arXiv subjects

Songhao Zhu

Publications and source records attributed to Songhao Zhu.

8 recordsLinked to original sources

arXiv2026

Super Krawtchouk Polynomials via Lie Superalgebras

Multivariate extensions of the Krawtchouk polynomials have been studied by numerous authors in recent decades by exploring new connections to probability, representation theory and quantum integrability. We develop a theory of multivariate super Krawtchouk polynomials using the representation theory of the general linear Lie superalgebra, extending results of the first author in the classical setting. Specifically, in the present work we generalize the classical Krawtchouk polynomials, prove their orthogonality, construct certain recurrence relations, and discuss their connections with zonal spherical functions arising from a fermionic Fock-space framework in quantum mechanics.

math.RT↗

arXiv2024

Growth of root multiplicities along imaginary root strings in Kac--Moody algebras

Let $\mathfrak{g}$ be a symmetrizable Kac--Moody algebra. Given a root $\alpha$ and a real root $\beta$ of $\mathfrak{g}$, it is known that the $\beta$-string through $\alpha$, denoted $R_\alpha(\beta)$, is finite. Given an imaginary root $\beta$, we show that $R_\alpha(\beta)=\{\beta\}$ or $R_\alpha(\beta)$ is infinite. If $(\beta,\beta)<0$, we also show that the multiplicity of the root ${\alpha+n\beta}$ grows at least exponentially as $n\to\infty$. If $(\beta,\beta)=(\alpha, \beta) = 0$, we show that $R_\alpha(\beta)$ is bi-infinite and the multiplicities of $\alpha+n\beta$ are bounded. If $(\beta,\beta)=0$ and $(\alpha, \beta) \neq 0$, we show that $R_\alpha(\beta)$ is semi-infinite and the muliplicity of $\alpha+n\beta$ or $\alpha-n\beta$ grows faster than every polynomial as $n\to\infty$. We also prove that $\dim \mathfrak{g}_{\alpha+\beta} \geq \dim \mathfrak{g}_\alpha + \dim \mathfrak{g}_\beta -1$ whenever $\alpha \neq \beta$ with $(\alpha, \beta)<0$.

math.RT↗

arXiv2023

Eigenvalues of supersymmetric Shimura operators and interpolation polynomials

The Shimura operators are a certain distinguished basis for invariant differential operators on a Hermitian symmetric space. Answering a question of Shimura, Sahi and Zhang showed that the Harish-Chandra images of these operators are specializations of certain $BC$-symmetric interpolation polynomials that were defined by Okounkov. We consider the analogs of Shimura operators for the Hermitian symmetric superpair $(\mathfrak{g},\mathfrak{k})$ where $\mathfrak{g}= \mathfrak{gl}(2p|2q)$ and $\mathfrak{k}= \mathfrak{gl}(p|q)\oplus \mathfrak{gl}(p|q)$ and we prove their Harish-Chandra images are specializations of certain $BC$-supersymmetric interpolation polynomials introduced by Sergeev--Veselov.

math.RT↗

arXiv2022

Shimura operators for certain Hermitian symmetric superpairs

We give a partial super analog of a result obtained by S. Sahi and G. Zhang relating Shimura operators and certain interpolation symmetric polynomials. In particular, we study the pair $(\mathfrak{gl}(2p|2q), \mathfrak{gl}(p|q)\oplus\mathfrak{gl}(p|q))$, define the super Shimura operators in $\mathfrak{U}(\mathfrak{g})^\mathfrak{k}$, and using a new method, prove that their images under the Harish-Chandra homomorphism are proportional to Sergeev and Veselov's Type $BC$ interpolation supersymmetric polynomials, under the assumption that a family of irreducible $\mathfrak{g}$-modules are spherical. We prove this conjecture using the notion of quasi-sphericity for Kac modules when $p=q=1$, and give explicit coordinates of (quasi-)spherical vectors.

math.RT↗

arXiv2022

Short Range Correlation Transformer for Occluded Person Re-Identification

Occluded person re-identification is one of the challenging areas of computer vision, which faces problems such as inefficient feature representation and low recognition accuracy. Convolutional neural network pays more attention to the extraction of local features, therefore it is difficult to extract features of occluded pedestrians and the effect is not so satisfied. Recently, vision transformer is introduced into the field of re-identification and achieves the most advanced results by constructing the relationship of global features between patch sequences. However, the performance of vision transformer in extracting local features is inferior to that of convolutional neural network. Therefore, we design a partial feature transformer-based person re-identification framework named PFT. The proposed PFT utilizes three modules to enhance the efficiency of vision transformer. (1) Patch full dimension enhancement module. We design a learnable tensor with the same size as patch sequences, which is full-dimensional and deeply embedded in patch sequences to enrich the diversity of training samples. (2) Fusion and reconstruction module. We extract the less important part of obtained patch sequences, and fuse them with original patch sequence to reconstruct the original patch sequences. (3) Spatial Slicing Module. We slice and group patch sequences from spatial direction, which can effectively improve the short-range correlation of patch sequences. Experimental results over occluded and holistic re-identification datasets demonstrate that the proposed PFT network achieves superior performance consistently and outperforms the state-of-the-art methods.

cs.CV↗

arXiv2021

Learning to Amend Facial Expression Representation via De-albino and Affinity

Facial Expression Recognition (FER) is a classification task that points to face variants. Hence, there are certain affinity features between facial expressions, receiving little attention in the FER literature. Convolution padding, despite helping capture the edge information, causes erosion of the feature map simultaneously. After multi-layer filling convolution, the output feature map named albino feature definitely weakens the representation of the expression. To tackle these challenges, we propose a novel architecture named Amending Representation Module (ARM). ARM is a substitute for the pooling layer. Theoretically, it can be embedded in the back end of any network to deal with the Padding Erosion. ARM efficiently enhances facial expression representation from two different directions: 1) reducing the weight of eroded features to offset the side effect of padding, and 2) decomposing facial features to simplify representation learning. Experiments on public benchmarks prove that our ARM boosts the performance of FER remarkably. The validation accuracies are respectively 90.42% on RAF-DB, 65.2% on Affect-Net, and 58.71% on SFEW, exceeding current state-of-the-art methods. Our implementation and trained models are available at https://github.com/JiaweiShiCV/Amend-Representation-Module.

cs.CV↗

arXiv2021

A vanishing theorem for the canonical blow-ups of Grassmann manifolds

Let $\mathcal T_{s,p,n}$ be the canonical blow-up of the Grassmann manifold $G(p,n)$ constructed by blowing up the Plücker coordinate subspaces associated with the parameter $s$. We prove that the higher cohomology groups of the tangent bundle of $\mathcal T_{s,p,n}$ vanish. As an application, $\mathcal T_{s,p,n}$ is locally rigid in the sense of Kodaira-Spencer.

math.AG↗

arXiv2019

Joint Learning of Self-Representation and Indicator for Multi-View Image Clustering

Multi-view subspace clustering aims to divide a set of multisource data into several groups according to their underlying subspace structure. Although the spectral clustering based methods achieve promotion in multi-view clustering, their utility is limited by the separate learning manner in which affinity matrix construction and cluster indicator estimation are isolated. In this paper, we propose to jointly learn the self-representation, continue and discrete cluster indicators in an unified model. Our model can explore the subspace structure of each view and fusion them to facilitate clustering simultaneously. Experimental results on two benchmark datasets demonstrate that our method outperforms other existing competitive multi-view clustering methods.

cs.CV↗