Justin W. L. Wan

Kaikwan Lau, Andrew S. Na, Justin W. L. Wan

This paper presents a novel framework to accelerate score-based diffusion models. It first converts the standard stable diffusion model into the Fokker-Planck formulation which results in solving large linear systems for each image. For training involving many images, it can lead to a high computational cost. The core innovation is a cross-matrix Krylov projection method that exploits mathematical similarities between matrices, using a shared subspace built from ``seed" matrices to rapidly solve for subsequent ``target" matrices. Our experiments show that this technique achieves a 15.8\% to 43.7\% time reduction over standard sparse solvers. Additionally, we compare our method against DDPM baselines in denoising tasks, showing a speedup of up to 115$\times$. Furthermore, under a fixed computational budget, our model is able to produce high-quality images while DDPM fails to generate recognizable content, illustrating our approach is a practical method for efficient generation in resource-limited settings.

Andrew S. Na, William Gao, Justin W. L. Wan

It is well known that training a denoising score-based diffusion models requires tens of thousands of epochs and a substantial number of image data to train the model. In this paper, we propose to increase the efficiency in training score-based diffusion models. Our method allows us to decrease the number of epochs needed to train the diffusion model. We accomplish this by solving the log-density Fokker-Planck (FP) Equation numerically to compute the score \textit{before} training. The pre-computed score is embedded into the image to encourage faster training under slice Wasserstein distance. Consequently, it also allows us to decrease the number of images we need to train the neural network to learn an accurate score. We demonstrate through our numerical experiments the improved performance of our proposed method compared to standard score-based diffusion models. Our proposed method achieves a similar quality to the standard method meaningfully faster.

Yangang Chen, Justin W. L. Wan

We propose a deep neural network framework for computing prices and deltas of American options in high dimensions. The architecture of the framework is a sequence of neural networks, where each network learns the difference of the price functions between adjacent timesteps. We introduce the least squares residual of the associated backward stochastic differential equation as the loss function. Our proposed framework yields prices and deltas on the entire spacetime, not only at a given point. The computational cost of the proposed approach is quadratic in dimension, which addresses the curse of dimensionality issue that state-of-the-art approaches suffer. Our numerical simulations demonstrate these contributions, and show that the proposed neural network framework outperforms state-of-the-art approaches in high dimensions.