Showing papers by "Hongkai Zhao published in 2019"

Instability of an Inverse Problem for the Stationary Radiative Transport Near the Diffusion Limit

Abstract: In this work, we study the instability of an inverse problem of radiative transport equation with angularly independent source and angularly averaged measurement near the diffusion limit, i.e., the...

Neural-Response-Based Extreme Learning Machine for Image Classification

Abstract: This paper proposes a novel and simple multilayer feature learning method for image classification by employing the extreme learning machine (ELM). The proposed algorithm is composed of two stages: the multilayer ELM (ML-ELM) feature mapping stage and the ELM learning stage. The ML-ELM feature mapping stage is recursively built by alternating between feature map construction and maximum pooling operation. In particular, the input weights for constructing feature maps are randomly generated and hence need not be trained or tuned, which makes the algorithm highly efficient. Moreover, the maximum pooling operation enables the algorithm to be invariant to certain transformations. During the ELM learning stage, elastic-net regularization is proposed to learn the output weight. Elastic-net regularization helps to learn more compact and meaningful output weight. In addition, we preprocess the input data with the dense scale-invariant feature transform operation to improve both the robustness and invariance of the algorithm. To evaluate the effectiveness of the proposed method, several experiments are conducted on three challenging databases. Compared with the conventional deep learning methods and other related ones, the proposed method achieves the best classification results with high computational efficiency.

Marchenko-Pastur law with relaxed independence conditions.

Abstract: We prove the Marchenko-Pastur law for the eigenvalues of $p \times p$ sample covariance matrices in two new situations where the data does not have independent coordinates. In the first scenario - the block-independent model - the $p$ coordinates of the data are partitioned into blocks in such a way that the entries in different blocks are independent, but the entries from the same block may be dependent. In the second scenario - the random tensor model - the data is the homogeneous random tensor of order $d$, i.e. the coordinates of the data are all $\binom{n}{d}$ different products of $d$ variables chosen from a set of $n$ independent random variables. We show that Marchenko-Pastur law holds for the block-independent model as long as the size of the largest block is $o(p)$ and for the random tensor model as long as $d = o(n^{1/3})$. Our main technical tools are new concentration inequalities for quadratic forms in random variables with block-independent coordinates, and for random tensors.

A Hybrid Adaptive Phase Space Method for Reflection Traveltime Tomography

Abstract: We present a hybrid imaging method for a challenging traveltime tomography problem which includes both unknown medium and unknown scatterers in a bounded domain. The goal is to recover both the med...

A data-driven approach for multiscale elliptic PDEs with random coefficients based on intrinsic dimension reduction

Abstract: We propose a data-driven approach to solve multiscale elliptic PDEs with random coefficients based on the intrinsic low dimension structure of the underlying elliptic differential operators. Our method consists of offline and online stages. At the offline stage, a low dimension space and its basis are extracted from the data to achieve significant dimension reduction in the solution space. At the online stage, the extracted basis will be used to solve a new multiscale elliptic PDE efficiently. The existence of low dimension structure is established by showing the high separability of the underlying Green's functions. Different online construction methods are proposed depending on the problem setup. We provide error analysis based on the sampling error and the truncation threshold in building the data-driven basis. Finally, we present numerical examples to demonstrate the accuracy and efficiency of the proposed method.

Separability of the kernel function in an integral formulation for anisotropic radiative transfer equation

Abstract: We study in this work an integral formulation for the radiative transfer equation (RTE) in anisotropic media with truncated approximation to the scattering phase function. The integral formulation consists of a coupled system of integral equations for the angular moments of the transport solution. We analyze the approximate separability of the kernel functions in these integral formulations, deriving asymptotic lower and upper bounds on the number of terms needed in a separable approximation of the kernel functions as the moment grows. Our analysis provides the mathematical understanding on when low-rank approximations to the discretized integral kernels can be used to develop fast numerical algorithms for the corresponding system of integral equations.

Intrinsic Complexity and Scaling Laws: From Random Fields to Random Vectors

Abstract: Random fields are commonly used for modeling of spatially (or timely) dependent stochastic processes. In this study, we provide a characterization of the intrinsic complexity of a random field in t...

Solving Phase Retrieval via Graph Projection Splitting

Abstract: Phase retrieval with prior information can be cast as a nonsmooth and nonconvex optimization problem. We solve the problem by graph projection splitting (GPS), where the two proximity subproblems and the graph projection step can be solved efficiently. With slight modification, we also propose a robust graph projection splitting (RGPS) method to stabilize the iteration for noisy measurements. Contrary to intuition, RGPS outperforms GPS with fewer iterations to locate a satisfying solution even for noiseless case. Based on the connection between GPS and Douglas-Rachford iteration, under mild conditions on the sampling vectors, we analyze the fixed point sets and provide the local convergence of GPS and RGPS applied to noiseless phase retrieval without prior information. For noisy case, we provide the error bound of the reconstruction. Compared to other existing methods, thanks for the splitting approach, GPS and RGPS can efficiently solve phase retrieval with prior information regularization for general sampling vectors which are not necessarily isometric. For Gaussian phase retrieval, compared to existing gradient flow approaches, numerical results show that GPS and RGPS are much less sensitive to the initialization. Thus they markedly improve the phase transition in noiseless case and reconstruction in the presence of noise respectively. GPS shows sharpest phase transition among existing methods including RGPS, while it needs more iterations than RGPS when the number of measurement is large enough. RGPS outperforms GPS in terms of stability for noisy measurements. When applying RGPS to more general non-Gaussian measurements with prior information, such as support, sparsity and TV minimization, RGPS either outperforms state-of-the-art solvers or can be combined with state-of-the-art solvers to improve their reconstruction quality.

Solving PDEs on manifolds represented as point clouds and applications

Abstract: This paper serves as a review of our series work of solving partial differential equations (PDEs) on manifolds represented as point clouds and using their solutions to conduct geometric understanding of point clouds. We first review our two systemic methods of discretizing PDEs including the moving least square method and the local mesh method. These methods of approximating differential operator on manifold-structured point clouds are based only on local approximation using nearest neighbours and achieve high order numerical convergence of the desired equations including diffusion and nondiffusion types of equations. We further discuss extensions of these methods to approximate the committor function for understanding dynamic systems and to solve PDEs on manifolds represented as incomplete inter-point distance information by combining with low-rank matrix completion theory. As direct applications, we use solutions of PDEs on manifold-structured point clouds as bridges to link local and global information. With this strategy, we discuss a few key applications essential to geometric understanding for point clouds, including skeletons extraction from point clouds, the conformal structures construction from point clouds, and nonrigid intrinsic registration among point clouds.