arXiv subjects

Tao Qian

Publications and source records attributed to Tao Qian.

At least 55 records · Page 3Linked to original sources

arXiv2020

A sufficient condition for n-Best Kernel Approximation in Reproducing Kernel Hilbert Spaces

We show that if a reproducing kernel Hilbert space $H_K,$ consisting of functions defined on ${\bf E},$ enjoys Double Boundary Vanishing Condition (DBVC) and Linear Independent Condition (LIC), then for any preset natural number $n,$ and any function $f\in H_K,$ there exists a set of $n$ parameterized multiple kernels ${\tilde{K}}_{w_1},\cdots,{\tilde{K}}_{w_n}, w_k\in {\bf E}, k=1,\cdots,n,$ and real (or complex) constants $c_1,\cdots,c_n,$ giving rise to a solution of the optimization problem \[ \|f-\sum_{k=1}^n c_k{\tilde{K}}_{w_k}\|=\inf \{\|f-\sum_{k=1}^n d_k{\tilde{K}}_{v_k}\|\ |\ v_k\in {\bf E}, d_k\in {\bf R}\ ({\rm or}\ {\bf C}), k=1,\cdots,n\}.\] By applying the theorem of this paper we show that the Hardy space and the Bergman space, as well as all the weighted Bergman spaces in the unit disc all possess $n$-best approximations. In the Hardy space case this gives a new proof of a classical result. Based on the obtained results we further prove existence of $n$-best spherical Poisson kernel approximation to functions of finite energy on the real-spheres.

math.CV↗

arXiv2020

The Fourier Type Expansions on Tubes

In view of recent developments of the study of reproducing kernel Hilbert spaces, in particular with the context the Hardy spaces on tubes, aspects of rational approximation for functions of finite energy in several complex and several real variables are developed.

math.CV↗

arXiv2019

Sparse Solutions for Inverse Problems in Reproducing Kernel Hilbert Spaces

A linear operator in a Hilbert space defined through the form of Riesz representation naturally introduces a reproducing kernel Hilbert space structure over the range space. Such formulation, called H-HK formulation in this paper, possesses a built-in mechanism to solve some basic type problems in the formulation by using the basis method, that include identification of the range space, the inversion problem, and the Moore-Penrose pseudo- (generalized) inversion problem. After a quick survey of the existing theory, the aim of the article is to establish connection between this formulation with sparse series representation, and in particular with one called pre-orthogonal adaptive Fourier decomposition (POAFD), the latter being one, most recent and well developed, with great efficiency and wide and deep connections with traditional analysis. Within the matching pursuit methodology the optimality of POAFD is theoretically guaranteed. In practice POAFD offers fast converging numerical solutions.

math.AP↗

arXiv2019

Fourier Spectrum Characterizations of Clifford $H^{p}$ Spaces on $\mathbf{R}^{n+1}_+$ for $1\leq p \leq \infty$

This article studies the Fourier spectrum characterization of functions in the Clifford algebra-valued Hardy spaces $H^p(\mathbf R^{n+1}_+), 1\leq p\leq \infty.$ Namely, for $f\in L^p(\mathbf R^n)$, Clifford algebra-valued, $f$ is further the non-tangential boundary limit of some function in $H^p(\mathbf R^{n+1}_+),$ $1\leq p\leq \infty,$ if and only if $\hat{f}=χ_+\hat{f},$ where $χ_+(\underlineξ)=\frac{1}{2}(1+i\frac{\underline ξ}{|\underline ξ|}),$ where the Fourier transformation and the above relation are suitably interpreted (for some cases in the distribution sense). These results further develop the relevant context of Alan McIntosh. As a particular case of our results, the vector-valued Clifford Hardy space functions are identical with the conjugate harmonic systems in the work of Stein and Weiss. The latter proved the corresponding results in terms of the single integral form for the cases $1\leq p<\infty.$

math.CV↗

arXiv2019

A Novel Feature Representation for Single-Channel Heartbeat Classification based on Adaptive Fourier Decomposition

