Showing papers on "Product (mathematics) published in 2016"

Generative Image Modeling Using Style and Structure Adversarial Networks

Abstract: Current generative frameworks use end-to-end learning and generate images by sampling from uniform noise distribution. However, these approaches ignore the most basic principle of image formation: images are product of: (a) Structure: the underlying 3D model; (b) Style: the texture mapped onto structure. In this paper, we factorize the image generation process and propose Style and Structure Generative Adversarial Network (\({\text {S}^2}\)-GAN). Our \({\text {S}^2}\)-GAN has two components: the Structure-GAN generates a surface normal map; the Style-GAN takes the surface normal map as input and generates the 2D image. Apart from a real vs. generated loss function, we use an additional loss with computed surface normals from generated images. The two GANs are first trained independently, and then merged together via joint learning. We show our \({\text {S}^2}\)-GAN model is interpretable, generates more realistic images and can be used to learn unsupervised RGBD representations.

Neural Sentiment Classification with User and Product Attention

Abstract: Document-level sentiment classification aims to predict user’s overall sentiment in a document about a product. However, most of existing methods only focus on local text information and ignore the global user preference and product characteristics. Even though some works take such information into account, they usually suffer from high model complexity and only consider wordlevel preference rather than semantic levels. To address this issue, we propose a hierarchical neural network to incorporate global user and product information into sentiment classification. Our model first builds a hierarchical LSTM model to generate sentence and document representations. Afterwards, user and product information is considered via attentions over different semantic levels due to its ability of capturing crucial semantic components. The experimental results show that our model achieves significant and consistent improvements compared to all state-of-theart methods. The source code of this paper can be obtained from https://github. com/thunlp/NSC.

Quantum gravity and inventory accumulation

Abstract: We begin by studying inventory accumulation at a LIFO (last-in-first-out) retailer with two products. In the simplest version, the following occur with equal probability at each time step: first product ordered, first product produced, second product ordered, second product produced. The inventory thus evolves as a simple random walk on $\mathbb{Z}^{2}$. In more interesting versions, a $p$ fraction of customers orders the “freshest available” product regardless of type. We show that the corresponding random walks scale to Brownian motions with diffusion matrices depending on $p$. We then turn our attention to the critical Fortuin–Kastelyn random planar map model, which gives, for each $q>0$, a probability measure on random (discretized) two-dimensional surfaces decorated by loops, related to the $q$-state Potts model. A longstanding open problem is to show that as the discretization gets finer, the surfaces converge in law to a limiting (loop-decorated) random surface. The limit is expected to be a Liouville quantum gravity surface decorated by a conformal loop ensemble, with parameters depending on $q$. Thanks to a bijection between decorated planar maps and inventory trajectories (closely related to bijections of Bernardi and Mullin), our results about the latter imply convergence of the former in a particular topology. A phase transition occurs at $p=1/2$, $q=4$.

Design-Efficient Approximate Multiplication Circuits Through Partial Product Perforation

Abstract: Approximate computing has received significant attention as a promising strategy to decrease power consumption of inherently error tolerant applications. In this paper, we focus on hardware-level approximation by introducing the partial product perforation technique for designing approximate multiplication circuits. We prove in a mathematically rigorous manner that in partial product perforation, the imposed errors are bounded and predictable, depending only on the input distribution. Through extensive experimental evaluation, we apply the partial product perforation method on different multiplier architectures and expose the optimal architecture–perforation configuration pairs for different error constraints. We show that, compared with the respective exact design, the partial product perforation delivers reductions of up to 50% in power consumption, 45% in area, and 35% in critical delay. In addition, the product perforation method is compared with the state-of-the-art approximation techniques, i.e., truncation, voltage overscaling, and logic approximation, showing that it outperforms them in terms of power dissipation and error.

