Showing papers on "Orthonormal basis published in 2019"

New Characterization and Parametrization of LCD Codes

Abstract: Linear complementary dual (LCD) cyclic codes were referred historically to as reversible cyclic codes, which had applications in data storage. Due to a newly discovered application in cryptography, there has been renewed interest in LCD codes. In particular, it has been shown that binary LCD codes play an important role in implementations against side-channel attacks and fault injection attacks. In this paper, we first present a new characterization of binary LCD codes in terms of their orthogonal or symplectic basis. Using such a characterization, we solve a conjecture proposed by Galvez et al. on the minimum distance of binary LCD codes. Next, we consider the action of the orthogonal group on the set of all LCD codes, determine all possible orbits of this action, derive simple closed formulas of the size of the orbits, and present some asymptotic results on the size of the corresponding orbits. Our results show that almost all binary LCD codes are odd-like codes with odd-like duals, and about half of $q$ -ary LCD codes have orthonormal basis, where $q$ is a power of an odd prime.

Multipole expansion for magnetic structures: A generation scheme for a symmetry-adapted orthonormal basis set in the crystallographic point group

Abstract: Identifying the order parameter for a magnetic phase is useful for a deeper understanding of physical phenomena, such as the anomalous Hall effect, electromagnetic effects, and optical responses. Here, the authors provide an intuitive and general procedure to deduce the proper order parameters of magnetic phases. The procedure is based on multipole expansion and the introduction of a novel concept, the virtual cluster. The method facilitates the understanding of order parameters in antiferromagnets and provides themissing link between order parameters and macroscopic phenomena.

Graph Topology Inference Based on Sparsifying Transform Learning

Abstract: Graph-based representations play a key role in machine learning. The fundamental step in these representations is the association of a graph structure to a dataset. In this paper, we propose a method that finds a block sparse representation of the data by associating a graph, whose Laplacian matrix admits the sparsifying dictionary as its eigenvectors. The main idea is to associate a graph topology to the data in order to make the observed signals band-limited over the inferred graph. The proposed strategy is composed of the following two optimization steps: first, learning an orthonormal sparsifying transform from the data; and second, recovering the Laplacian matrix, and then topology, from the transform. The first step is achieved through an iterative algorithm whose alternating intermediate solutions are expressed in closed form. The second step recovers the Laplacian matrix from the sparsifying transform through a convex optimization method. Numerical results corroborate the effectiveness of the proposed methods over both synthetic and real data. Specifically, we consider two real-world applications of our methods: the inference of the brain functional activity map from electrocorticography signals taken from patients affected by epilepsy, and the reconstruction of the radio environment map from sparse measurements of the electromagnetic field in an urban area.

A direct numerical solution of time-delay fractional optimal control problems by using Chelyshkov wavelets:

Abstract: The aim of the present study is to present a numerical algorithm for solving time-delay fractional optimal control problems (TDFOCPs). First, a new orthonormal wavelet basis, called Chelyshkov wave...

Resource Theory of Coherence Based on Positive-Operator-Valued Measures.

Abstract: Quantum coherence is a fundamental feature of quantum mechanics and an underlying requirement for most quantum information tasks. In the resource theory of coherence, incoherent states are diagonal with respect to a fixed orthonormal basis; i.e., they can be seen as arising from a von Neumann measurement. Here, we introduce and study a generalization to a resource theory of coherence defined with respect to the most general quantum measurements, i.e., to arbitrary positive-operator-valued measures (POVMs). We establish POVM-based coherence measures and POVM-incoherent operations that coincide for the case of von Neumann measurements with their counterparts in standard coherence theory. We provide a semidefinite program that allows us to characterize interconversion properties of resource states and exemplify our framework by means of the qubit trine POVM, for which we also show analytical results.

