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Weiyang Ding

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math.NA math.SP Artificial Intelligence Machine Learning math.OC Numerical Analysis Quantitative Methods

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preprint2026arXiv

A Universal Space of Brain Dynamics for Unveiling Cognitive Transitions and Individual Differences

Representing dynamical systems through data-driven universal spaces has proven effective; however, achieving this universality for human brain activity remains a significant challenge, further aggravated by diverse cognitive states and individual subjects. Recognizing that spatial properties reflect physical wiring while temporal properties reflect brain function, we develop Universal Brain Dynamics (UBD) to construct a universal space tailored to brain activity and quantify corresponding dynamics using a model-derived Jacobian matrix. Crucially, we validate UBD's universality by accurately predicting functional magnetic resonance imaging (fMRI) signals (Pearson's r > 0.9) across eight states and 963 subjects in the Human Connectome Project (HCP). Through evaluating resting-state fMRI represented within UBD, we gain insight into how infra-slow fluctuation (ISF) underpins brain activity. Furthermore, we reveal a new perspective on structure-function coupling (SFC) by analyzing the temporal sequence of brain dynamics. Extending UBD to task-evoked states, we derive brain dynamics across various cognitive conditions, elucidating the neural mechanisms driving cognitive transitions at a finer granularity. For individual differences, we compare brain dynamics across subjects to identify the neural underpinnings of these variations. Our findings suggest that synergistically integrating spatial and temporal properties of brain activity establishes a universal space for its unfolding, enabling the precise numerical analysis of underlying neural mechanisms across varying conditions.

preprint2026arXiv

Complex Diffusion Maps with $ω$-Parameterized Kernels Revealing Inherent Harmonic Representations

In this paper, we propose Complex Diffusion Maps (CDM), a novel diffusion mapping framework that aims to reveal the dominant complex harmonics of high-dimensional data. Inspired by the local Gaussian kernel relevant to the heat equation and the nonlocal Schrödinger kernel relevant to the Schrödinger equation, we propose a unified family of $ω$-parameterized complex-valued kernels for the trade-off between local and nonlocal connections. We establish the theoretical foundation based on the operator spectrum theory, where the corresponding diffusion operator, diffusion distance, and complex harmonic maps are well-defined. An optimization-based interpretation of the maps is also developed, aiming to preserve angular structure in the complex diffusion space rather than relying solely on real-valued magnitude. We extensively evaluate CDM on both synthetic and real-world datasets. The complex-valued kernel amplifies differences among easily confusable samples, improving discriminative power over both linear and nonlinear methods based on real-valued kernels. CDM remains robust in high-noise settings, yielding a clearer eigengap that enhances spectral separation. For resting-state fMRI data, CDM captures more strongly correlated and nonlocal spatiotemporal dynamics. Without task-specific tuning, CDM achieves competitive performance on a public EEG sleep dataset, while maintaining high computational efficiency compared with both traditional machine learning and deep neural network approaches, highlighting its generality and practical value.

preprint2020arXiv

Low Rank Pure Quaternion Approximation for Pure Quaternion Matrices

Quaternion matrices are employed successfully in many color image processing applications. In particular, a pure quaternion matrix can be used to represent red, green and blue channels of color images. A low-rank approximation for a pure quaternion matrix can be obtained by using the quaternion singular value decomposition. However, this approximation is not optimal in the sense that the resulting low-rank approximation matrix may not be pure quaternion, i.e., the low-rank matrix contains real component which is not useful for the representation of a color image. The main contribution of this paper is to find an optimal rank-$r$ pure quaternion matrix approximation for a pure quaternion matrix (a color image). Our idea is to use a projection on a low-rank quaternion matrix manifold and a projection on a quaternion matrix with zero real component, and develop an alternating projections algorithm to find such optimal low-rank pure quaternion matrix approximation. The convergence of the projection algorithm can be established by showing that the low-rank quaternion matrix manifold and the zero real component quaternion matrix manifold has a non-trivial intersection point. Numerical examples on synthetic pure quaternion matrices and color images are presented to illustrate the projection algorithm can find optimal low-rank pure quaternion approximation for pure quaternion matrices or color images.

