arXiv subjects

Cailing Yao

Publications and source records attributed to Cailing Yao.

4 recordsLinked to original sources

arXiv2026

The $ϕ$-conjugation of quaternionic matrices and generalized Autonne-Takagi factorization

Let $ϕ$ be a quaternion of modulus $1$. In this article, we study some topics related to $ϕ$-conjugation for quaternionic matrices, including $ϕ$-Hermitian matrices, $ϕ$-conjugate normal matrices, unitary $ϕ$-congruence and $ϕ$-HSH decomposition (decomposition of a $ϕ$-Hermitian matrix and a skew $ϕ$-Hermitian matrix). In particular, we generalize the Autonne-Takagi factorization of quaternion $ϕ$-Hermitian matrices for all unit quaternion $ϕ$. This gives an affirmative answer to a problem proposed by R. Horn and F. Zhang in the paper ``A generalization of the complex Autonne-Takagi factorization to quaternion matrices, Linear Multilinear A. 60: 1239--1244, 2012''.

math.RA↗

arXiv2026

$\boldsymbol{i}$-conjugate for quaternionic matrices and related properties

Motivated by the result that a complex $n\times n$ matrix $A$ being unitarily equivalent to a real matrix, we extend the conclusion to the quaternion skew field in this paper, we present a necessary and sufficient condition for that a quaternion $n\times n$ matrix $A$ is unitarily equivalent to a complex matrix. To state the truth more clearly, we put forward the concept which we call $\boldsymbol{i}$-conjugate. Furthermore, we study the concepts related to $\boldsymbol{i}$-conjugate and their properties, such as unitary $\boldsymbol{i}$-congruence, $\boldsymbol{i}$-conjugate normality and $\boldsymbol{i}$-Hermicity in $M_{n}(\mathbb{H})$ as generalizations of the conventional unitary congruence, conjugate normality and Hermicity of matrices in $M_{n}(\mathbb{C})$. Finally, we present a new type of polar decomoposition of quaternion matrices.

math.RA↗

arXiv2026

G-LoG Bi-filtration for Medical Image Classification

Building practical filtrations on objects to detect topological and geometric features is an important task in the field of Topological Data Analysis (TDA). In this paper, leveraging the ability of the Laplacian of Gaussian operator to enhance the boundaries of medical images, we define the G-LoG (Gaussian-Laplacian of Gaussian) bi-filtration to generate the features more suitable for multi-parameter persistence module. By modeling volumetric images as bounded functions, then we prove the interleaving distance on the persistence modules obtained from our bi-filtrations on the bounded functions is stable with respect to the maximum norm of the bounded functions. Finally, we conduct experiments on the MedMNIST dataset, comparing our bi-filtration against single-parameter filtration and the established deep learning baselines, including Google AutoML Vision, ResNet, AutoKeras and auto-sklearn. Experiments results demonstrate that our bi-filtration significantly outperforms single-parameter filtration. Notably, a simple Multi-Layer Perceptron (MLP) trained on the topological features generated by our bi-filtration achieves performance comparable to complex deep learning models trained on the original dataset.

cs.CV↗

arXiv2023

Some invariants of $U(1,1;\mathbb{H})$ and diagonalization

Denote by $\mathbb{H}$ the set of all quaternions. We are interested in the group $U(1,1;\mathbb{H})$, which is a subgroup of $2\times 2$ quaternionic matrix group and is sometimes called $Sp(1,1)$. As well known, $U(1,1;\mathbb{H})$ corresponds to the quaternionic Möbius transformations on the unit ball in $\mathbb{H}$. In this article, some similar invariants on $U(1,1;\mathbb{H})$ are discussed. Our main result shows that each matrix $T\in U(1,1;\mathbb{H})$, which corresponds to an elliptic quaternionic Möbius transformation $g_T(z)$, could be $U(1,1;\mathbb{H})$-similar to a diagonal matrix.

math.RA↗