We design a new unconstrained coordinate system where a $p\times p$ symmetric positive definite (SPD) matrix $Θ$ is represented by a reverse telescoping map $Θ(x)=\rm{RT}(x)$, with $x=(v,d,r)\in\mathbb{R}\times\mathbb{R}^{(p-1)}\times\mathbb{R}^{p(p-1)/2}$, representing respectively the log volume or log determinant; and the shape, as encoded by log relative diagonal scales and partial covariances among the nodes. This construction results in important properties not available in other charts, e.g., matrix logarithm, such as Jacobian depending on only the log-determinant. A useful feature of our construction is $x$ contains a lossless symbolic representation of both the matrix and its inverse. Many important computations involving a matrix and its inverse can be performed in $O(p^2)$ in the transformed domain, while it is the rendering of results in matrix forms (on demand) that must incur an $O(p^3)$ cost. Moreover, two unit-determinant matrices in the transformed domain can be joined by a straight line with pathwise unit determinant. For generative modeling, this allows designing a split volume-shape flow model trained by conditional flow matching for transporting the shape over the unit-determinant path, with a separate one-dimensional flow for transporting the volume or the determinant. The forbidding SPD constraint, tamed thus into a powerful guiding force, leads to the surprising insight that it is in some sense easier to design a volume-normalized shape flow for SPD compared to the unconstrained $\mathbb{R}^{p\times p}$, with no intrinsic notion of volume to aid normalization, unlike the determinant of SPD matrices. We apply our construction for up to $p=200$ in generative modeling of SPD matrices on a difficult synthetic bimodal target, and in generating brain connectivity networks by models trained on fMRI data; as well as in intrinsic diffusion on the SPD manifold.

翻译：我们设计了一种新的无约束坐标系，其中 $p\times p$ 对称正定（SPD）矩阵 $\Theta$ 通过反向伸缩映射 $\Theta(x)=\rm{RT}(x)$ 表示，其中 $x=(v,d,r)\in\mathbb{R}\times\mathbb{R}^{(p-1)}\times\mathbb{R}^{p(p-1)/2}$，分别代表对数体积或对数行列式；以及由对数相对对角线尺度和节点间部分协方差编码的形状。该构造产生了其他坐标系（例如矩阵对数）所不具备的重要性质，例如雅可比矩阵仅依赖于对数行列式。我们构造的一个有用特性是 $x$ 包含矩阵及其逆的无损符号表示。涉及矩阵及其逆的许多重要计算可在变换域中以 $O(p^2)$ 完成，而只有以矩阵形式呈现结果（按需）才需承担 $O(p^3)$ 的计算成本。此外，变换域中两个单位行列式矩阵可以通过一条路径上保持单位行列式的直线连接。对于生成式建模，这允许设计一个分离的体积-形状流模型，通过条件流匹配训练，在单位行列式路径上传输形状，同时用独立的一维流传输体积或行列式。严苛的SPD约束被驯服为强大的引导力量，带来了令人惊讶的洞见：在某些意义上，为SPD设计体积归一化的形状流比无约束的 $\mathbb{R}^{p\times p}$ 更容易，后者没有像SPD矩阵行列式那样的内在体积概念来辅助归一化。我们将我们的构造应用于高达 $p=200$ 的SPD矩阵生成式建模，在困难的合成双峰目标上，以及通过fMRI数据训练的模型生成大脑连接网络；此外还应用于SPD流形上的本征扩散。