Partial Matrix Research Articles

Bus OD matrix reconstruction based on clustering Wi-Fi probe data

The estimation of citywide passenger demand plays a vital role in system planning, operation, and management of the urban transit system. The Wi-Fi probe data, one of the emerging crowdsourcing data, is utilized to collect traces of smartphone users in this study. We establish a framework for OD matrix reconstruction, including extracting features for transit patronage and distinguishing them from non-transit users based on K-means clustering. Such a framework makes partial OD matrix more reliable. A probabilistic estimation method of bus OD matrix reconstruction is then proposed based on the partial OD matrix and the number of boarding and alighting passengers. A field study was carried out on bus line 5 in Suzhou, China. Compared to the measured ground truth, the difference in OD-level is 0.5–1.5 passengers per stop, showing that the proposed method for OD matrix reconstruction is reliable.

A partial solution operator discretization-based method for small signal stability analysis of time-delay power systems

To analyze the small signal stability of large-scale time-delay power systems (TDPSs), critical oscillation modes are required to be reliably computed. The solution operator discretization with pseudo-spectral collocation (SOD-PS)-based method has been proposed to indirectly calculate critical eigenvalues from the solution operator’s discretization matrix. However, dimension of the discretization matrix is dozens of times the number of system state variables, and thus limits the efficiency of SOD-PS. To resolve this, SOD-PS is evolved into partial SOD-PS (PSOD-PS) in this paper by applying the idea of partial spectral discretization (PSD). Specifically, only the discretization of retarded states is reserved instead of discretizing all system states in delay interval. The resultant partial discretization matrix exhibits very low order, which is close to the number of system state variables. In addition, the rotation-and-multiplication preconditioning technique is studied to efficiently capture oscillation modes with damping ratios less than the given threshold. Specifically, the preconditioning technique is implemented by two ways and their consistency is derived in detail. Both theoretical analyses and intensive tests on the 16-machine 68-bus test system, a real-life 516-bus provincial transmission system of China and the 33028-bus ultra-high-voltage (UHV) interconnected system of China verify the accuracy and efficiency of the presented PSOD-PS.

Conditional Functional Graphical Models

Graphical modeling of multivariate functional data is becoming increasingly important in a wide variety of applications. The changes of graph structure can often be attributed to external variables, such as the diagnosis status or time, the latter of which gives rise to the problem of dynamic graphical modeling. Most existing methods focus on estimating the graph by aggregating samples, but largely ignore the subject-level heterogeneity due to the external variables. In this article, we introduce a conditional graphical model for multivariate random functions, where we treat the external variables as conditioning set, and allow the graph structure to vary with the external variables. Our method is built on two new linear operators, the conditional precision operator and the conditional partial correlation operator, which extend the precision matrix and the partial correlation matrix to both the conditional and functional settings. We show that their nonzero elements can be used to characterize the conditional graphs, and develop the corresponding estimators. We establish the uniform convergence of the proposed estimators and the consistency of the estimated graph, while allowing the graph size to grow with the sample size, and accommodating both completely and partially observed data. We demonstrate the efficacy of the method through both simulations and a study of brain functional connectivity network.

The Truncated Hamburger Moment Problems with Gaps in the Index Set

In this article we solve four special cases of the truncated Hamburger moment problem (THMP) of degree $2k$ with one or two missing moments in the sequence. As corollaries we obtain, by using appropriate substitutions, the solutions to bivariate truncated moment problems of degree $2k$ for special curves. Namely, for the curves $y=x^3$ (first solved by Fialkow), $y^2=x^3$, $y=x^4$ where a certain moment of degree $2k+1$ is known and $y^3=x^4$ with a certain moment given. The main technique is the completion of the partial positive semidefinite matrix (ppsd) such that the conditions of Curto and Fialkow's solution of the THMP are satisfied. The main tools are the use of the properties of positive semidefinite Hankel matrices and a result on all completions of a ppsd matrix with one unknown entry, proved by the use of the Schur complements for $2\times 2$ and $3\times 3$ block matrices.