This paper proposes a novel approach for heartbeat classification from single-lead electrocardiogram (ECG) signals based on the novel adaptive Fourier decomposition (AFD). AFD is a recently developed signal processing tool that provides useful morphological features, referred to as AFD-derived instantaneous frequency (IF) features, that are different from those provided by traditional tools. A support vector machine (SVM) classifier is trained with the AFD-derived IF features, ECG landmark features, and RR interval features. To evaluate the performance of the trained classifier, the Association for the Advancement of Medical Instrumentation (AAMI) standard is applied to the publicly available benchmark databases, including MIT-BIH arrhythmia database and MIT-BIH supraventricular arrhythmia database, to classify heartbeats from single-lead ECG. The overall performance in terms of sensitivities and positive predictive values is comparable to the state-of-the-art automatic heartbeat classification algorithms based on two-leads ECG.

eess.SP↗

arXiv2018

A Novel Fourier Theory on Non-linear Phases and Applications

Positive time varying frequency representation for transient signals has been a hearty desire of signal analysts due to its theoretical and practical importance. During approximately the last two decades there has formulated a signal decomposition and reconstruction method rooted in harmonic and complex analysis giving rise to the desired signal representation. The method decomposes any signal into a few basic signals that possess positive instantaneous frequencies. The theory has profound relations with classical mathematics and can be generalized to signals defined in higher dimensional manifolds with vector and matrix values, and in particular, promotes rational approximation in higher dimensions. This article mainly serves as a survey. It also gives a new proof for a general convergence result, as well as a proof for the necessity of multiple selection of the parameters. Mono-components are crucial to understand the concept instantaneous frequency. We will present several most important mono-component function classes. Decompositions of signals into mono-components are called adaptive Fourier decompositions (AFDs). We note that some scopes of the studies on the 1D mono-components and AFDs can be extended to vector-valued or even matrix-valued signals defined on higher dimensional manifolds. We finally provide an account of related studies in pure and applied mathematics, and in signal analysis, as well as applications of the theory found in the literature.

math.CV↗

arXiv2018

An Enhancement Algorithm of Cyclic Adaptive Fourier Decomposition

The paper investigates the complex gradient descent method (CGD) for the best rational approximation of a given order to a function in the Hardy space on the unit disk. It is equivalent to finding the best Blaschke form with free poles. The adaptive Fourier decomposition (AFD) and the cyclic AFD methods in literature are based on the grid search technique. The precision of these methods is limited by the grid spacing. The proposed method employs a fast search algorithm to find the initial for CGD, then finds the target poles by gradient descent optimization. Hence, it can reach higher precision with less computation cost. Its validity and effectiveness are confirmed by several examples.

math.CV↗

arXiv2018

Further Aspects of the Fueter Mapping Theorem

In this paper we first show that for $n$ being even the images of the Fueter mapping, as monogenic functions, like for the odd $n$ cases proved through computation on the pointwise differential operator, are also of the axial type (the Axial Form Theorem). Due to a recent result of B. Dong, K. I. Kou, T. Qian, I. Sabadini (2016), we know that the Fueter mapping is surjective on the set of all left- and right-monogenic functions of the axial type in axial domains (the Surjectivity Theorem). The second part results of this paper address the action of the Fueter mapping on the monomials $f^{(l)}_0(z)=z^l, l=0, \pm 1, \pm 2,...,$ in one complex variable. In generalizing the Fueter and the Sce theorems to the even dimensions $n,$ Qian used the following mapping $τ$ (applicable for odd dimensions as well): $$τ(f^{(-k)}_0)=\mathcal{F}^{-1}((-2π|\cdot |)^{n-1}\mathcal{F}({\vec{f_0^{(-k)}}})), \ \ \ τ(f^{(n-2+k)}_0)=I(τ(f^{(-k)}_0)), $$ where $k$ is any positive integer, $\mathcal{F}$ is the Fourier transformation in $\mathbb{R}^{n+1},$ $I$ is the Kelvin inversion. We show that on the monomials the action of the mapping $τ$ defined through the Kelvin inversion coincides with that of the Fueter mapping defined through the Fourier transform (the Monomial Theorem). In the mentioned 1997 paper this result is proved for the case $n$ being odd integers. The Monomial Theorem further implies that the extended mapping $τ$ on Laurent series of real coefficients is identical with the Fueter mapping defined through pointwise differentiation or the Fourier multiplier.

