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.

本研究提出了一种新的生成模型SPD-DDPM，通过在SPD空间中引入高斯分布来估计E(X|y)，从而无条件和有条件地学习数据分布，并提供准确的预测。

SPD-DDPM：在对称正定空间中去噪的扩散概率模型