arXiv subjects

Tao Qian

Publications and source records attributed to Tao Qian.

At least 19 recordsLinked to original sources

arXiv2026

Kernel Takenaka-Malmquist Systems and Adaptive Approximation in Boundary Weighted Hardy Spaces

Takenaka-Malmquist (TM) systems, as generalizations of the trigonometric function system, play a decisive role in approximation theory on the open unit disc and the upper-half complex plane. In the present paper with the setting of boundary weighted Hardy spaces we develop what we call kernel-TM systems and related approximation theory aspects. In such framework, besides adaptive Fourier decomposition (AFD), we also establish double-zero AFD (DAFD)as an advanced sparse representation in the generalized Hardy spaces. The paper also includes brief reviews of several commonly studied RKHSs as subclasses of the classical Hardy space with particular attention to admissibility of kernel-TM systems.

math.FA↗

arXiv2026

Selecting the optimal Parameters Results in Double Interpolation: Double AFD

Let $f$ belong to the Hardy space $H^2(\mathbb{D})$ of the unit disc, and $e_a$ the normalized Szegö (reproducing) kernel of $H^2(\mathbb{D}).$ It is well known that, due to the reproducing kernel property, for any distinct $n$ points $a_1,\cdots,a_n$ in $\mathbb{D}$ the orthogonal projection of $f$ into ${\rm span}\{e_{a_1},\cdots,e_{a_n}\},$ denoted as $P_{{\rm span}\{e_{a_1},\cdots,e_{a_n}\}}(f),$ interpolates $f$ at the points $a_k$'s. The present study further proves that if the $a_k$'s are optimally selected according to certain energy matching pursuit principle, then $P_{{\rm span}\{e_{a_1},\cdots,e_{a_n}\}}(f)$ double interpolates $f$ at the points $a_k$'s, or order $m=2$ interpolation, that is, \[ P_{{\rm span}\{e_{a_1},\cdots,e_{a_n}\}}(f)(a_k)=f(a_k), \quad {\rm and}\quad P_{{\rm span}\{e_{a_1},\cdots,e_{a_n}\}}'(f)(a_k)=f'(a_k),\quad k=1,\cdots,n.\] With the accordingly newly defined double Takenaka-Malmquist system, the norm convergence for $n\to \infty,$ the $n$-best approximation for $n$ being fixed, and the related boundary function interpolation are studied. The such generated new sparse representation, named as double AFD, is shown to outperform the classical AFD. Pointwise interpolations for orders $m>2,$ meaning to simultaneously interpolates all functions $f,f',\cdots,f^{(m-1)}$ at a set of $a_k$'s are, additionally, discussed. For the Hardy space of the upper-half complex plane there exists a counterpart theory.

math.CV↗

arXiv2026

Low-rank Optimization Trajectories Modeling for LLM RLVR Acceleration

Recently, scaling reinforcement learning with verifiable rewards (RLVR) for large language models (LLMs) has emerged as an effective training paradigm for significantly improving model capabilities, which requires guiding the model to perform extensive exploration and learning, leading to substantial computational overhead and becoming a key challenge. To reduce the number of training steps, Prior work performs linear extrapolation of model parameters. However, the dynamics of model parameter updates during RLVR training remain insufficiently understood. To further investigate the evolution of LLMs during RLVR training, we conduct empirical experiments and find that the rank-1 subspace of the model does not evolve linearly, and its dominance over the original parameters is further amplified during LoRA training. Based on the above insights, we propose the \textbf{N}onlinear \textbf{Ext}rapolation of low-rank trajectories (\textbf{NExt}), a novel framework that models and extrapolates low-rank parameter trajectories in a nonlinear manner. Concretely, we first train the model using LoRA and extract the rank-1 subspace of parameter differences at multiple training steps, which is then used for the subsequent nonlinear extrapolation. Afterward, we utilized the extracted rank-1 subspace to train a predictor, which can model the trajectory of parameter updates during RLVR, and then perform the predict-extend process to extrapolate model parameters, achieving the acceleration of RLVR. To further study and understand NExt, we conduct comprehensive experiments that demonstrate the effectiveness and robustness of the method. Our method reduces computational overhead by approximately 37.5\% while remaining compatible with a wide range of RLVR algorithms and tasks. We release our code in https://github.com/RUCAIBox/NExt.

