arXiv subjects

Chao Ding

Publications and source records attributed to Chao Ding.

At least 55 records · Page 3Linked to original sources

arXiv2022

Robins-Monro Augmented Lagrangian Method for Stochastic Convex Optimization

In this paper, we propose a Robbins-Monro augmented Lagrangian method (RMALM) to solve a class of constrained stochastic convex optimization, which can be regarded as a hybrid of the Robbins-Monro type stochastic approximation method and the augmented Lagrangian method of convex optimizations. Under mild conditions, we show that the proposed algorithm exhibits a linear convergence rate. Moreover, instead of verifying a computationally intractable stopping criteria, we show that the RMALM with the increasing subproblem iteration number has a global complexity $\mathcal{O}(1/\varepsilon^{1+q})$ for the $\varepsilon$-solution (i.e., $\mathbb{E}\left(\|x^k-x^*\|^2\right) < \varepsilon$), where $q$ is any positive number. Numerical results on synthetic and real data demonstrate that the proposed algorithm outperforms the existing algorithms.

math.OC↗

arXiv2022

Local convergence analysis of augmented Lagrangian method for nonlinear semidefinite programming

The augmented Lagrangian method (ALM) has gained tremendous popularity for its elegant theory and impressive numerical performance since it was proposed by Hestenes and Powell in 1969. It has been widely used in numerous efficient solvers to improve numerical performance to solve many problems. In this paper, without requiring the uniqueness of multipliers, the local (asymptotic Q-superlinear) Q-linear convergence rate of the primal-dual sequences generated by ALM for the nonlinear semidefinite programming (NLSDP) is established by assuming the second-order sufficient condition (SOSC) and the semi-isolated calmness of the Karush-Kuhn-Tucker (KKT) solution under some mild conditions.

math.OC↗

arXiv2022

A Semismooth Newton based Augmented Lagrangian Method for Nonsmooth Optimization on Matrix Manifolds

This paper is devoted to studying an augmented Lagrangian method for solving a class of manifold optimization problems, which have nonsmooth objective functions and nonlinear constraints. Under the constant positive linear dependence condition on manifolds, we show that the proposed method converges to a stationary point of the nonsmooth manifold optimization problem. Moreover, we propose a globalized semismooth Newton method to solve the augmented Lagrangian subproblem on manifolds efficiently. The local superlinear convergence of the manifold semismooth Newton method is also established under some suitable conditions. We also prove that the semismoothness on submanifolds can be inherited from that in the ambient manifold. Finally, numerical experiments on compressed modes and (constrained) sparse principal component analysis illustrate the advantages of the proposed method.

math.OC↗

arXiv2021

A Semantic Segmentation Network for Urban-Scale Building Footprint Extraction Using RGB Satellite Imagery

Urban areas consume over two-thirds of the world's energy and account for more than 70 percent of global CO2 emissions. As stated in IPCC's Global Warming of 1.5C report, achieving carbon neutrality by 2050 requires a clear understanding of urban geometry. High-quality building footprint generation from satellite images can accelerate this predictive process and empower municipal decision-making at scale. However, previous Deep Learning-based approaches face consequential issues such as scale invariance and defective footprints, partly due to ever-present class-wise imbalance. Additionally, most approaches require supplemental data such as point cloud data, building height information, and multi-band imagery - which has limited availability and are tedious to produce. In this paper, we propose a modified DeeplabV3+ module with a Dilated Res-Net backbone to generate masks of building footprints from three-channel RGB satellite imagery only. Furthermore, we introduce an F-Beta measure in our objective function to help the model account for skewed class distributions and prevent false-positive footprints. In addition to F-Beta, we incorporate an exponentially weighted boundary loss and use a cross-dataset training strategy to further increase the quality of predictions. As a result, we achieve state-of-the-art performances across three public benchmarks and demonstrate that our RGB-only method produces higher quality visual results and is agnostic to the scale, resolution, and urban density of satellite imagery.

cs.CV↗

arXiv2020

Matrix optimization based Euclidean embedding with outliers

Euclidean embedding from noisy observations containing outlier errors is an important and challenging problem in statistics and machine learning. Many existing methods would struggle with outliers due to a lack of detection ability. In this paper, we propose a matrix optimization based embedding model that can produce reliable embeddings and identify the outliers jointly. We show that the estimators obtained by the proposed method satisfy a non-asymptotic risk bound, implying that the model provides a high accuracy estimator with high probability when the order of the sample size is roughly the degree of freedom up to a logarithmic factor. Moreover, we show that under some mild conditions, the proposed model also can identify the outliers without any prior information with high probability. Finally, numerical experiments demonstrate that the matrix optimization-based model can produce configurations of high quality and successfully identify outliers even for large networks.