preprint2015arXiv

Inheritance Properties and Sum-of-Squares Decomposition of Hankel Tensors: Theory and Algorithms

In this paper, we show that if a lower-order Hankel tensor is positive semi-definite (or positive definite, or negative semi-definite, or negative definite, or SOS), then its associated higher-order Hankel tensor with the same generating vector, where the higher order is a multiple of the lower order, is also positive semi-definite (or positive definite, or negative semi-definite, or negative definite, or SOS, respectively). Furthermore, in this case, the extremal H-eigenvalues of the higher order tensor are bounded by the extremal H-eigenvalues of the lower order tensor, multiplied with some constants. Based on this inheritance property, we give a concrete sum-of-squares decomposition for each strong Hankel tensor. Then we prove the second inheritance property of Hankel tensors, i.e., a Hankel tensor has no negative (or non-positive, or positive, or nonnegative) H-eigenvalues if the associated Hankel matrix of that Hankel tensor has no negative (or non-positive, or positive, or nonnegative, respectively) eigenvalues. In this case, the extremal H-eigenvalues of the Hankel tensor are also bounded by the extremal eigenvalues of the associated Hankel matrix, multiplied with some constants. The third inheritance property of Hankel tensors is raised as a conjecture.

preprint2015arXiv

P-Tensors, P$_0$-Tensors, and Tensor Complementarity Problem

The concepts of P- and P$_0$-matrices are generalized to P- and P$_0$-tensors of even and odd orders via homogeneous formulae. Analog to the matrix case, our P-tensor definition encompasses many important classes of tensors such as the positive definite tensors, the nonsingular M-tensors, the nonsingular H-tensors with positive diagonal entries, the strictly diagonally dominant tensors with positive diagonal entries, etc. As even-order symmetric PSD tensors are exactly even-order symmetric P$_0$-tensors, our definition of P$_0$-tensors, to some extent, can be regarded as an extension of PSD tensors for the odd-order case. Along with the basic properties of P- and P$_0$-tensors, the relationship among P$_0$-tensors and other extensions of PSD tensors are then discussed for comparison. Many structured tensors are also shown to be P- and P$_0$-tensors. As a theoretical application, the P-tensor complementarity problem is discussed and shown to possess a nonempty and compact solution set.

preprint2014arXiv

Fast Hankel Tensor-Vector Products and Application to Exponential Data Fitting

This paper is contributed to a fast algorithm for Hankel tensor-vector products. For this purpose, we first discuss a special class of Hankel tensors that can be diagonalized by the Fourier matrix, which is called \emph{anti-circulant} tensors. Then we obtain a fast algorithm for Hankel tensor-vector products by embedding a Hankel tensor into a larger anti-circulant tensor. The computational complexity is about $\mathcal{O}(m^2 n \log mn)$ for a square Hankel tensor of order $m$ and dimension $n$, and the numerical examples also show the efficiency of this scheme. Moreover, the block version for multi-level block Hankel tensors is discussed as well. Finally, we apply the fast algorithm to exponential data fitting and the block version to 2D exponential data fitting for higher performance.

preprint2013arXiv

M-Tensors and Nonsingular M-Tensors

The M-matrix is an important concept in matrix theory, and has many applications. Recently, this concept has been extended to higher order tensors [18]. In this paper, we establish some important properties of M-tensors and nonsingular M-tensors. An M-tensor is a Z-tensor. We show that a Z-tensor is a nonsingular M-tensor if and only if it is semi-positive. Thus, a nonsingular M-tensor has all positive diagonal entries; and an M-tensor, regarding as the limitation of a series of nonsingular M-tensors, has all nonnegative diagonal entries. We introduce even-order monotone tensors and present their spectral properties. In matrix theory, a Z-matrix is a nonsingular M-matrix if and only if it is monotone. This is no longer true in the case of higher order tensors. We show that an even-order monotone Z-tensor is an even-order nonsingular M-tensor but not vice versa. An example of an even-order nontrivial monotone Z-tensor is also given.

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