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Jun Xian

Publications and source records attributed to Jun Xian.

17 recordsLinked to original sources
arXiv2026

Stable phase retrieval from short-time linear canonical transforms of signals in Gaussian shift-invariant spaces

Gabor phase retrieval for signals has attracted considerable attention in recent years. For the more general short-time linear canonical transform (STLCT), which arises naturally in optical systems and canonical time--frequency analysis, existing work has so far focused mainly on uniqueness and sampling conditions. Explicit reconstruction formulas, quantitative stability estimates, and robust reconstruction algorithms, however, are still missing. In this paper, we study uniqueness, stability, and robust reconstruction for phase retrieval from phaseless STLCT measurements in the complex Gaussian shift-invariant space $V_\beta^\infty(\varphi)$. We first prove that every signal in $V_\beta^\infty(\varphi)$ is uniquely determined, up to a global unimodular constant, by its phaseless STLCT measurements on the semi-discrete set $\frac{\beta}{2}\mathbb Z\times\mathbb R$, and we derive an explicit reconstruction formula. We then establish stability on intervals under an anchor-point condition, showing that the stability constant is governed by the maximal spacing between adjacent anchor points rather than by the radius of the whole interval. This prevents exponential deterioration with respect to the interval size. Motivated by the practical setting in which only finitely many discrete noisy magnitude samples are available, we further develop an explicit reconstruction algorithm with quantitative robustness guarantees, where the reconstruction error is controlled by the discretization parameters, the noise level, and the conditioning induced by the anchor points. In the Fourier case, our results recover the corresponding Gabor phase retrieval results of Grohs and Liehr and provide improved stability constants.

math.FA↗
arXiv2025

On a Class of Partitions with Lower Expected Star Discrepancy and Its Upper Bound than Jittered Sampling

We investigate the expected star discrepancy under a newly designed class of convex equivolume partition models. The main contributions are two-fold. First, we establish a strong partition principle for the star discrepancy, showing that our newly designed partitions yield stratified sampling point sets with lower expected star discrepancy than both classical jittered sampling and simple random sampling. Specifically, we prove that $\mathbb{E}(D^{*}_{N}(Z))\leq\mathbb{E}(D^{*}_{N}(Y))<\mathbb{E}(D^{*}_{N}(X))$, where $X$, $Y$, and $Z$ represent simple random sampling, jittered sampling, and our new partition sampling, respectively. Second, we derive explicit upper bounds for the expected star discrepancy under our partition models, which improve upon existing bounds for jittered sampling. Our results resolve Open Question 2 posed in Kiderlen and Pausinger (2021) regarding the strong partition principle for star discrepancy.

math.PR↗
arXiv2025

Sharp Non-Asymptotic Bounds for the Star Discrepancy of Double-Infinite Random Matrices via Optimal Covering Numbers

We establish sharp non-asymptotic probabilistic bounds for the star discrepancy of double-infinite random matrices -- a canonical model for sequences of random point sets in high dimensions. By integrating the recently proved \textbf{optimal covering numbers for axis-parallel boxes} (Gnewuch, 2024) into the dyadic chaining framework, we achieve \textbf{explicitly computable constants} that improve upon all previously known bounds. For dimension $d \ge 3$, we prove that with high probability, \[ D_N^d \le \sqrt{\alpha A_d + \beta B \frac{\ln \log_2 N}{d}} \sqrt{\frac{d}{N}}, \] where $A_d$ is given by an explicit series and satisfies $A_3 \le 745$, a \textbf{14\% improvement} over the previous best constant of 868 (Fiedler et al., 2023). For $d=2$, we obtain the currently smallest known constant $A_2 \le 915$. Our analysis reveals a \textbf{precise trade-off} between the dimensional dependence and the logarithmic factor in $N$, highlighting how optimal covering estimates directly translate to tighter discrepancy bounds. These results immediately yield improved error guarantees for \textbf{quasi-Monte Carlo integration, uncertainty quantification, and high-dimensional sampling}, and provide a new benchmark for the probabilistic analysis of geometric discrepancy. \textbf{Keywords:} Star discrepancy, double-infinite random matrices, covering numbers, dyadic chaining, high-dimensional integration, quasi-Monte Carlo, probabilistic bounds.

math.ST↗
arXiv2025

Expected star discrepancy based on stratified sampling

We present two main contributions to the expected star discrepancy theory. First, we derive a sharper expected upper bound for jittered sampling, improving the leading constants and logarithmic terms compared to the state-of-the-art [Doerr, 2022]. Second, we prove the strong partition principle for star discrepancy, showing that any equal-measure stratified sampling yields a strictly smaller expected discrepancy than simple random sampling, thereby resolving an open question in [Kiderlen and Pausinger, 2022]. Numerical simulations confirm our theoretical advances and illustrate the superiority of stratified sampling in low to moderate dimensions.

