arXiv subjects

Pengtao Li

Publications and source records attributed to Pengtao Li.

At least 19 recordsLinked to original sources

arXiv2026

Central limit theorems and bootstrap for sparse regularized multimarginal optimal transport and barycenters

Wasserstein barycenters are a fundamental tool for the analysis of distribution-valued data, but their computation and statistical analysis become challenging in high-dimensional and multi-sample settings. The unregularized barycenter problem is computationally demanding and suffers severely from the curse of dimensionality, making regularization essential in many applications. While regularized optimal transport methods are widely used, approaches based on entropic penalization tend to overspread mass and may blur the geometric structure of the solution. Motivated by barycenter estimation, we propose a more general methodology based on sparse regularized multimarginal optimal transport (RMOT). Our formulation creates sparse transport plans and is designed to avoid this overspreading effect. We establish consistency, central limit theorems, and bootstrap validity for the optimal potentials and couplings for the empirical RMOT problem. Specializing to sparse regularized barycenters, our results yield a uniform weak limit and bootstrap consistency. Numerical simulations show that sparsity plays a fundamental role in preserving the structure of the solution and avoiding the overspreading behavior typical of entropic regularization.

math.ST↗

arXiv2026

Tree-of-Ideas: Automated Research Ideation via Cross-Trajectory Reasoning over Scholarly Evolution

Effective research ideation requires moving beyond a static understanding of prior work to trace how research problems and solutions evolve across the literature. Existing methods either treat papers as unstructured context or model scholarly evolution as isolated citation chains, overlooking interactions among research trajectories. We propose Tree-of-Ideas (ToI), a two-stage framework. EvoTrace reconstructs branching scholarly trajectories from citations, tracking evolving methods, resolved problems, and gaps. EvoAgent then reasons across trajectories to identify convergent problems and complementary solutions, generating grounded research ideas. Across six AI research topics, ToI achieves the highest score among automatic methods (6.27 vs. 5.36 for the strongest baseline on a 10-point scale), with strong Novelty (6.36) and Groundedness (7.00). Also, its score approaches that of human-paper references (6.29), demonstrating the value of cross-path evolutionary reasoning.

cs.AI↗

arXiv2026

Smoothed estimation of Wasserstein barycenters

This paper studies the statistical estimation of exact Wasserstein barycenters. Existing non-asymptotic results for empirical barycenters exhibit a severe curse of dimensionality. Motivated by the semi-dual formulation of the barycenter problem and its associated Sobolev optimization geometry, we develop a smoothness-aware approach that combines density estimation with Sobolev geometric structure to estimate the population barycenter. We establish nonparametric convergence rates for estimating both the barycenter functional and its minimizer, demonstrating how smoothness can substantially improve statistical performance.

math.ST↗

arXiv2026

Hierarchy of discriminative power and complexity in learning quantum ensembles

Distance metrics are central to machine learning, yet distances between ensembles of quantum states remain poorly understood due to fundamental quantum measurement constraints. We introduce a hierarchy of integral probability metrics, termed MMD-$k$, which generalizes the maximum mean discrepancy to quantum ensembles and exhibit a strict trade-off between discriminative power and statistical efficiency as the moment order $k$ increases. For pure-state ensembles of size $N$, estimating MMD-$k$ using experimentally feasible SWAP-test-based estimators requires $Θ(N^{2-2/k})$ samples for constant $k$, and $Θ(N^3)$ samples to achieve full discriminative power at $k = N$. In contrast, the quantum Wasserstein distance attains full discriminative power with $Θ(N^2 \log N)$ samples. These results provide principled guidance for the design of loss functions in quantum machine learning, which we illustrate in the training quantum denoising diffusion probabilistic models.

quant-ph↗

arXiv2025

Sample complexity and weak limits of nonsmooth multimarginal Schrödinger system with application to optimal transport barycenter

Multimarginal optimal transport (MOT) has emerged as a useful framework for many applied problems. However, compared to the well-studied classical two-marginal optimal transport theory, analysis of MOT is far more challenging and remains much less developed. In this paper, we study the statistical estimation and inference problems for the entropic MOT (EMOT), whose optimal solution is characterized by the multimarginal Schrödinger system. Assuming only boundedness of the cost function, we derive sharp sample complexity for estimating several key quantities pertaining to EMOT (cost functional and Schrödinger coupling) from point clouds that are randomly sampled from the input marginal distributions. Moreover, with substantially weaker smoothness assumption on the cost function than the existing literature, we derive distributional limits and bootstrap validity of various key EMOT objects. As an application, we propose the multimarginal Schrödinger barycenter as a new and natural way to regularize the exact Wasserstein barycenter and demonstrate its statistical optimality.

