method to a class of event-triggered control for finite evolutionary networked games

Abstract: Using the approach of semi-tensor product of matrices, this study studies a class of event-triggered control for finite evolutionary networked games, where the control only works at some certain individual states First, by identifying `control does not work' as a new specific control strategy, the controlled game dynamics is converted into an algebraic form Second, to make the game converge globally, two necessary and sufficient conditions for the existence of event-triggered control are obtained Meanwhile, a constructive procedure is proposed to design state feedback control strategy and an adjustment method is presented to minimise the control times Finally, the developed theory results are illustrated by a numerical method

Secure and traceable manufactured parts

Abstract: A method for the verification and authentication of additive manufactured product, comprising the steps of receiving, from a customer, at least one customer requirement for a product, deriving at least one manufacturing requirement and generating a product geometry file for the product, recording, by a first computing device, to a distributed transaction register, a first transaction reflecting certification of the product geometry file, obtaining a first output reflecting the first transaction, printing the product with a 3D printer, recording, by a second computing device, to the distributed transaction register, a second transaction reflecting the printing of the product and the first output, obtaining a second output reflecting the second transaction, embedding within the product a unique code reflecting the second output, whereby the product geometry file and the printing of said product may be verified with the unique code such that the product may be authenticated.

Semicrossed Products of Operator Algebras by Semigroups

Abstract: We examine the semicrossed products of a semigroup action by $*$-endomorphisms on a C*-algebra, or more generally of an action on an arbitrary operator algebra by completely contractive endomorphisms. The choice of allowable representations affects the corresponding universal algebra. We seek quite general conditions which will allow us to show that the C*-envelope of the semicrossed product is (a full corner of) a crossed product of an auxiliary C*-algebra by a group action. Our analysis concerns a case-by-case dilation theory on covariant pairs. In the process we determine the C*-envelope for various semicrossed products of (possibly nonselfadjoint) operator algebras by spanning cones and lattice-ordered abelian semigroups. In particular, we show that the C*-envelope of the semicrossed product of C*-dynamical systems by doubly commuting representations of $\mathbb{Z}^n_+$ (by generally non-injective endomorphisms) is the full corner of a C*-crossed product. In consequence we connect the ideal structure of C*-covers to properties of the actions. In particular, when the system is classical, we show that the C*-envelope is simple if and only if the action is injective and minimal. The dilation methods that we use may be applied to non-abelian semigroups. We identify the C*-envelope for actions of the free semigroup $\mathbb{F}_+^n$ by automorphisms in a concrete way, and for injective systems in a more abstract manner. We also deal with C*-dynamical systems over Ore semigroups when the appropriate covariance relation is considered.

Galois self-dual constacyclic codes

Abstract: Generalizing the Euclidean inner product and the Hermitian inner product, we introduce Galois inner products, and study Galois self-dual constacyclic codes in a very general setting by a uniform method. The conditions for existence of Galois self-dual and isometrically Galois self-dual constacyclic codes are obtained. As consequences, results on self-dual, iso-dual and Hermitian self-dual constacyclic codes are derived.

Geometrical product specification and verification in additive manufacturing

Abstract: The geometric freedom associated with additive manufacturing (AM) processes create new challenges in defining, communicating, and assessing the dimensional and geometric accuracy of parts. Starting from a review of the ASME-GD&T and ISO-GPS current practices, a new approach is proposed in this paper. The new approach combines current tolerancing practices with an enriched voxel-based volumetric representation scheme to overcome the limitations of standard methods. Moreover, the new approach enables a linkage between product design optimization and product verification with respect to the AM process chain. A case study is considered to demonstrate the concept.

