arXiv subjects

Xingyu Xu

Publications and source records attributed to Xingyu Xu.

16 recordsLinked to original sources

arXiv2026

Approximate Message Passing with Random Initialization for Phase Retrieval

We analyze approximate message passing (AMP) with an independent Gaussian initialization for noiseless phase retrieval in the proportional asymptotic regime. A random initialization has overlap of order $d^{-1/2}$ with the signal, and AMP requires a growing number of iterations to attain non-vanishing overlap. Thus, its precise behavior cannot be characterized by classical fixed-time state evolution. We prove a Gaussian decomposition of the AMP trajectory and control its error over the horizons required for recovery. The resulting analysis shows that random initialization attains the weak-recovery threshold $\delta_{\rm weak}=1/2$. For $\delta\in(\delta_{\rm weak},\delta_{\rm str})$, where $\delta_{\rm str}\approx1.13$, the signal strength follows state evolution and approaches its stable finite fixed point uniformly for $n^{1/3}/\operatorname{polylog}(n)$ iterations. For $\delta>\delta_{\rm str}$, AMP reaches any prescribed fixed recovery accuracy within $O_{\delta,\varepsilon}(\log n)$ iterations. The majority of our analysis applies more generally to generalized AMP for single-index models.

math.ST↗

arXiv2026

Polynomial Convergence of Riemannian Diffusion Models

Diffusion models have demonstrated remarkable empirical success in the recent years and are considered one of the state-of-the-art generative models in modern AI. These models consist of a forward process, which gradually diffuses the data distribution to a noise distribution spanning the whole space, and a backward process, which inverts this transformation to recover the data distribution from noise. Most of the existing literature assumes that the underlying space is Euclidean. However, in many practical applications, the data are constrained to lie on a submanifold of Euclidean space. Addressing this setting, De Bortoli et al. (2022) introduced Riemannian diffusion models and proved that using an exponentially small step size yields a small sampling error in the Wasserstein distance, provided the data distribution is smooth and strictly positive, and the score estimate is $L_\infty$-accurate. In this paper, we greatly strengthen this theory by establishing that, under $L_2$-accurate score estimate, a {\em polynomially small stepsize} suffices to guarantee small sampling error in the total variation distance, without requiring smoothness or positivity of the data distribution. Our analysis only requires mild and standard curvature assumptions on the underlying manifold. The main ingredients in our analysis are Li-Yau estimate for the log-gradient of heat kernel, and Minakshisundaram-Pleijel parametrix expansion of the perturbed heat equation. Our approach opens the door to a sharper analysis of diffusion models on non-Euclidean spaces.

cs.LG↗

arXiv2025

Adaptive Multimodal Protein Plug-and-Play with Diffusion-Based Priors

In an inverse problem, the goal is to recover an unknown parameter (e.g., an image) that has typically undergone some lossy or noisy transformation during measurement. Recently, deep generative models, particularly diffusion models, have emerged as powerful priors for protein structure generation. However, integrating noisy experimental data from multiple sources to guide these models remains a significant challenge. Existing methods often require precise knowledge of experimental noise levels and manually tuned weights for each data modality. In this work, we introduce Adam-PnP, a Plug-and-Play framework that guides a pre-trained protein diffusion model using gradients from multiple, heterogeneous experimental sources. Our framework features an adaptive noise estimation scheme and a dynamic modality weighting mechanism integrated into the diffusion process, which reduce the need for manual hyperparameter tuning. Experiments on complex reconstruction tasks demonstrate significantly improved accuracy using Adam-PnP.

cs.LG↗

arXiv2025

Provably Efficient and Agile Randomized Q-Learning

While Bayesian-based exploration often demonstrates superior empirical performance compared to bonus-based methods in model-based reinforcement learning (RL), its theoretical understanding remains limited for model-free settings. Existing provable algorithms either suffer from computational intractability or rely on stage-wise policy updates which reduce responsiveness and slow down the learning process. In this paper, we propose a novel variant of Q-learning algorithm, refereed to as RandomizedQ, which integrates sampling-based exploration with agile, step-wise, policy updates, for episodic tabular RL. We establish an $\widetilde{O}(\sqrt{H^5SAT})$ regret bound, where $S$ is the number of states, $A$ is the number of actions, $H$ is the episode length, and $T$ is the total number of episodes. In addition, we present a logarithmic regret bound under a mild positive sub-optimality condition on the optimal Q-function. Empirically, RandomizedQ exhibits outstanding performance compared to existing Q-learning variants with both bonus-based and Bayesian-based exploration on standard benchmarks.

