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Liang Song

Publications and source records attributed to Liang Song.

72 records · Page 4Linked to original sources
arXiv2020

Generalized Adaptation for Few-Shot Learning

Many Few-Shot Learning research works have two stages: pre-training base model and adapting to novel model. In this paper, we propose to use closed-form base learner, which constrains the adapting stage with pre-trained base model to get better generalized novel model. Following theoretical analysis proves its rationality as well as indication of how to train a well-generalized base model. We then conduct experiments on four benchmarks and achieve state-of-the-art performance in all cases. Notably, we achieve the accuracy of 87.75% on 5-shot miniImageNet which approximately outperforms existing methods by 10%.

cs.CV↗
arXiv2020

Improving BLE Beacon Proximity Estimation Accuracy through Bayesian Filtering

The interconnectedness of all things is continuously expanding which has allowed every individual to increase their level of interaction with their surroundings. Internet of Things (IoT) devices are used in a plethora of context-aware application such as Proximity-Based Services (PBS), and Location-Based Services (LBS). For these systems to perform, it is essential to have reliable hardware and predict a user's position in the area with high accuracy in order to differentiate between individuals in a small area. A variety of wireless solutions that utilize Received Signal Strength Indicators (RSSI) have been proposed to provide PBS and LBS for indoor environments, though each solution presents its own drawbacks. In this work, Bluetooth Low Energy (BLE) beacons are examined in terms of their accuracy in proximity estimation. Specifically, a mobile application is developed along with three Bayesian filtering techniques to improve the BLE beacon proximity estimation accuracy. This includes a Kalman filter, a particle filter, and a Non-parametric Information (NI) filter. Since the RSSI is heavily influenced by the environment, experiments were conducted to examine the performance of beacons from three popular vendors in two different environments. The error is compared in terms of Mean Absolute Error (MAE) and Root Mean Squared Error (RMSE). According to the experimental results, Bayesian filters can improve proximity estimation accuracy up to 30 % in comparison with traditional filtering, when the beacon and the receiver are within 3 m.

cs.NI↗
arXiv2019

A q-atomic decomposition of weighted tent spaces on spaces of homogeneous type and its application

The theory of tent spaces on $\mathbb{R}^n$ was introduced by Coifman, Meyer and Stein, including atomic decomposition, duality theory and so on. Russ generalized the atomic decomposition for tent spaces to the case of spaces of homogeneous type $(X,d,μ)$. The main purpose of this paper is to extend the results of Coifman, Meyer, Stein and Russ to weighted version. More precisely, we obtain a $q$-atomic decomposition for the weighted tent spaces $T^p_{2,w}(X)$, where $0<p\leq 1, 1<q<\infty,$ and $w\in A_\infty$. As an application, we give an atomic decomposition for weighted Hardy spaces associated to nonnegative self-adjoint operators on $X$.

math.FA↗
arXiv2019

A New Deep Learning Method for Image Deblurring in Optical Microscopic Systems

Deconvolution is the most commonly used image processing method to remove the blur caused by the point-spread-function (PSF) in optical imaging systems. While this method has been successful in deblurring, it suffers from several disadvantages including being slow, since it takes many iterations, suboptimal, in cases where experimental operator chosen to represent PSF is not optimal. In this paper, we are proposing a deep-learning-based deblurring method applicable to optical microscopic imaging systems. We tested the proposed method in database data, simulated data, and experimental data (include 2D optical microscopic data and 3D photoacoustic microscopic data), all of which showed much improved deblurred results compared to deconvolution. To quantify the improved performance, we compared our results against several deconvolution methods. Our results are better than conventional techniques and do not require multiple iterations or pre-determined experimental operator. Our method has the advantages of simple operation, short time to compute, good deblur results and wide application in all types of optical microscopic imaging systems. The deep learning approach opens up a new path for deblurring and can be applied in various biomedical imaging fields.

eess.IV↗
arXiv2019

BMO spaces associated to operators with generalised Poisson bounds on non-doubling manifolds with ends