Large $N$ topologically twisted index: necklace quivers, dualities, and Sasaki-Einstein spaces

Abstract: In this paper, we calculate the topological free energy for a number of $$ \mathcal{N} $$ ≥ 2 Yang-Mills-Chern-Simons-matter theories at large N and fixed Chern-Simons levels. The topological free energy is defined as the logarithm of the partition function of the theory on S 2 × S 1 with a topological A-twist along S 2 and can be reduced to a matrix integral by exploiting the localization technique. The theories of our interest are dual to a variety of Calabi-Yau four-fold singularities, including a product of two asymptotically locally Euclidean singularities and the cone over various well-known homogeneous Sasaki-Einstein seven-manifolds, N 0,1,0, V 5,2, and Q 1,1,1. We check that the large N topological free energy can be matched for theories which are related by dualities, including mirror symmetry and $$ \mathrm{S}\mathrm{L}\left(2,\mathbb{Z}\right) $$ duality.

Quantum nonlocality of multipartite orthogonal product states

Abstract: Local distinguishability of orthogonal quantum states is an area of active research in quantum information theory. However, most of the relevant results are about local distinguishability in bipartite Hilbert space and very little is known about the multipartite case. In this paper we present a generic method to construct a completable $n$-partite $(n\ensuremath{\ge}3)$ product basis with only $2n$ members, which exhibits nonlocality without entanglement with $n$ parties, each holding a system of any finite dimension. We give an effective proof of the nonlocality of the completable multipartite product basis. In addition, we construct another incomplete multipartite product basis with a smaller number of members that cannot be distinguished by local operations and classical communication in a ${d}_{1}\ensuremath{\bigotimes}{d}_{2}\ensuremath{\bigotimes}\ensuremath{\cdots}\ensuremath{\bigotimes}{d}_{n}$ quantum system, where $n\ensuremath{\ge}3$ and ${d}_{i}\ensuremath{\ge}2$ for $i=1,2,...,n$. The results can lead to a better understanding of the phenomenon of nonlocality without entanglement in any multipartite quantum system.

Low-tubal-rank Tensor Completion using Alternating Minimization

Abstract: The low-tubal-rank tensor model has been recently proposed for real-world multidimensional data In this paper, we study the low-tubal-rank tensor completion problem, ie, to recover a third-order tensor by observing a subset of its elements selected uniformly at random We propose a fast iterative algorithm, called {\em Tubal-Alt-Min}, that is inspired by a similar approach for low-rank matrix completion The unknown low-tubal-rank tensor is represented as the product of two much smaller tensors with the low-tubal-rank property being automatically incorporated, and Tubal-Alt-Min alternates between estimating those two tensors using tensor least squares minimization First, we note that tensor least squares minimization is different from its matrix counterpart and nontrivial as the circular convolution operator of the low-tubal-rank tensor model is intertwined with the sub-sampling operator Second, the theoretical performance guarantee is challenging since Tubal-Alt-Min is iterative and nonconvex in nature We prove that 1) Tubal-Alt-Min guarantees exponential convergence to the global optima, and 2) for an $n \times n \times k$ tensor with tubal-rank $r \ll n$, the required sampling complexity is $O(nr^2k \log^3 n)$ and the computational complexity is $O(n^2rk^2 \log^2 n)$ Third, on both synthetic data and real-world video data, evaluation results show that compared with tensor-nuclear norm minimization (TNN-ADMM), Tubal-Alt-Min improves the recovery error dramatically (by orders of magnitude) It is estimated that Tubal-Alt-Min converges at an exponential rate $10^{-04423 \text{Iter}}$ where $\text{Iter}$ denotes the number of iterations, which is much faster than TNN-ADMM's $10^{-00332 \text{Iter}}$, and the running time can be accelerated by more than $5$ times for a $200 \times 200 \times 20$ tensor

Sets of Lengths

Abstract: Oftentimes the elements of a ring or semigroup can be written as finite products of irreducible elements. An element a can be a product of k irreducibles and a product of l irreducibles. The set L(a) of all possible factorization lengths of a is called the set of lengths of a, and the system consisting of all these sets L(a) is a well-studied means of describing the nonuniqueness of factorizations of a ring or semigroup. We provide a friendly introduction, which is largely self-contained, to what is known about systems of sets of lengths for rings of integers of algebraic number fields and for transfer Krull monoids of finite type as their generalization.

