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.

Few-shot Non-line-of-sight Imaging with Signal-surface Collaborative Regularization

The non-line-of-sight imaging technique aims to reconstruct targets from multiply reflected light. For most existing methods, dense points on the relay surface are raster scanned to obtain high-quality reconstructions, which requires a long acquisition time. In this work, we propose a signal-surface collaborative regularization (SSCR) framework that provides noise-robust reconstructions with a minimal number of measurements. Using Bayesian inference, we design joint regularizations of the estimated signal, the 3D voxel-based representation of the objects, and the 2D surface-based description of the targets. To our best knowledge, this is the first work that combines regularizations in mixed dimensions for hidden targets. Experiments on synthetic and experimental datasets illustrated the efficiency and robustness of the proposed method under both confocal and non-confocal settings. We report the reconstruction of the hidden targets with complex geometric structures with only $5 \times 5$ confocal measurements from public datasets, indicating an acceleration of the conventional measurement process by a factor of 10000. Besides, the proposed method enjoys low time and memory complexities with sparse measurements. Our approach has great potential in real-time non-line-of-sight imaging applications such as rescue operations and autonomous driving.

Non-line-of-sight imaging with arbitrary illumination and detection pattern

Non-line-of-sight (NLOS) imaging aims at reconstructing targets obscured from the direct line of sight. Existing NLOS imaging algorithms require dense measurements at rectangular grid points in a large area of the relay surface, which severely hinders their availability to variable relay scenarios in practical applications such as robotic vision, autonomous driving, rescue operations and remote sensing. In this work, we propose a Bayesian framework for NLOS imaging with no specific requirements on the spatial pattern of illumination and detection points. By introducing virtual confocal signals, we design a confocal complemented signal-object collaborative regularization (CC-SOCR) algorithm for high quality reconstructions. Our approach is capable of reconstructing both albedo and surface normal of the hidden objects with fine details under the most general relay setting. Moreover, with a regular relay surface, coarse rather than dense measurements are enough for our approach such that the acquisition time can be reduced significantly. As demonstrated in multiple experiments, the new framework substantially enhances the applicability of NLOS imaging.

Point Normal Orientation and Surface Reconstruction by Incorporating Isovalue Constraints to Poisson Equation

Oriented normals are common pre-requisites for many geometric algorithms based on point clouds, such as Poisson surface reconstruction. However, it is not trivial to obtain a consistent orientation. In this work, we bridge orientation and reconstruction in implicit space and propose a novel approach to orient point clouds by incorporating isovalue constraints to the Poisson equation. Feeding a well-oriented point cloud into a reconstruction approach, the indicator function values of the sample points should be close to the isovalue. Based on this observation and the Poisson equation, we propose an optimization formulation that combines isovalue constraints with local consistency requirements for normals. We optimize normals and implicit functions simultaneously and solve for a globally consistent orientation. Owing to the sparsity of the linear system, an average laptop can be used to run our method within reasonable time. Experiments show that our method can achieve high performance in non-uniform and noisy data and manage varying sampling densities, artifacts, multiple connected components, and nested surfaces.

Semi-Supervised Clustering via Dynamic Graph Structure Learning

Most existing semi-supervised graph-based clustering methods exploit the supervisory information by either refining the affinity matrix or directly constraining the low-dimensional representations of data points. The affinity matrix represents the graph structure and is vital to the performance of semi-supervised graph-based clustering. However, existing methods adopt a static affinity matrix to learn the low-dimensional representations of data points and do not optimize the affinity matrix during the learning process. In this paper, we propose a novel dynamic graph structure learning method for semi-supervised clustering. In this method, we simultaneously optimize the affinity matrix and the low-dimensional representations of data points by leveraging the given pairwise constraints. Moreover, we propose an alternating minimization approach with proven convergence to solve the proposed nonconvex model. During the iteration process, our method cyclically updates the low-dimensional representations of data points and refines the affinity matrix, leading to a dynamic affinity matrix (graph structure). Specifically, for the update of the affinity matrix, we enforce the data points with remarkably different low-dimensional representations to have an affinity value of 0. Furthermore, we construct the initial affinity matrix by integrating the local distance and global self-representation among data points. Experimental results on eight benchmark datasets under different settings show the advantages of the proposed approach.