arXiv subjects

Qun Mo

Publications and source records attributed to Qun Mo.

9 recordsLinked to original sources

arXiv2024

The Tammes Problem in $\mathbb{R}^{n}$ and Linear Programming Method

The Tammes problem delves into the optimal arrangement of $N$ points on the surface of the $n$-dimensional unit sphere (denoted as $\mathbb{S}^{n-1}$), aiming to maximize the minimum distance between any two points. In this paper, we articulate the sufficient conditions requisite for attaining the optimal value of the Tammes problem for arbitrary $n, N \in \mathbb{N}^{+}$, employing the linear programming framework pioneered by Delsarte et al. Furthermore, we showcase several illustrative examples across various dimensions $n$ and select values of $N$ that yield optimal configurations. The findings illuminate the intricate structure of optimal point distributions on spheres, thereby enriching the existing body of research in this domain.

math.MG↗

arXiv2024

A New Linear Programming Method in Sphere Packing

Inspired by the linear programming method developed by Cohn and Elkies (Ann. Math. 157(2): 689-714, 2003), we introduce a new linear programming method to solve the sphere packing problem. More concretely, we consider sequences of auxiliary functions $\{g_m\}_{m\in \mathbb{N}^{+}}$, where $g_m$ is a $m\Lambda$-periodic auxiliary function defined on $\mathbb{R}^n$, with $\Lambda$ being a given full-rank lattice in $\mathbb{R}^n$. This new method extends the original approach and offers a greater flexibility. Furthermore, using this new linear programming framework, we construct several effective auxiliary functions for dimensions $n=1,2,3$. We hope this approach provides valuable insights into solving sphere packing problems for $n=2,3$ and even higher dimensions.

math.MG↗

arXiv2018

A Sharp Condition for Exact Support Recovery of Sparse Signals With Orthogonal Matching Pursuit

Support recovery of sparse signals from noisy measurements with orthogonal matching pursuit (OMP) has been extensively studied in the literature. In this paper, we show that for any $K$-sparse signal $\x$, if the sensing matrix $\A$ satisfies the restricted isometry property (RIP) of order $K + 1$ with restricted isometry constant (RIC) $\delta_{K+1} < 1/\sqrt {K+1}$, then under some constraint on the minimum magnitude of the nonzero elements of $\x$, the OMP algorithm exactly recovers the support of $\x$ from the measurements $\y=\A\x+\v$ in $K$ iterations, where $\v$ is the noise vector. This condition is sharp in terms of $\delta_{K+1}$ since for any given positive integer $K\geq 2$ and any $1/\sqrt{K+1}\leq t<1$, there always exist a $K$-sparse $\x$ and a matrix $\A$ satisfying $\delta_{K+1}=t$ for which OMP may fail to recover the signal $\x$ in $K$ iterations. Moreover, the constraint on the minimum magnitude of the nonzero elements of $\x$ is weaker than existing results.

cs.IT↗

arXiv2015

A Sharp Condition for Exact Support Recovery of with Orthogonal Matching Pursuit

Support recovery of sparse signals from noisy measurements with orthogonal matching pursuit (OMP) has been extensively studied. In this paper, we show that for any $K$-sparse signal $\x$, if a sensing matrix $\A$ satisfies the restricted isometry property (RIP) with restricted isometry constant (RIC) $\delta_{K+1} < 1/\sqrt {K+1}$, then under some constraints on the minimum magnitude of nonzero elements of $\x$, OMP exactly recovers the support of $\x$ from its measurements $\y=\A\x+\v$ in $K$ iterations, where $\v$ is a noise vector that is $\ell_2$ or $\ell_{\infty}$ bounded. This sufficient condition is sharp in terms of $\delta_{K+1}$ since for any given positive integer $K$ and any $1/\sqrt{K+1}\leq \delta<1$, there always exists a matrix $\A$ satisfying the RIP with $\delta_{K+1}=\delta$ for which OMP fails to recover a $K$-sparse signal $\x$ in $K$ iterations. Also, our constraints on the minimum magnitude of nonzero elements of $\x$ are weaker than existing ones. Moreover, we propose worst-case necessary conditions for the exact support recovery of $\x$, characterized by the minimum magnitude of the nonzero elements of $\x$.