math.ST↗

arXiv2025

Regularity estimates of fractional heat semigroups related with uniformly elliptic operators

Let $L = -{\rm div}( A(x) \cdot \nabla ) + V(x)$ be a second-order uniformly elliptic operator on $\mathbb{ R }^{n}$ $(n\geq 3)$, where $A(x)$ is a real symmetric matrix satisfying standard ellipticity conditions, and $V$ is a nonnegative potential belonging to the reverse Hölder class. For $ α\in (0,1) $, we study regularity estimates of the fractional heat semigroups $ \{ exp (-tL^ {α} )\} _ { t > 0 }$, via the subordination formula and the fundamental solution of the associated uniformly parabolic equation $ \partial_t u + Lu = 0 $. This approach avoids the use of Fourier transforms and is applicable to second-order differential operators whose heat kernels satisfy Gaussian upper bounds. As an application, we characterize the Campanato-type space $Λ_{ L , γ} \left( \mathbb{R}^n \right)$ via the fractional heat semigroups $\{exp ( - t L ^ {α} ) \} _ { t > 0 } $.

math.FA↗

arXiv2025

On Concentration Inequality of the Laplacian Matrix of Erdős-Rényi Graphs

This paper focuses on the concentration properties of the spectral norm of the normalized Laplacian matrix for Erdős-Rényi random graphs. First, We achieve the optimal bound that can be attained in the further question posed by Le et al. [24] for the regularized Laplacian matrix. Beyond that, we also establish a uniform concentration inequality for the spectral norm of the Laplacian matrix in the homogeneous case, relying on a key tool: the uniform concentration property of degrees, which may be of independent interest. Additionally, we prove that after normalizing the eigenvector corresponding to the largest eigenvalue, the spectral norm of the Laplacian matrix concentrates around 1, which may be useful in special cases.

math.PR↗

arXiv2024

Several functional capacities and Carleson type embeddings of fractional Sobolev sapces on stratified Lie groups

In this paper, we focus on the functional and geometrical aspects of the fractional Sobolev capacity, the Besov capacity and the Riesz capacity on stratified lie groups, respectively. Firstly, we provide a new Carleson characterization of the extension of fractional Sobolev spaces to $L^{q}(\X\times\mathbb{R}_{+},μ)$ with $q\in\mathbb{R}_{+}$ using the fractional heat semigroup and the Caffarelli-Silvestre type extension on stratified Lie groups $\X$. Secondly, a characterization of $ν$ on $\X$ which ensures the continuity of the fractional Sobolev space belonging to $L^{q}(\X,ν)$ is also obtained via taking $t\rightarrow 0$. Finally, with the help of inequalities related to the Besov capacity and its properties, we also obtain a characterization of $ν$ on $\X$ which ensures the continuity of the Besov type space belonging to $L^{q}(\X,ν)$.

math.AP↗

arXiv2024

Geometric topics related to Besov type spaces on the Grushin setting

The Grushin spaces, as one of the most important models in the Carnot-Carathéodory space, are a class of locally compact and geodesic metric spaces which admit a dilation. Function spaces on Grushin spaces and some related geometric problems are always the research hotspots in this field. Firstly, we investigate two classes of Besov type spaces based on the Grushin semigroup and the fractional Grushin semigroup, respectively, and prove some important properties of these two Besov type spaces. Moreover, we also reveal the relationship between them. Secondly, we establish the isoperimetric inequality for the fractional perimeter, which is defined by the Grushin-Laplace operator on Grushin spaces. Finally, we combine the semigroup theory with a nonlocal calculus for the Grushin-Laplace operator to obtain the Sobolev type inequality. As a corollary, we also obtain the embedding theorem for Besov type spaces.

math.AP↗

arXiv2023

On Talagrand's functional and generic chaining

In the study of the supremum of stochastic processes, Talagrand's chaining functionals and his generic chaining method are heavily related to the distribution of stochastic processes. In the present paper, we construct Talagrand's type functionals in the general distribution case and obtain the upper bound for the suprema of all $p$-th moments of the stochastic process using the generic chaining method. As applications, we obtained the Johnson-Lindenstrauss lemma, the upper bound for the supremum of all $p$-th moment of order 2 Gaussian chaos, and convex signal recovery in our setting.

