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Riemannian Geometry of Symmetric Positive Definite Matrices via Cholesky Decomposition
Zhenhua Lin
TL;DR
SPD metrics must avoid determinant swelling while supporting useful geometric computations. The paper constructs the Log-Cholesky metric by transporting a bi-invariant structure from Cholesky space to SPD matrices, yielding closed-form averages and efficient parallel transport. Its average has determinant behavior free of swelling, though the metric does not provide congruence invariance and may be less suitable where that property is central.
Problem
Euclidean and Cholesky-based SPD geometry can suffer determinant swelling, motivating metrics that preserve SPD geometry without sacrificing computational usefulness.
Method
The paper constructs a bi-invariant metric and Lie group structure on positive-diagonal lower triangular matrices, then pushes them to SPD matrices through Cholesky decomposition.
Results
The Log-Cholesky average has determinant equal to the geometric mean of the input determinants and avoids the swelling affecting Euclidean and classic Cholesky metrics.
Takeaways & Limitations
Log-Cholesky provides a computationally efficient alternative with closed-form Fréchet averages and parallel transport along geodesics.
Takeaways & Limitations
For applications where congruence invariance is central, the affine-invariant metric is recommended because Log-Cholesky does not have that property.
Abstract
from arXiv · showhide
We present a new Riemannian metric, termed Log-Cholesky metric, on the manifold of symmetric positive definite (SPD) matrices via Cholesky decomposition. We first construct a Lie group structure and a bi-invariant metric on Cholesky space, the collection of lower triangular matrices whose diagonal elements are all positive. Such group structure and metric are then pushed forward to the space of SPD matrices via the inverse of Cholesky decomposition that is a bijective map between Cholesky space and SPD matrix space. This new Riemannian metric and Lie group structure fully circumvent swelling effect, in the sense that the determinant of the Fréchet average of a set of SPD matrices under the presented metric, called Log-Cholesky average, is between the minimum and the maximum of the determinants of the original SPD matrices. Comparing to existing metrics such as the affine-invariant metric and Log-Euclidean metric, the presented metric is simpler, more computationally efficient and numerically stabler. In particular, parallel transport along geodesics under Log-Cholesky metric is given in a closed and easy-to-compute form.
1. Introduction.
SPD matrices arise across scientific applications, but standard Euclidean and Cholesky-based approaches can produce undesirable swelling of determinants. The paper introduces the Log-Cholesky metric to retain useful geometric and computational properties while avoiding this effect.
- The Euclidean metric is inadequate because it places some indefinite symmetric matrices at finite distance and causes swelling in SPD averages.For covariance matrices, inflated determinants represent artificially introduced overall diffusion.
- Existing affine-invariant and Log-Euclidean metrics avoid swelling, but their computational properties differ: Log-Euclidean maps require infinite series, while affine-invariant averages lack a known closed form.
- The Cholesky distance is simple to compute but still suffers from swelling under its geodesic interpolation.
- The proposed Log-Cholesky metric pushes a metric from positive-diagonal lower triangular matrices to SPD matrices through Cholesky decomposition.
- It combines simple computation with bi-invariance, closed-form averages, and easy-to-compute exponential, logarithmic, and parallel-transport expressions.
2. Lower triangular matrices and SPD matrices.
This section establishes the matrix spaces and Cholesky correspondence underlying the construction. Positive-diagonal lower triangular matrices form the Cholesky space, and Cholesky decomposition provides a smooth bijection with SPD matrices.
- 2.1. Notations and basic properties: The paper uses the Frobenius inner product and norm, matrix exponential and logarithm, and the spaces L and L+ of lower triangular matrices.L+ consists of lower triangular matrices with strictly positive diagonal elements.
- 2.1. Notations and basic properties: Lower triangular matrices are closed under addition and multiplication, while L+ is also closed under inversion and multiplication.For triangular matrices, determinants equal the product of diagonal entries.
- 2.2. SPD matrices: SPD matrices have positive eigenvalues and determinants, and products of the form XX⊤ are SPD for invertible X.
- 2.3. Cholesky decomposition: Cholesky decomposition uniquely represents each SPD matrix as P = LL⊤ with L lower triangular and positive diagonal.