cs.LG↗

arXiv2026

Hilbertian Hardy--Sobolev Spaces on Tube Domains over Convex Cones

We introduce Hilbertian Hardy--Sobolev spaces on tube domains over convex cones and develop their structural theory from a Fourier-analytic point of view. We first establish a Paley--Wiener type representation, which identifies these spaces with weighted $L^2$ spaces on the dual cone and reveals their intrinsic Fourier structure. This representation leads naturally to a Hardy--Sobolev decomposition theorem for boundary Sobolev spaces on $\mathbb{R}^d$. Building on these structural results, we derive explicit reproducing kernels and characterize Carleson measures for the Hilbertian Hardy--Sobolev spaces. As a preliminary operator-theoretic application, we also derive basic consequences for multipliers and weighted composition operators on these spaces.

math.FA↗

arXiv2026

Banach Hardy-Sobolev Spaces on the Upper Half-plane and Operator Theory

We study Hardy--Sobolev spaces H_n^p(C^+) on the upper half-plane for 1<=p<=infty and n is a nonnegative integer, from both function-theoretic and operator-theoretic viewpoints. We establish an isometric boundary characterization of H_n^p(C^+) via nontangential limits, together with a Sobolev-type embedding theorem, a Cauchy integral representation, a direct-sum decomposition of W_n^p(R) for 1<p<infty, and a generalized Banach algebra structure under pointwise multiplication. We also obtain a finer Fourier-analytic description in the Hilbert case p=2 by proving a Paley--Wiener theorem and deriving the reproducing kernel of H_n^2(C^+).On the operator-theoretic side, we prove the spectral formula for multiplication operators and establish two verifiable sufficient conditions for the boundedness of weighted composition operators. These results provide a systematic theory of Hardy--Sobolev spaces on the upper half-plane beyond the Hilbert setting.

math.FA↗

arXiv2025

Matrix Pre-orthogonal Matching Pursuit and Pseudo-Inverse

We introduce a new fundamental algorithm called Matrix-POAFD to solve the matrix least square problem. The method is based on the matching pursuit principle. The method directly extracts, among the given features as column vectors of the measurement matrix, in the order of their importance, the decisive features for the observing vector. With competitive computational efficiency to the existing sophisticated least square solutions the proposed method, due to its explicit and iterative algorithm process, has the advantage of trading off minimum norms with tolerable error scales. The method inherits recently developed studies in functional space contexts. The second main contribution, also in the algorithm aspect, is to present a two-step iterative computation method for pseudo-inverse. We show that consecutively performing two least square solutions, of which one is to $X$ and the other to $X^*,$ results in the minimum norm least square solution. The two-step algorithm can also be combined into one solving a single least square problem but with respect to $XX^\ast.$ The result is extended to the functional formulation as well. To better explain the idea, as well as for the self-containing purpose, we give short surveys with proofs of key results on closely relevant subjects, including solutions with reproducing kernel Hilbert space setting, AFD type sparse representation in terms of matching pursuit, the general ${\mathcal H}$-$H_K$ formulation and pseudo-inverse of bounded linear operator in Hilbert spaces.

cs.IT↗

arXiv2025

Dirichlet Spaces In Balls And Half-spaces of $\R^n$

The present paper studies the Dirichlet spaces in balls and upper-half Euclidean spaces. As main results, we give identical characterizations of the Dirichlet norms in the respective contexts as for the classical 2-D disc case proved by Douglas and Ahlfors.

math.FA↗

arXiv2024

V2Sim: An Open-Source Microscopic V2G Simulation Platform in Urban Power and Transportation Network