cs.LG↗

arXiv2024

Sparsity of Quadratically Regularized Optimal Transport: Bounds on concentration and bias

We study the quadratically regularized optimal transport (QOT) problem for quadratic cost and compactly supported marginals $\mu$ and $\nu$. It has been empirically observed that the optimal coupling $\pi_\epsilon$ for the QOT problem has sparse support for small regularization parameter $\epsilon>0.$ In this article we provide the first quantitative description of this phenomenon in general dimension: we derive bounds on the size and on the location of the support of $\pi_\epsilon$ compared to the Monge coupling. Our analysis is based on pointwise bounds on the density of $\pi_\epsilon$ together with Minty's trick, which provides a quadratic detachment from the optimal transport duality gap. In the self-transport setting $\mu=\nu$ we obtain optimal rates of order $\epsilon^{\frac{1}{2+d}}.$

math.OC↗

arXiv2024

Agnostic Active Learning of Single Index Models with Linear Sample Complexity

We study active learning methods for single index models of the form $F({\mathbf x}) = f(\langle {\mathbf w}, {\mathbf x}\rangle)$, where $f:\mathbb{R} \to \mathbb{R}$ and ${\mathbf x,\mathbf w} \in \mathbb{R}^d$. In addition to their theoretical interest as simple examples of non-linear neural networks, single index models have received significant recent attention due to applications in scientific machine learning like surrogate modeling for partial differential equations (PDEs). Such applications require sample-efficient active learning methods that are robust to adversarial noise. I.e., that work even in the challenging agnostic learning setting. We provide two main results on agnostic active learning of single index models. First, when $f$ is known and Lipschitz, we show that $\tilde{O}(d)$ samples collected via {statistical leverage score sampling} are sufficient to learn a near-optimal single index model. Leverage score sampling is simple to implement, efficient, and already widely used for actively learning linear models. Our result requires no assumptions on the data distribution, is optimal up to log factors, and improves quadratically on a recent ${O}(d^{2})$ bound of \cite{gajjar2023active}. Second, we show that $\tilde{O}(d)$ samples suffice even in the more difficult setting when $f$ is \emph{unknown}. Our results leverage tools from high dimensional probability, including Dudley's inequality and dual Sudakov minoration, as well as a novel, distribution-aware discretization of the class of Lipschitz functions.

cs.LG↗

arXiv2024

Provably Robust Score-Based Diffusion Posterior Sampling for Plug-and-Play Image Reconstruction

In a great number of tasks in science and engineering, the goal is to infer an unknown image from a small number of measurements collected from a known forward model describing certain sensing or imaging modality. Due to resource constraints, this task is often extremely ill-posed, which necessitates the adoption of expressive prior information to regularize the solution space. Score-based diffusion models, due to its impressive empirical success, have emerged as an appealing candidate of an expressive prior in image reconstruction. In order to accommodate diverse tasks at once, it is of great interest to develop efficient, consistent and robust algorithms that incorporate unconditional score functions of an image prior distribution in conjunction with flexible choices of forward models. This work develops an algorithmic framework for employing score-based diffusion models as an expressive data prior in general nonlinear inverse problems. Motivated by the plug-and-play framework in the imaging community, we introduce a diffusion plug-and-play method (DPnP) that alternatively calls two samplers, a proximal consistency sampler based solely on the likelihood function of the forward model, and a denoising diffusion sampler based solely on the score functions of the image prior. The key insight is that denoising under white Gaussian noise can be solved rigorously via both stochastic (i.e., DDPM-type) and deterministic (i.e., DDIM-type) samplers using the unconditional score functions. We establish both asymptotic and non-asymptotic performance guarantees of DPnP, and provide numerical experiments to illustrate its promise in solving both linear and nonlinear image reconstruction tasks. To the best of our knowledge, DPnP is the first provably-robust posterior sampling method for nonlinear inverse problems using unconditional diffusion priors.

eess.IV↗

arXiv2023

Provably Accelerating Ill-Conditioned Low-rank Estimation via Scaled Gradient Descent, Even with Overparameterization

