Showing papers on "Operator (computer programming) published in 2017"

Axiomatic Attribution for Deep Networks

Abstract: We study the problem of attributing the prediction of a deep network to its input features, a problem previously studied by several other works. We identify two fundamental axioms---Sensitivity and Implementation Invariance that attribution methods ought to satisfy. We show that they are not satisfied by most known attribution methods, which we consider to be a fundamental weakness of those methods. We use the axioms to guide the design of a new attribution method called Integrated Gradients. Our method requires no modification to the original network and is extremely simple to implement; it just needs a few calls to the standard gradient operator. We apply this method to a couple of image models, a couple of text models and a chemistry model, demonstrating its ability to debug networks, to extract rules from a network, and to enable users to engage with models better.

Modal Analysis of Fluid Flows: An Overview

Abstract: Simple aerodynamic configurations under even modest conditions can exhibit complex flows with a wide range of temporal and spatial features. It has become common practice in the analysis of these flows to look for and extract physically important features, or modes, as a first step in the analysis. This step typically starts with a modal decomposition of an experimental or numerical dataset of the flowfield, or of an operator relevant to the system. We describe herein some of the dominant techniques for accomplishing these modal decompositions and analyses that have seen a surge of activity in recent decades [1–8]. For a nonexpert, keeping track of recent developments can be daunting, and the intent of this document is to provide an introduction to modal analysis that is accessible to the larger fluid dynamics community. In particular, we present a brief overview of several of the well-established techniques and clearly lay the framework of these methods using familiar linear algebra. The modal analysis techniques covered in this paper include the proper orthogonal decomposition (POD), balanced proper orthogonal decomposition (balanced POD), dynamic mode decomposition (DMD), Koopman analysis, global linear stability analysis, and resolvent analysis.

From the Quantum Approximate Optimization Algorithm to a Quantum Alternating Operator Ansatz.

Abstract: The next few years will be exciting as prototype universal quantum processors emerge, enabling implementation of a wider variety of algorithms. Of particular interest are quantum heuristics, which require experimentation on quantum hardware for their evaluation, and which have the potential to significantly expand the breadth of quantum computing applications. A leading candidate is Farhi et al.'s Quantum Approximate Optimization Algorithm, which alternates between applying a cost-function-based Hamiltonian and a mixing Hamiltonian. Here, we extend this framework to allow alternation between more general families of operators. The essence of this extension, the Quantum Alternating Operator Ansatz, is the consideration of general parametrized families of unitaries rather than only those corresponding to the time-evolution under a fixed local Hamiltonian for a time specified by the parameter. This ansatz supports the representation of a larger, and potentially more useful, set of states than the original formulation, with potential long-term impact on a broad array of application areas. For cases that call for mixing only within a desired subspace, refocusing on unitaries rather than Hamiltonians enables more efficiently implementable mixers than was possible in the original framework. Such mixers are particularly useful for optimization problems with hard constraints that must always be satisfied, defining a feasible subspace, and soft constraints whose violation we wish to minimize. More efficient implementation enables earlier experimental exploration of an alternating operator approach to a wide variety of approximate optimization, exact optimization, and sampling problems. Here, we introduce the Quantum Alternating Operator Ansatz, lay out design criteria for mixing operators, detail mappings for eight problems, and provide brief descriptions of mappings for diverse problems.

Ten equivalent definitions of the fractional laplace operator

Abstract: This article reviews several definitions of the fractional Laplace operator (-Delta)^{alpha/2} (0 < alpha < 2) in R^d, also known as the Riesz fractional derivative operator, as an operator on Lebesgue spaces L^p, on the space C_0 of continuous functions vanishing at infinity and on the space C_{bu} of bounded uniformly continuous functions. Among these definitions are ones involving singular integrals, semigroups of operators, Bochner's subordination and harmonic extensions. We collect and extend known results in order to prove that all these definitions agree: on each of the function spaces considered, the corresponding operators have common domain and they coincide on that common domain.

