arXiv subjects
Publications and source records attributed to Xiaojing Ye.
This draft book offers a comprehensive and rigorous treatment of the mathematical principles underlying modern deep learning. The book spans core theoretical topics, from the approximation capabilities of deep neural networks, the theory and algorithms of optimal control and reinforcement learning integrated with deep learning techniques, to contemporary generative models that drive today's advances in artificial intelligence.
We develop a novel transfer learning framework to tackle the challenge of limited training data in image reconstruction problems. The proposed framework consists of two training steps, both of which are formed as bi-level optimizations. In the first step, we train a powerful universal feature-extractor that is capable of learning important knowledge from large, heterogeneous data sets in various domains. In the second step, we train a task-specific domain-adapter for a new target domain or task with only a limited amount of data available for training. Then the composition of the adapter and the universal feature-extractor effectively explores feature which serve as an important component of image regularization for the new domains, and this leads to high-quality reconstruction despite the data limitation issue. We apply this framework to reconstruct under-sampled MR images with limited data by using a collection of diverse data samples from different domains, such as images of other anatomies, measurements of various sampling ratios, and even different image modalities, including natural images. Experimental results demonstrate a promising transfer learning capability of the proposed method.
We develop a general theoretical framework for optimal probability density control on standard measure spaces, aimed at addressing large-scale multi-agent control problems. In particular, we establish a maximum principle (MP) for control problems posed on infinite-dimensional spaces of probability distributions and control vector fields. We further derive the Hamilton--Jacobi--Bellman equation for the associated value functional defined on the space of probability distributions. Both results are presented in a concise form and supported by rigorous mathematical analysis, enabling efficient numerical treatment of these problems. Building on the proposed MP, we introduce a scalable numerical algorithm that leverages deep neural networks to handle high-dimensional settings. The effectiveness of the approach is demonstrated through several multi-agent control examples involving domain obstacles and inter-agent interactions.
In this paper, we propose a new method to compute the solution of time-dependent Schrödinger equation (TDSE). Using push-forward maps and Wasserstein Hamiltonian flow, we reformulate the TDSE as a Hamiltonian system in terms of push-forward maps. The new formulation can be viewed as a generative model in the Wasserstein space, which is a manifold of probability density functions. Then we parameterize the push-forward maps by reduce-order models such as neural networks. This induces a new metric in the parameter space by pulling back the Wasserstein metric on density manifold, which further results in a system of ordinary differential equations (ODEs) for the parameters of the reduce-order model. Leveraging the computational techniques from deep learning, such as Neural ODE, we design an algorithm to solve the TDSE in the parameterized push-forward map space, which provides an alternative approach with the potential to scale up to high-dimensional problems. Several numerical examples are presented to demonstrate the performance of this algorithm.
Inverse problems arise in many applications, especially tomographic imaging. We develop a Learned Alternating Minimization Algorithm (LAMA) to solve such problems via two-block optimization by synergizing data-driven and classical techniques with proven convergence. LAMA is naturally induced by a variational model with learnable regularizers in both data and image domains, parameterized as composite functions of neural networks trained with domain-specific data. We allow these regularizers to be nonconvex and nonsmooth to extract features from data effectively. We minimize the overall objective function using Nesterov's smoothing technique and residual learning architecture. It is demonstrated that LAMA reduces network complexity, improves memory efficiency, and enhances reconstruction accuracy, stability, and interpretability. Extensive experiments show that LAMA significantly outperforms state-of-the-art methods on popular benchmark datasets for Computed Tomography.
We propose a versatile model with a flexible choice of control for an early-pandemic outbreak prevention when vaccine/drug is not yet available. At that stage, control is often limited to non-medical interventions like social distancing and other behavioral changes. For the SIR optimal control problem, we show that the running cost of control satisfying mild, practically justified conditions generates an optimal strategy, $u(t)$, $t \in [0, T]$, that is sustainable up until some moment $τ\in [0 ,T)$. However, for any $t \in [τ, T]$, the function $u(t)$ will decline as $t$ approaches $T$, which may cause the number of newly infected people to increase. So, the window from $0$ to $τ$ is the time for public health officials to prepare alternative mitigation measures, such as vaccines, testing, antiviral medications, and others. In addition to theoretical study, we develop a fast and stable computational method for solving the proposed optimal control problem. The efficiency of the new method is illustrated with numerical examples of optimal control trajectories for various cost functions and weights. Simulation results provide a comprehensive demonstration of the effects of control on the epidemic spread and mitigation expenses, which can serve as invaluable references for public health officials.