Many problems encountered in science and engineering can be formulated as estimating a low-rank object (e.g., matrices and tensors) from incomplete, and possibly corrupted, linear measurements. Through the lens of matrix and tensor factorization, one of the most popular approaches is to employ simple iterative algorithms such as gradient descent (GD) to recover the low-rank factors directly, which allow for small memory and computation footprints. However, the convergence rate of GD depends linearly, and sometimes even quadratically, on the condition number of the low-rank object, and therefore, GD slows down painstakingly when the problem is ill-conditioned. This chapter introduces a new algorithmic approach, dubbed scaled gradient descent (ScaledGD), that provably converges linearly at a constant rate independent of the condition number of the low-rank object, while maintaining the low per-iteration cost of gradient descent for a variety of tasks including sensing, robust principal component analysis and completion. In addition, ScaledGD continues to admit fast global convergence to the minimax-optimal solution, again almost independent of the condition number, from a small random initialization when the rank is over-specified in the presence of Gaussian noise. In total, ScaledGD highlights the power of appropriate preconditioning in accelerating nonconvex statistical estimation, where the iteration-varying preconditioners promote desirable invariance properties of the trajectory with respect to the symmetry in low-rank factorization without hurting generalization.

cs.LG↗

arXiv2023

The Predictability of Stock Price: Empirical Study onTick Data in Chinese Stock Market

Whether or not stocks are predictable has been a topic of concern for decades.The efficient market hypothesis (EMH) says that it is difficult for investors to make extra profits by predicting stock prices, but this may not be true, especially for the Chinese stock market. Therefore, we explore the predictability of the Chinese stock market based on tick data, a widely studied high-frequency data. We obtain the predictability of 3, 834 Chinese stocks by adopting the concept of true entropy, which is calculated by Limpel-Ziv data compression method. The Markov chain model and the diffusion kernel model are used to compare the upper bounds on predictability, and it is concluded that there is still a significant performance gap between the forecasting models used and the theoretical upper bounds.Our work shows that more than 73% of stocks have prediction accuracy greater than 70% and RMSE less than 2 CNY under different quantification intervals with different models. We further take Spearman's correlation to reveal that the average stock price and price volatility may have a negative impact on prediction accuracy, which may be helpful for stock investors.

math.NA↗

arXiv2023

The Power of Preconditioning in Overparameterized Low-Rank Matrix Sensing

We propose $\textsf{ScaledGD($\lambda$)}$, a preconditioned gradient descent method to tackle the low-rank matrix sensing problem when the true rank is unknown, and when the matrix is possibly ill-conditioned. Using overparametrized factor representations, $\textsf{ScaledGD($\lambda$)}$ starts from a small random initialization, and proceeds by gradient descent with a specific form of damped preconditioning to combat bad curvatures induced by overparameterization and ill-conditioning. At the expense of light computational overhead incurred by preconditioners, $\textsf{ScaledGD($\lambda$)}$ is remarkably robust to ill-conditioning compared to vanilla gradient descent ($\textsf{GD}$) even with overprameterization. Specifically, we show that, under the Gaussian design, $\textsf{ScaledGD($\lambda$)}$ converges to the true low-rank matrix at a constant linear rate after a small number of iterations that scales only logarithmically with respect to the condition number and the problem dimension. This significantly improves over the convergence rate of vanilla $\textsf{GD}$ which suffers from a polynomial dependency on the condition number. Our work provides evidence on the power of preconditioning in accelerating the convergence without hurting generalization in overparameterized learning.

cs.LG↗

arXiv2021

Phase Transitions in Frequency Agile Radar Using Compressed Sensing

FAR has improved anti-jamming performance over traditional pulse-Doppler radars under complex electromagnetic circumstances. To reconstruct the range-Doppler information in FAR, many compressed sensing (CS) methods including standard and block sparse recovery have been applied. In this paper, we study phase transitions of range-Doppler recovery in FAR using CS. In particular, we derive closed-form phase transition curves associated with block sparse recovery and complex Gaussian matrices, based on prior results of standard sparse recovery under real Gaussian matrices. We further approximate the obtained curves with elementary functions of radar and target parameters, facilitating practical applications of these curves. Our results indicate that block sparse recovery outperforms the standard counterpart when targets occupy more than one range cell, which are often referred to as extended targets. Simulations validate the availability of these curves and their approximations in FAR, which benefit the design of the radar parameters.

eess.SP↗

arXiv2020

Multi-Carrier Agile Phased Array Radar

Modern radar systems are expected to operate reliably in congested environments. A candidate technology for meeting these demands is frequency agile radar (FAR), which randomly changes its carrier frequencies. FAR is known to improve the electronic counter-countermeasures (ECCM) performance while facilitating operation in congested setups. To enhance the target recovery performance of FAR in complex electromagnetic environments, we propose two radar schemes extending FAR to multi-carrier waveforms. The first is Wideband Multi-carrier Agile Radar (WMAR), which transmits/receives wideband waveforms simultaneously with every antenna. To mitigate the demanding hardware requirements associated with wideband waveforms used by WMAR, we next propose multi-Carrier AgilE phaSed Array Radar (CAESAR). CAESAR uses narrowband monotone waveforms, thus facilitating ease of implementation of the system, while introducing {\em spatial agility}. We characterize the transmitted and received signals of the proposed schemes, and develop an algorithm for recovering the targets, based on concepts from compressed sensing to estimate the range-Doppler parameters of the targets. We then derive conditions which guarantee their accurate reconstruction. Our numerical study demonstrates that both multi-carrier schemes improve performance compared to FAR while maintaining its practical benefits. We also demonstrate that the performance of CAESAR, which uses monotone waveforms, is within a small gap from the wideband radar.