math.ST↗
arXiv2025

Radon random sampling and reconstruction in local shift-invariant signal space

In this paper, we deal with the problem of reconstruction from Radon random samples in local shift-invariant signal space. Different from sampling after Radon transform, we consider sampling before Radon transform, where the sample set is randomly selected from a square domain with a general probability distribution. First, we prove that the sampling set is stable with high probability under a sufficiently large sample size. Second, we address the problem of signal reconstruction in two-dimensional computed tomography. We demonstrate that the sample values used for this reconstruction process can be determined completely from its Radon transform data. Consequently, we develop an explicit formula to reconstruct the signal using Radon random samples.

math.OC↗
arXiv2025

Conjugate phase retrieval on graphs and with applications in shift-invariant spaces

In this paper, we study the conjugate phase retrieval for complex-valued \mbox{signals} residing on graphs, and explore its applications to shift-invariant spaces. Given a complex-valued graph signal $\bf f$ residing on the graph $\mathcal G$, we introduce a graph ${\mathcal G}_{\bf f}$ and show that its connectivity is sufficient to determine $\bf f$ up to a global unimodular constant and conjugation. We then construct two explicit graph models and show that graph signals residing on them can be recovered, up to a unimodular constant and conjugation, from its absolute values on the vertices and the relative magnitudes between neighboring vertices. Building on this graph-based framework, we apply our results to shift-invariant spaces generated by real-valued functions. For signals in the Paley-Wiener space, we show that any complex-valued function can be recovered, up to a unimodular constant and conjugation, from structured phaseless samples taken at three times the Nyquist rate. For more general shift invariant spaces, we establish the conjugate phase retrievability of signals from phaseless samples collected on a discrete sampling set, in conjunction with relative magnitude measurements between neighboring sample points. Two numerical reconstruction algorithms are introduced to recover the signals in the Paley-Wiener space and general shift-invariant spaces, up to a unimodular constant and conjugation, from the given phaseless measurements.

math.FA↗
arXiv2025

Random uniform approximation under weighted importance sampling of a class of stratified input

We consider random discrepancy under weighted importance sampling of a class of stratified input. We give the expected $L_p-$discrepancy($2\leq p<\infty$) upper bound in weighted form under a class of stratified sampling. This result contributes to the error estimate of the upper bound of the integral approximation under weighted importance sampling, and and our sampling pattern is a stratified input.

math.PR↗
arXiv2024

Graph Convolutional Network For Semi-supervised Node Classification With Subgraph Sketching

In this paper, we propose the Graph-Learning-Dual Graph Convolutional Neural Network called GLDGCN based on the classic Graph Convolutional Neural Network(GCN) by introducing dual convolutional layer and graph learning layer. We apply GLDGCN to the semi-supervised node classification task. Compared with the baseline methods, we achieve higher classification accuracy on three citation networks Citeseer, Cora and Pubmed, and we also analyze and discussabout selection of the hyperparameters and network depth. GLDGCN also perform well on the classic social network KarateClub and the new Wiki-CS dataset. For the insufficient ability of our algorithm to process large graphs during the experiment, we also introduce subgraph clustering and stochastic gradient descent methods into GCN and design a semi-supervised node classification algorithm based on the CLustering Graph Convolutional neural Network, which enables GCN to process large graph and improves its application value. We complete semi-supervised node classification experiments on two classic large graph which are PPI dataset (more than 50,000 nodes) and Reddit dataset (more than 200,000 nodes), and also perform well.

cs.LG↗
arXiv2023

Random Sampling of Bandlimited Graph Signals from Local Measurements

The random sampling on graph signals is one of the fundamental topics in graph signal processing. In this letter, we consider the random sampling of k-bandlimited signals from the local measurements and show that no more than O(klogk) measurements with replacement are sufficient for the accurate and stable recovery of any k-bandlimited graph signals. We propose two random sampling strategies based on the minimum measurements, i.e., the optimal sampling and the estimated sampling. The geodesic distance between vertices is introduced to design the sampling probability distribution. Numerical experiments are included to show the effectiveness of the proposed methods.

cs.IT↗
arXiv2023

Star discrepancy bounds based on Hilbert space filling curve stratified sampling and its applications

In this paper, we consider the upper bound of the probabilistic star discrepancy based on Hilbert space filling curve sampling. This problem originates from the multivariate integral approximation, but the main result removes the strict conditions on the sampling number of the classical grid-based jittered sampling. The main content has three parts. First, we inherit the advantages of this new sampling and achieve a better upper bound of the random star discrepancy than the use of Monte Carlo sampling. In addition, the convergence order of the upper bound is improved from $O(N^{-\frac{1}{2}})$ to $O(N^{-\frac{1}{2}-\frac{1}{2d}})$. Second, a better uniform integral approximation error bound of the function in the weighted space is obtained. Third, other applications will be given. Such as the sampling theorem in Hilbert spaces and the improvement of the classical Koksma-Hlawka inequality. Finally, the idea can also be applied to the proof of the strong partition principle of the star discrepancy version.

