Representation Coefficients Research Articles

Improving the sparse coding model via hybrid Gaussian priors

Sparse Coding (SC) imposes a sparse prior on the representation coefficients under a dictionary or a sensing matrix. However, the sparse regularization, approximately expressed as the L1-norm, is not strongly convex. The uniqueness of the optimal solution requires the dictionary to be of low mutual coherence. As a specialized form of SC, Convolutional Sparse Coding (CSC) encounters the same issue. Inspired by the Elastic Net, this paper proposes to learn an additional anisotropic Gaussian prior for the sparse codes, thus improving the convexity of the SC problem and enabling the modeling of feature correlation. As a result, the SC problem is modified by the proposed elastic projection. We thereby analyze the effectiveness of the proposed method under the framework of LISTA and demonstrate that this simple technique has the potential to correct bad codes and reduce the error bound, especially in noisy scenarios. Furthermore, we extend this technique to the CSC model for the vision practice of image denoising. Extensive experimental results show that the learned Gaussian prior significantly improves the performance of both the SC and CSC models. Source codes are available at https://github.com/eeejyang/EPCSCNet.

Construction and non-vanishing of a family of vector-valued Siegel Poincaré series

Using Poincaré series of K-finite matrix coefficients of integrable antiholomorphic discrete series representations of Sp2n(R), we construct a spanning set for the space Sρ(Γ) of Siegel cusp forms of weight ρ for Γ, where ρ is an irreducible polynomial representation of GLn(C) of highest weight ω∈Zn with ω1≥…≥ωn>2n, and Γ is a discrete subgroup of Sp2n(R) commensurable with Sp2n(Z). Moreover, using a variant of Muić's integral non-vanishing criterion for Poincaré series on unimodular locally compact Hausdorff groups, we prove a result on the non-vanishing of constructed Siegel Poincaré series.

Bearing fault diagnosis by sparse frequency spiral spectrum driven NAF-LDM under strong noise and small samples

Abstract The complexity of background noise and the scarcity of real fault samples seriously affect the diagnostic accuracy of the model. To address this, a noise-robust two-dimensional feature map, the sparse frequency spiral spectrum (SFSM), based on sparse representation theory, is proposed. A bridge penalty coefficient is applied to the sparse representation model to accurately select impact components, and the fast iterative shrinkage threshold algorithm is used to solve for sparse representation coefficients. Sparse reconstructed signals are obtained by convolving the impact patterns with these coefficients, leading to a sparse reconstruction algorithm with reduced computational complexity. Furthermore, the novel non-linear activation-free blocks (NAF Blocks) are embedded into the latent diffusion model to augment small samples, significantly improving image generation speed and quality. The integration of the Swin transformer for feature extraction and classification further enhances diagnostic performance. The superiority of this method is validated on the XJTU-SY dataset, a bearing experimental platform dataset, and enterprise engineering dataset. Experimental results demonstrate that the structural and generalization advantages of NAF Blocks are crucial for improving image quality and inference speed. The noise suppression capability of the proposed method, facilitated by the SFSM feature processing technique, is confirmed through ablation and noise robustness tests. Finally, the Swin transformer’s excellent feature extraction and classification capabilities for SFSM are verified. The proposed method achieves diagnostic accuracies of 99.10% and 98.7% on the XJTU-SY and experimental platform datasets, respectively.

Beyond the root: Geometric characterization for the diagnosis of syndromic heritable thoracic aortic diseases