Consider a non-doubling manifold with ends $M = \mathfrak{R}^{n}\sharp\, {\mathbb R}^{m}$ where $\mathfrak{R}^n=\mathbb{R}^n\times \mathbb{S}^{m-n}$ for $m> n \ge 3$. We say that an operator $L$ has a generalised Poisson kernel if $\sqrt{ L}$ generates a semigroup $e^{-t\sqrt{L}}$ whose kernel $p_t(x,y)$ has an upper bound similar to the kernel of $e^{-t\sqrtΔ}$ where $Δ$ is the Laplace-Beltrami operator on $M$. An example for operators with generalised Gaussian bounds is the Schrödinger operator $L = Δ+ V$ where $V$ is an arbitrary non-negative locally integrable potential. In this paper, our aim is to introduce the BMO space ${\rm BMO}_L(M)$ associated to operators with generalised Poisson bounds which serves as an appropriate setting for certain singular integrals with rough kernels to be bounded from $L^{\infty}(M)$ into this new ${\rm BMO}_L(M)$. On our ${\rm BMO}_L(M)$ spaces, we show that the John--Nirenberg inequality holds and we show an interpolation theorem for a holomorphic family of operators which interpolates between $L^q(M)$ and ${\rm BMO}_L(M)$. As an application, we show that the holomorphic functional calculus $m(\sqrt{L})$ is bounded from $L^{\infty}(M)$ into ${\rm BMO}_L(M)$, and bounded on $L^p(M)$ for $1 < p < \infty$.

math.AP↗
arXiv2019

Weak type $(1,1)$ bounds for Schrödinger groups

Let $L$ be a non-negative self-adjoint operator acting on $L^2(X)$ where $X$ is a space of homogeneous type with a dimension $n$. Suppose that the heat kernel of $L$ satisfies a Gaussian upper bound. It is known that the operator $(I+L)^{-s } e^{itL}$ is bounded on $L^p(X)$ for $s> n|{1/ 2}-{1/p}| $ and $ p\in (1, \infty)$ (see for example, \cite{CCO, H, Sj}). The index $s= n|{1/ 2}-{1/p}|$ was only obtained recently in \cite{CDLY, CDLY2}, and this range of $s$ is sharp since it is precisely the range known in the case when $L$ is the Laplace operator $Δ$ on $X=\mathbb R^n$ (\cite{Mi1}). In this paper we establish that for $p=1,$ the operator $(1+L)^{-n/2}e^{itL}$ is of weak type $(1, 1)$, that is, there is a constant $C$, independent of $t$ and $f$ so that \begin{eqnarray*} μ\Big(\Big\{x: \big|(I+L)^{-n/2 }e^{itL} f(x)\big|>λ\Big\}\Big) \leq Cλ^{-1}(1+|t|)^{n/2} {\|f\|_{L^1(X)} }, \ \ \ t\in{\mathbb R} \end{eqnarray*} (for $λ> 0$ when $μ(X) = \infty$ and $λ>μ(X)^{-1}\|f\|_{L^1(X)}$ when $μ(X) < \infty$). Moreover, we also show the index $n/2$ is sharp when $L$ is the Laplacian on ${\mathbb R^n}$ by providing an example. Our results are applicable to Schrödinger group for large classes of operators including elliptic operators on compact manifolds, Schrödinger operators with non-negative potentials and Laplace operators acting on Lie groups of polynomial growth or irregular non-doubling domains of Euclidean spaces.

math.AP↗
arXiv2019

Frame decomposition and radial maximal semigroup characterization of Hardy spaces associated to operators

Let $L$ be the generator of an analytic semigroup whose kernels satisfy Gaussian upper bounds and Hölder's continuity. Also assume that $L$ has a bounded holomorphic functional calculus on $L^2(\mathbb{R}^n)$. In this paper, we construct a frame decomposition for the functions belonging to the Hardy space $H_{L}^{1}(\mathbb{R}^n)$ associated to $L$, and for functions in the Lebesgue spaces $L^p$, $1<p<\infty$. We then show that the corresponding $H_{L}^{1}(\mathbb{R}^n)$-norm (resp. $L^p(\mathbb{R}^n)$-norm) of a function $f$ in terms of the frame coefficients is equivalent to the $H_{L}^{1}(\mathbb{R}^n)$-norm (resp. $L^p(\mathbb{R}^n)$-norm) of $f$. As an application of the frame decomposition, we establish the radial maximal semigroup characterization of the Hardy space $H_{L}^{1}(\mathbb{R}^n)$ under the extra condition of Gaussian upper bounds on the gradient of the heat kernels of $L$.

