Inner Product Matrix Research Articles

Damage identification based on the inner product matrix and parallel convolution neural network for frame structure

Structural health monitoring based on vibration signal analysis has been extensively employed for damage identification. Mainstream machine learning techniques, such as convolutional neural networks (CNN), often rely on single-domain inputs, which may provide limited information for accurate damage identification. To overcome this limitation, this study proposes a novel approach that combines an inner product matrix (IPM) with a parallel CNN (IPM-PCNN) to extract multidimensional features for detecting structural damage in a steel frame structure. The proposed IPM-PCNN framework consists of a one-dimensional (1D) CNN branch for processing time series data, a two-dimensional (2D) CNN branch for handling structural modal data, and several fully connected layers. This unique combination leverages the strengths of both 1D and 2D CNNs to capture temporal and modal features of the signal effectively. To validate the effectiveness and superiority of the proposed method, a five-story steel frame model is used as the research object, and five comparative methods are evaluated under the same experimental conditions. The results demonstrate that the IPM-PCNN model can automatically extract relevant features from the signals to accurately identify structural damage, achieving an accuracy of 96.60% on the test set, outperforming machine learning methods in performance. Furthermore, the internal inference processes of these methods are explored and visualized to provide insights into their decision-making mechanisms.

SFIM Detector Based on Joint-Sparse Index Removal for MIMO-OFDM-CR System

This letter proposes a joint-sparse index removal (JSIR) based algorithm to detect the index of space-frequency index modulation (SFIM) signal for MIMO-OFDM cognitive radio (MIMO-OFDM-CR) system. At first, the proposed algorithm calculates the inner-product matrix of the received signal and channel gain to measure the index information of each subcarrier on each antenna. Then, an antenna index metric is computed through a definition which is given by summing the metrics of all subcarriers on the antenna. According to the metric, the silent antenna indices are acquired by comparing the metric with a threshold derived from statistical distribution. Next, the index detection problem is simplified by removing the silent antenna indices, i.e., the joint-sparse indices of SFIM signal matrix. At last, the indices of active subcarriers on active antennas are obtained through solving the simplified problem via a compressed sensing reconstruction algorithm. The complexity analysis shows that the proposed JSIR algorithm has lower complexity under certain conditions, compared with other related methods. The simulation results verify the accuracy of the proposed method in terms of bit error rate (BER).

Bootstrapping massive quantum field theories

We propose a new non-perturbative method for studying UV complete unitary quantum field theories (QFTs) with a mass gap in general number of spacetime dimensions. The method relies on unitarity formulated as positive semi-definiteness of the matrix of inner products between asymptotic states (in and out) and states created by the action of local operators on the vacuum. The corresponding matrix elements involve scattering amplitudes, form factors and spectral densities of local operators. We test this method in two-dimensional QFTs by setting up a linear optimization problem that gives a lower bound on the central charge of the UV CFT associated to a QFT with a given mass spectrum of stable particles (and couplings between them). Some of our numerical bounds are saturated by known form factors in integrable theories like the sine-Gordon, E8 and O(N) models.

Computational color constancy from maximal projections mean assumption

In this paper, we propose an efficient algorithm for illuminant estimation. The 2D estimator is derived based on the maximal projections’ mean assumption according to which the estimated illuminant in a chromaticity subspace is the vector maximizing the sum of orthogonal projections of image colors. We show that, the illuminant estimation solution is the eigenvector corresponding to the maximal eigenvalue of the inner-product matrix of data. The 2D estimator is extended to the 3D space by minimizing the reconstruction errors from the 2D estimates obtained in the RGB color space. The evaluation of the 3D resulting estimator on three well-known image datasets generates lower estimation errors.

Intensive numerical studies of optimal sufficient dimension reduction with singularity

Yoo (2015, Statistics and Probability Letters, 99, 109-113) derives theoretical results in an optimal sufficient dimension reduction with singular inner-product matrix. The results are promising, but Yoo (2015) only presents one simulation study. So, an evaluation of its practical usefulness is necessary based on numerical studies. This paper studies the asymptotic behaviors of Yoo (2015) through various simulation models and presents a real data example that focuses on ordinary least squares. Intensive numerical studies show that the x2 test by Yoo (2015) outperforms the existing optimal sufficient dimension reduction method. The basis estimation by the former can be theoretically sub-optimal; however, there are no notable differences from that by the latter. This investigation confirms the practical usefulness of Yoo (2015).

A theoretical note on optimal sufficient dimension reduction with singularity

In this paper, we discuss an optimal sufficient dimension reduction through minimizing a quadratic objective function proposed by Cook and Ni (2005) with singular inner-product matrix. Within a less restrictive class of inner-product matrices, a generalized inverse of the consistent estimator of the asymptotic covariance matrix gives us the benefit of χ2 statistic and asymptotic efficiency within a subclass.

Quantum support vector machine for big data classification.

Supervised machine learning is the classification of new data based on already classified training examples. In this work, we show that the support vector machine, an optimized binary classifier, can be implemented on a quantum computer, with complexity logarithmic in the size of the vectors and the number of training examples. In cases where classical sampling algorithms require polynomial time, an exponential speedup is obtained. At the core of this quantum big data algorithm is a nonsparse matrix exponentiation technique for efficiently performing a matrix inversion of the training data inner-product (kernel) matrix.

