Analysis and Applications

We study reproducing kernel Hilbert spaces (RKHS) on a Riemannian manifold. In particular, we discuss under which condition Sobolev spaces are RKHS and characterize their reproducing kernels. Further, we introduce and discuss a class of smoother RKHS that we call diffusion spaces. We illustrate the general results with a number of detailed examples. While connections between Sobolev spaces, differential operators and RKHS are well known in the Euclidean setting, here we present a self-contained study of analogous connections for Riemannian manifolds. By collecting a number of results in unified a way, we think our study can be useful for researchers interested in the topic.

In this paper, we prove that, in the deep limit, the stochastic gradient descent on a ResNet type deep neural network, where each layer shares the same weight matrix, converges to the stochastic gradient descent for a Neural ODE and that the corresponding value/loss functions converge. Our result gives, in the context of minimization by stochastic gradient descent, a theoretical foundation for considering Neural ODEs as the deep limit of ResNets. Our proof is based on certain decay estimates for associated Fokker–Planck equations.

Forward-backward stochastic differential equation (FBSDE) systems were introduced as a probabilistic description for parabolic type partial differential equations. Although the probabilistic behavior of the FBSDE system makes it a natural mathematical model in many applications, the stochastic integrals contained in the system generate uncertainties in the solutions which makes the solution estimation a challenging task. In this paper, we assume that we could receive partial noisy observations on the solutions and introduce an optimal filtering method to make a data informed solution estimation for FBSDEs.

We consider a class of convex decentralized consensus optimization problems over connected multi-agent networks. Each agent in the network holds its local objective function privately, and can only communicate with its directly connected agents during the computation to find the minimizer of the sum of all objective functions. We propose a randomized incremental primal-dual method to solve this problem, where the dual variable over the network in each iteration is only updated at a randomly selected node, whereas the dual variables elsewhere remain the same as in the previous iteration. Thus, the communication only occurs in the neighborhood of the selected node in each iteration and hence can greatly reduce the chance of communication delay and failure in the standard fully synchronized consensus algorithms. We provide comprehensive convergence analysis including convergence rates of the primal residual and consensus error of the proposed algorithm, and conduct numerical experiments to show its performance using both uniform sampling and important sampling as node selection strategy.

Recent studies have revealed that deep networks can learn transferable features that generalize well to novel tasks with little or unavailable labeled data for domain adaptation. However, justifying which components of the feature representations can reason about original joint distributions using JMMD within the regime of deep architecture remains unclear. We present a new backpropagation algorithm for JMMD called the Balanced Joint Maximum Mean Discrepancy (B-JMMD) to further reduce the domain discrepancy. B-JMMD achieves the effect of balanced distribution adaptation for deep network architecture, and can be treated as an improved version of JMMD’s backpropagation algorithm. The proposed method leverages the importance of marginal and conditional distributions behind multiple domain-specific layers across domains adaptively to get a good match for the joint distributions in a second-order reproducing kernel Hilbert space. The learning of the proposed method can be performed technically by a special form of stochastic gradient descent, in which the gradient is computed by backpropagation with a strategy of balanced distribution adaptation. Theoretical analysis shows that the proposed B-JMMD is superior to JMMD method. Experiments confirm that our method yields state-of-the-art results with standard datasets.

Sparse additive machines (SAMs) have attracted increasing attention in high dimensional classification due to their representation flexibility and interpretability. However, most of existing methods are formulated under Tikhonov regularization scheme with the hinge loss, which are susceptible to outliers. To circumvent this problem, we propose a sparse additive machine with ramp loss (called ramp-SAM) to tackle classification and variable selection simultaneously. Misclassification error bound is established for ramp-SAM with the help of detailed error decomposition and constructive hypothesis error analysis. To solve the nonsmooth and nonconvex ramp-SAM, a proximal block coordinate descent method is presented with convergence guarantees. The empirical effectiveness of our model is confirmed on simulated and benchmark datasets.

In this paper, we establish a simultaneous decomposition for three quaternion tensors via Einstein product. This simultaneous decomposition transforms the given three quaternion tensors into nice forms which have only 1 and 0. We conclude with an application in the color video signal processing. This new approach only needs to store four keys to realize the simultaneous encryption and decryption of three videos.