Optimal Graph-Filter Design and Applications to Distributed Linear Network Operators

Abstract: We study the optimal design of graph filters (GFs) to implement arbitrary linear transformations between graph signals GFs can be represented by matrix polynomials of the graph-shift operator (GSO) Since this operator captures the local structure of the graph, GFs naturally give rise to distributed linear network operators In most setups, the GSO is given so that GF design consists fundamentally in choosing the (filter) coefficients of the matrix polynomial to resemble desired linear transformations We determine spectral conditions under which a specific linear transformation can be implemented perfectly using GFs For the cases where perfect implementation is infeasible, we address the optimization of the filter coefficients to approximate the desired transformation Additionally, for settings where the GSO itself can be modified, we study its optimal design as well After this, we introduce the notion of a node-variant GF, which allows the simultaneous implementation of multiple (regular) GFs in different nodes of the graph This additional flexibility enables the design of more general operators without undermining the locality in implementation Perfect and approximate designs are also studied for this new type of GFs To showcase the relevance of the results in the context of distributed linear network operators, this paper closes with the application of our framework to two particular distributed problems: finite-time consensus and analog network coding

A Mellin space approach to the conformal bootstrap

Abstract: We describe in more detail our approach to the conformal bootstrap which uses the Mellin representation of CFT d four point functions and expands them in terms of crossing symmetric combinations of AdS d+1 Witten exchange functions. We consider arbitrary external scalar operators and set up the conditions for consistency with the operator product expansion. Namely, we demand cancellation of spurious powers (of the cross ratios, in position space) which translate into spurious poles in Mellin space. We discuss two contexts in which we can immediately apply this method by imposing the simplest set of constraint equations. The first is the epsilon expansion. We mostly focus on the Wilson-Fisher fixed point as studied in an epsilon expansion about d = 4. We reproduce Feynman diagram results for operator dimensions to O(ϵ 3) rather straightforwardly. This approach also yields new analytic predictions for OPE coefficients to the same order which fit nicely with recent numerical estimates for the Ising model (at ϵ = 1). We will also mention some leading order results for scalar theories near three and six dimensions. The second context is a large spin expansion, in any dimension, where we are able to reproduce and go a bit beyond some of the results recently obtained using the (double) light cone expansion. We also have a preliminary discussion about numerical implementation of the above bootstrap scheme in the absence of a small parameter.

Averaged null energy condition from causality

Abstract: Unitary, Lorentz-invariant quantum field theories in flat spacetime obey mi-crocausality: commutators vanish at spacelike separation. For interacting theories in more than two dimensions, we show that this implies that the averaged null energy, ∫ duT uu , must be non-negative. This non-local operator appears in the operator product expansion of local operators in the lightcone limit, and therefore contributes to n-point functions. We derive a sum rule that isolates this contribution and is manifestly positive. The argument also applies to certain higher spin operators other than the stress tensor, generating an infinite family of new constraints of the form ∫ duX uuu···u ≥ 0. These lead to new inequalities for the coupling constants of spinning operators in conformal field theory, which include as special cases (but are generally stronger than) the existing constraints from the lightcone bootstrap, deep inelastic scattering, conformal collider methods, and relative entropy. We also comment on the relation to the recent derivation of the averaged null energy condition from relative entropy, and suggest a more general connection between causality and information-theoretic inequalities in QFT.

Conformal Bootstrap in Mellin Space

Abstract: We propose a new approach towards analytically solving for the dynamical content of conformal field theories (CFTs) using the bootstrap philosophy. This combines the original bootstrap idea of Polyakov with the modern technology of the Mellin representation of CFT amplitudes. We employ exchange Witten diagrams with built-in crossing symmetry as our basic building blocks rather than the conventional conformal blocks in a particular channel. Demanding consistency with the operator product expansion (OPE) implies an infinite set of constraints on operator dimensions and OPE coefficients. We illustrate the power of this method in the. expansion of the Wilson-Fisher fixed point by reproducing anomalous dimensions and, strikingly, obtaining OPE coefficients to higher orders in. than currently available using other analytic techniques (including Feynman diagram calculations). Our results enable us to get a somewhat better agreement between certain observables in the 3D Ising model and the precise numerical values that have been recently obtained.

