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.

翻译：对称正定矩阵在统计学和机器学习中展现出重要价值与应用，例如fMRI分析与交通预测。现有关于SPD矩阵的研究主要集中于判别模型，即直接对$E(X|y)$进行预测（其中$y$为向量，$X$为SPD矩阵）。然而，这些方法在处理大规模数据时面临挑战，因为它们需要访问并处理全部数据。受去噪扩散概率模型启发，本文提出一种名为SPD-DDPM的新型生成模型，通过在SPD空间中引入高斯分布来估计$E(X|y)$。此外，该模型能够无条件且灵活地估计$p(X)$而无需给定$y$。一方面，模型通过条件学习$p(X|y)$，并利用样本均值获取$E(X|y)$作为预测结果；另一方面，模型无条件学习数据概率分布$p(X)$，并生成符合该分布的样本。进一步地，我们提出一种深度远超现有网络的新型SPD网络，该网络支持条件因子的引入。在玩具数据集与真实出租车数据集上的实验结果表明，我们的模型既能有效拟合数据分布（包括无条件与条件两种情况），又能提供准确的预测结果。