This paper proposes V2Sim, an open source Pythonbased simulation platform designed for advanced vehicle-to-grid (V2G) analysis in coupled urban power and transportation networks. By integrating a microscopic urban transportation network (MUTN) with a power distribution network (PDN), V2Sim enables precise modeling of electric vehicle charging loads (EVCL) and dynamic V2G operations. The platform uniquely combines SUMO for MUTN simulations and an optimized DistFlow model for PDN analysis, with dedicated models for fast charging stations (FCS) and slow charging stations (SCS), capturing detailed charging dynamics often overlooked in existing simulation tools. V2Sim supports a range of customizable V2G strategies, advanced fault-sensing in charging stations, and parallel simulation through multi-processing to accelerate large-scale case studies. Case studies using a real-world MUTN from Nanjing, China, demonstrate V2Sim's capability to analyze the spatial-temporal distribution of EVCL and evaluate V2G impacts, such as fault dissemination and pricing variations, in unprecedented detail. Unlike traditional equilibrium models, V2Sim captures single-vehicle behavior and charging interactions at the microscopic level, offering unparalleled accuracy in assessing the operational and planning needs of V2G-compatible systems. This platform serves as a comprehensive tool for researchers and urban planners aiming to optimize integrated power and transportation networks.

eess.SY↗

arXiv2024

Higher Dimensional Versions of the Douglas-Ahlfors Identities

Denote by ${\mathcal D}$ the open unit disc in the complex plane and $\partial {\mathcal D}$ its boundary. Douglas showed through an identical quantity represented by the Fourier coefficients of the concerned function $u$ that \begin{eqnarray}\label{abs} A(u)=\int_{\mathcal D}|\bigtriangledown U|^2dxdy&=&\frac{1}{2π}\int\int_{\partial {\mathcal D}\times \partial {\mathcal D}} \left|\frac{u(z_1)-u(z_2)}{z_1-z_2}\right|^2|dz_1||dz_2|,\end{eqnarray} \end{abstract} where $u\in L^2(\partial {\mathcal D}), U$ is the harmonic extension of $u$ into ${\mathcal D}$. Ahlfors gave a fourth equivalence form of $A(u)$ in (\ref{more}) via a different proof. The present article studies relations between the counterpart quantities in higher dimensional spheres with several different but commonly adopted settings, namely, harmonic functions in the Euclidean ${\mathbb R}^n, n\ge 2,$ regular functions in the quaternionic algebra, and Clifford monogenic functions with the real-Clifford algebra ${\mathcal{CL}}_{0, n-1},$ the latter being generated by the multiplication anti-commutative basic imaginary units ${\e}_1, {\e}_2, \cdots , {\e}_{n-1}$ with ${\e}_j^2=-1, j=1, 2, \cdots, n-1.$ It is noted that, while exactly the same equivalence relations hold for harmonic functions in ${\mathbb R}^n$ and regular functions in the quaternionic algebra, for the Clifford algebra setting $n>2,$ the relation (\ref{more}) has to be replaced by essentially a different rule.

math.CV↗

arXiv2024

Muskits-ESPnet: A Comprehensive Toolkit for Singing Voice Synthesis in New Paradigm

This research presents Muskits-ESPnet, a versatile toolkit that introduces new paradigms to Singing Voice Synthesis (SVS) through the application of pretrained audio models in both continuous and discrete approaches. Specifically, we explore discrete representations derived from SSL models and audio codecs and offer significant advantages in versatility and intelligence, supporting multi-format inputs and adaptable data processing workflows for various SVS models. The toolkit features automatic music score error detection and correction, as well as a perception auto-evaluation module to imitate human subjective evaluating scores. Muskits-ESPnet is available at \url{https://github.com/espnet/espnet}.

cs.SD↗

arXiv2024

Multi-modal Mood Reader: Pre-trained Model Empowers Cross-Subject Emotion Recognition

Emotion recognition based on Electroencephalography (EEG) has gained significant attention and diversified development in fields such as neural signal processing and affective computing. However, the unique brain anatomy of individuals leads to non-negligible natural differences in EEG signals across subjects, posing challenges for cross-subject emotion recognition. While recent studies have attempted to address these issues, they still face limitations in practical effectiveness and model framework unity. Current methods often struggle to capture the complex spatial-temporal dynamics of EEG signals and fail to effectively integrate multimodal information, resulting in suboptimal performance and limited generalizability across subjects. To overcome these limitations, we develop a Pre-trained model based Multimodal Mood Reader for cross-subject emotion recognition that utilizes masked brain signal modeling and interlinked spatial-temporal attention mechanism. The model learns universal latent representations of EEG signals through pre-training on large scale dataset, and employs Interlinked spatial-temporal attention mechanism to process Differential Entropy(DE) features extracted from EEG data. Subsequently, a multi-level fusion layer is proposed to integrate the discriminative features, maximizing the advantages of features across different dimensions and modalities. Extensive experiments on public datasets demonstrate Mood Reader's superior performance in cross-subject emotion recognition tasks, outperforming state-of-the-art methods. Additionally, the model is dissected from attention perspective, providing qualitative analysis of emotion-related brain areas, offering valuable insights for affective research in neural signal processing.