Group Decision Making Based on Heronian Aggregation Operators of Intuitionistic Fuzzy Numbers

Abstract: Archimedean ${t}$ -conorm and ${t}$ -norm provide the general operational rules for intuitionistic fuzzy numbers (IFNs). The aggregation operators based on them can generalize most of the existing aggregation operators. At the same time, the Heronian mean (HM) has a significant advantage of considering interrelationships between the attributes. Therefore, it is very necessary to extend the HM based on IFNs and to construct intuitionistic fuzzy HM operators based on the Archimedean ${t}$ -conorm and ${t}$ -norm. In this paper, we first discuss intuitionistic fuzzy operational rules based on the Archimedean ${t}$ -conorm and ${t}$ -norm. Then, we propose the intuitionistic fuzzy Archimedean Heronian aggregation (IFAHA) operator and the intuitionistic fuzzy weight Archimedean Heronian aggregation (IFWAHA) operator. We also further discuss some properties and some special cases of these new operators. Moreover, we also propose a new multiple attribute group decision making (MAGDM) method based on the proposed IFAHA operator and the proposed IFWAHA operator. Finally, we use an illustrative example to show the MAGDM processes and to illustrate the effectiveness of the developed method.

Bulk locality from modular flow

Abstract: We study the reconstruction of bulk operators in the entanglement wedge in terms of low energy operators localized in the respective boundary region. To leading order in N, the dual boundary operators are constructed from the modular flow of single trace operators in the boundary subregion. The appearance of modular evolved boundary operators can be understood due to the equality between bulk and boundary modular flows and explicit formulas for bulk operators can be found with a complete understanding of the action of bulk modular flow, a difficult but in principle solvable task. We also obtain an expression when the bulk operator is located on the Ryu-Takayanagi surface which only depends on the bulk to boundary correlator and does not require the explicit use of bulk modular flow. This expression generalizes the geodesic operator/OPE block dictionary to general states and boundary regions.

Fast Image Processing with Fully-Convolutional Networks

Abstract: We present an approach to accelerating a wide variety of image processing operators. Our approach uses a fully-convolutional network that is trained on input-output pairs that demonstrate the operator’s action. After training, the original operator need not be run at all. The trained network operates at full resolution and runs in constant time. We investigate the effect of network architecture on approximation accuracy, runtime, and memory footprint, and identify a specific architecture that balances these considerations. We evaluate the presented approach on ten advanced image processing operators, including multiple variational models, multiscale tone and detail manipulation, photographic style transfer, nonlocal dehazing, and nonphoto- realistic stylization. All operators are approximated by the same model. Experiments demonstrate that the presented approach is significantly more accurate than prior approximation schemes. It increases approximation accuracy as measured by PSNR across the evaluated operators by 8.5 dB on the MIT-Adobe dataset (from 27.5 to 36 dB) and reduces DSSIM by a multiplicative factor of 3 com- pared to the most accurate prior approximation scheme, while being the fastest. We show that our models general- ize across datasets and across resolutions, and investigate a number of extensions of the presented approach.

Slow scrambling in disordered quantum systems

Abstract: Understanding the spread of quantum entanglement and scrambling of information across a quantum many-body system is a fundamental problem in quantum dynamics. A necessary part of the spread of the entanglement is the growth in time of Heisenberg operators of initially localized operators. This operator growth can be diagnosed by studying the space-time structure of commutators of well-separated operators. Previous studies in this area focused on translation invariant models, but the effects of disorder, which can radically alter the motion of heat and charge, had not been studied. Here, the authors study the growth of operators in interacting systems with quenched disorder. In the localized phase, they find that operator sizes grow logarithmically in time similar to other measures of entanglement spreading. In the disordered metal, they find that operator sizes grow linearly with time, i.e., ballistically, in contrast to the diffusive motion of charge and heat. The ballistic growth of operators is quantified by the butterfly velocity which the authors relate to the charge diffusion constant and the interaction-induced inelastic scattering rate. When the diffusion of charge is slow, the resulting butterfly velocity is much smaller than the maximum speed allowed by microscopic causality constraints.

