arXiv subjects

Justin W. L. Wan

Publications and source records attributed to Justin W. L. Wan.

4 recordsLinked to original sources

arXiv2025

Efficient Score Pre-computation for Diffusion Models via Cross-Matrix Krylov Projection

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.

cs.CV↗

arXiv2024

Efficient Denoising using Score Embedding in Score-based Diffusion Models

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.

cs.CV↗

arXiv2019

Monotone Mixed Finite Difference Scheme for Monge-Ampère Equation

In this paper, we propose a monotone mixed finite difference scheme for solving the two-dimensional Monge-Ampère equation. In order to accomplish this, we convert the Monge-Ampère equation to an equivalent Hamilton-Jacobi-Bellman (HJB) equation. Based on the HJB formulation, we apply the standard 7-point stencil discretization, which is second order accurate, to the grid points wherever monotonicity holds, and apply semi-Lagrangian wide stencil discretization elsewhere to ensure monotonicity on the entire computational domain. By dividing the admissible control set into six regions and optimizing the sub-problem in each region, the computational cost of the optimization problem at each grid point is reduced from $O(M^2)$ to $O(1)$ when the standard 7-point stencil discretization is applied and to $O(M)$ otherwise, where the discretized control set is $M \times M$. We prove that our numerical scheme satisfies consistency, stability, monotonicity and strong comparison principle, and hence is convergent to the viscosity solution of the Monge-Ampère equation. In the numerical results, second order convergence rate is achieved when the standard 7-point stencil discretization is applied monotonically on the entire computation domain, and up to order one convergence is achieved otherwise. The proposed mixed scheme yields a smaller discretization error and a faster convergence rate compared to the pure semi-Lagrangian wide stencil scheme.

math.NA↗

arXiv2019

Deep Neural Network Framework Based on Backward Stochastic Differential Equations for Pricing and Hedging American Options in High Dimensions

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.

q-fin.CP↗