arXiv subjects

Han Cheng

Publications and source records attributed to Han Cheng.

15 recordsLinked to original sources

arXiv2026

Endpoint Mapping Properties of Wave Operators for Two-Dimensional Schr\"odinger Operators

We establish sharp endpoint mapping properties for the wave operators $W_\pm(H,-\Delta)$ of two-dimensional Schr\"odinger operators $H=-\Delta+V$ with real-valued decaying potentials $V$. Together with the known non-endpoint $L^p$ theory, our results give a complete classification of the $L^p$ mapping properties of the two-dimensional wave operators, and reveal an unexpected reversal of the usual threshold paradigm at the endpoints $p=1$ and $p=\infty$. When zero is a regular point of $H$, the wave operators fail to be bounded on $L^1(\mathbb{R}^2)$ and on $L^\infty(\mathbb{R}^2)$, but they satisfy the atural substitute estimates of Calder\'on--Zygmund type: $$ L^1(\mathbb{R}^2)\longrightarrow L^{1,\infty}(\mathbb{R}^2),\ \ \ \mathcal{H}^1(\mathbb{R}^2)\longrightarrow L^1(\mathbb{R}^2),\ \ \ L^\infty(\mathbb{R}^2)\longrightarrow \mathrm{BMO}(\mathbb{R}^2). $$ When zero is instead a threshold singularity of the first kind---an s-wave resonance with no other threshold obstruction, the wave operators are bounded on both endpoint spaces $L^1(\mathbb{R}^2)$ and $L^\infty(\mathbb{R}^2)$. Thus, in dimension two, an s-wave resonance improves the endpoint behavior of the wave operators, in sharp contrast with dimensions $n\ge3$, where the only regular case is the favorable one. We also determine the endpoint behavior in the remaining zero-energy spectral configurations of $H$. A p-wave resonance obstructs both the $L^1$- and the $L^\infty$-boundedness of the wave operators, while in the zero-eigenvalue case we obtain necessary and sufficient conditions for endpoint boundedness, expressed in terms of the presence of s- and p-wave resonances and of explicit second-order harmonic moment cancellations satisfied by the zero-energy eigenfunctions.

math.AP↗

arXiv2026

Stabilizing Short Duration Speaker Verification through Neural Re-scoring with Hybrid Enrollment

Short-duration speaker verification (SDSV) is crucial for personalized keyword spotting, where test utterances are typically shorter than three seconds. Limited speech duration results in unstable speaker representations and increased sensitivity to noise and phoneme variations, thereby degrading performance. To investigate this issue, we construct VoxPhrase, a large-scale SDSV corpus automatically segmented from the VoxCeleb dataset. Our analysis shows that text-dependent (TD) enrollment is constrained by duration and yields unstable speaker representations. In contrast, although text-independent (TI) enrollment introduces content mismatch, its representations become more stable as the enrollment duration increases. Accordingly, we propose a hybrid-enrollment neural re-scoring framework that combines TD and TI enrollment and performs frame-level comparison via parallel cross-attention. Experiments on VoxPhrase demonstrate consistent improvements across multiple speaker models.