AUTOMAT[R]IX: learning simple matrix pipelines

Matrices are a very common way of representing and working with data in data science and artificial intelligence. Writing a small snippet of code to make a simple matrix transformation is frequently frustrating, especially for those people without an extensive programming expertise. We present AUTOMAT[R]IX, a system that is able to induce R program snippets from a single (and possibly partial) matrix transformation example provided by the user. Our learning algorithm is able to induce the correct matrix pipeline snippet by composing primitives from a library. Because of the intractable search space—exponential on the size of the library and the number of primitives to be combined in the snippet, we speed up the process with (1) a typed system that excludes all combinations of primitives with inconsistent mapping between input and output matrix dimensions, and (2) a probabilistic model to estimate the probability of each sequence of primitives from their frequency of use and a text hint provided by the user. We validate AUTOMAT[R]IX with a set of real programming queries involving matrices from Stack Overflow, showing that we can learn the transformations efficiently, from just one partial example.

Free vibration analysis of rotating composite Timoshenko beams with bending-torsion couplings

The free vibration of rotating bending-torsional composite Timoshenko beams (CTBs) with arbitrary boundary conditions is analyzed. The composite material coupled rigidity, Coriolis effects and the separation of the cross-section’s mass and shear centers are considered and they will cause the bending-torsion coupled vibration of the beam. Based on the Hamilton’s principle, the governing partial differential matrix equation of motion of the beam are formulated with variable coefficients. Those equations containing variable coefficients are expressed in a special matrix form in terms of the flexural translation, rotation of cross section and torsional rotation of the beam. The differential transform matrix method (DTMM), which is an improved approach from the traditional differential transform method (DTM), is proposed to deal with the governing matrix equation with corresponding boundary conditions. The mode shapes and natural frequencies of the beam are obtained. Cantilevers are calculated as examples to verify the present theory and investigate the dynamic characteristics of the rotating CTB. The present results are consistent with those reported in the literature and those calculated from COMSOL. The influence of the composite material rigidity, rotation speed, hub radius and axial load on the natural frequencies and mode shapes of the beam are studied, while the frequency veering and mode shift phenomena are observed. Furthermore, the beam’s critical buckling loads corresponding to the axial load are found.

Disrupted Brain Network Topology in Drug-naïve Essential Tremor Patients with and Without Depression : AResting State Functional Magnetic Resonance Imaging Study.

This study was carried out to investigate brain functional connectome and its potential relationships with the disease severity and emotion function in patients with essential tremor with and without depressive symptoms by using resting-state functional magnetic resonance imaging and graph theory approaches. In this study 33 essential tremor patients with depression, 45 essential tremor patients without depression and 79age and gender-matched healthy controls were recruited to undergo a3.0‑T imaging scan. The whole brain functional connectome was constructed by thresholding the partial correlation matrices of 116 brain regions, and the topologic properties were analyzed by using graph theory approaches and network-based statistic approaches. Nonparametric permutation test was also used for group comparisons of topological metrics. Correlation analyses between topographic features and the clinical characteristics were performed. The functional connectome in both essential tremor patients with and without depression showed abnormalities at the global level (decrease in clustering coefficient, global efficiency, and local efficiency but increase in characteristic path length) and at the nodal level (decrease nodal centralities in the cerebellum, motor cortex, prefrontal-limbic regions, default mode network) (p < 0.05, false discovery rate corrected). Moreover, essential tremor patients with depression showed higher node efficiency in superior frontal gyrus and posterior cingulate gyrus compared to essential tremor without depression. Our results may provide insights into the underlying pathophysiology of essential tremor patients with and without depression and aid the development of some potential biomarkers of the depressive symptoms in patients with essential tremor.