stat.ML↗

arXiv2020

3DPVNet: Patch-level 3D Hough Voting Network for 6D Pose Estimation

In this paper, we focus on estimating the 6D pose of objects in point clouds. Although the topic has been widely studied, pose estimation in point clouds remains a challenging problem due to the noise and occlusion. To address the problem, a novel 3DPVNet is presented in this work, which utilizes 3D local patches to vote for the object 6D poses. 3DPVNet is comprised of three modules. In particular, a Patch Unification (\textbf{PU}) module is first introduced to normalize the input patch, and also create a standard local coordinate frame on it to generate a reliable vote. We then devise a Weight-guided Neighboring Feature Fusion (\textbf{WNFF}) module in the network, which fuses the neighboring features to yield a semi-global feature for the center patch. WNFF module mines the neighboring information of a local patch, such that the representation capability to local geometric characteristics is significantly enhanced, making the method robust to a certain level of noise. Moreover, we present a Patch-level Voting (\textbf{PV}) module to regress transformations and generates pose votes. After the aggregation of all votes from patches and a refinement step, the final pose of the object can be obtained. Compared to recent voting-based methods, 3DPVNet is patch-level, and directly carried out on point clouds. Therefore, 3DPVNet achieves less computation than point/pixel-level voting scheme, and has robustness to partial data. Experiments on several datasets demonstrate that 3DPVNet achieves the state-of-the-art performance, and is also robust against noise and occlusions.

cs.CV↗

arXiv2020

Green's Formulas and Poisson's Equation for Bosonic Laplacians

A bosonic Laplacian is a conformally invariant second order differential operator acting on smooth functions defined on domains in Euclidean space and taking values in higher order irreducible representations of the special orthogonal group. In this paper, we firstly introduce the motivation for study of the generalized Maxwell operators and bosonic Laplacians (also known as the higher spin Laplace operators). Then, with the help of connections between Rarita-Schwinger type operators and bosonic Laplacians, we solve Poisson's equation for bosonic Laplacians. A representation formula for bounded solutions to Poisson's equation in Euclidean space is also provided. In the end, we provide Green's formulas for bosonic Laplacians in scalar-valued and Clifford-valued cases, respectively. These formulas reveal that bosonic Laplacians are self-adjoint with respect to a given $L^2$ inner product on certain compact supported function spaces.

math.AP↗

arXiv2020

Boundary value problems in Euclidean space for Bosonic Laplacians

A bosonic Laplacian is a conformally invariant second order differential operator acting on smooth functions defined on domains in Euclidean space and taking values in higher order irreducible representations of the special orthogonal group. In this paper, we study boundary value problems involving bosonic Laplacians in the upper-half space and the unit ball. Poisson kernels in the upper-half space and the unit ball are constructed, which give us solutions to the Dirichlet problems with $L^p$ boundary data, $1 \leq p \leq \infty$. We also prove the uniqueness for solutions to the Dirichlet problems with continuous data for bosonic Laplacians and provide analogs of some properties of harmonic functions for null solutions of bosonic Laplacians, for instance, Cauchy's estimates, the mean-value property, Liouville's Theorem, etc.

math-ph↗

arXiv2020

The linear unicyclic hypergraph with the second or third largest spectral radius

The spectral radius of a uniform hypergraph is defined to be that of the adjacency tensor the hypergraph. It is known that the unique unicyclic hypergraph with the largest spectral radius is a nonlinear hypergraph, and the unique linear unicyclic hypergraph with the largest spectral radius is a power hypergraph. In this paper we determine the the unique linear unicyclic hypergraph with the second or third largest spectral radius, where the former hypergraph is a power hypergraph and the latter hypergraph is a non-power hypergraph.

math.CO↗

arXiv2019

A proximal DC approach for quadratic assignment problem

In this paper, we show that the quadratic assignment problem (QAP) can be reformulated to an equivalent rank constrained doubly nonnegative (DNN) problem. Under the framework of the difference of convex functions (DC) approach, a semi-proximal DC algorithm (DCA) is proposed for solving the relaxation of the rank constrained DNN problem whose subproblems can be solved by the semi-proximal augmented Lagrangian method (sPALM). We show that the generated sequence converges to a stationary point of the corresponding DC problem, which is feasible to the rank constrained DNN problem. Moreover, numerical experiments demonstrate that for most QAP instances, the proposed approach can find the global optimal solutions efficiently, and for others, the proposed algorithm is able to provide good feasible solutions in a reasonable time.

math.OC↗

arXiv2019

Nonsmooth Composite Matrix Optimization: Strong Regularity, Constraint Nondegeneracy and Beyond

The nonsmooth composite matrix optimization problem (CMatOP), in particular, the matrix norm minimization problem, is a generalization of the matrix conic programming problem with wide applications in numerical linear algebra, computational statistics and engineering. This paper is devoted to the characterization of the strong regularity for the CMatOP via the generalized strong second order sufficient condition and constraint nondegeneracy for problems with nonsmooth objective functions. The derived result supplements the existing characterization of the strong regularity for the constrained optimization problems with twice continuously differentiable data.

math.OC↗

arXiv2019

Integral Formulas for Higher Order Conformally Invariant Fermionic Operators

In this paper, we establish higher order Borel-Pompeiu formulas for conformally invariant fermionic operators in higher spin theory, which is the theory of functions on m-dimensional Euclidean space taking values in arbitrary irreducible representations of the Spin group. As applications, we provide higher order Cauchy's integral formulas for those fermionic operators. This paper continues the work of building up basic integral formulas for conformally invariant differential operators in higher spin theory.