eess.AS↗

arXiv2026

Decay estimates for the two-dimensional Beam equation with potentials

This paper establishes time decay estimates for the following two-dimensional beam (plate) equation with a decaying real-valued potential $V$: \begin{equation*} \partial_t^2 u + (\Delta^2 + V) u = 0, \qquad u(0,x)=f(x),\quad \partial_t u(0,x)=g(x). \end{equation*} When zero is a regular point or a first-kind resonance of $H=\Delta^2+V$, we first prove sharp $L^1\to L^\infty$ estimates for the solution operators: \begin{align*} \left\|\cos(t\sqrt{H})P_{\mathrm{ac}}(H)\right\|_{L^1\to L^\infty} + \left\|\frac{\sin(t\sqrt{H})}{t\sqrt{H}}P_{\mathrm{ac}}(H)\right\|_{L^1\to L^\infty} \lesssim \frac{1}{|t|}, \end{align*} and obtain an enhanced decay $(|t|\log|t|)^{-1}$ in logarithmically weighted spaces $L^1_\omega\to L^\infty_{-\omega}$ with $\omega(x)=\log(2+|x|)$. For second-kind resonances of $H$ (the bi-Laplacian $\Delta^2$ belongs to this class), a non-zero trace moment $\langle |x|^2V,\phi\rangle\neq0$ for some second-kind resonance function $\phi$ induces severe threshold singularities, worsening the $L^1\to L^\infty$ estimate to $|t|^{-1}(\log|t|)^2$. Finally, for third-kind resonances or a zero eigenvalue, we prove that the presence of $d$-wave resonance leads to the worst $L^1\to L^\infty$ decay rate $\sim(\log|t|)^{-1}$. Several improved estimates are also obtained without a $d$-wave resonance. In particular, in the pure eigenvalue case (i.e., neither $d$-wave nor $p$-wave resonance), both propagators recover the optimal unweighted $L^1\to L^\infty$ estimate $|t|^{-1}.$

math.AP↗

arXiv2026

Effective User-defined Keyword Spotting with Dual-stage Matching, Multi-modal Enrollment, and Continual Adaptation

User-defined keyword spotting (KWS) is crucial for personalized voice interaction, yet existing methods face several challenges: (1) insufficient discriminability among confusable words, (2) performance inconsistency across speakers with varying pronunciations, and (3) high data cost to ensure reliable wake-word performance. In this paper, we introduce DMA-KWS, an efficient and robust framework for user-defined keyword spotting. First, it adopts a dual-stage matching pipeline: CTC decoding with streaming phoneme search to locate candidate segments, followed by QbyT with a phoneme matcher for fine-grained verification, enabling it to better distinguish confusable words. Next, multi-modal enrollment fuses user-specific speech with text embeddings to further improve accuracy for registered users. Finally, a parameter-efficient continual adaptation mechanism performs lightweight updates using synthetic and real data. Extensive experiments demonstrate the superior performance of DMA-KWS. On the LibriPhrase Hard subset, it achieves 97.85% AUC and 6.13% EER, reaching state-of-the-art performance. In speaker-dependent settings, DMA-KWS consistently outperforms text-only enrollment, demonstrating significant performance gains. Moreover, the proposed parameter-efficient fine-tuning mechanism adapts DMA-KWS with only 187k updated parameters, further enhancing KWS performance while ensuring suitability for on-device deployment.

eess.AS↗

arXiv2026

Step 3.5 Flash: Open Frontier-Level Intelligence with 11B Active Parameters

We introduce Step 3.5 Flash, a sparse Mixture-of-Experts (MoE) model that bridges frontier-level agentic intelligence and computational efficiency. We focus on what matters most when building agents: sharp reasoning and fast, reliable execution. Step 3.5 Flash pairs a 196B-parameter foundation with 11B active parameters for efficient inference. It is optimized with interleaved 3:1 sliding-window/full attention and Multi-Token Prediction (MTP-3) to reduce the latency and cost of multi-round agentic interactions. To reach frontier-level intelligence, we design a scalable reinforcement learning framework that combines verifiable signals with preference feedback, while remaining stable under large-scale off-policy training, enabling consistent self-improvement across mathematics, code, and tool use. Step 3.5 Flash demonstrates strong performance across agent, coding, and math tasks, achieving 85.4% on IMO-AnswerBench, 86.4% on LiveCodeBench-v6 (2024.08-2025.05), 88.2% on tau2-Bench, 69.0% on BrowseComp (with context management), and 51.0% on Terminal-Bench 2.0, comparable to frontier models such as GPT-5.2 xHigh and Gemini 3.0 Pro. By redefining the efficiency frontier, Step 3.5 Flash provides a high-density foundation for deploying sophisticated agents in real-world industrial environments.