We develop a fast and scalable numerical approach to solve Wasserstein gradient flows (WGFs), particularly suitable for high-dimensional cases. Our approach is to use general reduced-order models, like deep neural networks, to parameterize the push-forward maps such that they can push a simple reference density to the one solving the given WGF. The new dynamical system is called parameterized WGF (PWGF), and it is defined on the finite-dimensional parameter space equipped with a pullback Wasserstein metric. Our numerical scheme can approximate the solutions of WGFs for general energy functionals effectively, without requiring spatial discretization or nonconvex optimization procedures, thus avoiding some limitations of classical numerical methods and more recent deep-learning-based approaches. A comprehensive analysis of the approximation errors measured by Wasserstein distance is also provided in this work. Numerical experiments show promising computational efficiency and verified accuracy on various WGF examples using our approach.
We propose a finite-dimensional control-based method to approximate solution operators for evolutional partial differential equations (PDEs), particularly in high-dimensions. By employing a general reduced-order model, such as a deep neural network, we connect the evolution of the model parameters with trajectories in a corresponding function space. Using the computational technique of neural ordinary differential equation, we learn the control over the parameter space such that from any initial starting point, the controlled trajectories closely approximate the solutions to the PDE. Approximation accuracy is justified for a general class of second-order nonlinear PDEs. Numerical results are presented for several high-dimensional PDEs, including real-world applications to solving Hamilton-Jacobi-Bellman equations. These are demonstrated to show the accuracy and efficiency of the proposed method.
We develop a novel computational framework to approximate solution operators of evolution partial differential equations (PDEs). By employing a general nonlinear reduced-order model, such as a deep neural network, to approximate the solution of a given PDE, we realize that the evolution of the model parameter is a control problem in the parameter space. Based on this observation, we propose to approximate the solution operator of the PDE by learning the control vector field in the parameter space. From any initial value, this control field can steer the parameter to generate a trajectory such that the corresponding reduced-order model solves the PDE. This allows for substantially reduced computational cost to solve the evolution PDE with arbitrary initial conditions. We also develop comprehensive error analysis for the proposed method when solving a large class of semilinear parabolic PDEs. Numerical experiments on different high-dimensional evolution PDEs with various initial conditions demonstrate the promising results of the proposed method.
In this paper, we compute numerical approximations of the minimal surfaces, an essential type of Partial Differential Equation (PDE), in higher dimensions. Classical methods cannot handle it in this case because of the Curse of Dimensionality, where the computational cost of these methods increases exponentially fast in response to higher problem dimensions, far beyond the computing capacity of any modern supercomputers. Only in the past few years have machine learning researchers been able to mitigate this problem. The solution method chosen here is a model known as a Physics-Informed Neural Network (PINN) which trains a deep neural network (DNN) to solve the minimal surface PDE. It can be scaled up into higher dimensions and trained relatively quickly even on a laptop with no GPU. Due to the inability to view the high-dimension output, our data is presented as snippets of a higher-dimension shape with enough fixed axes so that it is viewable with 3-D graphs. Not only will the functionality of this method be tested, but we will also explore potential limitations in the method's performance.
We present a novel computational framework for density control in high-dimensional state spaces. The considered dynamical system consists of a large number of indistinguishable agents whose behaviors can be collectively modeled as a time-evolving probability distribution. The goal is to steer the agents from an initial distribution to reach (or approximate) a given target distribution within a fixed time horizon at minimum cost. To tackle this problem, we propose to model the drift as a nonlinear reduced-order model, such as a deep network, and enforce the matching to the target distribution at terminal time either strictly or approximately using the Wasserstein metric. The resulting saddle-point problem can be solved by an effective numerical algorithm that leverages the excellent representation power of deep networks and fast automatic differentiation for this challenging high-dimensional control problem. A variety of numerical experiments were conducted to demonstrate the performance of our method.
In this work, we propose a numerical method to compute the Wasserstein Hamiltonian flow (WHF), which is a Hamiltonian system on the probability density manifold. Many well-known PDE systems can be reformulated as WHFs. We use parameterized function as push-forward map to characterize the solution of WHF, and convert the PDE to a finite-dimensional ODE system, which is a Hamiltonian system in the phase space of the parameter manifold. We establish error analysis results for the continuous time approximation scheme in Wasserstein metric. For the numerical implementation, we use neural networks as push-forward maps. We apply an effective symplectic scheme to solve the derived Hamiltonian ODE system so that the method preserves some important quantities such as total energy. The computation is done by fully deterministic symplectic integrator without any neural network training. Thus, our method does not involve direct optimization over network parameters and hence can avoid the error introduced by stochastic gradient descent (SGD) methods, which is usually hard to quantify and measure. The proposed algorithm is a sampling-based approach that scales well to higher dimensional problems. In addition, the method also provides an alternative connection between the Lagrangian and Eulerian perspectives of the original WHF through the parameterized ODE dynamics.