Local indistinguishability of orthogonal product states

Abstract: In the general bipartite quantum system $m\ensuremath{\bigotimes}n\phantom{\rule{0.16em}{0ex}}(m,n\ensuremath{\ge}3)$, Y.-L. Wang et al. [Phys. Rev. A 92, 032313 (2015)] presented $3(m+n)\ensuremath{-}9$ orthogonal product states which cannot be distinguished by local operations and classical communication (LOCC). In this paper, we aim to construct less locally indistinguishable orthogonal product states in $m\ensuremath{\bigotimes}n$. First, in the $3\ensuremath{\bigotimes}n\phantom{\rule{0.28em}{0ex}}(3ln)$ quantum system, we construct $3n\ensuremath{-}2$ locally indistinguishable orthogonal product states which are not unextendible product bases. Then, for $m\ensuremath{\bigotimes}n\phantom{\rule{0.28em}{0ex}}(4\ensuremath{\le}m\ensuremath{\le}n)$, we present $3n+m\ensuremath{-}4$ orthogonal product states which cannot be perfectly distinguished by LOCC. Finally, in the general bipartite quantum system $m\ensuremath{\bigotimes}n\phantom{\rule{0.28em}{0ex}}(3\ensuremath{\le}m\ensuremath{\le}n)$, we show a smaller set with $2n\ensuremath{-}1$ orthogonal product states and prove that these states are LOCC indistinguishable using a very simple but quite effective method. All of the above results demonstrate the phenomenon of nonlocality without entanglement.

The median class and superrigidity of actions on CAT(0) cube complexes

Abstract: We define a bounded cohomology class, called the median class, in the second bounded cohomology, with appropriate coefficients, of the automorphism group of a finite-dimensional CAT(0) cube complex X. The median class of X behaves naturally with respect to taking products and appropriate subcomplexes and defines in turn the median class of an action by automorphisms of X. We show that the median class of a non-elementary action by automorphisms does not vanish and we show to what extent it does vanish if the action is elementary. We obtain as a corollary a superrigidity result and show, for example, that any irreducible lattice in the product of at least two locally compact connected groups acts on a finite-dimensional CAT(0) cube complex X with a finite orbit in the Roller compactification of X. In the case of a product of Lie groups, the appendix by Caprace allows us to deduce that the fixed point is in fact inside the complex X. In the course of the proof, we construct a Γequivariant measurable map from a Poisson boundary of Γ with values in the non-terminating ultrafilters on the Roller boundary of X.

IncLing: efficient product-line testing using incremental pairwise sampling

Abstract: A software product line comprises a family of software products that share a common set of features. It enables customers to compose software systems from a managed set of features. Testing every product of a product line individually is often infeasible due to the exponential number of possible products in the number of features. Several approaches have been proposed to restrict the number of products to be tested by sampling a subset of products achieving sufficient combinatorial interaction coverage. However, existing sampling algorithms do not scale well to large product lines, as they require a considerable amount of time to generate the samples. Moreover, samples are not available until a sampling algorithm completely terminates. As testing time is usually limited, we propose an incremental approach of product sampling for pairwise interaction testing (called IncLing), which enables developers to generate samples on demand in a step-wise manner. Furthermore, IncLing uses heuristics to efficiently achieve pairwise interaction coverage with a reasonable number of products. We evaluated IncLing by comparing it against existing sampling algorithms using feature models of different sizes. The results of our approach indicate efficiency improvements for product-line testing.