cs.CL↗

arXiv2025

Dual Data Scaling for Robust Two-Stage User-Defined Keyword Spotting

In this paper, we propose DS-KWS, a two-stage framework for robust user-defined keyword spotting. It combines a CTC-based method with a streaming phoneme search module to locate candidate segments, followed by a QbyT-based method with a phoneme matcher module for verification at both the phoneme and utterance levels. To further improve performance, we introduce a dual data scaling strategy: (1) expanding the ASR corpus from 460 to 1,460 hours to strengthen the acoustic model; and (2) leveraging over 155k anchor classes to train the phoneme matcher, significantly enhancing the distinction of confusable words. Experiments on LibriPhrase show that DS-KWS significantly outperforms existing methods, achieving 6.13\% EER and 97.85\% AUC on the Hard subset. On Hey-Snips, it achieves zero-shot performance comparable to full-shot trained models, reaching 99.13\% recall at one false alarm per hour.

cs.SD↗

arXiv2025

The $L^p$ boundedness of wave operators for the Laplace operator with finite rank perturbations

This paper investigates the $L^p$ boundedness of wave operators for the Laplace operator with finite rank perturbations \begin{equation*} H=-\Delta+\sum\limits_{i=1}^N\langle\cdot\,, \varphi_i\rangle \varphi_i \qquad \mbox{on}\,\,\, \R^d. \end{equation*} For dimensions $d\ge 3$, we prove that the wave operators $W_\pm(H,H_0)$ are bounded on $L^p$ for the full range $1\le p\le \infty$. This extends the work of Nier and the third author \cite{NS} by resolving the previously unexplored question of boundedness at the endpoint cases $p=1$ and $p=\infty$. In lower dimensions $d = 1, 2$, we establish the $L^p$-boundedness of the wave operators for the first time. Furthermore, we reveal an intriguing dichotomy in the endpoint case $p = 1$: \begin{itemize} \item If $\int_{\mathbb{R}^d} \varphi_i(x) \, \d x = 0$ holds for every $1\le i\le N$, then the wave operators are bounded on $L^p(\mathbb{R}^d)$ for all $1 \leq p \leq \infty$. \item If there exists at least one $i$ ($1\le i\le N$) such that $\int_{\mathbb{R}^d}\varphi_i(x)\d x\ne0$, then the wave operators remain bounded for $1 < p < \infty$ and satisfy weak type $(1,1)$ estimates, but fail to be bounded on $L^1(\mathbb{R}^d)$. \end{itemize}

math.AP↗

arXiv2025

The $L^p$-boundedness of wave operators for higher order Schr\"odinger operator with zero singularities in low odd dimensions

This paper investigates the $L^p$-bounds of wave operators for higher-order Schr\"odinger operators $H = (-\Delta)^m + V$ on $\mathbb{R}^n$, with $m \ge 2$ and real-valued decaying potentials $V$. Our main objective is to establish the sharp $L^p$-boundedness of the wave operators $W_\pm(H; (-\Delta)^m)$ in the presence of all types of zero-resonance singularities, for all odd dimensions $1 \le n \le 4m - 1$. Specifically, for odd $n$ with $1 \le n \le 4m - 1$, there exist $m_n$ types of zero resonances for $H$, along with a critical type $k_c$ (both depending on $n$ and $m$). If zero is a regular point of $H$ or a $\mathbf{k}$-th kind resonance with $1 \le \mathbf{k} \le k_c$, the wave operators $W_\pm(H; (-\Delta)^m)$ are bounded on $L^p(\mathbb{R}^n)$ for all $1 < p < \infty$. If zero is a $\mathbf{k}$-th kind resonance with $k_c < \mathbf{k} \le m_n$, we show that the range of $p$-boundedness for $W_\pm(H; (-\Delta)^m)$ narrows to $1 < p < p_{\mathbf{k}}$, where $$p_{\mathbf{k}} = \frac{n}{n - 2m + \mathbf{k} + k_c - 1}.$$ Additionally, if zero is an eigenvalue of $H$ (i.e., $\mathbf{k} = m_n + 1$), then $W_\pm(H; (-\Delta)^m)$ are bounded on $L^p(\mathbb{R}^n)$ for all $1 < p < \frac{2n}{n - 1}$. Furthermore, it is shown that the wave operators $W_\pm(H; (-\Delta)^m)$ are unbounded on $L^p(\mathbb{R}^n)$ for all $p_{\mathbf{k}} < p \le \infty$ if $k_c < \mathbf{k} \le m_n$, and for all $\frac{2n}{n - 1} < p \le \infty$ if zero is an eigenvalue of $H$ with a non-zero solution $\phi$ to $H\phi = 0$ in $\bigcap_{s < -\frac{1}{2}} L^{2}_{s}(\mathbb{R}^n) \setminus L^2(\mathbb{R}^n)$(referred to as a $p$-wave resonance). The key idea of the proof is to reduce the $L^p$-unboundedness to establishing the optimality of time-decay estimates for $e^{itH}P_{ac}(H)$ in weighted $L^2$ spaces.