Imaging through glass diffusers using densely connected convolutional networks

Abstract: Computational imaging through scatter generally is accomplished by first characterizing the scattering medium so that its forward operator is obtained; and then imposing additional priors in the form of regularizers on the reconstruction functional so as to improve the condition of the originally ill-posed inverse problem. In the functional, the forward operator and regularizer must be entered explicitly or parametrically (e.g. scattering matrices and dictionaries, respectively.) However, the process of determining these representations is often incomplete, prone to errors, or infeasible. Recently, deep learning architectures have been proposed to instead learn both the forward operator and regularizer through examples. Here, we propose for the first time, to our knowledge, a convolutional neural network architecture called "IDiffNet" for the problem of imaging through diffuse media and demonstrate that IDiffNet has superior generalization capability through extensive tests with well-calibrated diffusers. We found that the Negative Pearson Correlation Coefficient loss function for training is more appropriate for spatially sparse objects and strong scattering conditions. Our results show that the convolutional architecture is robust to the choice of prior, as demonstrated by the use of multiple training and testing object databases, and capable of achieving higher space-bandwidth product reconstructions than previously reported.

Hexagonalization of correlation functions

Abstract: We propose a nonperturbative framework to study general correlation functions of single-trace operators in $$ \mathcal{N} $$ = 4 supersymmetric Yang-Mills theory at large N . The basic strategy is to decompose them into fundamental building blocks called the hexagon form factors, which were introduced earlier to study structure constants using integrability. The decomposition is akin to a triangulation of a Riemann surface, and we thus call it hexagonalization. We propose a set of rules to glue the hexagons together based on symmetry, which naturally incorporate the dependence on the conformal and the R-symmetry cross ratios. Our method is conceptually different from the conventional operator product expansion and automatically takes into account multi-trace operators exchanged in OPE channels. To illustrate the idea in simple set-ups, we compute four-point functions of BPS operators of arbitrary lengths and correlation functions of one Konishi operator and three short BPS operators, all at one loop. In all cases, the results are in perfect agreement with the perturbative data. We also suggest that our method can be a useful tool to study conformal integrals, and show it explicitly for the case of ladder integrals.

Operator bases, $S$-matrices, and their partition functions

Abstract: Relativistic quantum systems that admit scattering experiments are quantitatively described by effective field theories, where S-matrix kinematics and symmetry considerations are encoded in the operator spectrum of the EFT. In this paper we use the S-matrix to derive the structure of the EFT operator basis, providing complementary descriptions in (i) position space utilizing the conformal algebra and cohomology and (ii) momentum space via an algebraic formulation in terms of a ring of momenta with kinematics implemented as an ideal. These frameworks systematically handle redundancies associated with equations of motion (on-shell) and integration by parts (momentum conservation). We introduce a partition function, termed the Hilbert series, to enumerate the operator basis — correspondingly, the S-matrix — and derive a matrix integral expression to compute the Hilbert series. The expression is general, easily applied in any spacetime dimension, with arbitrary field content and (linearly realized) symmetries. In addition to counting, we discuss construction of the basis. Simple algorithms follow from the algebraic formulation in momentum space. We explicitly compute the basis for operators involving up to n = 5 scalar fields. This construction universally applies to fields with spin, since the operator basis for scalars encodes the momentum dependence of n-point amplitudes. We discuss in detail the operator basis for non-linearly realized symmetries. In the presence of massless particles, there is freedom to impose additional structure on the S- matrix in the form of soft limits. The most na¨ive implementation for massless scalars leads to the operator basis for pions, which we confirm using the standard CCWZ formulation for non-linear realizations. Although primarily discussed in the language of EFT, some of our results — conceptual and quantitative — may be of broader use in studying conformal field theories as well as the AdS/CFT correspondence.