Arithmetic of commutative semigroups with a focus on semigroups of ideals and modules

Abstract: Let H be a commutative semigroup with unit element such that every non-unit can be written as a finite product of irreducible elements (atoms). For every k ∈ ℕ, let 𝒰k(H) denote the set of all l ∈ ℕ with the property that there are atoms u1,…,uk,v1,…,vl such that u1 ⋅… ⋅ uk = v1 ⋅… ⋅ vl (thus, 𝒰k(H) is the union of all sets of lengths containing k). The Structure Theorem for Unions states that, for all sufficiently large k, the sets 𝒰k(H) are almost arithmetical progressions with the same difference and global bound. We present a new approach to this result in the framework of arithmetic combinatorics, by deriving, for suitably defined families of subsets of the non-negative integers, a characterization of when the Structure Theorem holds. This abstract approach allows us to verify, for the first time, the Structure Theorem for a variety of possibly non-cancellative semigroups, including semigroups of (not necessarily invertible) ideals and semigroups of modules. Furthermore, we provide the very first exa...

Logarithmic conformal field theory, log-modular tensor categories and modular forms

Abstract: The two pillars of rational conformal field theory and rational vertex operator algebras are modularity of characters on the one hand and its interpretation of modules as objects in a modular tensor category on the other one. Overarching these pillars is the Verlinde formula. In this paper we consider the more general class of logarithmic conformal field theories and $C_2$-cofinite vertex operator algebras. We suggest that their modular pillar are trace functions with insertions corresponding to intertwiners of the projective cover of the vacuum, and that the categorical pillar are finite tensor categories $\mathcal C$ which are ribbon and whose double is isomorphic to the Deligne product $\mathcal C\otimes \mathcal C^{opp}$. Overarching these pillars is then a logarithmic variant of Verlinde's formula. Numerical data realizing this are the modular $S$-matrix and modified traces of open Hopf links. The representation categories of $C_2$-cofinite and logarithmic conformal field theories that are fairly well understood are those of the $\mathcal W_p$-triplet algebras and the symplectic fermions. We illustrate the ideas in these examples and especially make the relation between logarithmic Hopf links and modular transformations explicit.

Stochastic Cubic Regularization for Fast Nonconvex Optimization

Abstract: This paper proposes a stochastic variant of a classic algorithm---the cubic-regularized Newton method [Nesterov and Polyak 2006]. The proposed algorithm efficiently escapes saddle points and finds approximate local minima for general smooth, nonconvex functions in only $\mathcal{\tilde{O}}(\epsilon^{-3.5})$ stochastic gradient and stochastic Hessian-vector product evaluations. The latter can be computed as efficiently as stochastic gradients. This improves upon the $\mathcal{\tilde{O}}(\epsilon^{-4})$ rate of stochastic gradient descent. Our rate matches the best-known result for finding local minima without requiring any delicate acceleration or variance-reduction techniques.

Query-to-communication lifting for PNP

Abstract: We prove that the PNP-type query complexity (alternatively, decision list width) of any boolean function f is quadratically related to the PNP-type communication complexity of a lifted version of f. As an application, we show that a certain "product" lower bound method of Impagliazzo and Williams (CCC 2010) fails to capture PNP communication complexity up to polynomial factors, which answers a question of Papakonstantinou, Scheder, and Song (CCC 2014).

Local indistinguishability of multipartite orthogonal product bases

Abstract: So far, very little is known about local indistinguishability of multipartite orthogonal product bases except some special cases. We first give a method to construct an orthogonal product basis with n parties each holding a $$\frac{1}{2}(n+1)$$ -dimensional system, where $$n\ge 5$$ and n is odd. The proof of the local indistinguishability of the basis exhibits that it is a sufficient condition for the local indistinguishability of an orthogonal multipartite product basis that all the positive operator-valued measure elements of each party can only be proportional to the identity operator to make further discrimination feasible. Then, we construct a set of n-partite product states, which contains only 2n members and cannot be perfectly distinguished by local operations and classic communication. All the results lead to a better understanding of the phenomenon of quantum nonlocality without entanglement in multipartite and high-dimensional quantum systems.