math.AP↗

arXiv2025

Pointwise estimates for the fundamental solutions of higher order schr\"{o}dinger equations with finite rank perturbations

This paper is dedicated to studying pointwise estimates of the fundamental solution for the higher order Schr\"{o}dinger equation: % we investigate the fundamental solution of the higher order Schr\"{o}dinger equation $$i{\partial}_{t}u(x,t)=Hu(x,t),\ \ \ t\in \mathbb{R},\ x\in {\mathbb{R}}^{n},$$ where the Hamiltonian $H$ is defined as $$H={(-\Delta)}^{m}+\displaystyle\sum_{j=1}^{N} \langle\cdotp ,{\varphi }_{j} \rangle{\varphi }_{j},$$ with each $\varphi_j$ ($1\le j\le N$) satisfying certain smoothness and decay conditions. %Let ${P}_{ac}(H)$ denote the projection onto the absolutely continuous space of $H$. We show that for any positive integer $m>1$ and spatial dimension $n\ge 1$, %under a spectral assumption, the operator is sharp in the sense that it ${e}^{-i tH}P_{ac}(H)$ has an integral kernel $K(t,x,y)$ satisfying the following pointwise estimate: $$\left |K(t,x,y)\right |\lesssim |t|^{-\frac{n}{2m}}(1+|t|^{-\frac{1}{2m}}\left | x-y\right |)^{-\frac{n(m-1)}{2m-1}} ,\ \ t\ne 0,\ x,y\in {\mathbb{R}}^{n}.$$ This estimate is consistent with the upper bounds in the free case. As an application, we derive $L^p-L^q$ decay estimates for the propagator ${e}^{-\i tH}P_{ac}(H)$, where the pairs $(1/p, 1/q)$ lie within a quadrilateral region in the plane.

math.AP↗

arXiv2024

Pointwise estimates for the fundamental solutions of higher order Schr\"{o}dinger equations in odd dimensions II: high dimensional case

In this paper, for any odd $n$ and any integer $m\geq1$ with $n>4m$, we study the fundamental solution of the higher order Schr\"{o}dinger equation \begin{equation*} \mathrm{i}\partial_tu(x,t)=((-\Delta)^m+V(x))u(x,t),\quad t\in \mathbb{R},\,\,x\in \mathbb{R}^n, \end{equation*} where $V$ is a real-valued $C^{\frac{n+1}{2}-2m}$ potential with certain decay. Let $P_{ac}(H)$ denote the projection onto the absolutely continuous spectrum space of $H=(-\Delta)^m+V$, and assume that $H$ has no positive embedded eigenvalue. Our main result says that $e^{-\mathrm{i}tH}P_{ac}(H)$ has integral kernel $K(t,x,y)$ satisfying \begin{equation*} |K(t, x,y)|\le C(1+|t|)^{-(\frac{n}{2m}-\sigma)}(1+|t|^{-\frac{n}{2 m}})\left(1+|t|^{-\frac{1}{2 m}}|x-y|\right)^{-\frac{n(m-1)}{2 m-1}},\quad t\neq0,\,x,y\in\mathbb{R}^n, \end{equation*} where $\sigma=2$ if $0$ is an eigenvalue of $H$, and $\sigma=0$ otherwise. A similar result for smoothing operators $H^\frac{\alpha}{2m}e^{-\mathrm{i}tH}P_{ac}(H)$ is also given. The regularity condition $V\in C^{\frac{n+1}{2}-2m}$ is optimal in the second order case, and it also seems optimal when $m>1$.