UL-Isomap based nonlinear dimensionality reduction for hyperspectral imagery classification

The paper proposes an upgraded landmark-Isometric mapping (UL-Isomap) method to solve the two problems of landmark selection and computational complexity in dimensionality reduction using landmark Isometric mapping (LIsomap) for hyperspectral imagery (HSI) classification. First, the vector quantization method is introduced to select proper landmarks for HSI data. The approach considers the variations in local density of pixels in the spectral space. It locates the unique landmarks representing the geometric structures of HSI data. Then, random projections are used to reduce the bands of HSI data. After that, the new method incorporates the Recursive Lanczos Bisection (RLB) algorithm to construct the fast approximate k-nearest neighbor graph. The RLB algorithm accompanied with random projections improves the speed of neighbor searching in UL-Isomap. After constructing the geodesic distance graph between landmarks and all pixels, the method uses a fast randomized low-rank approximate method to speed up the eigenvalue decomposition of the inner-product matrix in multidimensional scaling. Manifold coordinates of landmarks are then computed. Manifold coordinates of non-landmarks are computed through the pseudo inverse transformation of landmark coordinates. Five experiments on two different HSI datasets are run to test the new UL-Isomap method. Experimental results show that UL-Isomap surpasses LIsomap, both in the overall classification accuracy (OCA) and in computational speed, with a speed over 5 times faster. Moreover, the UL-Isomap method, when compared against the Isometric mapping (Isomap) method, obtains only slightly lower OCAs.

Mean vector component analysis for visualization and clustering of nonnegative data.

Mean vector component analysis (MVCA) is introduced as a new method for visualization and clustering of nonnegative data. The method is based on dimensionality reduction by preserving the squared length, and implicitly also the direction, of the mean vector of the original data. The optimal mean vector preserving basis is obtained from the spectral decomposition of the inner-product matrix, and it is shown to capture clustering structure. MVCA corresponds to certain uncentered principal component analysis (PCA) axes. Unlike traditional PCA, these axes are in general not corresponding to the top eigenvalues. MVCA is shown to produce different visualizations and sometimes considerably improved clustering results for nonnegative data, compared with PCA.

A technique using principal component analysis to compare seasonal cycles of Earth radiation from CERES and model computations

A method for quantitatively comparing the seasonal cycles of two global data sets is presented. The seasonal cycles of absorbed solar radiation (ASR) and outgoing longwave radiation (OLR) have been computed from an eight‐year data set from the Clouds and Earth's Radiant Energy System (CERES) scanning radiometers and from a model data set produced by the NASA Goddard Space Flight Center Global Modeling and Assimilation Office. To compare the seasonal cycles from these two data sets, principal component (PC) analysis is used, where the PCs express the time variations and the corresponding empirical orthogonal functions (EOFs) describe the geographic variations. Ocean has a long thermal response time compared to land, so land and ocean are separated for the analysis. The root‐mean square values for the seasonal cycles of ASR and OLR are extremely close for the two data sets. The first three PCs are quite close, showing that the time responses and magnitudes over the globe are very similar. The agreement between the two sets of PCs is quantified by computing the matrix of inner products of the two sets. For ASR over land, the first PCs of CERES and the model agree to better than 99.9%. The EOF maps are similar for most of the globe, but differ in a few places, and the agreement of the EOF maps is likewise quantified. Maps of differences between the annual cycles show regions of agreement and disagreement.

The weighted dual functions for Wang–Bézier type generalized Ball bases and their applications

By introducing the inner-product matrix of two vector functions and using conversion matrix, explicit formulas for the dual basis functions of Wang–Bézier type generalized Ball bases (WBGB) with respect to the Jacobi weight function are given. The dual basis functions with and without boundary constraints are also considered. As a result, the paper includes the weighted dual basis functions of Bernstein basis, Wang–Ball basis and some intermediate bases. Dual functionals for WBGB and the least square approximation polynomials are also obtained.

Least-squares inner product shaping

We develop methods that construct an optimal set of vectors with a specified inner product structure, from a given set of vectors in a complex Hilbert space. The optimal vectors are chosen to minimize the sum of the squared norms of the errors between the constructed vectors and the given vectors. Four special cases are considered. In the first, the constructed vectors are orthonormal. In the second, they are orthogonal. In the third, the Gram matrix of inner products of the constructed vectors is a circulant matrix. As we show, the vectors form a cyclic set. In the fourth, the Gram matrix has the property that the rows are all permutations of each other. The constructed vectors are shown to be geometrically uniform.

Generalized canonical analysis based on optimizing matrix correlations and a relation with IDIOSCAL

Carroll's method for generalized canonical analysis of two or more sets of variables is shown to optimize the sum of squared inner-product matrix correlations between a consensus matrix and matrices with canonical variates for each set of variables. In addition, the method that analogously optimizes the sum of squared RV matrix correlations (proposed by Escoufier, 1973) between a consensus matrix and matrices with canonical variates, can be interpreted as an application of Carroll and Chang's IDIOSCAL. A simple algorithm is developed for this and other applications of IDIOSCAL where the similarity matrices are positive semi-definite.