All point correlation functions in SYK

Abstract: Large N melonic theories are characterized by two-point function Feynman diagrams built exclusively out of melons. This leads to conformal invariance at strong coupling, four-point function diagrams that are exclusively ladders, and higher-point functions that are built out of four-point functions joined together. We uncover an incredibly useful property of these theories: the six-point function, or equivalently, the three-point function of the primary O(N ) invariant bilinears, regarded as an analytic function of the operator dimensions, fully determines all correlation functions, to leading nontrivial order in 1/N , through simple Feynman-like rules. The result is applicable to any theory, not necessarily melonic, in which higher-point correlators are built out of four-point functions. We explicitly calculate the bilinear three-point function for q-body SYK, at any q. This leads to the bilinear four-point function, as well as all higher-point functions, expressed in terms of higher-point conformal blocks, which we discuss. We find universality of correlators of operators of large dimension, which we simplify through a saddle point analysis. We comment on the implications for the AdS dual of SYK.

Picture 2-Tuple Linguistic Bonferroni Mean Operators and Their Application to Multiple Attribute Decision Making

Abstract: In this paper, we investigate the multiple attribute decision-making problems with picture 2-tuple linguistic information. Then, we utilize Bonferroni mean and geometric Bonferroni mean operations to develop some picture 2-tuple linguistic aggregation operators: picture 2-tuple linguistic Bonferroni mean operator and picture 2-tuple linguistic geometric Bonferroni mean operator. Some desired properties and special cases of the developed operators are discussed in detail. Furthermore, considering the importance of the input arguments, we propose the picture 2-tuple linguistic weighted Bonferroni mean operator and picture 2-tuple linguistic weighted geometric Bonferroni mean operator. Finally, a practical example for selecting the service outsourcing provider of communications industry is given to verify the developed approach and to demonstrate its practicality and effectiveness.

B physics Beyond the Standard Model at One Loop: Complete Renormalization Group Evolution below the Electroweak Scale

Abstract: General analyses of B-physics processes beyond the Standard Model require accounting for operator mixing in the renormalization-group evolution from the matching scale down to the typical scale of B physics. For this purpose the anomalous dimensions of the full set of local dimension-six operators beyond the Standard Model are needed. We present here for the first time a complete and non-redundant set of dimension-six operators relevant for B-meson mixing and decay, together with the complete one-loop anomalous dimensions in QCD and QED. These results are an important step towards the automation of general New Physics analyses.

On the Shift Operator, Graph Frequency, and Optimal Filtering in Graph Signal Processing

Abstract: Defining a sound shift operator for graph signals, similar to the shift operator in classical signal processing, is a crucial problem in graph signal processing (GSP), since almost all operations, such as filtering, transformation, prediction, are directly related to the graph shift operator. We define a set of energy-preserving shift operators that satisfy many properties similar to their counterparts in classical signal processing, but are different from the shift operators defined in the literature, such as the graph adjacency matrix and Laplacian matrix based shift operators, which modify the energy of a graph signal. We decouple the graph structure represented by eigengraphs and the eigenvalues of the adjacency matrix or the Laplacian matrix. We show that the adjacency matrix of a graph is indeed a linear shift invariant (LSI) graph filter with respect to the defined shift operator. We further define autocorrelation and cross-correlation functions of signals on the graph, enabling us to obtain the solution to the optimal filtering on graphs, i.e., the corresponding Wiener filtering on graphs and the efficient spectra analysis and frequency domain filtering in parallel with those in classical signal processing. This new shift operator based GSP framework enables the signal analysis along a correlation structure defined by a graph shift manifold as opposed to classical signal processing operating on the assumption of the correlation structure with a linear time shift manifold. Several illustrative simulations are presented to validate the performance of the designed optimal LSI filters.