math.AP↗

arXiv2024

Towards Robustness Analysis of E-Commerce Ranking System

Information retrieval (IR) is a pivotal component in various applications. Recent advances in machine learning (ML) have enabled the integration of ML algorithms into IR, particularly in ranking systems. While there is a plethora of research on the robustness of ML-based ranking systems, these studies largely neglect commercial e-commerce systems and fail to establish a connection between real-world and manipulated query relevance. In this paper, we present the first systematic measurement study on the robustness of e-commerce ranking systems. We define robustness as the consistency of ranking outcomes for semantically identical queries. To quantitatively analyze robustness, we propose a novel metric that considers both ranking position and item-specific information that are absent in existing metrics. Our large-scale measurement study with real-world data from e-commerce retailers reveals an open opportunity to measure and improve robustness since semantically identical queries often yield inconsistent ranking results. Based on our observations, we propose several solution directions to enhance robustness, such as the use of Large Language Models. Note that the issue of robustness discussed herein does not constitute an error or oversight. Rather, in scenarios where there exists a vast array of choices, it is feasible to present a multitude of products in various permutations, all of which could be equally appealing. However, this extensive selection may lead to customer confusion. As e-commerce retailers use various techniques to improve the quality of search results, we hope that this research offers valuable guidance for measuring the robustness of the ranking systems.

cs.IR↗

arXiv2024

Pointwise estimates for the fundamental solutions of higher order Schr\"{o}dinger equations in low odd dimensions

In this paper, we study the fundamental solution of the higher order Schr\"odinger equation \begin{equation*} \mathrm{i}\partial_t u(x,t) = \big((-\Delta)^m + V(x)\big)u(x,t), \quad t \in \mathbb{R}, \ x \in \mathbb{R}^n, \end{equation*} for any odd dimension $n$ and integer $m \geq 1$ satisfying $n < 4m$, where $V$ is a real-valued bounded potential with suitable decay. Let $P_{ac}(H)$ denote the projection onto the absolutely continuous spectral subspace of $H = (-\Delta)^m + V$, and assume $H$ has no positive embedded eigenvalues. Our main result says that the evolution operator $e^{-\mathrm{i}tH}P_{ac}(H)$ has an integral kernel $K(t,x,y)$ satisfying the pointwise estimate \begin{equation*} |K(t,x,y)| \leq C (1 + |t|)^{-h} (1 + |t|^{-\frac{n}{2m}}) \left(1 + |t|^{-\frac{1}{2m}}|x - y|\right)^{-\frac{n(m-1)}{2m-1}}, \quad t \neq 0, \ x,y \in \mathbb{R}^n, \end{equation*} where the exponent $h$ depends on $m$, $n$, and the zero energy resonance structure of $H$. We also prove analogous estimates for smoothing operators of the form $H^{\frac{\alpha}{2m}}e^{-\mathrm{i}tH}P_{ac}(H)$. The key innovation of this paper is a unified approach to deriving asymptotic expansions of the perturbed resolvents around zero, which comprehensively addresses all possible resonance types.