cs.IT↗

arXiv2015

A new method on deterministic construction of the measurement matrix in compressed sensing

Construction on the measurement matrix $A$ is a central problem in compressed sensing. Although using random matrices is proven optimal and successful in both theory and applications. A deterministic construction on the measurement matrix is still very important and interesting. In fact, it is still an open problem proposed by T. Tao. In this paper, we shall provide a new deterministic construction method and prove it is optimal with regard to the mutual incoherence.

cs.IT↗

arXiv2015

A Sharp Restricted Isometry Constant Bound of Orthogonal Matching Pursuit

We shall show that if the restricted isometry constant (RIC) $\delta_{s+1}(A)$ of the measurement matrix $A$ satisfies $$ \delta_{s+1}(A) < \frac{1}{\sqrt{s + 1}}, $$ then the greedy algorithm Orthogonal Matching Pursuit(OMP) will succeed. That is, OMP can recover every $s$-sparse signal $x$ in $s$ iterations from $b = Ax$. Moreover, we shall show the upper bound of RIC is sharp in the following sense. For any given $s \in \N$, we shall construct a matrix $A$ with the RIC $$ \delta_{s+1}(A) = \frac{1}{\sqrt{s + 1}} $$ such that OMP may not recover some $s$-sparse signal $x$ in $s$ iterations.

cs.IT↗

arXiv2013

Compactly Supported Tensor Product Complex Tight Framelets with Directionality

Although tensor product real-valued wavelets have been successfully applied to many high-dimensional problems, they can only capture well edge singularities along the coordinate axis directions. As an alternative and improvement of tensor product real-valued wavelets and dual tree complex wavelet transform, recently tensor product complex tight framelets with increasing directionality have been introduced in [8] and applied to image denoising in [13]. Despite several desirable properties, the directional tensor product complex tight framelets constructed in [8,13] are bandlimited and do not have compact support in the space/time domain. Since compactly supported wavelets and framelets are of great interest and importance in both theory and application, it remains as an unsolved problem whether there exist compactly supported tensor product complex tight framelets with directionality. In this paper, we shall satisfactorily answer this question by proving a theoretical result on directionality of tight framelets and by introducing an algorithm to construct compactly supported complex tight framelets with directionality. Our examples show that compactly supported complex tight framelets with directionality can be easily derived from any given eligible low-pass filters and refinable functions. Several examples of compactly supported tensor product complex tight framelets with directionality have been presented.

cs.IT↗

arXiv2012

A remark on the Restricted Isometry Property in Orthogonal Matching Pursuit

This paper demonstrates that if the restricted isometry constant $\delta_{K+1}$ of the measurement matrix $A$ satisfies $$ \delta_{K+1} < \frac{1}{\sqrt{K}+1}, $$ then a greedy algorithm called Orthogonal Matching Pursuit (OMP) can recover every $K$--sparse signal $\mathbf{x}$ in $K$ iterations from $A\x$. By contrast, a matrix is also constructed with the restricted isometry constant $$ \delta_{K+1} = \frac{1}{\sqrt{K}} $$ such that OMP can not recover some $K$-sparse signal $\mathbf{x}$ in $K$ iterations. This result positively verifies the conjecture given by Dai and Milenkovic in 2009.

cs.IT↗

arXiv2011

Remarks on the Restricted Isometry Property in Orthogonal Matching Pursuit algorithm

This paper demonstrates theoretically that if the restricted isometry constant $\delta_K$ of the compressed sensing matrix satisfies $$ \delta_{K+1} < \frac{1}{\sqrt{K}+1}, $$ then a greedy algorithm called Orthogonal Matching Pursuit (OMP) can recover a signal with $K$ nonzero entries in $K$ iterations. In contrast, matrices are also constructed with restricted isometry constant $$ \delta_{K+1} = \frac{1}{\sqrt{K}} $$ such that OMP can not recover $K$-sparse $x$ in $K$ iterations. This result shows that the conjecture given by Dai and Milenkovic is ture.

cs.IT↗