- 2.4. Construction basis: The construction first equips L+ with a group structure and bi-invariant metric, then pushes these structures forward to SPD matrices.
- 2.3. Cholesky decomposition: The Cholesky map and its inverse S(L) = LL⊤ are smooth, making them diffeomorphisms between L+ and the SPD manifold.
3. Lie group structure and bi-invariant metric.
The paper builds a Riemannian geometry on Cholesky space, then transfers it to SPD matrices through the Cholesky diffeomorphism. The resulting structure is an abelian Lie group with a bi-invariant metric and efficient geometric operations.
- Cholesky-space geometry: Positive-diagonal lower triangular matrices are parameterized by unconstrained off-diagonal entries and exponentiated diagonal entries.The metric uses the Euclidean Frobenius inner product for the strict lower-triangular component and a distinct relative inner product for the diagonal component.
- Cholesky-space geometry: The resulting metric on Cholesky space has geodesics, exponential maps, logarithmic maps, and distances characterized explicitly.A coordinate chart makes the metric coefficients constant and the Christoffel symbols vanish, yielding simple geodesic equations.
- Transfer to SPD matrices: Cholesky decomposition provides a diffeomorphism between positive-diagonal lower triangular matrices and SPD matrices, enabling the metric to be pushed forward.The induced SPD metric is defined through the inverse Cholesky map and makes the map an isometry.
- Transfer to SPD matrices: Geodesics, exponential maps, distances, and connections on SPD matrices are obtained by mapping the corresponding Cholesky-space objects through the isometry.The resulting formulas require Cholesky decomposition, lower-triangular inversion, and related differentials that can be computed efficiently.
- Lie group structure: The SPD space becomes an abelian Lie group, while the induced Log-Cholesky metric is bi-invariant and compatible with the Cholesky-space group structure.Both the Cholesky map and its inverse are Riemannian isometries and Lie-group isomorphisms.
4. Mean of distributions on S+
The Log-Cholesky framework establishes existence, uniqueness, and closed-form computation for means and finite averages of SPD matrices. Its averages preserve determinant bounds, avoid swelling effect, remain stable in difficult numerical regimes, and support efficient computation.
- Geometric structure: The Log-Cholesky metric is flat, with constantly zero sectional curvature on both Cholesky space and SPD space.The construction uses an isometry between the two spaces, transferring the curvature property.
- Random means: Under finite expected squared distance, Fréchet means of random Cholesky factors and Log-Cholesky means of random SPD matrices exist and are unique.The SPD result follows from the corresponding result on Cholesky space and the isometry.
- Closed forms: The Log-Cholesky mean of a random SPD matrix is obtained by mapping the Fréchet mean of its Cholesky factor back to SPD space.For a finite collection, the same principle yields the average as the product of the mean Cholesky factor and its transpose.
- Determinant behavior: For finite SPD averages, the determinant equals the geometric mean of the input determinants and therefore lies between their minimum and maximum.This establishes the determinant-based no-swelling property for Log-Cholesky averages.
- Interpolation: Geodesic interpolation shows no swelling effect for the Log-Cholesky metric, while Euclidean and Cholesky-distance interpolation exhibit swelling.The comparison includes affine-invariant and Log-Euclidean metrics, which also avoid swelling in the illustration.
- Computation and stability: On synthetic 3×3 SPD matrices, Log-Cholesky remains stable when eigenvalue ratios reach 10^10 and 10^15, while Log-Euclidean deteriorates and then numerically collapses.The paper also reports simpler computation, numerical stability, and parallel-transport speedups of approximately 45× versus Log-Euclidean and 4× versus affine-invariant transport.
5. Concluding remark.
The Log-Cholesky metric offers an alternative to existing SPD-matrix metrics, with computational advantages for big datasets but without congruence invariance. Future work will examine additional properties and applications to brain functional connectivities.
- The Log-Cholesky metric provides an alternative to the Log-Euclidean and affine-invariant metrics, with computational advantages that are attractive for big datasets.
- Applications requiring congruence invariance should use the affine-invariant metric, because numerical experiments suggest that Log-Cholesky lacks this property.
- Future work will investigate anisotropy, relations to other geometric or Fréchet means, and performance in brain functional-connectivity studies.