Conformational sampling by a general linearized embedding algorithm

AbstractLinearized embedding is a variant on the usual distance geometry methods for finding atomic Cartesian coordinates given constraints on interatomic distances. Instead of dealing primarily with the matrix of interatomic distances, linearized embedding concentrates on properties of the metric matrix, the matrix of inner products between pairs of vectors defining local coordinate systems within the molecule. We developed a pair of general computer programs that first convert a given arbitrary conformation of any covalent molecule from atomic Cartesian coordinates representation to internal local coordinate systems enforcing rigid valence geometry and then generate a random sampling of conformers in terms of atomic Cartesian coordinates that satisfy the rigid local geometry and a given list of interatomic distance constraints. We studied the sampling properties of this linearized embedding algorithm vs. a standard metric matrix embedding program, DGEOM, on cyclohexane, cycloheptane, and a cyclic pentapeptide. Linearized embedding always produces exactly correct bond lengths, bond angles, planarities, and chiralities; it runs at least two times faster per structure generated, and is successful as much as four times as often at refining these structures to full agreement with the constraints. It samples the full range of allowed conformations broadly, although not perfectly uniformly. Because local geometry is rigid, linearized embedding's sampling in terms of torsion angles is more restricted than that of DGEOM, but it finds in some instances conformations missed by DGEOM. © 1992 by John Wiley & Sons, Inc.

Exploring the conformation space of cycloalkanes by linearized embedding

AbstractLinearized embedding is a variant on the usual distance geometry methods for finding atomic Cartesian coordinates given constraints on interatomic distances. Instead of dealing primarily with the matrix of interatomic distances, linearized embedding concentrates on properties of the metric matrix, the matrix of inner products between pairs of vectors defining local coordinate systems within the molecule. Here, the approach is used to explore the full conformation space allowed to small cyclic alkanes, given the constraint of exact bond lenghts and bond angles. Useful general tools developed along the way are expressions for rotation matrices in any number of dimensions and a generalization of spherical coordinates to any number of dimensions. Analytical results give some novel views of the conformation spaces of cyclopropane, cyclobutane, cyclopentane, and eyclohexane. A combination of numerical and analytical approaches gives the most comprehensive description to date of the cycloheptane conformation space with fixed bond lengths and angles. In this representation, the pseudorotation paths of cyclohexane and cycloheptane are closed curved lines on the surfaces of spheres.

Weighted averaging of a set of noisy images for maximum signal-to-noise ratio

The problem of estimating a signal from a weighted average of N registered noisy observations is considered. A set of optimal weighting coefficients is determined by maximizing a signal-to-noise ratio criterion. This solution can be computed by first standardizing each observation with respect to its first and second moments and then evaluating the first eigenvector of the corresponding N*N inner-product matrix. The resulting average is shown to be proportional to the first basis vector of the Karhunen-Loeve transform provided that the data has been standardized in an appropriate fashion. The low sensitivity of this approach to the presence of outliers is illustrated by using real electron micrographs of ostensibly identical virus particles.< <ETX xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">&gt;</ETX>

The Conditional Wishart: Normal and Nonnormal

The response variable of a general multivariate model can be constructed as a positive affine transformation of a vector error variable. In the case of an error variable that is rotationally symmetric, the multivariate model has parameters that can be expressed as the mean vector and the variance matrix. In the case of an error variables that is standard normal, it becomes the ordinary multivariate normal. For the general multivariate model with rotationally-symmetric error the sample inner-product matrix is a conventional statistic for inference. The distribution of this statistics is derived: the distribution is a conditional distribution given observable characteristics of the error variable. The response variable of a more specialized multivariable model can be constructed as a positive linear transformation of a vector error variable. In the case of an error variable that is rotationally symmetric, the model has parameters that can be expressed in terms of the variance matrix, the mean vector being zero. In the case of an error variable that is standard normal; it becomes the ordinary central multivariate normal. For the multivariate model with rotationally-symmetric error, the distribution of the Wishart matrix is derived; the distribution is a conditional distribution given observable characteristics of the error variable. And for the multivariate model with standard normal error, the noncentral distribution of the Wishart matrix is also derived, again as the appropriate conditional distribution.

A contribution to the signal-design problem for incoherent-phase communication systems

In a previous study, a general expression for the detection probability was given for the transmission of an arbitrary M -ary signaling alphabet through the incoherent-phase channel which adds white Gaussian noise. Necessary and sufficient conditions were obtained demonstrating that with no dimensionality or bandwidth restrictions, the incoherent orthogonal signal structure locally minimizes the error probability for all signal-to-noise ratios. The previous expression is improved herein, from which it is verified that, when the signal dimensionality and bandwidth are restricted to be less than that necessary for the previous optimum, necessary conditions to locally extremize the probability of error are satisfied by the signal-waveform inner product matrix whose dimensionality is as large as possible. The form of the solution obtained is a 2 \times 2 block diagonal matrix with each block the regular simplex inner-product matrix.