The Maxwell Compactness Property in Bounded Weak Lipschitz Domains with Mixed Boundary Conditions

Abstract: Let $\Omega\subset\mathbb{R}^3$ be a bounded weak Lipschitz domain with boundary $\Gamma:=\operatorname{\partial}\Omega$ divided into two weak Lipschitz submanifolds $\Gamma_{\tau}$ and $\Gamma_{ u}$, and let $\varepsilon$ denote an $\mathsf{L}^{\infty}$-matrix field inducing an inner product in $\mathsf{L}^2(\Omega)$. The key result of this paper is the so-called Maxwell compactness property, i.e., the Hilbert space $\bigl\{E\in\mathsf{L}^2(\Omega):\operatorname{rot}E\in\mathsf{L}^2(\Omega),\,\operatorname{div}\varepsilon E\in\mathsf{L}^2(\Omega),\, u\times E|_{\Gamma_{\tau}}=0,\, u\cdot\varepsilon E|_{\Gamma_{ u}}=0\bigr\}$ is compactly embedded into $\mathsf{L}^2(\Omega)$. We will also prove some canonical applications, such as Maxwell estimates, Helmholtz decompositions, and a static solution theory. Furthermore, a Fredholm alternative for the underlying time-harmonic Maxwell problem and all corresponding and related results for exterior domains formulated in weighted Sobolev spaces are straightfo...

Fractional-Wavelet Analysis of Positive definite Distributions and Wavelets on $$\varvec{\mathscr {D'}}(\mathbb {C})$$

Abstract: In this chapter we describe a wavelet expansion theory for positive definite distributions over the real line and define a fractional derivative operator for complex functions in the distribution sense. In order to obtain a characterization of the complex fractional derivative through the distribution theory, the Ortigueira-Caputo fractional derivative operator \(_{\text {C}}\text {D}^{\alpha }\) [13] is rewritten as a convolution product according to the fractional calculus of real distributions [8]. In particular, the fractional derivative of the Gabor–Morlet wavelet is computed together with its plots and main properties.

Theory of (2+1)-dimensional fermionic topological orders and fermionic/bosonic topological orders with symmetries

Abstract: We propose a systematic framework to classify (2+1)-dimensional (2+1D) fermionic topological orders without symmetry and 2+1D fermionic/bosonic topological orders with symmetry $G$. The key is to use the so-called symmetric fusion category $\mathcal{E}$ to describe the symmetry. Here, $\mathcal{E}=\mathrm{sRep}({\mathbb{Z}}_{2}^{f})$ describing particles in a fermionic product state without symmetry, or $\mathcal{E}=\mathrm{sRep}({G}^{f}) [\mathcal{E}=\mathrm{Rep}(G)]$ describing particles in a fermionic (bosonic) product state with symmetry $G$. Then, topological orders with symmetry $\mathcal{E}$ are classified by nondegenerate unitary braided fusion categories over $\mathcal{E}$, plus their modular extensions and total chiral central charges. This allows us to obtain a list that contains all 2+1D fermionic topological orders without symmetry. For example, we find that, up to $p+\phantom{\rule{1.0pt}{0ex}}\mathrm{i}\phantom{\rule{1.0pt}{0ex}}p$ fermionic topological orders, there are only four fermionic topological orders with one nontrivial topological excitation: (1) the $K=\left(\begin{array}{cc}\hfill \ensuremath{-}1a \hfill 0\\ \hfill 0a \hfill 2\end{array}\right)$ fractional quantum Hall state, (2) a Fibonacci bosonic topological order stacking with a fermionic product state, (3) the time-reversal conjugate of the previous one, and (4) a fermionic topological order with chiral central charge $c=\frac{1}{4}$, whose only topological excitation has non-Abelian statistics with spin $s=\frac{1}{4}$ and quantum dimension $d=1+\sqrt{2}$.

The Full Classification of Orthogonal Easy Quantum Groups

Abstract: We study easy quantum groups, a combinatorial class of orthogonal quantum groups introduced by Banica–Speicher in 2009. We show that there is a countable descending chain of easy quantum groups interpolating between Bichon’s free wreath product with the permutation group Sn and a semi-direct product of a permutation action of Sn on a free product. This reveals a series of new commutation relations interpolating between a free product construction and the tensor product. Furthermore, we prove a dichotomy result saying that every hyperoctahedral easy quantum group is either part of our new interpolating series of quantum groups or belongs to a class of semi-direct product quantum groups recently studied by the authors. This completes the classification of easy quantum groups. We also study combinatorial and operator algebraic aspects of the new interpolating series.