math.CV↗

arXiv2017

Fast algorithm of adaptive Fourier series

Adaptive Fourier decomposition (AFD, precisely 1-D AFD or Core-AFD) was originated for the goal of positive frequency representations of signals. It achieved the goal and at the same time offered fast decompositions of signals. There then arose several types of AFDs. AFD merged with the greedy algorithm idea, and in particular, motivated the so-called pre-orthogonal greedy algorithm (Pre-OGA) that was proven to be the most efficient greedy algorithm. The cost of the advantages of the AFD type decompositions is, however, the high computational complexity due to the involvement of maximal selections of the dictionary parameters. The present paper offers one formulation of the 1-D AFD algorithm by building the FFT algorithm into it. Accordingly, the algorithm complexity is reduced, from the original $\mathcal{O}(M N^2)$ to $\mathcal{O}(M N\log_2 N)$, where $N$ denotes the number of the discretization points on the unit circle and $M$ denotes the number of points in $[0,1)$. This greatly enhances the applicability of AFD. Experiments are carried out to show the high efficiency of the proposed algorithm.

math.NA↗

arXiv2017

Hardy Space Decompositions of $L^p(\mathbb{R}^n)$ for $0<p<1$ with Rational Approximation

This paper aims to obtain decompositions of higher dimensional $L^p(\mathbb{R}^n)$ functions into sums of non-tangential boundary limits of the corresponding Hardy space functions on tubes for the index range $0<p<1$. In the one-dimensional case, Deng and Qian \cite{DQ} recently obtained such Hardy space decomposition result: for any function $f\in L^p(\mathbb{R}),\ 0<p<1$, there exist functions $f_1$ and $f_2$ such that $f=f_1+f_2$, where $f_1$ and $f_2$ are, respectively, the non-tangential boundary limits of some Hardy space functions in the upper-half and lower-half planes. In the present paper, we generalize the one-dimensional Hardy space decomposition result to the higher dimensions, and discuss the uniqueness issue of such decomposition.

math.CV↗

arXiv2017

Hilbert Transformation and $r\mathrm{Spin}(n)+\mathbb{R}^n$ Group

In this paper we study symmetry properties of the Hilbert transformation of several real variables in the Clifford algebra setting. In order to describe the symmetry properties we introduce the group $r\mathrm{Spin}(n)+\mathbb{R}^n, r>0,$ which is essentially an extension of the ax+b group. The study concludes that the Hilbert transformation has certain characteristic symmetry properties in terms of $r\mathrm{Spin}(n)+\mathbb{R}^n.$ In the present paper, for $n=2$ and $3$ we obtain, explicitly, the induced spinor representations of the $r\mathrm{Spin}(n)+\mathbb{R}^n$ group. Then we decompose the natural representation of $r\mathrm{Spin}(n)+\mathbb{R}^n$ into the direct sum of some two irreducible spinor representations, by which we characterize the Hilbert transformation in $\mathbb{R}^3$ and $\mathbb{R}^2.$ Precisely, we show that a nontrivial skew operator is the Hilbert transformation if and only if it is invariant under the action of the $r\mathrm{Spin}(n)+\mathbb{R}^n, n=2,3,$ group.