Syndromic heritable thoracic aortic diseases (sHTAD), such as Marfan (MFS) or Loeys–Dietz (LDS) syndromes, involve high risk of life threatening aortic events. Diagnosis of syndromic features alone is difficult, and negative genetic tests do not necessarily exclude a genetic or hereditary condition. Periodic 3D imaging of the aorta is recommended in patients with aortic disease. Thus, an imaging-based approach aimed at identifying unique features of aortic geometry can be highly effective for diagnosing sHTAD and assessing risk. In this study, we present a method that can help identify the manifestations of sHTAD by focusing on the entire geometry of the thoracic aorta, rather than only using measurements of dilation of the aortic root. We analyze the geometric phenotype of 97 patients with genetically confirmed sHTAD (79 MF and 18 LDS) and of 45 healthy volunteers, using 3D aorta meshes obtained from phase contrast-enhanced magnetic resonance angiograms computed from 4D flow cardiac magnetic resonance. We build a geometric encoding of the aorta, based on a vessel coordinate system, and use several mathematical models to discriminate between controls and patients with sHTAD: a baseline scenario, based on aortic root dimensions only, a descriptor typically used in sHTAD patients; a low dimensional scenario, with a reduce encoding using principal component analysis; and a high-dimensional scenario, which included the full coefficient representation for geometry encoding, aiming to capture finer geometric details. The results indicate that considering the anatomy of the whole thoracic aorta can improve predictive ability. We achieve precision and sensitivity values over 0.8, with a specificity of over 70% in all the models used, while a single value classifiers (based only on aortic root diameter) demonstrated a trade-off between sensitivity and specificity. Using the mathematical properties of the vessel coordinate system representation, feature importance is mapped onto a set of anatomical traits that are used by the models to do the classification, thus providing interpretability of the results. This analysis indicates that in addition to the diameter of the aortic root, aortic elongation and a narrowing of the descending thoracic aorta may be markers of positive sHTAD.

A clustered federated learning framework for collaborative fault diagnosis of wind turbines

Data-driven approaches demonstrate significant potential in accurately diagnosing faults in wind turbines. To enhance diagnostic performance and reduce communication costs in federated learning with data heterogeneity among different clients, we introduce a clustered federated learning framework to wind turbine fault diagnosis. Initially, a lightweight multiscale separable residual network (LMSRN) model is proposed for each local client. The LMSRN model integrates a multiscale spatial feature derivation unit and a depthwise separable feature extraction unit. Subsequently, to tackle data heterogeneity among clients, canonical correlation coefficients of representations are extracted from the intermediate layers of local LMSRN models, and a representational canonical correlation clustering (RCCC) method is proposed to assess the similarity of local LMSRN models and group them into clusters. Finally, a global model is trained for each cluster. Real-world wind turbine data experiments showcase the superior performance of the proposed clustered federated learning framework over traditional methods in terms of diagnostic accuracy and computational speed. Additionally, the optimal choice of the number of clusters is also discussed.

Fekete-Szegö and Hankel inequalities related to the sine function

ABSTRACT The Fekete-Szegö inequality is one of the inequalities for the coefficients which associated with the famous Bieberbach conjecture. Other issues associated with this inequality are determining the Hankel determinant denoted as Hd inequalities which are involved in the study of the singularities and integral coefficients in the Taylor series representations. In this investigation, we discuss such inequalities for certain mappings g for which ( g ′ ( z ) ) α [ 2 z g ′ ( z ) g ( z ) − g ( − z ) ] ( 1 − α ) ≺ 1 + sin ⁡ z , ( 0 ≤ α ≤ 1 ) lies in the image domain starlike about 1 and symmetric about real axis. Particularly, we obtain certain inequality for functions involving Poisson distribution. Furthermore, we also find the second Hankel determinant and other inequalities related to this newly defined class.

Trapped acoustic waves and raindrops: High-order accurate integral equation method for localized excitation of a periodic staircase

We present a high-order boundary integral equation (BIE) method for the frequency-domain acoustic scattering of a point source by a singly-periodic, infinite, corrugated boundary. We apply it to the accurate numerical study of acoustic radiation in the neighborhood of a sound-hard two-dimensional staircase modeled after the El Castillo pyramid. Such staircases support trapped waves which travel along the surface and decay exponentially away from it. We use the array scanning method (Floquet–Bloch transform) to recover the scattered field as an integral over the family of quasiperiodic solutions parameterized by on-surface wavenumber. Each such BIE solution requires the quasiperiodic Green's function, which we evaluate using an efficient integral representation of lattice sum coefficients. We avoid the singularities and branch cuts present in the array scanning integral by complex contour deformation. For each frequency, this enables a solution accurate to around 10 digits in a few seconds. We propose a residue method to extract the limiting powers carried by trapped modes far from the source. Finally, by computing the trapped mode dispersion relation, we use a simple ray model to explain an acoustic chirp-like time-domain response that is referred to in the literature as the “raindrop effect.”