eess.SP↗

arXiv2024

Can TM system form an unconditional basis for Banach spaces?

The research on the algorithm of analytic signal has received much attention for a long time. Takenaka-Malmquist (TM) system was introduced to consider analytic functions in 1925. If TM system satisfies hyperbolic inseparability condition, then it is an orthogonal basis. It can form unconditional basis for Hilbert space $\mathbb{H}^{2}(D)$ and Schauder basis for Banach space $\mathbb{H}^{p}(D)(1 < p < \infty)$. In characterizing a function space, a necessary condition is whether the basis is unconditional. But since the introduction of TM systems in 1925, to the best of our knowledge, no one has proved the existence of a TM system capable of forming an unconditional basis for Banach space $\mathbb{H}^{p}(D) (p \neq 2)$. TM system has a simple and intuitive analytical structure. Hence it is applied also to the learning algorithms and systematically developed to the reproducing kernel Hilbert spaces (RKHS). Due to the lack of unconditional basis properties, it cannot be extended to the reproducing kernel Banach spaces (RKBS) algorithm. But the case of Banach space plays an important role in machine learning. In this paper, we prove that two TM systems can form unconditional basis for $\mathbb{H}^{p}(D) (1<p <\infty)$.

math.FA↗

arXiv2023

Stable Relay Learning Optimization Approach for Fast Power System Production Cost Minimization Simulation

Production cost minimization (PCM) simulation is commonly employed for assessing the operational efficiency, economic viability, and reliability, providing valuable insights for power system planning and operations. However, solving a PCM problem is time-consuming, consisting of numerous binary variables for simulation horizon extending over months and years. This hinders rapid assessment of modern energy systems with diverse planning requirements. Existing methods for accelerating PCM tend to sacrifice accuracy for speed. In this paper, we propose a stable relay learning optimization (s-RLO) approach within the Branch and Bound (B&B) algorithm. The proposed approach offers rapid and stable performance, and ensures optimal solutions. The two-stage s-RLO involves an imitation learning (IL) phase for accurate policy initialization and a reinforcement learning (RL) phase for time-efficient fine-tuning. When implemented on the popular SCIP solver, s-RLO returns the optimal solution up to 2 times faster than the default relpscost rule and 1.4 times faster than IL, or exhibits a smaller gap at the predefined time limit. The proposed approach shows stable performance, reducing fluctuations by approximately 50% compared with IL. The efficacy of the proposed s-RLO approach is supported by numerical results.

eess.SY↗

arXiv2023

Frequency Analysis with Multiple Kernels and Complete Dictionary

In signal analysis, among the effort of seeking for efficient representations of a signal into the basic ones of meaningful frequencies, to extract principal frequency components, consecutively one after another or $n$ at one time, is a fundamental strategy. For this goal, we define the concept of mean-frequency and develop the related frequency decomposition with the complete Szegö kernel dictionary, the latter consisting of the multiple kernels, being defined as the parameter-derivatives of the Szegö kernels. Several major energy matching pursuit type sparse representations, including greedy algorithm (GA), orthogonal greedy algorithm (OGA), adaptive Fourier decomposition (AFD), pre-orthogonal adaptive Fourier decomposition (POAFD), $n$-Best approximation and unwinding Blaschke expansion, are analyzed and compared. Of which an order in re-construction efficiency between the mentioned algorithms is given based on detailed study of their respective remainders. The study spells out the natural connections between the multiple kernels and the related Laguerre system, and in particular shows that both, like the Fourier series, extract out the $O(n^{-σ})$ order convergence rate from the functions in the Hardy-Sobolev space of order $σ>0.$ Existence of the $n$-Best approximation with the complete Szegö dictionary is proved and the related algorithm aspects are discussed. The included experiments form a significant integration part of the study, for they not only illustrate the theoretical results, but also provide cross comparison between various ways of combination between the matching pursuit algorithms and the dictionaries in use. Experiments show that the complete dictionary remarkably improves approximation efficiency.

