Neumann Series-based Neural Operator for Solving Inverse Medium Problem

The inverse medium problem, inherently ill-posed and nonlinear, presents significant computational challenges. This study introduces a novel approach by integrating a Neumann series structure within a neural network framework to effectively handle multiparameter inputs. Experiments demonstrate that our methodology not only accelerates computations but also significantly enhances generalization performance, even with varying scattering properties and noisy data. The robustness and adaptability of our framework provide crucial insights and methodologies, extending its applicability to a broad spectrum of scattering problems. These advancements mark a significant step forward in the field, offering a scalable solution to traditionally complex inverse problems.

Fast and Scalable Semi-Supervised Learning for Multi-View Subspace Clustering

In this paper, we introduce a Fast and Scalable Semi-supervised Multi-view Subspace Clustering (FSSMSC) method, a novel solution to the high computational complexity commonly found in existing approaches. FSSMSC features linear computational and space complexity relative to the size of the data. The method generates a consensus anchor graph across all views, representing each data point as a sparse linear combination of chosen landmarks. Unlike traditional methods that manage the anchor graph construction and the label propagation process separately, this paper proposes a unified optimization model that facilitates simultaneous learning of both. An effective alternating update algorithm with convergence guarantees is proposed to solve the unified optimization model. Additionally, the method employs the obtained anchor graph and landmarks' low-dimensional representations to deduce low-dimensional representations for raw data. Following this, a straightforward clustering approach is conducted on these low-dimensional representations to achieve the final clustering results. The effectiveness and efficiency of FSSMSC are validated through extensive experiments on multiple benchmark datasets of varying scales.

Deep Learning with Data Privacy via Residual Perturbation

Protecting data privacy in deep learning (DL) is of crucial importance. Several celebrated privacy notions have been established and used for privacy-preserving DL. However, many existing mechanisms achieve privacy at the cost of significant utility degradation and computational overhead. In this paper, we propose a stochastic differential equation-based residual perturbation for privacy-preserving DL, which injects Gaussian noise into each residual mapping of ResNets. Theoretically, we prove that residual perturbation guarantees differential privacy (DP) and reduces the generalization gap of DL. Empirically, we show that residual perturbation is computationally efficient and outperforms the state-of-the-art differentially private stochastic gradient descent (DPSGD) in utility maintenance without sacrificing membership privacy.

Interface Laplace Learning: Learnable Interface Term Helps Semi-Supervised Learning

We introduce a novel framework, called Interface Laplace learning, for graph-based semi-supervised learning. Motivated by the observation that an interface should exist between different classes where the function value is non-smooth, we introduce a Laplace learning model that incorporates an interface term. This model challenges the long-standing assumption that functions are smooth at all unlabeled points. In the proposed approach, we add an interface term to the Laplace learning model at the interface positions. We provide a practical algorithm to approximate the interface positions using k-hop neighborhood indices, and to learn the interface term from labeled data without artificial design. Our method is efficient and effective, and we present extensive experiments demonstrating that Interface Laplace learning achieves better performance than other recent semi-supervised learning approaches at extremely low label rates on the MNIST, FashionMNIST, and CIFAR-10 datasets.

Convection-Diffusion Equation: A Theoretically Certified Framework for Neural Networks

In this paper, we study the partial differential equation models of neural networks. Neural network can be viewed as a map from a simple base model to a complicate function. Based on solid analysis, we show that this map can be formulated by a convection-diffusion equation. This theoretically certified framework gives mathematical foundation and more understanding of neural networks. Moreover, based on the convection-diffusion equation model, we design a novel network structure, which incorporates diffusion mechanism into network architecture. Extensive experiments on both benchmark datasets and real-world applications validate the performance of the proposed model.

Generalization Error Curves for Analytic Spectral Algorithms under Power-law Decay

The generalization error curve of certain kernel regression method aims at determining the exact order of generalization error with various source condition, noise level and choice of the regularization parameter rather than the minimax rate. In this work, under mild assumptions, we rigorously provide a full characterization of the generalization error curves of the kernel gradient descent method (and a large class of analytic spectral algorithms) in kernel regression. Consequently, we could sharpen the near inconsistency of kernel interpolation and clarify the saturation effects of kernel regression algorithms with higher qualification, etc. Thanks to the neural tangent kernel theory, these results greatly improve our understanding of the generalization behavior of training the wide neural networks. A novel technical contribution, the analytic functional argument, might be of independent interest.

Neural Born Series Operator for Biomedical Ultrasound Computed Tomography

Ultrasound Computed Tomography (USCT) provides a radiation-free option for high-resolution clinical imaging. Despite its potential, the computationally intensive Full Waveform Inversion (FWI) required for tissue property reconstruction limits its clinical utility. This paper introduces the Neural Born Series Operator (NBSO), a novel technique designed to speed up wave simulations, thereby facilitating a more efficient USCT image reconstruction process through an NBSO-based FWI pipeline. Thoroughly validated on comprehensive brain and breast datasets, simulated under experimental USCT conditions, the NBSO proves to be accurate and efficient in both forward simulation and image reconstruction. This advancement demonstrates the potential of neural operators in facilitating near real-time USCT reconstruction, making the clinical application of USCT increasingly viable and promising.

Reconstruction of dynamical systems from data without time labels

In this paper, we study the method to reconstruct dynamical systems from data without time labels. Data without time labels appear in many applications, such as molecular dynamics, single-cell RNA sequencing etc. Reconstruction of dynamical system from time sequence data has been studied extensively. However, these methods do not apply if time labels are unknown. Without time labels, sequence data becomes distribution data. Based on this observation, we propose to treat the data as samples from a probability distribution and try to reconstruct the underlying dynamical system by minimizing the distribution loss, sliced Wasserstein distance more specifically. Extensive experiment results demonstrate the effectiveness of the proposed method.

Addressing preferred orientation in single-particle cryo-EM through AI-generated auxiliary particles

The single-particle cryo-EM field faces the persistent challenge of preferred orientation, lacking general computational solutions. We introduce cryoPROS, an AI-based approach designed to address the above issue. By generating the auxiliary particles with a conditional deep generative model, cryoPROS addresses the intrinsic bias in orientation estimation for the observed particles. We effectively employed cryoPROS in the cryo-EM single particle analysis of the hemagglutinin trimer, showing the ability to restore the near-atomic resolution structure on non-tilt data. Moreover, the enhanced version named cryoPROS-MP significantly improves the resolution of the membrane protein NaX using the no-tilted data that contains the effects of micelles. Compared to the classical approaches, cryoPROS does not need special experimental or image acquisition techniques, providing a purely computational yet effective solution for the preferred orientation problem. Finally, we conduct extensive experiments that establish the low risk of model bias and the high robustness of cryoPROS.

An axiomatized PDE model of deep neural networks

Inspired by the relation between deep neural network (DNN) and partial differential equations (PDEs), we study the general form of the PDE models of deep neural networks. To achieve this goal, we formulate DNN as an evolution operator from a simple base model. Based on several reasonable assumptions, we prove that the evolution operator is actually determined by convection-diffusion equation. This convection-diffusion equation model gives mathematical explanation for several effective networks. Moreover, we show that the convection-diffusion model improves the robustness and reduces the Rademacher complexity. Based on the convection-diffusion equation, we design a new training method for ResNets. Experiments validate the performance of the proposed method.