Log-Cholesky Metric in Riemannian Geometry

• Log-Cholesky metric is a Riemannian metric on SPD matrices derived via unique Cholesky factorization, splitting lower-triangular and diagonal components.
• It provides closed-form expressions for distances, geodesics, and exponential/logarithm maps, streamlining principal Riemannian operations.
• The metric offers computational efficiency, numerical stability, and a Lie-group structure, distinguishing it from affine-invariant and Log-Euclidean metrics.

The Log-Cholesky metric is a Riemannian metric on the manifold of symmetric positive definite matrices obtained by transferring a simple product geometry from Cholesky factors to the SPD manifold through the unique factorization $P=LL^{\top}$ with $L$ lower-triangular and $\operatorname{diag}(L)>0$. In the formulation introduced by Lin and later recast in product-geometric and Lie-theoretic terms, the strictly lower-triangular entries of $L$ carry a flat Euclidean geometry, while the diagonal entries carry a logarithmic geometry equivalent to the one-dimensional affine-invariant metric on positive scalars (Lin, 2019, Chen et al., 2024). This construction makes the Cholesky map a global diffeomorphism and a Riemannian isometry, so the principal Riemannian operations on SPD matrices—distance, geodesics, exponential and logarithm maps, parallel transport, and Fréchet means—admit closed forms in Cholesky coordinates (Lin, 2019).

1. Cholesky realization of the SPD manifold

Let

$$C_n=\{L\in\mathbb{R}^{n\times n}: L \text{ is lower-triangular and } L_{ii}>0 \text{ for } i=1,\dots,n\}.$$

This is an open submanifold of the Euclidean space of lower-triangular matrices, and every $X\in SPD_n$ admits a unique Cholesky factorization

$X=LL^{\top}, \qquad L\in C_n.$

Hence $C_n$ is diffeomorphic to $SPD_n$ under the Cholesky map $L\mapsto X=LL^{\top}$ (Chen et al., 2024).

In Lin’s notation, the same space is written

$L$0

with $L$1 as a smooth manifold (Lin, 2019). At a point $L$2, the tangent space $L$3 consists of all lower-triangular matrices $L$4, and each such $L$5 decomposes uniquely into its strictly lower-triangular and diagonal parts,

$L$6

The entire Log-Cholesky construction is built from this splitting (Chen et al., 2024).

The same Cholesky-based description extends beyond $L$7. Khare and Vishwakarma define sign-pattern cones

$L$8

and show that every $L$9 can be uniquely written

$\operatorname{diag}(L)>0$0

with $\operatorname{diag}(L)>0$1 lower-triangular and $\operatorname{diag}(L)>0$2. On $\operatorname{diag}(L)>0$3, this reduces to the usual factorization $\operatorname{diag}(L)>0$4 (Khare et al., 31 Jul 2025). This broader setting clarifies that the classical Log-Cholesky geometry on $\operatorname{diag}(L)>0$5 is one instance of a geometry defined on multiple Cholesky-type cones.

2. Metric tensor and coordinate representations

The Log-Cholesky metric on the Cholesky manifold is defined at $\operatorname{diag}(L)>0$6 by

$\operatorname{diag}(L)>0$7

Equivalently,

$\operatorname{diag}(L)>0$8

Thus the coefficient matrix of the metric tensor is block-diagonal: it is $\operatorname{diag}(L)>0$9 on every strictly lower-triangular direction and $L$0 on the $L$1-th diagonal direction (Chen et al., 2024).

This decomposition is fundamental. The metric is explicitly the product of two geometries: a flat Euclidean inner product on the strict lower-triangular entries and the affine-invariant, equivalently log-Euclidean, metric on the positive diagonal vector $L$2 (Chen et al., 2024). If the diagonal is identified with $L$3, then on that factor

$L$4

A coordinate description makes the flatness transparent. Define

$L$5

listing first the logarithms of the diagonal entries and then the strict subdiagonal entries. In these coordinates, the induced metric is literally the standard Euclidean metric,

$L$6

where $L$7 and $L$8 for $L$9 (Khare et al., 31 Jul 2025).