Three-phase-lag analysis of transversely isotropic double porous thermoelastic waves with liquid medium

This manuscript presents a mathematical framework for a double porous, thermoelastic solid half-space with transverse isotropy in contact with an inviscid liquid half-space. The analysis takes into account the three-phase-lag model within thermoelasticity theory. To gain deeper insights, we formulate the governing equations in a two-dimensional representation and subsequently employ normal mode analysis for their rigorous solution. The governing equations are influenced by both types of voids, anisotropy, thermal effect, and inviscid liquid. It has been discovered that thermoelastic half-space exhibits five coupled longitudinal waves, and in the liquid half-space, there is one mechanical wave. Secular equations are obtained by using thermal and mechanical conditions, resulting in a compact representation of phase velocity, specific loss, penetrating depth, and attenuation coefficient. Analytical presentations have been made to showcase the effects of anisotropy, voids, and liquid on various wave parameters. These effects are compared by illustrating the results graphically using the MATLAB software, allowing for a visual understanding of the comparative outcomes. This study has practical applications in engineering, materials science, geophysics, and environmental studies, contributing to advancements in various industries and scientific fields.

Comparative Studies on the Radiative Heat Transfer in Arc Plasma and Its Impact in a Model of a Free-Burning Arc

The radiative heat transfer in arc plasma models is considered from the point of view of its description in terms of a net emission coefficient, the method of spherical harmonics in its lowest order, and the discrete ordinate method. Net emission coefficients are computed, applying approximate analytical and numerical approaches and a multi-band representation of the spectral absorption coefficient with three kinds of its averaging and two datasets. Self-consistent access to the radiative heat transfer is applied to a two-dimensional axisymmetric model of a free-burning arc in argon at atmospheric pressure. The results obtained from the models employing the net emission coefficient, the method of spherical harmonics, and the discrete ordinate method are compared.

Functional Bethe Ansatz for a sinh-Gordon Model with Real q

Recently, Bazhanov and Sergeev have described an Ising-type integrable model which can be identified as a sinh-Gordon-type model with an infinite number of states but with a real parameter q. This model is the subject of Sklyanin’s Functional Bethe Ansatz. We develop in this paper the whole technique of the FBA which includes: (1) Construction of eigenstates of an off-diagonal element of a monodromy matrix. The most important ingredients of these eigenstates are the Clebsh-Gordan coefficients of the corresponding representation. (2) Separately, we discuss the Clebsh-Gordan coefficients, as well as the Wigner’s 6j symbols, in details. The later are rather well known in the theory of 3D indices. Thus, the Sklyanin basis of the quantum separation of variables is constructed. The matrix elements of an eigenstate of the auxiliary transfer matrix in this basis are products of functions satisfying the Baxter equation. Such functions are called usually the Q-operators. We investigate the Baxter equation and Q-operators from two points of view. (3) In the model considered the most convenient Bethe-type variables are the zeros of a Wronskian of two well defined particular solutions of the Baxter equation. This approach works perfectly in the thermodynamic limit. We calculate the distribution of these roots in the thermodynamic limit, and so we reproduce in this way the partition function of the model. (4) The real parameter q, which is the standard quantum group parameter, plays the role of the absolute temperature in the model considered. Expansion with respect to q (tropical expansion) gives an alternative way to establish the structure of the eigenstates. In this way we classify the elementary excitations over the ground state.