Algebraic Decoding of Quasi-Reversible BCH Codes Using Band Matrices

A Bose-Chaudhuri-Hocquenghem (BCH) is called quasi-reversible if there are consecutive elements a <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">1</sub> ,⋯,a <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">2</sub> in the defining set, where a <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">1</sub> is a negative integer and a <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">2</sub> is a positive integer. A matrix in band form presented in this article involves the use of the Newton identities and symmetric polynomial identities. This matrix is the coefficient matrix of a linear system. The solution to such a linear system is the coefficients of an error-locator polynomial for quasi-reversible BCH codes. Its computational complexity is significantly lower than BCH decoding using the linear system with coefficient matrix in column echelon form. This article also proposes a new partial syndrome matrix with radical locators and a lot of zeros. Such a matrix is still in band form. The expansion of its matrix determinant is exactly a radical-locator polynomial. Compared to the former radical-locator polynomials in the literature, the newly discovered radical-locator polynomial not only requires less storage memory but also results in computational benefits. Surprisingly, the resulting polynomials are much sparser than those for the narrow-sense BCH codes. Finally, a complete algebraic decoding algorithm for quasi-reversible BCH codes is provided. The error-locator and radical-locator polynomials of degree less than or equal to 5 are listed.

Sparse recovery in bounded Riesz systems with applications to numerical methods for PDEs

We study sparse recovery with structured random measurement matrices having independent, identically distributed, and uniformly bounded rows and with a nontrivial covariance structure. This class of matrices arises from random sampling of bounded Riesz systems and generalizes random partial Fourier matrices. Our main result improves the currently available results for the null space and restricted isometry properties of such random matrices. The main novelty of our analysis is a new upper bound for the expectation of the supremum of a Bernoulli process associated with a restricted isometry constant. We apply our result to prove new performance guarantees for the CORSING method, a recently introduced numerical approximation technique for partial differential equations (PDEs) based on compressive sensing.

A note on the Cayley-Menger determinant and the Molecular Distance Geometry Problem

Using information from protein geometry and distance data provided by Nuclear Magnetic Resonance (NMR) experiments, the Molecular Distance Geometry Problem (MDGP) can be solved by a combinatorial approach, called Branch-and-Prune (BP). The primal version of BP algorithm seeks MDGP graph realizations, while the dual BP looks for completions of associated partial distance matrices. These two algorithms are very similar when distance values are precise. In the literature, there are some proposals for extending the primal BP to take care of NMR uncertainties. Using Cayley-Menger determinant, we present a global optimization approach that also allows the dual BP to deal with NMR uncertainties.

Time-evolving controllability of effective connectivity networks during seizure progression

Over one third of the estimated 3 million people with epilepsy in the United States are medication resistant. Responsive neurostimulation from chronically implanted electrodes provides a promising treatment alternative to resective surgery. However, determining optimal personalized stimulation parameters, including when and where to intervene to guarantee a positive patient outcome, is a major open challenge. Network neuroscience and control theory offer useful tools that may guide improvements in parameter selection for control of anomalous neural activity. Here we use a method to characterize dynamic controllability across consecutive effective connectivity (EC) networks based on regularized partial correlations between implanted electrodes during the onset, propagation, and termination regimes of 34 seizures. We estimate regularized partial correlation adjacency matrices from 1-s time windows of intracranial electrocorticography recordings using the Graphical Least Absolute Shrinkage and Selection Operator (GLASSO). Average and modal controllability metrics calculated from each resulting EC network track the time-varying controllability of the brain on an evolving landscape of conditionally dependent network interactions. We show that average controllability increases throughout a seizure and is negatively correlated with modal controllability throughout. Our results support the hypothesis that the energy required to drive the brain to a seizure-free state from an ictal state is smallest during seizure onset, yet we find that applying control energy at electrodes in the seizure onset zone may not always be energetically favorable. Our work suggests that a low-complexity model of time-evolving controllability may offer insights for developing and improving control strategies targeting seizure suppression.