Data-driven recovery of hidden physics in reduced order modeling of fluid flows

Abstract: In this article, we introduce a modular hybrid analysis and modeling (HAM) approach to account for hidden physics in reduced order modeling (ROM) of parameterized systems relevant to fluid dynamics. The hybrid ROM framework is based on using the first principles to model the known physics in conjunction with utilizing the data-driven machine learning tools to model remaining residual that is hidden in data. This framework employs proper orthogonal decomposition as a compression tool to construct orthonormal bases and Galerkin projection (GP) as a model to built the dynamical core of the system. Our proposed methodology hence compensates structural or epistemic uncertainties in models and utilizes the observed data snapshots to compute true modal coefficients spanned by these bases. The GP model is then corrected at every time step with a data-driven rectification using a long short-term memory (LSTM) neural network architecture to incorporate hidden physics. A Grassmannian manifold approach is also adapted for interpolating basis functions to unseen parametric conditions. The control parameter governing the system's behavior is thus implicitly considered through true modal coefficients as input features to the LSTM network. The effectiveness of the HAM approach is discussed through illustrative examples that are generated synthetically to take hidden physics into account. Our approach thus provides insights addressing a fundamental limitation of the physics-based models when the governing equations are incomplete to represent underlying physical processes.

Solution of fractional Volterra–Fredholm integro-differential equations under mixed boundary conditions by using the HOBW method

Abstract: A new approximate technique is introduced to find a solution of FVFIDE with mixed boundary conditions. This paper started from the meaning of Caputo fractional differential operator. The fractional derivatives are replaced by the Caputo operator, and the solution is demonstrated by the hybrid orthonormal Bernstein and block-pulse functions wavelet method (HOBW). We demonstrate the convergence analysis for this technique to emphasize its reliability. The applicability of the HOBW is demonstrated using three examples. The approximate results of this technique are compared with the correct solutions, which shows that this technique has approval with the correct solutions to the problems.

GODS: Generalized One-Class Discriminative Subspaces for Anomaly Detection

Abstract: One-class learning is the classic problem of fitting a model to data for which annotations are available only for a single class. In this paper, we propose a novel objective for one-class learning. Our key idea is to use a pair of orthonormal frames -- as subspaces -- to ``sandwich'' the labeled data via optimizing for two objectives jointly: i) minimize the distance between the origins of the two subspaces, and ii) to maximize the margin between the hyperplanes and the data, either subspace demanding the data to be in its positive and negative orthant respectively. Our proposed objective however leads to a non-convex optimization problem, to which we resort to Riemannian optimization schemes and derive an efficient conjugate gradient scheme on the Stiefel manifold. To study the effectiveness of our scheme, we propose a new dataset Dash-Cam-Pose, consisting of clips with skeleton poses of humans seated in a car, the task being to classify the clips as normal or abnormal; the latter is when any human pose is out-of-position with regard to say an airbag deployment. Our experiments on the proposed Dash-Cam-Pose dataset, as well as several other standard anomaly/novelty detection benchmarks demonstrate the benefits of our scheme, achieving state-of-the-art one-class accuracy.

A Directed Graph Fourier Transform With Spread Frequency Components

Abstract: We study the problem of constructing a graph Fourier transform (GFT) for directed graphs (digraphs), which decomposes graph signals into different modes of variation with respect to the underlying network. Accordingly, to capture low, medium, and high frequencies we seek a digraph (D)GFT such that the orthonormal frequency components are as spread as possible in the graph spectral domain. To that end, we advocate a two-step design whereby we 1) find the maximum directed variation (i.e., a novel notion of frequency on a digraph) a candidate basis vector can attain and 2) minimize a smooth spectral dispersion function over the achievable frequency range to obtain the desired spread DGFT basis. Both steps involve non-convex, orthonormality-constrained optimization problems, which are efficiently tackled via a feasible optimization method on the Stiefel manifold that provably converges to a stationary solution. We also propose a heuristic to construct the DGFT basis from Laplacian eigenvectors of an undirected version of the digraph. We show that the spectral-dispersion minimization problem can be cast as supermodular optimization over the set of candidate frequency components, whose orthonormality can be enforced via a matroid basis constraint. This motivates adopting a scalable greedy algorithm to obtain an approximate solution with quantifiable worst-case spectral dispersion. We illustrate the effectiveness of our DGFT algorithms through numerical tests on synthetic and real-world networks. We also carry out a graph-signal denoising task, whereby the DGFT basis is used to decompose and then low pass filter temperatures recorded across the United States.