eess.SP↗

arXiv2019

MAJoRCom: A Dual-Function Radar Communication System Using Index Modulation

Dual-function radar communication (DFRC) systems implement both sensing and communication using the same hardware. Such schemes are often more efficient in terms of size, power, and cost, over using distinct radar and communication systems. Since these functionalities share resources such as spectrum, power, and antennas, DFRC methods typically entail some degradation in both radar and communication performance. In this work we propose a DFRC scheme based on the carrier agile phased array radar (CAESAR), which combines frequency and spatial agility. The proposed DFRC system, referred to as multi-carrier agile joint radar communication (MAJoRCom), exploits the inherent spatial and spectral randomness of CAESAR to convey digital messages in the form of index modulation. The resulting communication scheme naturally coexists with the radar functionality, and thus does not come at the cost of reduced radar performance. We analyze the performance of MAJoRCom, quantifying its achievable bit rate. In addition, we develop a low complexity decoder and a codebook design approach, which simplify the recovery of the communicated bits. Our numerical results demonstrate that MAJoRCom is capable of achieving a bit rate which is comparable to utilizing independent communication modules without affecting the radar performance, and that our proposed low-complexity decoder allows the receiver to reliably recover the transmitted symbols with an affordable computational burden.

eess.SP↗

arXiv2019

Lower Bound for RIP Constants and Concentration of Sum of Top Order Statistics

Restricted Isometry Property (RIP) is of fundamental importance in the theory of compressed sensing and forms the base of many exact and robust recovery guarantees in this field. A quantitative description of RIP involves bounding the so-called RIP constants of measurement matrices. In this respect, it is noteworthy that most results in the literature concerning RIP are upper bounds of RIP constants, which can be interpreted as a theoretical guarantee of successful sparse recovery. On the contrary, the land of lower bounds for RIP constants remains uncultivated. Lower bounds of RIP constants, if exist, can be interpreted as the fundamental limit aspect of successful sparse recovery. In this paper, the lower bound of RIP constants Gaussian random matrices are derived, along with a guide for generalization to sub-Gaussian random matrices. This provides a new proof of the fundamental limit that the minimal number of measurements needed to enforce the RIP of order $s$ is $Ω(s\log({\rm e}N/s))$, which is more straight-forward than the classical Gelfand width argument. Furthermore, in the proof, we propose a useful technical tool featuring the concentration phenomenon for top-$k$ sum of a sequence of i.i.d. random variables, which is closely related to mainstream problems in statistics and is of independent interest.

cs.IT↗

arXiv2018

Analysis of Frequency Agile Radar via Compressed Sensing

Frequency agile radar (FAR) is known to have excellent electronic counter-countermeasures (ECCM) performance and the potential to realize spectrum sharing in dense electromagnetic environments. Many compressed sensing (CS) based algorithms have been developed for joint range and Doppler estimation in FAR. This paper considers theoretical analysis of FAR via CS algorithms. In particular, we analyze the properties of the sensing matrix, which is a highly structured random matrix. We then derive bounds on the number of recoverable targets. Numerical simulations and field experiments validate the theoretical findings and demonstrate the effectiveness of CS approaches to FAR.

cs.IT↗

arXiv2018

Accelerating vapor condensation with daytime radiative cooling

Vapor condensation plays a crucial role in solar water-purification technologies. Conventional condensers in solar water-purification systems do not provide sufficient cooling power for vapor condensation, limiting the water production rate to $0.4 L m^{-2} hour^{-1}$. On the other hand, radiative dew condensation, a technique used by existing radiative dew condensers, only works at nighttime and is incompatible with solar water-purification technologies. Here, we develop daytime radiative condensers that reflect almost all solar radiation, and can thus create dew water even in direct sunlight. Compared to state-of-art condensers, our daytime radiative condenser doubles the production of purified water over a 24-hour period. The integration of our daytime radiative condenser with solar water-purification systems can increase the water production rate in sunlight from $0.4 L m^{-2} hour^{-1}$ to more than $1 L m^{-2} hour^{-1}$.

physics.app-ph↗