SPD-DDPM: Denoising Diffusion Probabilistic Models in the Symmetric Positive Definite Space
This work addresses the challenge of handling large-scale SPD matrix data for generative tasks, offering a novel approach that enables both conditional and unconditional modeling, which is incremental as it adapts an existing method to a specific domain.
The paper tackles the problem of generative modeling for symmetric positive definite (SPD) matrices, which are used in applications like FMRI analysis and traffic prediction, by proposing SPD-DDPM, a denoising diffusion probabilistic model adapted to SPD space, and demonstrates its effectiveness in fitting data distributions and providing accurate predictions on toy and real taxi data.
Symmetric positive definite~(SPD) matrices have shown important value and applications in statistics and machine learning, such as FMRI analysis and traffic prediction. Previous works on SPD matrices mostly focus on discriminative models, where predictions are made directly on $E(X|y)$, where $y$ is a vector and $X$ is an SPD matrix. However, these methods are challenging to handle for large-scale data, as they need to access and process the whole data. In this paper, inspired by denoising diffusion probabilistic model~(DDPM), we propose a novel generative model, termed SPD-DDPM, by introducing Gaussian distribution in the SPD space to estimate $E(X|y)$. Moreover, our model is able to estimate $p(X)$ unconditionally and flexibly without giving $y$. On the one hand, the model conditionally learns $p(X|y)$ and utilizes the mean of samples to obtain $E(X|y)$ as a prediction. On the other hand, the model unconditionally learns the probability distribution of the data $p(X)$ and generates samples that conform to this distribution. Furthermore, we propose a new SPD net which is much deeper than the previous networks and allows for the inclusion of conditional factors. Experiment results on toy data and real taxi data demonstrate that our models effectively fit the data distribution both unconditionally and unconditionally and provide accurate predictions.