GODS: Generalized One-class Discriminative Subspaces for Anomaly Detection

Abstract: One-class learning is the classic problem of fitting a model to data for which annotations are available only for a single class. In this paper, we propose a novel objective for one-class learning. Our key idea is to use a pair of orthonormal frames -- as subspaces -- to "sandwich" the labeled data via optimizing for two objectives jointly: i) minimize the distance between the origins of the two subspaces, and ii) to maximize the margin between the hyperplanes and the data, either subspace demanding the data to be in its positive and negative orthant respectively. Our proposed objective however leads to a non-convex optimization problem, to which we resort to Riemannian optimization schemes and derive an efficient conjugate gradient scheme on the Stiefel manifold. To study the effectiveness of our scheme, we propose a new dataset~\emph{Dash-Cam-Pose}, consisting of clips with skeleton poses of humans seated in a car, the task being to classify the clips as normal or abnormal; the latter is when any human pose is out-of-position with regard to say an airbag deployment. Our experiments on the proposed Dash-Cam-Pose dataset, as well as several other standard anomaly/novelty detection benchmarks demonstrate the benefits of our scheme, achieving state-of-the-art one-class accuracy.

Hybrid Orthonormal Bernstein and Block-Pulse functions wavelet scheme for solving the 2D Bratu problem

Abstract: In this paper, an efficient numerical scheme is settled for solving two-dimensional Bratu–Gelfand problem, namely Hybrid Orthonormal Bernstein and Block-Pulse functions wavelet (HOBW) is presented for boundary value problems administered by nonlinear partial differential equations which effectively combines the Orthonormal Bernstein, Block-Pulse functions and the generalized wavelet. Operational Matrix of integration is utilized to provide an approximate result of the BG problems. By using the Operational Matrix, differentiation is changed to the nonlinear system of equations which can be disbanded via the Newton Raphson technique. As per our concentrated inquiry there is no exact solution of the problem and can solve the problem with higher accuracy than the methodologies used to solve this problem. The result is plotted for different values of λ then compared with the previous numerical results obtained.

randUTV: A Blocked Randomized Algorithm for Computing a Rank-Revealing UTV Factorization

Abstract: A randomized algorithm for computing a so-called UTV factorization efficiently is presented. Given a matrix A, the algorithm “randUTV” computes a factorization A = UTVa, where U and V have orthonormal columns, and T is triangular (either upper or lower, whichever is preferred). The algorithm randUTV is developed primarily to be a fast and easily parallelized alternative to algorithms for computing the Singular Value Decomposition (SVD). randUTV provides accuracy very close to that of the SVD for problems such as low-rank approximation, solving ill-conditioned linear systems, and determining bases for various subspaces associated with the matrix. Moreover, randUTV produces highly accurate approximations to the singular values of A. Unlike the SVD, the randomized algorithm proposed builds a UTV factorization in an incremental, single-stage, and noniterative way, making it possible to halt the factorization process once a specified tolerance has been met. Numerical experiments comparing the accuracy and speed of randUTV to the SVD are presented. Other experiments also demonstrate that in comparison to column-pivoted QR, which is another factorization that is often used as a relatively economic alternative to the SVD, randUTV compares favorably in terms of speed while providing far higher accuracy.

Solution of Nonlinear Volterra Integral Equations with Weakly Singular Kernel by Using the HOBW Method

Abstract: We present a new numerical technique to discover a new solution of Singular Nonlinear Volterra Integral Equations (SNVIE). The considered technique utilizes the Hybrid Orthonormal Bernstein and Block-Pulse functions wavelet method (HOBW) to solve the weakly SNVIE including Abel’s equations. We acquire the HOBW implementation matrix of the integration to derive the procedure of solving these kind integral equations. The explained technique is delineated with two numerical cases to demonstrate the benefit of the technique used by us. At last, the exchange uncovers the way that the strategy utilized here is basic in usage.