We propose a novel Learned Alternating Minimization Algorithm (LAMA) for dual-domain sparse-view CT image reconstruction. LAMA is naturally induced by a variational model for CT reconstruction with learnable nonsmooth nonconvex regularizers, which are parameterized as composite functions of deep networks in both image and sinogram domains. To minimize the objective of the model, we incorporate the smoothing technique and residual learning architecture into the design of LAMA. We show that LAMA substantially reduces network complexity, improves memory efficiency and reconstruction accuracy, and is provably convergent for reliable reconstructions. Extensive numerical experiments demonstrate that LAMA outperforms existing methods by a wide margin on multiple benchmark CT datasets.
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 Nesterov's 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 develop a versatile deep neural network architecture, called Lyapunov-Net, to approximate Lyapunov functions of dynamical systems in high dimensions. Lyapunov-Net guarantees positive definiteness, and thus it can be easily trained to satisfy the negative orbital derivative condition, which only renders a single term in the empirical risk function in practice. This significantly reduces the number of hyper-parameters compared to existing methods. We also provide theoretical justifications on the approximation power of Lyapunov-Net and its complexity bounds. We demonstrate the efficiency of the proposed method on nonlinear dynamical systems involving up to 30-dimensional state spaces, and show that the proposed approach significantly outperforms the state-of-the-art methods.
Generating multi-contrasts/modal MRI of the same anatomy enriches diagnostic information but is limited in practice due to excessive data acquisition time. In this paper, we propose a novel deep-learning model for joint reconstruction and synthesis of multi-modal MRI using incomplete k-space data of several source modalities as inputs. The output of our model includes reconstructed images of the source modalities and high-quality image synthesized in the target modality. Our proposed model is formulated as a variational problem that leverages several learnable modality-specific feature extractors and a multimodal synthesis module. We propose a learnable optimization algorithm to solve this model, which induces a multi-phase network whose parameters can be trained using multi-modal MRI data. Moreover, a bilevel-optimization framework is employed for robust parameter training. We demonstrate the effectiveness of our approach using extensive numerical experiments.
Goal: This work aims at developing a novel calibration-free fast parallel MRI (pMRI) reconstruction method incorporate with discrete-time optimal control framework. The reconstruction model is designed to learn a regularization that combines channels and extracts features by leveraging the information sharing among channels of multi-coil images. We propose to recover both magnitude and phase information by taking advantage of structured convolutional networks in image and Fourier spaces. Methods: We develop a novel variational model with a learnable objective function that integrates an adaptive multi-coil image combination operator and effective image regularization in the image and Fourier spaces. We cast the reconstruction network as a structured discrete-time optimal control system, resulting in an optimal control formulation of parameter training where the parameters of the objective function play the role of control variables. We demonstrate that the Lagrangian method for solving the control problem is equivalent to back-propagation, ensuring the local convergence of the training algorithm. Results: We conduct a large number of numerical experiments of the proposed method with comparisons to several state-of-the-art pMRI reconstruction networks on real pMRI datasets. The numerical results demonstrate the promising performance of the proposed method evidently. Conclusion: We conduct a large number of numerical experiments of the proposed method with comparisons to several state-of-the-art pMRI reconstruction networks on real pMRI datasets. The numerical results demonstrate the promising performance of the proposed method evidently. Significance: By learning multi-coil image combination operator and performing regularizations in both image domain and k-space domain, the proposed method achieves a highly efficient image reconstruction network for pMRI.
We study a matrix recovery problem with unknown correspondence: given the observation matrix $M_o=[A,\tilde P B]$, where $\tilde P$ is an unknown permutation matrix, we aim to recover the underlying matrix $M=[A,B]$. Such problem commonly arises in many applications where heterogeneous data are utilized and the correspondence among them are unknown, e.g., due to privacy concerns. We show that it is possible to recover $M$ via solving a nuclear norm minimization problem under a proper low-rank condition on $M$, with provable non-asymptotic error bound for the recovery of $M$. We propose an algorithm, $\text{M}^3\text{O}$ (Matrix recovery via Min-Max Optimization) which recasts this combinatorial problem as a continuous minimax optimization problem and solves it by proximal gradient with a Max-Oracle. $\text{M}^3\text{O}$ can also be applied to a more general scenario where we have missing entries in $M_o$ and multiple groups of data with distinct unknown correspondence. Experiments on simulated data, the MovieLens 100K dataset and Yale B database show that $\text{M}^3\text{O}$ achieves state-of-the-art performance over several baselines and can recover the ground-truth correspondence with high accuracy.