math.RT↗

arXiv2018

Spectral operators of matrices: semismoothness and characterizations of the generalized Jacobian

Spectral operators of matrices proposed recently in [C. Ding, D.F. Sun, J. Sun, and K.C. Toh, Math. Program. {\bf 168}, 509--531 (2018)] are a class of matrix valued functions, which map matrices to matrices by applying a vector-to-vector function to all eigenvalues/singular values of the underlying matrices. Spectral operators play a crucial role in the study of various applications involving matrices such as matrix optimization problems (MOPs) {that include semidefinite programming as one of the most important example classes}. In this paper, we will study more fundamental first- and second-order properties of spectral operators, including the Lipschitz continuity, $ρ$-order B(ouligand)-differentiability ($0<ρ\le 1$), $ρ$-order G-semismoothness ($0<ρ\le 1$), and characterization of generalized Jacobians.

math.OC↗

arXiv2017

Quadratic growth conditions for convex matrix optimization problems associated with spectral functions

In this paper, we provide two types of sufficient conditions for ensuring the quadratic growth conditions of a class of constrained convex symmetric and non-symmetric matrix optimization problems regularized by nonsmooth spectral functions. These sufficient conditions are derived via the study of the $\mathcal{C}^2$-cone reducibility of spectral functions and the metric subregularity of their subdifferentials, respectively. As an application, we demonstrate how quadratic growth conditions are used to guarantee the desirable fast convergence rates of the augmented Lagrangian methods (ALM) for solving convex matrix optimization problems. Numerical experiments on an easy-to-implement ALM applied to the fastest mixing Markov chain problem are also presented to illustrate the significance of the obtained results.

math.OC↗

arXiv2016

Some Properties of the Higher Spin Laplace Operator

The higher spin Laplace operator has been constructed recently as the generalization of the Laplacian in higher spin theory. This acts on functions taking values in arbitrary irreducible representations of the Spin group. In this paper, we first provide a decomposition of the higher spin Laplace operator in terms of Rrita-Schwinger operators. With such a decomposition, a connection between the fundamental solutions for the higher spin Laplace operator and the fundamental solutions for the Rarita-Schwinger operators is provided. Further, we show that the two components in this decomposition are conformally invariant differential operators. An alternative proof for the conformally invariance property is also pointed out, which can be connected to Knapp-Stein intertwining operators. Last but not least, we establish a Borel-Pompeiu type formula for the higher spin Laplace operator. As an application, we give a Green type integral formula.

math.CV↗

arXiv2016

On Some Conformally Invariant Operators in Euclidean Space

The aim of this paper is to correct a mistake in earlier work on the conformal invariance of Rarita-Schwinger operators and use the method of correction to develop properties of some conformally invariant operators in the Rarita-Schwinger setting. We also study properties of some other Rarita-Schwinger type operators, for instance, twistor operators and dual twistor operators. This work is also intended as an attempt to motivate the study of Rarita-Schwinger operators via some representation theory. This calls for a review of earlier work by Stein and Weiss.

math.CV↗

arXiv2016

Construction of Arbitrary Order Conformally Invariant Operators in Higher Spin Spaces

This paper completes the construction of arbitrary order conformally invariant differential operators in higher spin spaces. Jan Slovák has classified all conformally invariant differential operators on locally conformally flat manifolds. We complete his results in higher spin theory by giving explicit expressions for arbitrary order conformally invariant differential operators, where by conformally invariant we mean equivariant with respect to the conformal group of $\mathbb{S}^m$ acting in Euclidean space $\mathbb{R}^m$. We name these the fermionic operators when the order is odd and bosonic operators when the order is even. Our approach explicitly uses convolution type operators to construct conformally invariant differential operators. These convolution type operators are examples of Knapp-Stein operators and they can be considered as the inverses of the corresponding differential operators. Intertwining operators of these convolution type operators are provided and intertwining operators of differential operators follow immediately. This reveals that our convolution type operators and differential operators are all conformally invariant. This also gives us a class of conformally invariant convolution type operators in higher spin spaces. Their inverses, when they exist, are conformally invariant pseudo-differential operators. Further we use Stein Weiss gradient operators and representation theory for the Spin group to naturally motivate the Rarita-Schwinger operators.

math.DG↗

arXiv2016

Characterization of the Robust Isolated Calmness for a Class of Conic Programming Problems

This paper is devoted to studying the robust isolated calmness of the Karush-Kuhn-Tucker (KKT) solution mapping for a large class of interesting conic programming problems (including most commonly known ones arising from applications) at a locally optimal solution. Under the Robinson constraint qualification, we show that the KKT solution mapping is robustly isolated calm if and only if both the strict Robinson constraint qualification and the second order sufficient condition hold. This implies, among others, that at a locally optimal solution the second order sufficient condition is needed for the KKT solution mapping to have the Aubin property.

math.OC↗