Symplectic model order reduction with non-orthonormal bases

Abstract: Parametric high-fidelity simulations are of interest for a wide range of applications. However, the restriction of computational resources renders such models to be inapplicable in a real-time context or in multi-query scenarios. Model order reduction (MOR) is used to tackle this issue. Recently, MOR is extended to preserve specific structures of the model throughout the reduction, e.g., structure-preserving MOR for Hamiltonian systems. This is referred to as symplectic MOR. It is based on the classical projection-based MOR and uses a symplectic reduced order basis (ROB). Such an ROB can be derived in a data-driven manner with the Proper Symplectic Decomposition (PSD) in the form of a minimization problem. Due to the strong nonlinearity of the minimization problem, it is unclear how to efficiently find a global optimum. In our paper, we show that current solution procedures almost exclusively yield suboptimal solutions by restricting to orthonormal ROBs. As a new methodological contribution, we propose a new method which eliminates this restriction by generating non-orthonormal ROBs. In the numerical experiments, we examine the different techniques for a classical linear elasticity problem and observe that the non-orthonormal technique proposed in this paper shows superior results with respect to the error introduced by the reduction.

Investigation of power quality disturbances by using 2D discrete orthonormal S-transform, machine learning and multi-objective evolutionary algorithms

Abstract: The aim of this paper is to investigate the power quality analysis by using 2D discrete orthonormal S-transform, machine learning and multi-objective evolutionary algorithms. The fact that PQ signals are one-dimensional (1D) signals due to their nature leads to the search for feature extraction approaches based on 1D signal processing methods. Due to the electric network is getting more and more complicated day by day, it is necessary to determine effectively the disturbances events. In the proposed method, extraction of a new feature based on two-dimensional (2D) signal processing by 2D Fast Discrete Orthonormal Stockwell Transform (2D-FDOST) method and determination of the most suitable feature group by Non-dominated Sorting Genetic Algorithm II (NSGA-II) method are performed. Eleven different PQ events are synthetically produced based on mathematical modelling. 1D signals are transformed into 2D signals with equal row and column numbers. Statistical and image-based features are created on the amplitude and phase matrices obtained by 2D-FDOST method from 2D signals. The NSGA-II method, which is one of the multi-objective evolutionary optimization methods, is used to convert a large number of feature vectors into a small number of useful feature groups. NSGA-II produces the optimal solution for two different fitness functions that calculate the number of features and classifier performance. By using different machine learning classifiers for selected features, a model classifying PQ disturbances with high performance and robust structure against noisy situations is created.

Solution of delay fractional optimal control problems using a hybrid of block-pulse functions and orthonormal Taylor polynomials

Abstract: This paper introduces an efficient direct approach for solving delay fractional optimal control problems. The concepts of the fractional integral and the fractional derivative are considered in the Riemann–Liouville sense and the Caputo sense, respectively. The suggested framework is based on a hybrid of block-pulse functions and orthonormal Taylor polynomials. The convergence of the proposed hybrid functions with respect to the L2-norm is demonstrated. The operational matrix of fractional integration associated with the hybrid functions is constructed by using the Laplace transform method. The problem under consideration is transformed into a mathematical programming one. The method of Lagrange multipliers is then implemented for solving the resulting optimization problem. The performance and computational efficiency of the developed numerical scheme are assessed through various types of delay fractional optimal control problems. Our numerical findings are compared with either exact solutions or the existing results in the literature.

Low energy spectrum of SU(2) lattice gauge theory

Abstract: Prepotential formulation of gauge theories on honeycomb lattice yields local loop states, which are exact and orthonormal being free from any spurious loop degrees of freedom. We illustrate that, the dynamics of orthonormal loop states are exactly same in both the square and honeycomb lattices. We further extend this construction to arbitrary dimensions. Utilizing this result, we make a mean field ansatz for loop configurations for SU(2) lattice gauge theory in $$2+1$$ dimension contributing to the low energy sector of the spectrum. Using variational analysis, we show that, this type of mean loop configurations has two distinct phases in the strong and weak coupling regime and shows a first order transition at $$g=1$$ . We also propose a reduced Hamiltonian to describe the dynamics of the theory within the mean field ansatz. We further work with the mean loop configuration obtained towards the weak coupling limit and analytically calculate the spectrum of the reduced Hamiltonian. The spectrum matches with that of the existing literature in this regime, establishing our ansatz to be a valid alternate one which is far more easier to handle for computation.

