The Reverse Telescoping Coordinate System for Positive Definite Matrices: Geometry, Computation, and Generative Modeling
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 inv...
| Authors | Anindya Bhadra |
|---|---|
| Primary category | stat.ML |
| Published | 13 June 2026 |
| Last updated at source | 13 June 2026 |
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