math.PR↗

arXiv2023

Transformer-Based Denoising of Mechanical Vibration Signals

Mechanical vibration signal denoising is a pivotal task in various industrial applications, including system health monitoring and failure prediction. This paper introduces a novel deep learning transformer-based architecture specifically tailored for denoising mechanical vibration signals. The model leverages a Multi-Head Attention layer with 8 heads, processing input sequences of length 128, embedded into a 64-dimensional space. The architecture also incorporates Feed-Forward Neural Networks, Layer Normalization, and Residual Connections, resulting in enhanced recognition and extraction of essential features. Through a training process guided by the Mean Squared Error loss function and optimized using the Adam optimizer, the model demonstrates remarkable effectiveness in filtering out noise while preserving critical information related to mechanical vibrations. The specific design and choice of parameters offer a robust method adaptable to the complex nature of mechanical systems, with promising applications in industrial monitoring and maintenance. This work lays the groundwork for future exploration and optimization in the field of mechanical signal analysis and presents a significant step towards advanced and intelligent mechanical system diagnostics.

eess.SY↗

arXiv2022

On Catoni's M-Estimation

Catoni proposed a robust M-estimator and gave the deviation inequality for one fixed test function. The present paper is devoted to the uniform concentration inequality for a family of test functions. As an application, we consider empirical risk minimization for heavy-tailed losses.

math.ST↗

arXiv2022

The sparse representation related with fractional heat equations

This study introduces pre-orthogonal adaptive Fourier decomposition (POAFD) to obtain approximations and numerical solutions to the fractional Laplacian initial value problem and the extension problem of Caffarelli and Silvestre (generalized Poisson equation). The method, as the first step, expands the initial data function into a sparse series of the fundamental solutions with fast convergence, and, as the second step, makes use the semigroup or the reproducing kernel property of each of the expanding entries. Experiments show effectiveness and efficiency of the proposed series solutions.

math.NA↗

arXiv2022

BV spaces and the perimeters related to Schrodinger operators with inverse-square potentials and applications to the rank-one theorem

For $a \ge - {( \frac{d}{2}- 1)^2} $ and $2σ= {d - 2}-( {{{(d - 2)}^2} + 4a})^{1/2}$, let $$\begin{cases}\mathcal{H}_{a}= - Δ+ \frac{a} {{{{ | x |}^2}}},\\ \mathcal{\widetilde{H}}_σ= 2\big( { - Δ+ \frac{σ^2} {{{{ | x |}^2}}}}\big)\end{cases}$$ be two Schrödinger operators with inverse-square potentials. In this paper, on the domain $Ω\subset {\mathbb {R}^d}\backslash \{ 0\}, d\geq 2,$ %apart from the origin, the ${\mathcal{H} _a}$-BV space $\mathcal{B} {\mathcal{V} _{{\mathcal{H} _a}}}(Ω)$ and the ${\mathcal{\widetilde{H}}_σ}$-BV space $\mathcal{B} {\mathcal{V} _{{\mathcal{\widetilde H} _σ}}}(Ω)$ related to $\mathcal{H}_{a}$ and $\mathcal{\widetilde{H}}_σ$ are introduced, respectively. We investigate a series of basic properties of $\mathcal{B} {\mathcal{V} _{{\mathcal{H} _a}}}(Ω)$ and $\mathcal{B} {\mathcal{V} _{{\mathcal{\widetilde H} _σ}}}(Ω)$. Furthermore, we prove that ${\mathcal{\widetilde{H}}_σ}$-restricted BV functions can be characterized equivalently via their subgraphs. As applications, we derive the rank-one theorem for ${\mathcal{\widetilde{H}}_σ}$-restricted BV functions.

math.FA↗

arXiv2022

Strengthened Fractional Sobolev Type Inequalities in Besov Spaces

The purpose of this article is twofold. The first is to strengthen fractional Sobolev type inequalities in Besov spaces via the classical Lorentz space. In doing so, we show that the Sobolev inequality in Besov spaces is equivalent to the fractional Hardy inequality and the iso-capacitary type inequality. Secondly, we will strengthen fractional Sobolev type inequalities in Besov spaces via capacitary Lorentz spaces associated with Besov capacities. For this purpose, we first study the embedding of the associated capacitary Lorentz space to the classical Lorentz space. Then, the embedding of the Besov space to the capacitary Lorentz space is established. Meanwhile, we show that these embeddings are closely related to the iso-capacitary type inequalities in terms of a new-introduced fractional $(β, p, q)$-perimeter. Moreover, characterizations of more general Sobolev type inequalities in Besov spaces have also been established.

