SPD-DDPM: Denoising Diffusion Probabilistic Models in the Symmetric
  Positive Definite Space
Yunchen Li Zhou Yu Gaoqi He Yunhang Shen Ke Li Xing Sun Shaohui Lin
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.