Abstract. Point cloud data represents a crucial category of information for mathematical modeling, and surface reconstruction from such data is an important task across various disciplines. However, during the scanning process, the collected point cloud data may fail to cover the entire surface due to factors such as high light-absorption rate and occlusions, resulting in incomplete datasets. Inferring surface structures in data-missing regions and successfully reconstructing the surface poses a challenge. In this paper, we present a principal component analysis (PCA) based model for surface reconstruction from incomplete point cloud data. Initially, we employ PCA to estimate the normal information of the underlying surface from the available point cloud data. This estimated normal information serves as a regularizer in our model, guiding the reconstruction of the surface, particularly in areas with missing data. Additionally, we introduce an operator-splitting method to effectively solve the proposed model. Through systematic experimentation, we demonstrate that our model successfully infers surface structures in data-missing regions and well reconstructs the underlying surfaces, outperforming existing methodologies.

Deep learning has exhibited remarkable results across diverse areas. To understand its success, substantial research has been directed towards its theoretical foundations. Nevertheless, the majority of these studies examine how well deep neural networks can model functions with uniform regularities. In this paper, we explore a different angle: how deep neural networks can adapt to varying degrees of smoothness in functions and nonuniform data distributions across different locations and scales. More precisely, we focus on a broad class of functions defined by nonlinear tree-based approximation methods. This class encompasses a range of function types, such as functions with uniform regularities and discontinuous functions. We develop nonparametric approximation and estimation theories for this class using deep ReLU networks. Our results show that deep neural networks are adaptive to the nonuniform smoothness of functions and nonuniform data distributions at different locations and scales. We apply our results to several function classes, and derive the corresponding approximation and generalization errors. The validity of our results is demonstrated through numerical experiments.

Learning operators between infinitely dimensional spaces is an important learning task arising in machine learning, imaging science, mathematical modeling and simulations, etc. This paper studies the nonparametric estimation of Lipschitz operators using deep neural networks. Non-asymptotic upper bounds are derived for the generalization error of the empirical risk minimizer over a properly chosen network class. Under the assumption that the target operator exhibits a low dimensional structure, our error bounds decay as the training sample size increases, with an attractive fast rate depending on the intrinsic dimension in our estimation. Our assumptions cover most scenarios in real applications and our results give rise to fast rates by exploiting low dimensional structures of data in operator estimation. We also investigate the influence of network structures (e.g., network width, depth, and sparsity) on the generalization error of the neural network estimator and propose a general suggestion on the choice of network structures to maximize the learning efficiency quantitatively.

Abstract. For problems in image processing and many other fields, a large class of effective neural networks has encoder-decoder-based architectures. Although these networks have shown impressive performance, mathematical explanations of their architectures are still underdeveloped. In this paper, we study the encoder-decoder-based network architecture from the algorithmic perspective and provide a mathematical explanation. We use the two-phase Potts model for image segmentation as an example for our explanations. We associate the segmentation problem with a control problem in the continuous setting. Then, the continuous control model is time discretized by an operator-splitting scheme, the PottsMGNet, and space discretized by the multigrid method. We show that the resulting discrete PottsMGNet is equivalent to an encoder-decoder-based network. With minor modifications, it is shown that a number of the popular encoder-decoder-based neural networks are just instances of the proposed PottsMGNet. By incorporating the soft-threshold-dynamics into the PottsMGNet as a regularizer, the PottsMGNet has shown to be robust with the network parameters such as network width and depth and has achieved remarkable performance on datasets with very large noise. In nearly all our experiments, the new network always performs better than or as well as on accuracy and dice score compared to existing networks for image segmentation.

Abstract. The choice of a proper regularization measure plays an important role in the field of image processing. One classical approach treats color images as two- dimensional surfaces embedded in a five-dimensional spatial-chromatic space. In this case, a natural regularization term arises as the image surface area. Choosing the chromatic coordinates as dominating over the spatial ones, we can think of the image spatial coordinates could as a parameterization of the image surface manifold in a three-dimensional color space. Minimizing the area of the image manifold leads to the Beltrami flow or mean curvature flow of the image surface in the three-dimensional color space, while minimizing the elastica of the image surface yields an additional interesting regularization. Recently, we proposed a color elastica model, which minimizes both the surface area and the elastica of the image manifold. In this paper, we propose to modify the color elastica and introduce two new models for color image regularization. The revised measures are motivated by the relations between the color elastica model, Euler’s elastica model, and the total variation model for gray level images. Compared to our previous color elastica model, the new models are direct extensions of Euler’s elastica model to color images. The proposed models are nonlinear and challenging to minimize. To overcome this difficulty, two operator-splitting methods are suggested. Specifically, nonlinearities are decoupled by the introduction of new vector- and matrix-valued variables. Then, the minimization problems are converted to initial value problems which are time-discretized by operator splitting. Each subproblem, after splitting, either has a closed-form solution or can be solved efficiently. The effectiveness and advantages of the proposed models are demonstrated by comprehensive experiments. The benefits of incorporating the elastica of the image surface as regularization terms compared to common alternatives are empirically validated.

