# The Reverse Telescoping Coordinate System for Positive Definite Matrices: Geometry, Computation, and Generative Modeling

arXiv: https://arxiv.org/abs/2606.15442 · HTML: https://mlsanity.com/paper/2606.15442/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 15 of its 47 references that are in the map.
- First posted on arXiv 13 Jun 2026.

## Builds on (most cited first, 15 of 15)

- [Adam: A Method for Stochastic Optimization](https://mlsanity.com/paper/1412.6980/) (2014)
- [Score-Based Generative Modeling through Stochastic Differential Equations](https://mlsanity.com/paper/2011.13456/) (2020)
- [Flow Matching for Generative Modeling](https://mlsanity.com/paper/2210.02747/) (2022)
- [Stochastic Interpolants: A Unifying Framework for Flows and Diffusions](https://mlsanity.com/paper/2303.08797/) (2023)
- [Flow Matching on General Geometries](https://mlsanity.com/paper/2302.03660/) (2023)
- [Riemannian Score-Based Generative Modelling](https://mlsanity.com/paper/2202.02763/) (2022)
- [High-dimensional Bayesian inference via the Unadjusted Langevin Algorithm](https://mlsanity.com/paper/1605.01559/) (2016)
- [Deep Generative Models through the Lens of the Manifold Hypothesis: A Survey and New Connections](https://mlsanity.com/paper/2404.02954/) (2024)
- [Variational Rectified Flow Matching](https://mlsanity.com/paper/2502.09616/) (2025)
- [Towards Hierarchical Rectified Flow](https://mlsanity.com/paper/2502.17436/) (2025)
- [SPD-DDPM: Denoising Diffusion Probabilistic Models in the Symmetric Positive Definite Space](https://mlsanity.com/paper/2312.08200/) (2023)
- [Riemannian Flow Matching for Brain Connectivity Matrices via Pullback Geometry](https://mlsanity.com/paper/2505.18193/) (2025)
- [Graph-Regularized Manifold-Aware Conditional Wasserstein GAN for Brain Functional Connectivity Generation](https://mlsanity.com/paper/2212.05316/) (2022)
- [Evidence Estimation in Gaussian Graphical Models Using a Telescoping Block Decomposition of the Precision Matrix](https://mlsanity.com/paper/2205.01016/) (2022)
- [An Order of Magnitude Time Complexity Reduction for Gaussian Graphical Model Posterior Sampling Using a Reverse Telescoping Block Decomposition](https://mlsanity.com/paper/2509.26385/) (2025)

## Abstract (arXiv)

We design a new unconstrained coordinate system where a p× p symmetric positive definite (SPD) matrix Theta is represented by a reverse telescoping map Theta(x)=rmRT(x), with x=(v,d,r)∈ℝ×ℝ(p-1)×ℝp(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(p2) in the transformed domain, while it is the rendering of results in matrix forms (on demand) that must incur an O(p3) 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 ℝp× 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.