Organizational capacity, motivation, or intervention capacity?: Are these affects online teaching readiness and need satisfaction of students

In this pandemic situation, educational institutions are trying to co-operate students in continuing learning without face-to-face interaction. Teachers have online teaching readiness with their own organizational capacities, motivation, and interactive capacities and use online teaching strategies to satisfy students' needs and desires. In fact, the study aimed to assess the effect of organizational capacity, motivation, or intervention capacity on online teaching readiness and satisfaction of students’ learning during COVID-19 broke out or future unforeseen circumstances. 90 teachers and 135 students participated in an online survey and responded the online teaching readiness questionnaire and e-learning satisfaction scale. Structural equation modeling (SEM) was performed to evaluate the online teaching readiness and online learning satisfaction models. A partial correlation matrix, CMIN, CFI, PCFI; NCP, FMIN, RMSEA, AIC, and ECVI was generated by the AMOS 21.0 program to calculate path coefficients and the overall model fit. The authors claimed that organizational capacity, motivation, and intervention capacity significantly relate to teachers' online teaching readiness. Synchronous e learning (e.g., Skype/Zoom/WebEx) results of the direct effects of students’ learning performance, need satisfaction, openness to learning, self-concept, self-pace, problem solving, self-assessing, love of learning, creativity, and motivation

Matrix Completion with Deterministic Sampling: Theories and Methods.

In some significant applications such as data forecasting, the locations of missing entries cannot obey any non-degenerate distributions, questioning the validity of the prevalent assumption that the missing data is randomly chosen according to some probabilistic model. To break through the limits of random sampling, we explore in this paper the problem of real-valued matrix completion under the setup of deterministic sampling. We propose two conditions, isomeric condition and relative well-conditionedness, for guaranteeing an arbitrary matrix to be recoverable from a sampling of the matrix entries. It is provable that the proposed conditions are weaker than the assumption of uniform sampling and, most importantly, it is also provable that the isomeric condition is necessary for the completions of any partial matrices to be identifiable. Equipped with these new tools, we prove a collection of theorems for missing data recovery as well as convex/nonconvex matrix completion. Among other things, we study in detail a Schatten quasi-norm induced method termed isomeric dictionary pursuit (IsoDP), and we show that IsoDP exhibits some distinct behaviors absent in the traditional bilinear programs.

Pseudococyclic Partial Hadamard Matrices over Latin Rectangles

The classical design of cocyclic Hadamard matrices has recently been generalized by means of both the notions of the cocycle of Hadamard matrices over Latin rectangles and the pseudococycle of Hadamard matrices over quasigroups. This paper delves into this topic by introducing the concept of the pseudococycle of a partial Hadamard matrix over a Latin rectangle, whose fundamentals are comprehensively studied and illustrated.

Color image encryption scheme based on quaternion discrete multi-fractional random transform and compressive sensing

A color image compression-encryption algorithm by combining quaternion discrete multi-fractional random transform with compressive sensing is investigated, in which the chaos-based fractional orders greatly improve key sensitivity. The original color image is compressed and encrypted with the assistance of compressive sensing, in which the partial Hadamard matrix adopted as a measurement matrix is constructed by iterating Chebyshev map instead of utilizing the entire Guassian matrix as a key. The sparse images are divided into 12 sub-images and then represented as three quaternion signals, which are modulated by the quaternion discrete multi-fractional random transform. The image blocking and the quaternion representation make the proposed cryptosystem avoid additional data extension existing in many transform-based methods. To further improve the level of security, the plaintext-related key streams generated by the 2D logistic-sine-coupling map are adopted to diffuse and confuse the intermediate results simultaneously. Consequently, the final ciphertext image is attained. Simulation results reveal that the proposed cryptosystem is feasible with high security and has strong robustness against various attacks.

On Proving Parameterized Size Lower Bounds for Multilinear Algebraic Models

We consider the problem of obtaining parameterized lower bounds for the size of arithmetic circuits computing polynomials with the degree of the polynomial as the parameter. We consider the following special classes of multilinear algebraic branching programs: 1) Read Once Oblivious Branching Programs (ROABPs), 2) Strict interval branching programs, 3) Sum of read once formulas with restricted ordering. We obtain parameterized lower bounds (i.e., nΩ(t(k)) lower bound for some function t of k) on the size of the above models computing a multilinear polynomial that can be computed by a depth four circuit of size g(k)nO(1) for some computable function g. Further, we obtain a parameterized separation between ROABPs and read-2 ABPs. This is obtained by constructing a degree k polynomial that can be computed by a read-2 ABP of small size such that the rank of the partial derivative matrix under any partition of the variables is large.