On the coefficients of automorphic representations over polynomials

On the coefficients of automorphic representations over polynomials

Non-negative consistency affinity graph learning for unsupervised feature selection and clustering

Feature selection plays a crucial role in data mining and pattern recognition tasks. This paper proposes an efficient method for robust unsupervised feature selection, called joint local preserving and low-rank representation with nonnegative and symmetric constraint (JLPLRNS). This approach utilizes an indicator matrix instead of the row sparsity of the projection matrix to directly select some significant features from the original data and adaptively preserves the local geometric structure of original data into the low-dimensional embedded feature subspace via learned indicator matrix. By simultaneously imposing nonnegative symmetric and low-rank constraints on the representation coefficient matrix, it cannot only make this matrix discriminative, sparse and weight consistency for each pair of data, but also uncover the global structure of original data. These effectively will improve clustering performance. An algorithm based on the augmented Lagrange multiplier method with an alternating direction strategy is designed to resolve this model. Experimental results on various real datasets show that the proposed method can effectively identify some important features in data and outperforms many state-of-the-art unsupervised feature selection methods in terms of clustering performance.

On the resonance frequency of the irregular shaped visco-acoustic Helmholtz cavity problem

In this paper we develop a closed-form mathematical solution for the resonance frequency of the axisymmetric irregular shaped visco-acoustic Helmholtz cavity problem. The solution is based on a conformal mapping that transforms the governing visco-acoustic equations of motion for the irregular cavity geometry into a new domain described by a set of orthogonal curvilinear coordinates. These orthogonal curvilinear coordinates have an important characteristic relative to the transformation of the physical boundary: the mapped domain is constructed such that the locus of all points for the length-wise profile of the physical boundary coincides with a constant value for a (quasi radial) coordinate in the new mapped domain. In these orthogonal curvilinear coordinates we obtain an analytical solution for the visco-acoustic equations of motion and exactly satisfy the no-slip fluid boundary conditions over the irregular shaped cavity walls by matching the coefficients of an orthogonal Legendre polynomial series representation for the non-uniform velocity on the cavity walls. The mathematical solution is an approximation to the extent that the scale factors chosen for the mapping consist of strictly the mutual-coupling terms of the true scale factors. Thus we analyze some results to understand the correlation of the analytical solution frequency response spectra with numerical finite element analyses of the idealized boundary conditions with various irregular shaped cavity geometries. The results for fundamental resonance frequency were found to exhibit good correlation with the results of finite element analyses over a significant range of fluid properties with different irregular cavity geometries. As the primary motivation for the derivation stems from the lack of a rapid yet accurate alternative to computationally intensive MultiPhysics FEA in the formulation of acoustic trendlines for sensor development, the solution’s acoustic impedance was applied as dispersive loading in a simple lumped parameter model of a fluid identification sensor for comparison. The resulting acoustic trendlines of the reduced (5 DOF) model were found to exhibit good correlation with the results of the MultiPhysics FEA (>105 DOF). Thus the solution was found to enable a closed-form analytical method for rapid construction of parametric studies in the practical design of fluid analysis sensors based on complex resonant cavity geometries.

On linear combinations of binomial OWA functions

We consider the binomial decomposition of ordered weighted averaging (OWA) functions proposed by Calvo and De Baets (1998) in the framework of Choquet integration. Our aim in the paper is to further investigate the equivalence between the two representations of OWA functions involved in the binomial decomposition: the usual canonical representation in terms of the order statistics, and the binomial representation in terms of the binomial OWA functions. We describe and discuss in detail the linear transformations that relate the coefficients of these two equivalent representations: the original expression of the weights in terms of the coefficients of the binomial representation, and its inverse, the expression of those coefficients in terms of the weights. In both cases simple and direct proofs are presented. Moreover, we consider the linear transformations between the two representations in the general linear algebra framework of unconstrained linear combinations of binomial OWA functions. In this perspective we obtain compact matrix expressions for the linear transformations, which also offer new insight on the geometry of the coefficient constraints in the binomial representation of OWA functions.