math.ST↗
arXiv2022

Expected $L_2-$discrepancy bound for a class of new stratified sampling models

We introduce a class of convex equivolume partitions. Expected $L_2-$discrepancy are discussed under these partitions. There are two main results. First, under this kind of partitions, we generate random point sets with smaller expected $L_2-$discrepancy than classical jittered sampling for the same sampling number. Second, an explicit expected $L_2-$discrepancy upper bound under this kind of partitions is also given. Further, among these new partitions, there is optimal expected $L_2-$discrepancy upper bound.

math.ST↗
arXiv2022

Star discrepancy for new stratified random sampling I: optimal expected star discrepancy

We introduce a class of convex equivolume partitions. Expected star discrepancy results are compared for stratified samples under these partitions, including simple random samples. There are four main parts of our results. First, among these newly designed partitions, there is one that minimizes the expected star discrepancy, thus we partly answer an open question in [F. Pausinger, S. Steinerberger, J. Complex. 2016]. Second, there are an infinite number of such class of partitions, which generate point sets with smaller expected discrepancy than classical jittered sampling for large sampling number, leading to an open question in [M. Kiderlen, F. Pausinger, Monatsh. Math. 2021] being solved. Third, we prove a strong partition principle and generalize the expected star discrepancy under these partition models from $L_2-$discrepancy to star discrepancy, hence an open question in [M. Kiderlen, F. Pausinger, J. Complex. 2021] is answered. In the end, optimal expected star discrepancy upper bound under this class of partitions is given, which is better than using jittered sampling.

math.ST↗
arXiv2021

Expected uniform integration approximation under general equal measure partition

In this paper, we study bounds of expected $L_2-$discrepancy to give mean square error of uniform integration approximation for functions in Sobolev space $\mathcal{H}^{\mathbf{1}}(K)$, where $\mathcal{H}$ is a reproducing Hilbert space with kernel $K$. Better order $O(N^{-1-\frac{1}{d}})$ of approximation error is obtained, comparing with previously known rate $O(N^{-1})$ using crude Monte Carlo method. Secondly, we use expected $L_{p}-$discrepancy bound($p\ge 1$) of stratified samples to give several upper bounds of $p$-moment of integral approximation error in general Sobolev space $F_{d,q}^{*}$.

math.NA↗
arXiv2020

Random sampling and reconstruction of concentrated signals in a reproducing kernel space

In this paper, we consider (random) sampling of signals concentrated on a bounded Corkscrew domain $Ω$ of a metric measure space, and reconstructing concentrated signals approximately from their (un)corrupted sampling data taken on a sampling set contained in $Ω$. We establish a weighted stability of bi-Lipschitz type for a (random) sampling scheme on the set of concentrated signals in a reproducing kernel space. The weighted stability of bi-Lipschitz type provides a weak robustness to the sampling scheme, however due to the nonconvexity of the set of concentrated signals, it does not imply the unique signal reconstruction. From (un)corrupted samples taken on a finite sampling set contained in $Ω$, we propose an algorithm to find approximations to signals concentrated on a bounded Corkscrew domain $Ω$. Random sampling is a sampling scheme where sampling positions are randomly taken according to a probability distribution. Next we show that, with high probability, signals concentrated on a bounded Corkscrew domain $Ω$ can be reconstructed approximately from their uncorrupted (or randomly corrupted) samples taken at i.i.d. random positions drawn on $Ω$, provided that the sampling size is at least of the order $μ(Ω) \ln (μ(Ω))$, where $μ(Ω)$ is the measure of the concentrated domain $Ω$. Finally, we demonstrate the performance of proposed approximations to the original concentrated signal when the sampling procedure is taken either with small density or randomly with large size.

cs.IT↗
arXiv2014

Relevant sampling in finitely generated shift-invariant spaces

We consider random sampling in finitely generated shift-invariant spaces $V(Φ) \subset {\rm L}^2(\mathbb{R}^n)$ generated by a vector $Φ= (φ_1,\ldots,φ_r) \in {\rm L}^2(\mathbb{R}^n)^r$. Following the approach introduced by Bass and Gröchenig, we consider certain relatively compact subsets $V_{R,δ}(Φ)$ of such a space, defined in terms of a concentration inequality with respect to a cube with side lengths $R$. Under very mild assumptions on the generators, we show that for $R$ sufficiently large, taking $O(R^n log(R^{n^2/α'}))$ many random samples (taken independently uniformly distributed within $C_R$) yields a sampling set for $V_{R,δ}(Φ)$ with high probability. Here $α' \le n$ is a suitable constant.We give explicit estimates of all involved constants in terms of the generators $φ_1, \ldots, φ_r$.

math.FA↗