math.AP↗

arXiv2023

Human Semantic Segmentation using Millimeter-Wave Radar Sparse Point Clouds

This paper presents a framework for semantic segmentation on sparse sequential point clouds of millimeter-wave radar. Compared with cameras and lidars, millimeter-wave radars have the advantage of not revealing privacy, having a strong anti-interference ability, and having long detection distance. The sparsity and capturing temporal-topological features of mmWave data is still a problem. However, the issue of capturing the temporal-topological coupling features under the human semantic segmentation task prevents previous advanced segmentation methods (e.g PointNet, PointCNN, Point Transformer) from being well utilized in practical scenarios. To address the challenge caused by the sparsity and temporal-topological feature of the data, we (i) introduce graph structure and topological features to the point cloud, (ii) propose a semantic segmentation framework including a global feature-extracting module and a sequential feature-extracting module. In addition, we design an efficient and more fitting loss function for a better training process and segmentation results based on graph clustering. Experimentally, we deploy representative semantic segmentation algorithms (Transformer, GCNN, etc.) on a custom dataset. Experimental results indicate that our model achieves mean accuracy on the custom dataset by $\mathbf{82.31}\%$ and outperforms the state-of-the-art algorithms. Moreover, to validate the model's robustness, we deploy our model on the well-known S3DIS dataset. On the S3DIS dataset, our model achieves mean accuracy by $\mathbf{92.6}\%$, outperforming baseline algorithms.

cs.CV↗

arXiv2022

Dispersive estimates for the Schr\"{o}dinger equation with finite rank perturbations

In this paper, we investigate dispersive estimates for the time evolution of Hamiltonians $$ H=-\Delta+\sum_{j=1}^N\langle\cdot\,, \varphi_j\rangle \varphi_j\quad\,\,\,\text{in}\,\,\,\mathbb{R}^d,\,\, d\ge 1, $$ where each $\varphi_j$ satisfies certain smoothness and decay conditions. We show that, under a spectral assumption, there exists a constant $C=C(N, d, \varphi_1,\ldots, \varphi_N)>0$ such that $$ \|e^{-itH}\|_{L^1-L^{\infty}}\leq C t^{-\frac{d}{2}}, \,\,\,\text{for}\,\,\, t>0. $$ As far as we are aware, this seems to provide the first study of $L^1-L^{\infty}$ estimates for finite rank perturbations of the Laplacian in any dimension. We first deal with rank one perturbations ($N=1$). Then we turn to the general case. The new idea in our approach is to establish the Aronszajn-Krein type formula for finite rank perturbations. This allows us to reduce the analysis to the rank one case and solve the problem in a unified manner. Moreover, we show that in some specific situations, the constant $C(N, d, \varphi_1,\ldots, \varphi_N)$ grows polynomially in $N$. Finally, as an application, we are able to extend the results to $N=\infty$ and deal with some trace class perturbations.

math.AP↗

arXiv2021

Mass-Velocity Dispersion Relation in MaNGA Brightest Cluster Galaxies

We investigate a kinematic scaling relation between the baryonic mass and the flat velocity dispersion, i.e. mass-velocity dispersion relation (MVDR), from the brightest cluster galaxies (BCGs) to the galaxy clusters. In our studies, the baryonic mass of BCGs is mainly estimated by photometry. The velocity dispersion profiles are explored with the integrated field unit (IFU) by Mapping Nearby Galaxies at Apache Point Observatory (MaNGA). For the first time, we reveal two significant results with 54 MaNGA BCGs: (1) the flat velocity dispersion profiles; (2) a tight empirical relation on the BCG-cluster scale together with cluster samples, i.e., MVDR, $\log(M_\mathrm{bar}/M_\odot)=4.1^{+0.1}_{-0.1}\log(\sigma_{\mathrm{los}}/\mathrm{km}\,\mathrm{s}^{-1})+1.6^{+0.3}_{-0.3}$, with a tiny lognormal intrinsic scatter of $10^{+2}_{-1}\%$. This slope is identical to the acceleration relation in galaxy clusters, which is reminiscent of the spiral galaxies, albeit at a larger characteristic acceleration scale. The residuals of the MVDR represent a Gaussian distribution, displaying no correlations with four properties: baryonic mass, scale length, surface density, and redshift. Notably, the MVDR on the BCG-cluster scale provides a strict test, which disfavors the general prediction of the slope of three in the dark matter model.

astro-ph.GA↗