Robust sparse representation based on fitting error decomposition

Sparse representation (SR) of a signal aims at finding the minimum number of atoms for its representation. In several practical scenarios, the signal is vulnerable to outliers and thus robustness is required for SR based algorithms. However, most existing robust schemes are designed on the assumption that the anomalies are independently distributed, which may not perform well encountering complex interferences, especially the correlated ones. To deal with this drawback, a robust SR model is proposed, where both independent and correlated outliers are considered. Specifically, the fitting error is decomposed as the combination of a low-rank component and a sparse part, corresponding to the correlated gross error and independent outlier, respectively. Then, the group sparsity of the representation coefficient is utilized. Moreover, ℓ0-norm and ℓ2,0-norm are adopted as the sparsity regularization for the sparse outlier and representation coefficients, respectively. The solutions to ℓ0/ℓ2,0-norm minimization are generated by the hard-thresholding strategy, where the decision threshold is adaptively determined using median absolute deviation operator. As for the low-rank regularization, due to the NP-hardness of rank minimization, we employ matrix logarithmic norm as the rank surrogate to lessen the approximation gap. Finally, we apply the proposed model to face recognition task, and the excellent performance demonstrates its effectiveness.

Multivariate Meixner polynomials related to holomorphic discrete series representations of [formula omitted]

We show that Griffiths’ multivariate Meixner polynomials occur as matrix coefficients of holomorphic discrete series representations of the group SU(1,d). Using this interpretation we derive several fundamental properties of the multivariate Meixner polynomials, such as orthogonality relations and difference equations. Furthermore, we also show that matrix coefficients for specific group elements lead to degenerate versions of the multivariate Meixner polynomials and their properties.

Joint Metric Learning-Based Class-Specific Representation for Image Set Classification.

With the rapid advances in digital imaging and communication technologies, recently image set classification has attracted significant attention and has been widely used in many real-world scenarios. As an effective technology, the class-specific representation theory-based methods have demonstrated their superior performances. However, this type of methods either only uses one gallery set to measure the gallery-to-probe set distance or ignores the inner connection between different metrics, leading to the learned distance metric lacking robustness, and is sensitive to the size of image sets. In this article, we propose a novel joint metric learning-based class-specific representation framework (JMLC), which can jointly learn the related and unrelated metrics. By iteratively modeling probe set and related or unrelated gallery sets as affine hull, we reconstruct this hull sparsely or collaboratively over another image set. With the obtained representation coefficients, the combined metric between the query set and the gallery set can then be calculated. In addition, we also derive the kernel extension of JMLC and propose two new unrelated set constituting strategies. Specifically, kernelized JMLC (KJMLC) embeds the gallery sets and probe sets into the high-dimensional Hilbert space, and in the kernel space, the data become approximately linear separable. Extensive experiments on seven benchmark databases show the superiority of the proposed methods to the state-of-the-art image set classifiers.

On the asymptotics of matrix coefficients of representations of [formula omitted]

We study matrix coefficients of the unitary (and also the completely bounded) representations of SL(2,R) and its universal covering group. We describe the asymptotic distribution of column vectors in terms of Whittaker functions, exhibiting also a relationship to the Fourier transform.

Local representation‐based neighbourhood for robust classification

AbstractRepresentation‐based classification (RC) is an effective gauge of data similarity between a single instance and the whole dataset, which extends traditional individual‐wise distance metrics using representation coefficients. These coefficients show remarkable discrimination nature via various regularisation terms, but the interference from potentially uncorrelated objects involved in this single‐to‐global relation can degrade the effectiveness of the coefficients. In order to filter out those unproductive, or even counter‐productive, information from the decision making processes, this paper proposes a local representation‐based classification (LRC) algorithm to improve the classification accuracy or the RC approach. LRC uses a single‐to‐local relation induced by the local representation‐based neighbourhood (LRN) of each object, rather than the single‐to‐global relationship used by RC. Thanks to LRN, a compact and relevant dataset can be formed by selecting the most relevant data instances in the original dataset, to render a robust representation of a query. LRC was applied to multiple publicly available datasets, and the experimental results demonstrate the superiority of the proposed LRC algorithm as evidenced by the higher classification accuracy and more noise‐tolerant capability in reference to alternative RC approaches. Moreover, the sampling ability of LRN is also verified via a comparative study.

The Design of FIR Filter Based on improved DA Algorithm and its FPGA implementation: REVIEW

The Design of FIR Filter Based on improved DA Algorithm and its FPGA implementation: REVIEW