cs.IT↗

arXiv2023

A Systematic Exploration of Joint-training for Singing Voice Synthesis

There has been a growing interest in using end-to-end acoustic models for singing voice synthesis (SVS). Typically, these models require an additional vocoder to transform the generated acoustic features into the final waveform. However, since the acoustic model and the vocoder are not jointly optimized, a gap can exist between the two models, leading to suboptimal performance. Although a similar problem has been addressed in the TTS systems by joint-training or by replacing acoustic features with a latent representation, adopting corresponding approaches to SVS is not an easy task. How to improve the joint-training of SVS systems has not been well explored. In this paper, we conduct a systematic investigation of how to better perform a joint-training of an acoustic model and a vocoder for SVS. We carry out extensive experiments and demonstrate that our joint-training strategy outperforms baselines, achieving more stable performance across different datasets while also increasing the interpretability of the entire framework.

cs.SD↗

arXiv2023

PHONEix: Acoustic Feature Processing Strategy for Enhanced Singing Pronunciation with Phoneme Distribution Predictor

Singing voice synthesis (SVS), as a specific task for generating the vocal singing voice from a music score, has drawn much attention in recent years. SVS faces the challenge that the singing has various pronunciation flexibility conditioned on the same music score. Most of the previous works of SVS can not well handle the misalignment between the music score and actual singing. In this paper, we propose an acoustic feature processing strategy, named PHONEix, with a phoneme distribution predictor, to alleviate the gap between the music score and the singing voice, which can be easily adopted in different SVS systems. Extensive experiments in various settings demonstrate the effectiveness of our PHONEix in both objective and subjective evaluations.

cs.SD↗

arXiv2022

Sampling Gaussian Stationary Random Fields: A Stochastic Realization Approach

Generating large-scale samples of stationary random fields is of great importance in the fields such as geomaterial modeling and uncertainty quantification. Traditional methodologies based on covariance matrix decomposition have the diffculty of being computationally expensive, which is even more serious when the dimension of the random field is large. This paper proposes an effcient stochastic realization approach for sampling Gaussian stationary random fields from a systems and control point of view. Specifically, we take the exponential and Gaussian covariance functions as examples and make a decoupling assumption when there are multiple dimensions. Then a rational spectral density is constructed in each dimension using techniques from covariance extension, and the corresponding autoregressive moving-average (ARMA) model is obtained via spectral factorization. As a result, samples of the random field with a specific covariance function can be generated very effciently in the space domain by implementing the ARMA recursion using a white noise input. Such a procedure is computationally cheap due to the fact that the constructed ARMA model has a low order. Furthermore, the same method is integrated to multiscale simulations where interpolations of the generated samples are achieved when one zooms into finer scales. Both theoretical analysis and simulation results show that our approach performs favorably compared with covariance matrix decomposition methods.

stat.ME↗

arXiv2022

AFD Types Sparse Representations vs. the Karhunen-Loeve Expansion for Decomposing Stochastic Processes

This article introduces adaptive Fourier decomposition (AFD) type methods, emphasizing on those that can be applied to stochastic processes and random fields, mainly including stochastic adaptive Fourier decomposition and stochastic pre-orthogonal adaptive Fourier decomposition. We establish their algorithms based on the covariant function and prove that they enjoy the same convergence rate as the Karhunen-Loève (KL) decomposition. The AFD type methods are compared with the KL decomposition. In contrast with the latter, the AFD type methods do not need to compute eigenvalues and eigenfunctions of the kernel-integral operator induced by the covariance function, and thus considerably reduce the computation complexity and computer consumes. Various kinds of dictionaries offer AFD flexibility to solve problems of a great variety, including different types of deterministic and stochastic equations. The conducted experiments show, besides the numerical convenience and fast convergence, that the AFD type decompositions outperform the KL type in describing local details, in spite of the proven global optimality of the latter.

math.ST↗