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Diffusion models have quickly become some of the most popular and powerful generative models for high-dimensional data. The key insight that enabled their development was the realization that access to the score -- the gradient of the…
We systematically develop a learning-based treatment of stochastic optimal control (SOC), relying on direct optimization of parametric control policies. We propose a derivation of adjoint sensitivity results for stochastic differential…
Diffusion-based generative models learn to iteratively transfer unstructured noise to a complex target distribution as opposed to Generative Adversarial Networks (GANs) or the decoder of Variational Autoencoders (VAEs) which produce samples…
Diffusion models have emerged as a dominant framework for generative modeling, but their mathematical foundations are often presented separately through diffusion probabilistic models, score-based modeling, stochastic differential…
Diffusion models have demonstrated empirical successes in various applications and can be adapted to task-specific needs via guidance. This paper studies a form of gradient guidance for adapting a pre-trained diffusion model towards…
We develop a new continuous-time stochastic gradient descent method for optimizing over the stationary distribution of stochastic differential equation (SDE) models. The algorithm continuously updates the SDE model's parameters using an…
Asynchronous stochastic gradient descent (ASGD) is a popular parallel optimization algorithm in machine learning. Most theoretical analysis on ASGD take a discrete view and prove upper bounds for their convergence rates. However, the…
Diffusion models have demonstrated great potential in generating high-quality content for images, natural language, protein domains, etc. However, how to perform user-preferred targeted generation via diffusion models with only black-box…
We propose a novel numerical approach for nonlocal diffusion equations [8] with integrable kernels, based on the relationship between the backward Kolmogorov equation and backward stochastic differential equations (BSDEs) driven by L\`{e}vy…
Recent advancements in diffusion models have been effective in learning data priors for solving inverse problems. They leverage diffusion sampling steps for inducing a data prior while using a measurement guidance gradient at each step to…
Inverse problems describe the process of estimating the causal factors from a set of measurements or data. Mapping of often incomplete or degraded data to parameters is ill-posed, thus data-driven iterative solutions are required, for…
Stochastic optimal control (SOC) aims to direct the behavior of noisy systems and has widespread applications in science, engineering, and artificial intelligence. In particular, reward fine-tuning of diffusion and flow matching models and…
Sampling from Diffusion Models can alternatively be seen as solving differential equations, where there is a challenge in balancing speed and image visual quality. ODE-based samplers offer rapid sampling time but reach a performance limit,…
In recent years we have witnessed a growth in mathematics for deep learning, which has been used to solve inverse problems of partial differential equations (PDEs). However, most deep learning-based inversion methods either require paired…
In this paper, we consider solving the distributed optimization problem over a multi-agent network under the communication restricted setting. We study a compressed decentralized stochastic gradient method, termed ``compressed exact…
We introduce a novel paradigm for learning non-parametric drift and diffusion functions for stochastic differential equation (SDE). The proposed model learns to simulate path distributions that match observations with non-uniform time…
Existing customization methods require access to multiple reference examples to align pre-trained diffusion probabilistic models (DPMs) with user-provided concepts. This paper aims to address the challenge of DPM customization when the only…
Score-based diffusion models learn to reverse a stochastic differential equation that maps data to noise. However, for complex tasks, numerical error can compound and result in highly unnatural samples. Previous work mitigates this drift…
In this paper, we establish the almost sure convergence of two-timescale stochastic gradient descent algorithms in continuous time under general noise and stability conditions, extending well known results in discrete time. We analyse…
Stochastic differential equations (SDEs) or diffusions are continuous-valued continuous-time stochastic processes widely used in the applied and mathematical sciences. Simulating paths from these processes is usually an intractable problem,…