math.CV↗

arXiv2017

Hilbert Transformation and Representation of ax+b Group

In this paper we study the Hilbert transformations over $L^2(\mathbb{R})$ and $L^2(\mathbb{T})$ from the viewpoint of symmetry. For a linear operator over $L^2(\mathbb{R})$ commutative with the ax+b group we show that the operator is of the form $λI+ηH, $ where $I$ and $H$ are the identity operator and Hilbert transformation respectively, and $λ,η$ are complex numbers. In the related literature this result was proved through first invoking the boundedness result of the operator, proved though a big machinery. In our setting the boundedness is a consequence of the boundedness of the Hilbert transformation. The methodology that we use is Gelfand-Naimark's representation of the ax+b group. Furthermore we prove a similar result on the unit circle. Although there does not exist a group like ax+b on the unit circle, we construct a semigroup to play the same symmetry role for the Hilbert transformations over the circle $L^2(\mathbb{T}).$

math.CV↗

arXiv2017

Aspects of 2D-Adaptive Fourier Decompositions

As a new type of series expansion, the so-called one-dimensional adaptive Fourier decomposition (AFD) and its variations (1D-AFDs) have effective applications in signal analysis and system identification. The 1D-AFDs have considerable influence to the rational approximation of one complex variable and phase retrieving problems, etc. In a recent paper, Qian developed 2D-AFDs for treating square images as the essential boundary of the 2-torus embedded into the space of two complex variables. This paper studies the numerical aspects of multi-dimensional AFDs, and in particular 2D-AFDs, which mainly include (i) Numerical algorithms of several types of 2D-AFDs in relation to image representation; (ii) Perform experiments for the algorithms with comparisons between 5 types of image reconstruction methods in the Fourier category; and (iii) New and sharper estimations for convergence rates of orthogonal greedy algorithm and pre-orthogonal greedy algorithm. The comparison shows that the 2D-AFD methods achieve optimal results among the others.

math.NA↗

arXiv2017

Hardy-Hodge Decomposition of Vector Fields in $Rn$

We prove that a $IR n+1$-valued vector field on IR n is the sum of the traces of two harmonic gradients, one in each component of $IR n+1 \ IR n$ , and of a $IR n$-valued divergence free vector field. We apply this to the description of vanishing potentials in divergence form. The results are stated in terms of Clifford Hardy spaces, the structure of which is important for our study.

math.CV↗

arXiv2017

The duality about function set and Fefferman-Stein Decomposition

Let $D\in\mathbb{N}$, $q\in[2,\infty)$ and $(\mathbb{R}^D,|\cdot|,dx)$ be the Euclidean space equipped with the $D$-dimensional Lebesgue measure. In this article, the authors establish the Fefferman-Stein decomposition of Triebel-Lizorkin spaces $\dot{F}^0_{\infty,\,q'}(\mathbb{R}^D)$ on basis of the dual on function set which has special topological structure. The function in Triebel-Lizorkin spaces $\dot{F}^0_{\infty,\,q'}(\mathbb{R}^D)$ can be written as the certain combination of $D+1$ functions in $\dot{F}^0_{\infty,\,q'}(\mathbb{R}^D) \bigcap L^{\infty}(\mathbb{R}^D)$. To get such decomposition, {\bf (i),} The authors introduce some auxiliary function space $\mathrm{WE}^{1,\,q}(\mathbb R^D)$ and $\mathrm{WE}^{\infty,\,q'}(\mathbb{R}^D)$ defined via wavelet expansions. The authors proved $\dot{F}^{0}_{1,q} \subsetneqq L^{1} \bigcup \dot{F}^{0}_{1,q}\subset {\rm WE}^{1,\,q}\subset L^{1} + \dot{F}^{0}_{1,q}$ and $\mathrm{WE}^{\infty,\,q'}(\mathbb{R}^D)$ is strictly contained in $\dot{F}^0_{\infty,\,q'}(\mathbb{R}^D)$. {\bf (ii),} The authors establish the Riesz transform characterization of Triebel-Lizorkin spaces $\dot{F}^0_{1,\,q}(\mathbb{R}^D)$ by function set $\mathrm{WE}^{1,\,q}(\mathbb R^D)$. {\bf (iii),} We also consider the dual of $\mathrm{WE}^{1,\,q}(\mathbb R^D)$. As a consequence of the above results, the authors get also Riesz transform characterization of Triebel-Lizorkin spaces $\dot{F}^0_{1,\,q}(\mathbb{R}^D)$ by Banach space $L^{1} + \dot{F}^{0}_{1,q}$. Although Fefferman-Stein type decomposition when $D=1$ was obtained by C.-C. Lin et al. [Michigan Math. J. 62 (2013), 691-703], as was pointed out by C.-C. Lin et al., the approach used in the case $D=1$ can not be applied to the cases $D\ge2$, which needs some new methodology.