Numerical Computation for Orthogonal Low-Rank Approximation of Tensors

Abstract: In this paper we study the orthogonal low-rank approximation problem of tensors in the general setting in the sense that more than one matrix factor is required to be mutually orthonormal, which in...

Template bank for compact binary coalescence searches in gravitational wave data: A general geometric placement algorithm

Abstract: We introduce an algorithm for placing template waveforms for the search of compact binary mergers in gravitational wave interferometer data. We exploit the smooth dependence of the amplitude and unwrapped phase of the frequency-domain waveform on the parameters of the binary. We group waveforms with similar amplitude profiles and perform a singular value decomposition of the phase profiles to obtain an orthonormal basis for the phase functions. The leading basis functions span a lower-dimensional linear space in which the unwrapped phase of any physical waveform is well approximated. The optimal template placement is given by a regular grid in the space of linear coefficients. The algorithm is applicable to any frequency-domain waveform model and detector sensitivity curve. It is computationally efficient and requires little tuning. Applying this method, we construct a set of template banks suitable for the search of aligned-spin binary neutron star, neutron-star--black-hole (NSBH) and binary black hole mergers in LIGO-Virgo data.

Sampling of Graph Signals via Randomized Local Aggregations

Abstract: Sampling of signals defined over the nodes of a graph is one of the crucial problems in graph signal processing, whereas in classical signal processing, sampling is a well-defined operation; when we consider a graph signal, many new challenges arise and defining an efficient sampling strategy is not straightforward. Recently, several works have addressed this problem. The most common techniques select a subset of nodes to reconstruct the entire signal. However, such methods often require the knowledge of the signal support and the computation of the sparsity basis before sampling. Instead, in this paper, we propose a new approach to this issue. We introduce a novel technique that combines localized sampling with compressed sensing. We first choose a subset of nodes and then, for each node of the subset, we compute random linear combinations of signal coefficients localized at the node itself and its neighborhood. The proposed method provides theoretical guarantees in terms of reconstruction and stability to noise for any graph and any orthonormal basis, even when the support is not known.

New Calderón reproducing formulae with exponential decay on spaces of homogeneous type

Abstract: Assume that (X, d, μ) is a space of homogeneous type in the sense of Coifman and Weiss (1971, 1977). In this article, motivated by the breakthrough work of Auscher and Hytonen (2013) on orthonormal bases of regular wavelets on spaces of homogeneous type, we introduce a new kind of approximations of the identity with exponential decay (for short, exp-ATI). Via such an exp-ATI, motivated by another creative idea of Han et al. (2018) to merge the aforementioned orthonormal bases of regular wavelets into the frame of the existed distributional theory on spaces of homogeneous type, we establish the homogeneous continuous/discrete Calderon reproducing formulae on (X, d, μ), as well as their inhomogeneous counterparts. The novelty of this article exists in that d is only assumed to be a quasi-metric and the underlying measure μ a doubling measure, not necessary to satisfy the reverse doubling condition. It is well known that Calderon reproducing formulae are the cornerstone to develop analysis and, especially, harmonic analysis on spaces of homogeneous type.

Rational Approximation in a Class of Weighted Hardy Spaces

Abstract: This study aims at rational approximation of a class of weighted Hardy spaces, including the classical Bergman space, the weighted Bergman spaces, the Hardy space, the Dirichlet space and the Hardy–Sobolev spaces. We will mainly concentrate in the Bergman cases in the unit disc context. The methodology of the approximation is a pre-orthogonal method, called Pre-Adaptive Fourier Decomposition. The new idea is that a function is not expanded into a basis but an orthonormal system adapted to the given function. In such way by using a unified method we obtain efficient approximations in our sequence of spaces while avoiding discussions on basis and uniqueness sets, etc. The type of function decompositions is related to direct sum decompositions of the underlying spaces into the closure of the span of a sequence of repeating reproducing kernels and the corresponding zero-based invariant subspaces that arises deep studies.