math.AP↗
arXiv2017

On characterization of Poisson integrals of Schrödinger operators with Morrey traces

Let $L$ be a Schrödinger operator of the form $L=-Δ+V$ acting on $L^2(\mathbb R^n)$ where the nonnegative potential $V$ belongs to the reverse Hölder class $B_q$ for some $q\geq n.$ In this article we will show that a function $f\in L^{2, λ}({\mathbb{R}^n}), 0<λ<n$ is the trace of the solution of ${\mathbb L}u=-u_{tt}+L u=0, u(x,0)= f(x),$ where $u$ satisfies a Carleson type condition \begin{eqnarray*} \sup_{x_B, r_B} r_B^{-λ}\int_0^{r_B}\int_{B(x_B, r_B)} t|\nabla u(x,t)|^2 {dx dt} \leq C <\infty. \end{eqnarray*} Its proof heavily relies on investigate the intrinsic relationship between the classical Morrey spaces and the new Campanato spaces $\mathscr{L}_L^{2,λ}({\mathbb{R}^n})$ associated to the operator $L$, i.e. $$\mathscr{L}_L^{2,λ}(\mathbb{R}^n)= {L}^{2,λ}(\mathbb{R}^n). $$ Conversely, this Carleson type condition characterizes all the ${\mathbb L}$-harmonic functions whose traces belong to the space $L^{2, λ}({\mathbb{R}^n})$ for all $ 0<λ<n$. This extends the previous results of [FJN, DYZ, JXY].

math.AP↗
arXiv2017

Carleson measures, BMO spaces and balayages associated to Schrodinger operators

Let $Ł$ be a Schrödinger operator of the form $Ł=-Δ+V$ acting on $L^2(\mathbb R^n)$, $n\geq3$, where the nonnegative potential $V$ belongs to the reverse Hölder class $B_q$ for some $q\geq n.$ Let ${\rm BMO}_{\mathcal{L}}(\RR)$ denote the BMO space associated to the Schrödinger operator $Ł$ on $\RR$. In this article we show that for every $f\in {\rm BMO}_{\mathcal{L}}(\RR)$ with compact support, then there exist $g\in L^{\infty}(\RR)$ and a finite Carleson measure $μ$ such that $$ f(x)=g(x) + S_{μ, {\mathcal P}}(x) $$ with $\|g\|_{\infty} +\||μ\||_{c}\leq C \|f\|_{{\rm BMO}_{\mathcal{L}}(\RR)},$ where $$ S_{μ, {\mathcal P}}=\int_{{\mathbb R}^{n+1}_+} {\mathcal P}_t(x,y) dμ(y, t), $$ and ${\mathcal P}_t(x,y)$ is the kernel of the Poisson semigroup $\{e^{-t\sqrtŁ}\}_{t> 0} $ on $L^2(\mathbb R^n)$. Conversely, if $μ$ is a Carleson measure, then $S_{μ, {\mathcal P}}$ belongs to the space ${\rm BMO}_{\mathcal{L}}(\RR)$. This extends the result for the classical John--Nirenberg BMO space by Carleson \cite{C} (see also \cite{U,GJ,W}) to the BMO setting associated to Schrödinger operators.

math.AP↗
arXiv2016

Boundary Korn Inequality and Neumann Problems in Homogenization of Systems of Elasticity

This paper concerns with a family of elliptic systems of linear elasticity with rapidly oscillating periodic coefficients, arising in the theory of homogenization. We establish uniform optimal regularity estimates for solutions of Neumann problems in a bounded Lipschitz domain with $L^2$ boundary data. The proof relies on a boundary Korn inequality for solutions of systems of linear elasticity and uses a large-scale Rellich estimate obtained in \cite{Shen-2016}.

math.AP↗
arXiv2016

Maximal function characterizations for Hardy spaces associated to nonnegative self-adjoint operators on spaces of homogeneous type