math.AP↗

arXiv2022

Fractional Besov Trace/Extension Type Inequalities via the Caffarelli-Silvestre extension

Let $u(\cdot,\cdot)$ be the Caffarelli-Silvestre extension of $f.$ The first goal of this article is to establish the fractional trace type inequalities involving the Caffarelli-Silvestre extension $u(\cdot,\cdot)$ of $f.$ In doing so, firstly, we establish the fractional Sobolev/ logarithmic Sobolev/ Hardy trace inequalities in terms of $\nabla_{(x,t)}u(x,t).$ Then, we prove the fractional anisotropic Sobolev/ logarithmic Sobolev/ Hardy trace inequalities in terms of $ {\partial_{t} u(x,t)}$ or $(-Δ)^{-γ/2}u(x,t)$ only. Moreover, based on an estimate of the Fourier transform of the Caffarelli-Silvestre extension kernel and the sharp affine weighted $L^p$ Sobolev inequality, we prove that the $\dot{H}^{-β/2}(\mathbb{R}^n)$ norm of $f(x)$ can be controlled by the product of the weighted $L^p-$affine energy and the weighted $L^p-$norm of ${\partial_{t} u(x,t)}.$ The second goal of this article is to characterize non-negative measures $μ$ on $\mathbb{R}^{n+1}_+$ such that the embeddings $$\|u(\cdot,\cdot)\|_{L^{q_0,p_0}_μ(\mathbb{R}^{n+1})}\lesssim \|f\|_{\dotΛ^{p,q}_β(\mathbb{R}^n)}$$ hold for some $p_0$ and $q_0$ depending on $p$ and $q$ which are classified in three different cases: (1). $p=q\in (n/(n+β),1];$ (2) $(p,q)\in (1,n/β)\times (1,\infty);$ (3). $(p,q)\in (1,n/β)\times\{\infty\}.$ For case (1), the embeddings can be characterized in terms of an analytic condition of the variational capacity minimizing function, the iso-capacitary inequality of open balls, and other weak type inequalities. For cases (2) and (3), the embeddings are characterized by the iso-capacitary inequality for fractonal Besov capacity of open sets.

math.AP↗

arXiv2021

Embeddings of Function Spaces via the Caffarelli-Silvestre Extension, Capacities and Wolff potentials

Let $P_α f(x,t)$ be the Caffarelli-Silvestre extension of a smooth function $f(x): \mathbb{R}^n \rightarrow \mathbb{R}^{n+1}_+:=\mathbb{R}^n\times (0,\infty).$ The purpose of this article is twofold. Firstly, we want to characterize a nonnegative measure $μ$ on $\mathbb{R}^{n+1}_+$ such that $f(x)\rightarrow P_α f(x,t)$ induces bounded embeddings from the Lebesgue spaces $L^p(\mathbb{R}^n)$ to the $L^q(\mathbb{R}^{n+1}_+,μ).$ On one hand, these embeddings will be characterized by using a newly introduced $L^p-$capacity associated with the Caffarelli-Silvestre extension. In doing so, the mixed norm estimates of $P_α f(x,t),$ the dual form of the $L^p-$capacity, the $L^p-$capacity of general balls, and a capacitary strong type inequality will be established, respectively. On the other hand, when $p>q>1,$ these embeddings will also be characterized in terms of the Hedberg-Wolff potential of $μ.$ Secondly, we characterize a nonnegative measure $μ$ on $\mathbb{R}^{n+1}_+$ such that $f(x)\rightarrow P_α f(x,t)$ induces bounded embeddings from the homogeneous Sobolev spaces $\dot{W}^{β,p}(\mathbb{R}^n)$ to the $L^q(\mathbb{R}^{n+1}_+,μ)$ in terms of the fractional perimeter of open sets for endpoint cases and the fractional capacity for general cases.

math.AP↗

arXiv2021

Boundedness of operators generated by fractional semigroups associated with Schrödinger operators on Campanato type spaces via $T1$ theorem

Let $\mathcal{L}=-Δ+V$ be a Schrödinger operator, where the nonnegative potential $V$ belongs to the reverse Hölder class $B_{q}$. By the aid of the subordinative formula, we estimate the regularities of the fractional heat semigroup, $\{e^{-t\mathcal{L}^α}\}_{t>0},$ associated with $\mathcal{L}$. As an application, we obtain the $BMO^γ_{\mathcal{L}}$-boundedness of the maximal function, and the Littlewood-Paley $g$-functions associated with $\mathcal{L}$ via $T1$ theorem, respectively.

math.CA↗