Compressive Sensing Data with Partial Canonical Identity Matrix For Image and Video Reconstruction Using Lifting Wavelet

Compressive Sensing Data with Partial Canonical Identity Matrix For Image and Video Reconstruction Using Lifting Wavelet

Adaptive Partial Multi-View Hashing for Efficient Social Image Retrieval

Social networks allow users to actively upload images and descriptive tags, which has led to an explosive growth in the number of social images. Multi-view hashing is an efficient technique for supporting large-scale social image retrieval because of its desirable capabilities of encoding multi-view features into compact binary hash codes with extremely low storage costs and fast retrieval speeds. However, existing methods require multi-view features to be fully paired at both the offline model training and online query stages. This requirement cannot be easily satisfied for social image retrieval, where social images that lack descriptive tags are common in social networks. In this paper, we propose an <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">Unsupervised Adaptive Partial Multi-view Hashing</i> (UAPMH) method to handle the partial-view hashing problem for efficient social image retrieval. Specifically, the shared and view-specific latent representations of fully paired and partial-view images, respectively, are learned separately by an adaptive partial multi-view matrix factorization module within the identical semantic space. In particular, instead of adopting simple fixed view combination weights, we develop a parameter-free weight learning scheme to adaptively learn the weights to capture the view variations and the discriminative capabilities of different views. With such a design, our model can sufficiently exploit the available partial-view samples with separate hash code learning and effectively preserve the latent relations of images and tags in hash codes with semantic space sharing. Moreover, to avoid relaxing errors and improve the learning efficiency, binary hash codes are directly learned in a fast mode with simple and efficient operations. Finally, we extend UAPMH to the supervised learning paradigm as <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">Supervised Adaptive Partial Multi-view Hashing</i> (SAPMH) with the supervision of pair-wise semantic labels to further enhance the discriminative capability of hash codes. The experiments demonstrate the state-of-the-art performance of the proposed approaches on public social image retrieval datasets. Our source codes and testing datasets can be obtained at <uri xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">https://github.com/ChaoqunZheng/APMH</uri> .

Typical ranks in symmetric matrix completion

We study the problem of low-rank matrix completion for symmetric matrices. The minimum rank of a completion of a generic partially specified symmetric matrix depends only on the location of the specified entries, and not their values, if complex entries are allowed. When the entries are required to be real, this is no longer the case and the possible minimum ranks are called typical ranks . We give a combinatorial description of the patterns of specified entries of n × n symmetric matrices that have n as a typical rank. Moreover, we describe exactly when such a generic partial matrix is minimally completable to rank n . We also characterize the typical ranks for patterns of entries with low maximal typical rank.

Joint Representation Learning and Clustering: A Framework for Grouping Partial Multiview Data

Partial multi-view clustering has attracted various attentions from diverse fields. Most existing methods adopt separate steps to obtain unified representations and extract clustering indicators. This separate manner prevents two learning processes to negotiate to achieve optimal performance. In this paper, we propose the Joint Representation Learning and Clustering (JRLC) framework to address this issue. The JRLC framework employs representation matrices to extract view-specific clustering information directly from the presence of partial similarity matrices, and rotates them to learn a common probability label matrix simultaneously, which connects representation learning and clustering seamlessly to achieve better clustering performance. Under the guidance of JRLC framework, several new incomplete multi-view clustering methods can be developed by extending existing single-view graph-based representation learning methods. For illustration, within the framework, we propose two specific methods, JRLC with spectral embedding (JRLC-SE) and JRLC via integrating nonnegative embedding and spectral embedding (JRLC-NS). Two iterative algorithms with guaranteed convergence are designed to solve the resultant optimization problems of JRLC-SE and JRLC-NS. Experimental results on various datasets and news topic clustering application demonstrate the effectiveness of the proposed algorithms.