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Deep learning is emerging as a new paradigm for solving inverse imaging problems. However, the deep learning methods often lack the assurance of traditional physics-based methods due to the lack of physical information considerations in…

The utilization of residual learning has become widespread in deep and scalable neural nets. However, the fundamental principles that contribute to the success of residual learning remain elusive, thus hindering effective training of plain…

Plug-and-Play Priors (PnP) and Regularisation by Denoising (RED) have established that image denoisers can effectively replace traditional regularisers in linear inverse problem solvers for tasks like super-resolution, demosaicing, and…

Low-rank structures play important role in recent advances of many problems in image science and data science. As a natural extension of low-rank structures for data with nonlinear structures, the concept of the low-dimensional manifold…

The use of deep learning has successfully solved several problems in the field of medical imaging. Deep learning has been applied to the CT denoising problem successfully. However, the use of deep learning requires large amounts of data to…

In this effort, we derive a formula for the integral representation of a shallow neural network with the ReLU activation function. We assume that the outer weighs admit a finite $L_1$-norm with respect to Lebesgue measure on the sphere. For…

Recently, deep learning methods such as the convolutional neural networks have gained prominence in the area of image denoising. This is owing to their proven ability to surpass state-of-the-art classical image denoising algorithms such as…

Despite impressive performance on numerous visual tasks, Convolutional Neural Networks (CNNs) --- unlike brains --- are often highly sensitive to small perturbations of their input, e.g. adversarial noise leading to erroneous decisions. We…

Layer normalization (LN) is a ubiquitous technique in deep learning but our theoretical understanding to it remains elusive. This paper investigates a new theoretical direction for LN, regarding to its nonlinearity and representation…

Image denoising is an important low-level computer vision task, which aims to reconstruct a noise-free and high-quality image from a noisy image. With the development of deep learning, convolutional neural network (CNN) has been gradually…

Neural networks have shown tremendous potential for reconstructing high-resolution images in inverse problems. The non-convex and opaque nature of neural networks, however, hinders their utility in sensitive applications such as medical…

The low-dimensional manifold hypothesis posits that the data found in many applications, such as those involving natural images, lie (approximately) on low-dimensional manifolds embedded in a high-dimensional Euclidean space. In this…

Image restoration has been an extensively researched topic in numerous fields. With the advent of deep learning, a lot of the current algorithms were replaced by algorithms that are more flexible and robust. Deep networks have demonstrated…

As PET imaging is accompanied by substantial radiation exposure and cancer risk, reducing radiation dose in PET scans is an important topic. However, low-count PET scans often suffer from high image noise, which can negatively impact image…

We introduce a paradigm for nonlocal sparsity reinforced deep convolutional neural network denoising. It is a combination of a local multiscale denoising by a convolutional neural network (CNN) based denoiser and a nonlocal denoising based…

Regularization plays a pivotal role when facing the challenge of solving ill-posed inverse problems, where the number of observations is smaller than the ambient dimension of the object to be estimated. A line of recent work has studied…

We use smoothed analysis techniques to provide guarantees on the training loss of Multilayer Neural Networks (MNNs) at differentiable local minima. Specifically, we examine MNNs with piecewise linear activation functions, quadratic loss and…

A dimension reduction method based on the "Nonlinear Level set Learning" (NLL) approach is presented for the pointwise prediction of functions which have been sparsely sampled. Leveraging geometric information provided by the Implicit…

Despite the popularity of transformers in practice, their architectures are empirically designed and neither mathematically justified nor interpretable. Moreover, as indicated by many empirical studies, some components of transformer…

Neural network training is usually accomplished by solving a non-convex optimization problem using stochastic gradient descent. Although one optimizes over the networks parameters, the main loss function generally only depends on the…