Every Matrix is a Product of Toeplitz Matrices

Abstract: We show that every $$n\,\times \,n$$n×n matrix is generically a product of $$\lfloor n/2 \rfloor + 1$$?n/2?+1 Toeplitz matrices and always a product of at most $$2n+5$$2n+5 Toeplitz matrices. The same result holds true if the word `Toeplitz' is replaced by `Hankel,' and the generic bound $$\lfloor n/2 \rfloor + 1$$?n/2?+1 is sharp. We will see that these decompositions into Toeplitz or Hankel factors are unusual: We may not, in general, replace the subspace of Toeplitz or Hankel matrices by an arbitrary $$(2n-1)$$(2n-1)-dimensional subspace of $${n\,\times \,n}$$n×n matrices. Furthermore, such decompositions do not exist if we require the factors to be symmetric Toeplitz or persymmetric Hankel, even if we allow an infinite number of factors.

Large $N$ topologically twisted index: necklace quivers, dualities, and Sasaki-Einstein spaces

Abstract: In this paper, we calculate the topological free energy for a number of ${\mathcal N} \geq 2$ Yang-Mills-Chern-Simons-matter theories at large $N$ and fixed Chern-Simons levels. The topological free energy is defined as the logarithm of the partition function of the theory on $S^2 \times S^1$ with a topological A-twist along $S^2$ and can be reduced to a matrix integral by exploiting the localization technique. The theories of our interest are dual to a variety of Calabi-Yau four-fold singularities, including a product of two asymptotically locally Euclidean singularities and the cone over various well-known homogeneous Sasaki-Einstein seven-manifolds, $N^{0,1,0}$, $V^{5,2}$, and $Q^{1,1,1}$. We check that the large $N$ topological free energy can be matched for theories which are related by dualities, including mirror symmetry and $\mathrm{SL}(2,\mathbb{Z})$ duality.

Quasilinear SPDEs via rough paths

Abstract: We are interested in (uniformly) parabolic PDEs with a nonlinear dependance of the leading-order coefficients, driven by a rough right hand side. For simplicity, we consider a space-time periodic setting with a single spatial variable: \begin{equation*} \partial_2u -P( a(u)\partial_1^2u - \sigma(u)f ) =0 \end{equation*} where $P$ is the projection on mean-zero functions, and $f$ is a distribution and only controlled in the low regularity norm of $ C^{\alpha-2}$ for $\alpha > \frac{2}{3}$ on the parabolic Holder scale. The example we have in mind is a random forcing $f$ and our assumptions allow, for example, for an $f$ which is white in the time variable $x_2$ and only mildly coloured in the space variable $x_1$; any spatial covariance operator $(1 + |\partial_1|)^{-\lambda_1 }$ with $\lambda_1 > \frac13$ is admissible. On the deterministic side we obtain a $C^\alpha$-estimate for $u$, assuming that we control products of the form $v\partial_1^2v$ and $vf$ with $v$ solving the constant-coefficient equation $\partial_2 v-a_0\partial_1^2v=f$. As a consequence, we obtain existence, uniqueness and stability with respect to $(f, vf, v \partial_1^2v)$ of small space-time periodic solutions for small data. We then demonstrate how the required products can be bounded in the case of a random forcing $f$ using stochastic arguments. For this we extend the treatment of the singular product $\sigma(u)f$ via a space-time version of Gubinelli's notion of controlled rough paths to the product $a(u)\partial_1^2u$, which has the same degree of singularity but is more nonlinear since the solution $u$ appears in both factors. The PDE ingredient mimics the (kernel-free) Krylov-Safanov approach to ordinary Schauder theory.

