arXiv subjects

Lihong Qiu

Publications and source records attributed to Lihong Qiu.

3 recordsLinked to original sources

arXiv2021

Smith Normal Form and the Generalized Spectral Characterization of Graphs

Spectral characterization of graphs is an important topic in spectral graph theory, which has received a lot of attention from researchers in recent years. It is generally very hard to show a given graph to be determined by its spectrum. Recently, Wang [10] gave a simple arithmetic condition for graphs being determined by their generalized spectra. Let $G$ be a graph with adjacency matrix $A$ on $n$ vertices, and $W=[e,Ae,\ldots,A^{n-1}e]$ ($e$ is the all-one vector) be the walk-matrix of $G$. A theorem of Wang [10] states that if $2^{-\lfloor n/2\rfloor}\det W$ (which is always an integer) is odd and square-free, then $G$ is determined by the generalized spectrum. In this paper, we find a new and short route which leads to a stronger version of the above theorem. The result is achieved by using the Smith Normal Form of the walk-matrix of $G$. The proposed method gives a new insight in dealing with the problem of generalized spectral characterization of graphs.

math.CO↗

arXiv2020

Diversified Arbitrary Style Transfer via Deep Feature Perturbation

Image style transfer is an underdetermined problem, where a large number of solutions can satisfy the same constraint (the content and style). Although there have been some efforts to improve the diversity of style transfer by introducing an alternative diversity loss, they have restricted generalization, limited diversity and poor scalability. In this paper, we tackle these limitations and propose a simple yet effective method for diversified arbitrary style transfer. The key idea of our method is an operation called deep feature perturbation (DFP), which uses an orthogonal random noise matrix to perturb the deep image feature maps while keeping the original style information unchanged. Our DFP operation can be easily integrated into many existing WCT (whitening and coloring transform)-based methods, and empower them to generate diverse results for arbitrary styles. Experimental results demonstrate that this learning-free and universal method can greatly increase the diversity while maintaining the quality of stylization.

cs.CV↗

arXiv2019

Generalized spectral characterization of mixed graphs

A mixed graph $G$ is a graph obtained from a simple undirected graph by orientating a subset of edges. $G$ is self-converse if it is isomorphic to the graph obtained from $G$ by reversing each directed edge. For two mixed graphs $G$ and $H$ with Hermitian adjacency matrices $A(G)$ and $A(H)$, we say $G$ is $\mathbb{R}$\emph{-cospectral} to $H$ if, for any $y\in \mathbb{R}$, $yJ-A(G)$ and $yJ-A(H)$ have the same spectrum, where $J$ is the all-one matrix. A self-converse mixed graph $G$ is said to be determined by its generalized spectrum, if any self-converse mixed graph that is $R$-cospectral with $G$ is isomorphic to $G$. Let $G$ be a self-converse mixed graph of order $n$ such that $2^{-\lfloor n/2\rfloor}\det W$ (which is always a real or pure imaginary Gaussian integer) is square-free in $\mathbb{Z}[i]$, where $W=[e,Ae,\ldots,A^{n-1}e]$, $A=A(G)$ and $e$ is the all-one vector. We prove that, for any self-converse mixed graph $H$ that is $\mathbb{R}$-cospectral to $G$, there exists a Gaussian rational unitary matrix $U$ such that $Ue=e$, $U^*A(G)U=A(H)$ and $(1+i)U$ is a Gaussian integral matrix. In particular, if $G$ is an ordinary graph (viewed as a mixed graph) satisfying the above condition, then any self-converse mixed graph $H$ that is $\mathbb{R}$-cospectral to $G$ is $G$ itself (in the sense of isomorphism). This strengthens a recent result of the first author.

math.CO↗