Let $X$ be a metric measure space with a doubling measure and $L$ be a nonnegative self-adjoint operator acting on $L^2(X)$. Assume that $L$ generates an analytic semigroup $e^{-tL}$ whose kernels $p_t(x,y)$ satisfy Gaussian upper bounds but without any assumptions on the regularity of space variables $x$ and $y$. In this article we continue a study in \cite{SY} to give an atomic decomposition for the Hardy spaces $ H^p_{L,max}(X)$ in terms of the nontangential maximal function associated with the heat semigroup of $L$, and hence we establish characterizations of Hardy spaces associated to an operator $L$, via an atomic decomposition or the nontangential maximal function. We also obtain an equivalence of $ H^p_{L, max}(X)$ in terms of the radial maximal function.

math.AP↗
arXiv2015

A maximal function characterization for Hardy spaces associated to nonnegative self-adjoint operators satisfying Gaussian estimates

Let $L$ be a nonnegative, self-adjoint operator satisfying Gaussian estimates on $L^2(\RR^n)$. In this article we give an atomic decomposition for the Hardy spaces $ H^p_{L,max}(\R)$ in terms of the nontangential maximal functions associated with the heat semigroup of $L$, and this leads eventually to characterizations of Hardy spaces associated to $L$, via atomic decomposition or the nontangential maximal functions. The proofs are based on a modification of technique due to A. Calderón \cite{C}.

math.AP↗
arXiv2014

Preduals of quadratic Campanato spaces associated to operators with heat kernel bounds

Let $L$ be a nonnegative, self-adjoint operator on $L^2(\mathbb{R}^n)$ with the Gaussian upper bound on its heat kernel. As a generalization of the square Campanato space $\mathcal{L}^{2,λ}_{-Δ}(\mathbb R^n)$, in \cite{DXY} the quadratic Campanato space $\mathcal{L}_L^{2,λ}(\mathbb{R}^n)$ is defined by a variant of the maximal function associated with the semigroup $\{e^{-tL}\}_{t\geq 0}$. On the basis of \cite{DX} and \cite{YY} this paper addresses the preduality of $\mathcal{L}_L^{2,λ}(\mathbb{R}^n)$ through an induced atom (or molecular) decomposition. Even in the case $L=-Δ$ the discovered predual result is new and natural.

math.AP↗
arXiv2013

A method for the separation and reconstructions of charged hadron and neutral hadron from their overlapped showers in electromagnetic calorimeter

The separation and reconstructions of charged hadron and neutral hadron from their overlapped showers in electromagnetic calorimeter is very important for the reconstructions of some particles with hadronic decays, for example the tau reconstruction in the searches for the Standard Model and supersymmetric Higgs bosons at the LHC. In this paper, a method combining the shower cluster in electromagnetic calorimeter and the parametric formula for hadron showers, was developed to separate the overlapped showers between charged hadron and neutral hadron. Taking the hadronic decay containing one charged pion and one neutral pion in the final status of tau for example, satisfied results of the separation of the overlapped showers, the reconstructions of the energy and positions of the hadrons were obtained. An improved result for the tau reconstruction with this decay model can be also achieved after the application of the proposed method.

physics.ins-det↗
arXiv2013

The bilinear Bochner-Riesz problem

Motivated by the problem of spherical summability of products of Fourier series, we study the boundedness of the bilinear Bochner-Riesz multipliers $(1-|ξ|^2-|η|^2)^δ_+$ and we make some advances in this investigation. We obtain an optimal result concerning the boundedness of these means from $L^2\times L^2 $ into $L^1$ with minimal smoothness, i.e., any $δ>0$, and we obtain estimates for other pairs of spaces for larger values of $δ$. Our study is broad enough to encompass general bilinear multipliers $m(ξ,η)$ radial in $ξ$ and $η$ with minimal smoothness, measured in Sobolev space norms. Our results are based on a variety of techniques, that include Fourier series expansions, orthogonality, and bilinear restriction and extension theorems.

math.CA↗
arXiv2012

On Lp Estimates in Homogenization of Elliptic Equations of Maxwell's Type

For a family of second-order elliptic systems of Maxwell's type with rapidly oscillating periodic coefficients in a $C^{1, α}$ domain $Ω$, we establish uniform estimates of solutions $u_\varep$ and $\nabla \times u_\varep$ in $L^p(Ω)$ for $1<p\le \infty$. The proof relies on the uniform $W^{1,p}$ and Lipschitz estimates for solutions of scalar elliptic equations with periodic coefficients.

math.AP↗