Method for tracking stock level within a store

Abstract: One variation of a method for tracking stock level within a store includes: dispatching a robotic system to image shelving structures within the store during a scan cycle; receiving images from the robotic system, each image recorded by the robotic system during the scan cycle and corresponding to one waypoint within the store; identifying, in the images, empty slots within the shelving structures; identifying a product assigned to each empty slot based on product location assignments defined in a planogram of the store; for a first product of a first product value and assigned to a first empty slot, generating a first prompt to restock the first empty slot with a unit of the first product during the scan cycle; and, upon completion of the scan cycle, generating a global restocking list specifying restocking of a set of empty slots associated with product values less than the first product value.

Function-Hiding Inner Product Encryption is Practical.

Abstract: In a functional encryption scheme, secret keys are associated with functions and ciphertexts are associated with messages. Given a secret key for a function f, and a ciphertext for a message x, a decryptor learns f(x) and nothing else about x. Inner product encryption is a special case of functional encryption where both secret keys and ciphertext are associated with vectors. The combination of a secret key for a vector \({\mathbf {x}}\) and a ciphertext for a vector \(\mathbf {y}\) reveal \(\langle {\mathbf {x}}, \mathbf {y}\rangle \) and nothing more about \(\mathbf {y}\). An inner product encryption scheme is function-hiding if the keys and ciphertexts reveal no additional information about both \({\mathbf {x}}\) and \(\mathbf {y}\) beyond their inner product.

Global solutions to the oldroyd-b model with a class of large initial data

Abstract: Consider a global well-posed problem for the incompressible Oldroyd-B model. It is shown that this set of equations admits a unique global solution provided the initial horizontal velocity $u^h_0$, the product $\omega u^d_0$ of the coupling parameter $\omega$ and initial vertical velocity $u^d_0$, and initial symmetric tensor of constraints $\tau_0$ are sufficiently small in the scaling invariant Besov space $\dot{B}^{\tfrac{d}{2}-1}_{2,1}\times\dot{B}^{\tfrac{d}{2}}_{2,1}, d\ge2$. In particular, the result implies the global well-posedness of the Oldroyd-B model with large initial vertical velocity $u_0^d$.

Class of exact memory-kernel master equations

Abstract: A well-known situation in which a non-Markovian dynamics of an open quantum system $S$ arises is when this is coherently coupled to an auxiliary system $M$ in contact with a Markovian bath. In such cases, while the joint dynamics of $S\text{\ensuremath{-}}M$ is Markovian and obeys a standard (bipartite) Lindblad-type master equation (ME), this is in general not true for the reduced dynamics of $S$. Furthermore, there are several instances (e.g., the dissipative Jaynes-Cummings model) in which a closed ME for the $S$'s state cannot even be worked out. Here, we find a class of bipartite Lindblad-type MEs such that the reduced ME of $S$ can be derived exactly and in a closed form for any initial product state of $S\text{\ensuremath{-}}M$. We provide a detailed microscopic derivation of our result in terms of a mapping between two collision models.

Computing a Nonnegative Matrix Factorization---Provably

Abstract: In the nonnegative matrix factorization (NMF) problem we are given an $n \times m$ nonnegative matrix $M$ and an integer $r > 0$. Our goal is to express $M$ as $A W$, where $A$ and $W$ are nonnegative matrices of size $n \times r$ and $r \times m$, respectively. In some applications, it makes sense to ask instead for the product $AW$ to approximate $M$, i.e. (approximately) minimize $\left\lVert{M - AW}_F\right\rVert$, where $\left\lVert\right\rVert_F$, denotes the Frobenius norm; we refer to this as approximate NMF. This problem has a rich history spanning quantum mechanics, probability theory, data analysis, polyhedral combinatorics, communication complexity, demography, chemometrics, etc. In the past decade NMF has become enormously popular in machine learning, where $A$ and $W$ are computed using a variety of local search heuristics. Vavasis recently proved that this problem is NP-complete. (Without the restriction that $A$ and $W$ be nonnegative, both the exact and approximate problems can be solved ...

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.