Abbas Mammadov

We present a novel framework for solving inverse problems in function spaces using diffusion-based generative models. Unlike traditional methods, which often discretize the domain and operate on fixed grids, our approach is discretization-agnostic, allowing for flexibility during sampling and generalization across different resolutions. Built upon function space diffusion models with neural operator architectures, we adapt the denoising process of pre-trained diffusion models to efficiently sample from posterior distributions in function spaces. This framework can be applied to a variety of problems, such as recovery of initial conditions and coefficient functions in noisy or partially observed PDE-based inverse problems like Darcy flows and Navier–Stokes equations. To the best of our knowledge, this is the first diffusion-based plug-and-play solver for inverse problems that operates in a discretization-agnostic manner, providing a new perspective on inverse problems with functional data, as typically arising in the context of PDEs.

We propose a variational inference approach to sample from the posterior distribution for solving inverse problems. From a pre-trained diffusion model, our approach trains a conditional flow model to minimize the divergence between the proposal variational distribution and the posterior distribution implicitly defined through the diffusion model. Once trained, the flow model is capable of sampling from the posterior distribution with a single NFE, amortized with respect to the measurement. The proposed method paves a new path for distilling a diffusion prior for efficient posterior sampling. We show that our method is applicable to standard signals in Euclidean space, as well as signals on manifold.

Current theoretical and empirical research in neural networks suggests that complex datasets require large network architectures for thorough classification, yet the precise nature of this relationship remains unclear. This paper tackles this issue by defining upper and lower bounds for neural network widths, which are informed by the polytope structure of the dataset in question. We also delve into the application of these principles to simplicial complexes and specific manifold shapes, explaining how the requirement for network width varies in accordance with the geometric complexity of the dataset. Moreover, we develop an algorithm to investigate a converse situation where the polytope structure of a dataset can be inferred from its corresponding trained neural networks. Through our algorithm, it is established that popular datasets such as MNIST, Fashion-MNIST, and CIFAR10 can be efficiently encapsulated using no more than two polytopes with a small number of faces.

Despite the recent success of face image generation and conditional hair editing with GANs, there are several limitationsdue to its complex geometry and appearance. These models are quite heavy which leads to late inference time. In addition,those models are only considering to change color to some existing hair color but not dealing with customized color. Tocontinue, taking appropriate image for inference (correct angle and alignment) is also cumbersome. Lastly, those models arequite complex to figure out and use due to absence of convenient user interface. In this paper, we present Artificial Barber,a novel conditional and lightweight machine learning pipeline deployed on Qt based Graphical User Interface to change haircolor and style in a fast, accurate, and convenient manner. All the functionalities can be controlled through the GUI. The code is available at https://github.com/abbasmammadov/Artificial-Barber.