math.FA↗

arXiv2016

Clifford Coherent State Transforms on Spheres

We introduce a one-parameter family of transforms, $U^t_{(m)}$, $t>0$, from the Hilbert space of Clifford algebra valued square integrable functions on the $m$--dimensional sphere, $L^2(S^{m},dσ_{m})\otimes \mathbb{C}_{m+1}$, to the Hilbert spaces, ${\mathcal M}L^2(\mathbb{R}^{m+1} \setminus \{0\},dμ_t)$, of monogenic functions on $\mathbb{R}^{m+1}\setminus \{0\}$ which are square integrable with respect to appropriate measures, $dμ_t$. We prove that these transforms are unitary isomorphisms of the Hilbert spaces and are extensions of the Segal-Bargman coherent state transform, $U_{(1)} : L^2(S^{1},dσ_{1}) \longrightarrow {\mathcal H}L^2({\mathbb{C} \setminus \{0\}},dμ)$, to higher dimensional spheres in the context of Clifford analysis. In Clifford analysis it is natural to replace the analytic continuation from $S^m$ to $S^m_{\mathbb{C}}$ as in \cite{Ha1, St, HM} by the Cauchy--Kowalewski extension from $S^m$ to $\mathbb{R}^{m+1}\setminus \{0\}$. One then obtains a unitary isomorphism from an $L^2$--Hilbert space to an Hilbert space of solutions of the Dirac equation, that is to a Hilbert space of monogenic functions.

math.FA↗

arXiv2016

Extending coherent state transforms to Clifford analysis

We introduce two extensions of the Segal-Bargmann coherent state transform from $L^2({\mathbb R},dx)$ to Hilbert spaces of slice monogenic and axial monogenic functions and study their properties. These two transforms are related by the dual Radon transform. Representation theoretic and quantum mechanical aspects of the new representations are studied.

math-ph↗

arXiv2016

A Bi-LSTM-RNN Model for Relation Classification Using Low-Cost Sequence Features

Relation classification is associated with many potential applications in the artificial intelligence area. Recent approaches usually leverage neural networks based on structure features such as syntactic or dependency features to solve this problem. However, high-cost structure features make such approaches inconvenient to be directly used. In addition, structure features are probably domain-dependent. Therefore, this paper proposes a bi-directional long-short-term-memory recurrent-neural-network (Bi-LSTM-RNN) model based on low-cost sequence features to address relation classification. This model divides a sentence or text segment into five parts, namely two target entities and their three contexts. It learns the representations of entities and their contexts, and uses them to classify relations. We evaluate our model on two standard benchmark datasets in different domains, namely SemEval-2010 Task 8 and BioNLP-ST 2016 Task BB3. In the former dataset, our model achieves comparable performance compared with other models using sequence features. In the latter dataset, our model obtains the third best results compared with other models in the official evaluation. Moreover, we find that the context between two target entities plays the most important role in relation classification. Furthermore, statistic experiments show that the context between two target entities can be used as an approximate replacement of the shortest dependency path when dependency parsing is not used.

cs.CL↗