We propose a novel learning framework using neural mean-field (NMF) dynamics for inference and estimation problems on heterogeneous diffusion networks. Our new framework leverages the Mori-Zwanzig formalism to obtain an exact evolution equation of th
e individual node infection probabilities, which renders a delay differential equation with memory integral approximated by learnable time convolution operators. Directly using information diffusion cascade data, our framework can simultaneously learn the structure of the diffusion network and the evolution of node infection probabilities. Connections between parameter learning and optimal control are also established, leading to a rigorous and implementable algorithm for training NMF. Moreover, we show that the projected gradient descent method can be employed to solve the challenging influence maximization problem, where the gradient is computed extremely fast by integrating NMF forward in time just once in each iteration. Extensive empirical studies show that our approach is versatile and robust to variations of the underlying diffusion network models, and significantly outperform existing approaches in accuracy and efficiency on both synthetic and real-world data.
We propose a novel deep neural network architecture by mapping the robust proximal gradient scheme for fast image reconstruction in parallel MRI (pMRI) with regularization function trained from data. The proposed network learns to adaptively combine
the multi-coil images from incomplete pMRI data into a single image with homogeneous contrast, which is then passed to a nonlinear encoder to efficiently extract sparse features of the image. Unlike most of existing deep image reconstruction networks, our network does not require knowledge of sensitivity maps, which can be difficult to estimate accurately, and have been a major bottleneck of image reconstruction in real-world pMRI applications. The experimental results demonstrate the promising performance of our method on a variety of pMRI imaging data sets.
We propose a general learning based framework for solving nonsmooth and nonconvex image reconstruction problems. We model the regularization function as the composition of the $l_{2,1}$ norm and a smooth but nonconvex feature mapping parametrized as
a deep convolutional neural network. We develop a provably convergent descent-type algorithm to solve the nonsmooth nonconvex minimization problem by leveraging the Nesterovs smoothing technique and the idea of residual learning, and learn the network parameters such that the outputs of the algorithm match the references in training data. Our method is versatile as one can employ various modern network structures into the regularization, and the resulting network inherits the guaranteed convergence of the algorithm. We also show that the proposed network is parameter-efficient and its performance compares favorably to the state-of-the-art methods in a variety of image reconstruction problems in practice.
We propose a novel learning framework based on neural mean-field dynamics for inference and estimation problems of diffusion on networks. Our new framework is derived from the Mori-Zwanzig formalism to obtain an exact evolution of the node infection
probabilities, which renders a delay differential equation with memory integral approximated by learnable time convolution operators, resulting in a highly structured and interpretable RNN. Directly using cascade data, our framework can jointly learn the structure of the diffusion network and the evolution of infection probabilities, which are cornerstone to important downstream applications such as influence maximization. Connections between parameter learning and optimal control are also established. Empirical study shows that our approach is versatile and robust to variations of the underlying diffusion network models, and significantly outperform existing approaches in accuracy and efficiency on both synthetic and real-world data.
Optimization algorithms for solving nonconvex inverse problem have attracted significant interests recently. However, existing methods require the nonconvex regularization to be smooth or simple to ensure convergence. In this paper, we propose a nove
l gradient descent type algorithm, by leveraging the idea of residual learning and Nesterovs smoothing technique, to solve inverse problems consisting of general nonconvex and nonsmooth regularization with provable convergence. Moreover, we develop a neural network architecture intimating this algorithm to learn the nonlinear sparsity transformation adaptively from training data, which also inherits the convergence to accommodate the general nonconvex structure of this learned transformation. Numerical results demonstrate that the proposed network outperforms the state-of-the-art methods on a variety of different image reconstruction problems in terms of efficiency and accuracy.
We consider a weak adversarial network approach to numerically solve a class of inverse problems, including electrical impedance tomography and dynamic electrical impedance tomography problems. We leverage the weak formulation of PDE in the given inv
erse problem, and parameterize the solution and the test function as deep neural networks. The weak formulation and the boundary conditions induce a minimax problem of a saddle function of the network parameters. As the parameters are alternatively updated, the network gradually approximates the solution of the inverse problem. We provide theoretical justifications on the convergence of the proposed algorithm. Our method is completely mesh-free without any spatial discretization, and is particularly suitable for problems with high dimensionality and low regularity on solutions. Numerical experiments on a variety of test inverse problems demonstrate the promising accuracy and efficiency of our approach.
Inverse optimal transport (OT) refers to the problem of learning the cost function for OT from observed transport plan or its samples. In this paper, we derive an unconstrained convex optimization formulation of the inverse OT problem, which can be f
urther augmented by any customizable regularization. We provide a comprehensive characterization of the properties of inverse OT, including uniqueness of solutions. We also develop two numerical algorithms, one is a fast matrix scaling method based on the Sinkhorn-Knopp algorithm for discrete OT, and the other one is a learning based algorithm that parameterizes the cost function as a deep neural network for continuous OT. The novel framework proposed in the work avoids repeatedly solving a forward OT in each iteration which has been a thorny computational bottleneck for the bi-level optimization in existing inverse OT approaches. Numerical results demonstrate promising efficiency and accuracy advantages of the proposed algorithms over existing state-of-the-art methods.
Solving general high-dimensional partial differential equations (PDE) is a long-standing challenge in numerical mathematics. In this paper, we propose a novel approach to solve high-dimensional linear and nonlinear PDEs defined on arbitrary domains b
y leveraging their weak formulations. We convert the problem of finding the weak solution of PDEs into an operator norm minimization problem induced from the weak formulation. The weak solution and the test function in the weak formulation are then parameterized as the primal and adversarial networks respectively, which are alternately updated to approximate the optimal network parameter setting. Our approach, termed as the weak adversarial network (WAN), is fast, stable, and completely mesh-free, which is particularly suitable for high-dimensional PDEs defined on irregular domains where the classical numerical methods based on finite differences and finite elements suffer the issues of slow computation, instability and the curse of dimensionality. We apply our method to a variety of test problems with high-dimensional PDEs to demonstrate its promising performance.
We propose a novel problem formulation of continuous-time information propagation on heterogenous networks based on jump stochastic differential equations (SDE). The structure of the network and activation rates between nodes are naturally taken into
account in the SDE system. This new formulation allows for efficient and stable algorithm for many challenging information propagation problems, including estimations of individual activation probability and influence level, by solving the SDE numerically. To this end, we develop an efficient numerical algorithm incorporating variance reduction; furthermore, we provide theoretical bounds for its sample complexity. Moreover, we show that the proposed jump SDE approach can be applied to a much larger class of critical information propagation problems with more complicated settings. Numerical experiments on a variety of synthetic and real-world propagation networks show that the proposed method is more accurate and efficient compared with the state-of-the-art methods.
We propose a new joint image reconstruction method by recovering edge directly from observed data. More specifically, we reformulate joint image reconstruction with vectorial total-variation regularization as an $l_1$ minimization problem of the Jaco
bian of the underlying multi-modality or multi-contrast images. Derivation of data fidelity for Jacobian and transformation of noise distribution are also detailed. The new minimization problem yields an optimal $O(1/k^2)$ convergence rate, where $k$ is the iteration number, and the per-iteration cost is low thanks to the close-form matrix-valued shrinkage. We conducted numerical tests on a number multi-contrast magnetic resonance image (MRI) datasets, which show that the proposed method significantly improves reconstruction efficiency and accuracy compared to the state-of-the-arts.