2, 84, 30, 993, 560, 15456, 11962, 261485, . . .: higher dimension operators in the SM EFT

Abstract: In a companion paper [1], we show that operator bases for general effective field theories are controlled by the conformal algebra. Equations of motion and integration by parts identities can be systematically treated by organizing operators into irreducible representations of the conformal group. In the present work, we use this result to study the standard model effective field theory (SM EFT), determining the content and number of higher dimension operators up to dimension 12, for an arbitrary number of fermion generations. We find additional operators to those that have appeared in the literature at dimension 7 (specifically in the case of more than one fermion generation) and at dimension 8. (The title sequence is the total number of independent operators in the SM EFT with one fermion generation, including hermitian conjugates, ordered in mass dimension, starting at dimension 5.)

Correlation functions of Coulomb branch operators

Abstract: We consider the correlation functions of Coulomb branch operators in four-dimensional $$ \mathcal{N} $$ = 2 Superconformal Field Theories (SCFTs) involving exactly one antichiral operator. These extremal correlators are the “minimal” non-holomorphic local observables in the theory. We show that they can be expressed in terms of certain determinants of derivatives of the four-sphere partition function of an appropriate deformation of the SCFT. This relation between the extremal correlators and the deformed four-sphere partition function is non-trivial due to the presence of conformal anomalies, which lead to operator mixing on the sphere. Evaluating the deformed four-sphere partition function using supersymmetric localization, we compute the extremal correlators explicitly in many interesting examples. Additionally, the representation of the extremal correlators mentioned above leads to a system of integrable differential equations. We compare our exact results with previous perturbative computations and with the four-dimensional tt ∗ equations. We also use our results to study some of the asymptotic properties of the perturbative series expansions we obtain in $$ \mathcal{N} $$ = 2 SQCD.

The Role of Fractional Time-Derivative Operators on Anomalous Diffusion

Abstract: The generalized diffusion equations with fractional order derivatives have shown be quite efficient to describe the diffusion in complex systems, with the advantage of producing exact expressions for the underlying diffusive properties. Recently, researchers have proposed different fractional-time operators (namely: the Caputo-Fabrizio and Atangana-Baleanu) which, differently from the well-known Riemann-Liouville operator, are defined by non-singular memory kernels. Here we proposed to use these new operators to generalize the usual diffusion equation. By analyzing the corresponding fractional diffusion equations within the continuous time random walk framework, we obtained waiting time distributions characterized by exponential, stretched exponential, and power-law functions, as well as a crossover between two behaviors. For the mean square displacement, we found crossovers between usual and confined diffusion, and between usual and sub-diffusion. We obtained the exact expressions for the probability distributions, where non-Gaussian and stationary distributions emerged. This former feature is remarkable because the fractional diffusion equation is solved without external forces and subjected to the free diffusion boundary conditions. We have further shown that these new fractional diffusion equations are related to diffusive processes with stochastic resetting, and to fractional diffusion equations with derivatives of distributed order. Thus, our results suggest that these new operators may be a simple and efficient way for incorporating different structural aspects into the system, opening new possibilities for modeling and investigating anomalous diffusive processes.

Perturbative renormalization of quasi-parton distribution functions

Abstract: In this paper we present results for the renormalization of gauge invariant nonlocal fermion operators which contain a Wilson line, to one-loop level in lattice perturbation theory Our calculations have been performed for Wilson/clover fermions and a wide class of Symanzik improved gluon actions The extended nature of such ``long-link'' operators results in a nontrivial renormalization, including contributions which diverge linearly as well as logarithmically with the lattice spacing, along with additional finite factors On the lattice there is also mixing among certain subsets of these nonlocal operators; we calculate the corresponding finite mixing coefficients, which are necessary in order to disentangle individual matrix elements for each operator from lattice simulation data Finally, extending our perturbative setup, we present nonperturbative prescriptions to extract the linearly divergent contributions