Overparameterized neural networks enjoy great representation power on complex data, and more importantly yield sufficiently smooth output, which is crucial to their generalization and robustness. Most existing function approximation theories suggest that with sufficiently many parameters, neural networks can well approximate certain classes of functions in terms of the function value. The neural network themselves, however, can be highly nonsmooth. To bridge this gap, we take convolutional residual networks (ConvResNets) as an example, and prove that large ConvResNets can not only approximate a target function in terms of function value, but also exhibit sufficient first-order smoothness. Moreover, we extend our theory to approximating functions supported on a low-dimensional manifold. Our theory partially justifies the benefits of using deep and wide networks in practice. Numerical experiments on adversarial robust image classification are provided to support our theory.

Most of existing statistical theories on deep neural networks have sample complexities cursed by the data dimension and therefore cannot well explain the empirical success of deep learning on high-dimensional data. To bridge this gap, we propose to exploit the low-dimensional structures of the real world datasets and establish theoretical guarantees of convolutional residual networks (ConvResNet) in terms of function approximation and statistical recovery for binary classification problem. Specifically, given the data lying on a $d$-dimensional manifold isometrically embedded in $\mathbb{R}^D$, we prove that if the network architecture is properly chosen, ConvResNets can (1) approximate {\it Besov functions} on manifolds with arbitrary accuracy, and (2) learn a classifier by minimizing the empirical logistic risk, which gives an {\it excess risk} in the order of $n^{-\frac{s}{2s+2(s\vee d)}}$, where $s$ is a smoothness parameter. This implies that the sample complexity depends on the intrinsic dimension $d$, instead of the data dimension $D$. Our results demonstrate that ConvResNets are adaptive to low-dimensional structures of data sets.

Models related to the Euler's elastica energy have proven to be useful for many applications including image processing. Extending elastica models to color images and multichannel data is a challenging task, as stable and consistent numerical solvers for these geometric models often involve high order derivatives. Like the single channel Euler's elastica model and the total variation models, geometric measures that involve high order derivatives could help when considering image formation models that minimize elastic properties. In the past, the Polyakov action from high energy physics has been successfully applied to color image processing. Here, we introduce an addition to the Polyakov action for color images that minimizes the color manifold curvature. The color image curvature is computed by applying the Laplace--Beltrami operator to the color image channels. When reduced to gray-scale images, while selecting appropriate scaling between space and color, the proposed model minimizes Euler's elastica operating on the image level sets. Finding a minimizer for the proposed nonlinear geometric model is a challenge we address in this paper. Specifically, we present an operator-splitting method to minimize the proposed functional. The nonlinearity is decoupled by introducing three vector-valued and matrix-valued variables. The problem is then converted into solving for the steady state of an associated initial-value problem. The initial-value problem is time split into three fractional steps, such that each subproblem has a closed form solution, or can be solved by fast algorithms. The efficiency and robustness of the proposed method are demonstrated by systematic numerical experiments.

We propose a variational functional with a curvature constraint to reconstruct implicit surfaces from point cloud data. In the point cloud data, only locations are assumed to be given, without any normal direction or any curvature estimation. The minimizing functional balances two terms: the distance function from the point cloud to the surface and the mean curvature of the surface itself. We explore both the $L_1$ and $L_2$ norms for the curvature constraint. With the added curvature constraint, the computation becomes particularly challenging. We propose two efficient algorithms. The first algorithm is a novel operator splitting method. It replaces the original high-order PDEs by a decoupled PDE system, which is solved by a semi-implicit method. We also discuss an approach based on an augmented Lagrangian method. The proposed model shows robustness against noise and recovers concave features and corners better compared to models without curvature constraint. Numerical experiments on two- and three-dimensional data sets, noisy data and sparse data, are presented to validate the model. Experiments show that the operator splitting semi-implicit method is flexible and robust.

Reinforcement learning (RL) is an efficient learning approach to solving control problems for a robot by interacting with the environment to acquire the optimal control policy. However, there are many challenges for RL to execute continuous control tasks. In this article, without the need to know and learn the dynamic model of a robotic manipulator, a kernel-based dynamic model for RL is proposed. In addition, a new tuple is formed through kernel function sampling to describe a robotic RL control problem. In this algorithm, a reward function is defined according to the features of tracking control in order to speed up the learning process, and then an RL tracking controller with a kernel-based transition dynamic model is proposed. Finally, a critic system is presented to evaluate the policy whether it is good or bad to the RL control tasks. The simulation results illustrate that the proposed method can fulfill the robotic tracking tasks effectively and achieve similar and even better tracking performance with much smaller inputs of force/torque compared with other learning algorithms, demonstrating the effectiveness and efficiency of the proposed RL algorithm.

In recent years, reinforcement learning has been developed dramatically and is widely used to solve control problems, e.g., playing games. However, there are still some problems for reinforcement learning to perform robotic control tasks. Fortunately, the kernel trick-based methods provide a chance to deal with those challenges. This work aims at developing a kernel trick-based learning control method to carry out robotic tracking control tasks. A reward system, in this work, is presented in order to speed up the learning processes. And then, a kernel trick-based reinforcement learning tracking controller is presented to perform tracking control tasks on a robotic manipulator system. To evaluate the policy and assist the reward system to accelerate the speed of finding the optimal control policy, a critic system is introduced. Finally, from the comparison with the benchmark, the simulation results illustrate that our algorithm has faster convergence rate and can execute tracking control tasks effectively, the reward function and the critic system proposed in this work is efficient.