Transporting the metric from Cholesky space to the SPD manifold uses the map

$$C_n=\{L\in\mathbb{R}^{n\times n}: L \text{ is lower-triangular and } L_{ii}>0 \text{ for } i=1,\dots,n\}.$$0

whose differential at $$C_n=\{L\in\mathbb{R}^{n\times n}: L \text{ is lower-triangular and } L_{ii}>0 \text{ for } i=1,\dots,n\}.$$1 is

$$C_n=\{L\in\mathbb{R}^{n\times n}: L \text{ is lower-triangular and } L_{ii}>0 \text{ for } i=1,\dots,n\}.$$2

Its inverse is

$$C_n=\{L\in\mathbb{R}^{n\times n}: L \text{ is lower-triangular and } L_{ii}>0 \text{ for } i=1,\dots,n\}.$$3

where $$C_n=\{L\in\mathbb{R}^{n\times n}: L \text{ is lower-triangular and } L_{ii}>0 \text{ for } i=1,\dots,n\}.$$4 means “take strict-lower-triangular + $$C_n=\{L\in\mathbb{R}^{n\times n}: L \text{ is lower-triangular and } L_{ii}>0 \text{ for } i=1,\dots,n\}.$$5 diag.” The metric on the SPD manifold is then defined by

$$C_n=\{L\in\mathbb{R}^{n\times n}: L \text{ is lower-triangular and } L_{ii}>0 \text{ for } i=1,\dots,n\}.$$6

making $$C_n=\{L\in\mathbb{R}^{n\times n}: L \text{ is lower-triangular and } L_{ii}>0 \text{ for } i=1,\dots,n\}.$$7 isometric to $$C_n=\{L\in\mathbb{R}^{n\times n}: L \text{ is lower-triangular and } L_{ii}>0 \text{ for } i=1,\dots,n\}.$$8 (Lin, 2019).

3. Distance, geodesics, and closed-form Riemannian operations

Because the metric is a product metric, the squared geodesic distance between two SPD matrices $$C_n=\{L\in\mathbb{R}^{n\times n}: L \text{ is lower-triangular and } L_{ii}>0 \text{ for } i=1,\dots,n\}.$$9 and $X\in SPD_n$0 is

$X\in SPD_n$1

Equivalently,

$X\in SPD_n$2

where $X\in SPD_n$3 and $X\in SPD_n$4 (Chen et al., 2024, Lin, 2019).

Geodesics also split into strict-lower and diagonal components. The unique geodesic in Cholesky space through $X\in SPD_n$5 in direction $X\in SPD_n$6 is

$X\in SPD_n$7

or equivalently

$X\in SPD_n$8

Hence the strictly lower-triangular part evolves linearly, while each diagonal entry follows the one-dimensional affine-invariant geodesic $X\in SPD_n$9 (Chen et al., 2024).

For endpoints $X=LL^{\top}, \qquad L\in C_n.$0 with $X=LL^{\top}, \qquad L\in C_n.$1 and $X=LL^{\top}, \qquad L\in C_n.$2, the unique geodesic from $X=LL^{\top}, \qquad L\in C_n.$3 to $X=LL^{\top}, \qquad L\in C_n.$4 is

$X=LL^{\top}, \qquad L\in C_n.$5

with

$X=LL^{\top}, \qquad L\in C_n.$6

In coordinates, $X=LL^{\top}, \qquad L\in C_n.$7 for $X=LL^{\top}, \qquad L\in C_n.$8, while

$X=LL^{\top}, \qquad L\in C_n.$9

on the diagonal (Lin, 2019).

The exponential and logarithm maps are correspondingly explicit. In Cholesky space,

$C_n$0

and

$C_n$1

On the SPD manifold,

$C_n$2

and

$C_n$3

(Lin, 2019).

Parallel transport is likewise unusually explicit in this geometry: Lin states that parallel transport along geodesics under the Log-Cholesky metric is given in a closed and easy-to-compute form (Lin, 2019). A plausible implication is that the metric’s usefulness is tied not only to closed-form distances and geodesics, but also to closed-form connection-level operations.

4. Flatness, Lie-group structure, and Fréchet means

A distinctive feature of the Log-Cholesky metric is that it is not merely a pullback metric but part of a Lie-group geometry on Cholesky space. On

$C_n$4

Lin defines the commutative group law

$C_n$5

with identity $C_n$6 and inverse

$C_n$7

The metric $C_n$8 is both left- and right-invariant under $C_n$9, hence bi-invariant; in these log-coordinates the Christoffel symbols vanish and the sectional curvature is $SPD_n$0 (Lin, 2019).

Khare and Vishwakarma reformulate the same geometry in terms of a coordinate isomorphism $SPD_n$1, showing that the log-Cholesky metric descends from the flat Euclidean metric on $SPD_n$2. Transporting $SPD_n$3 to $SPD_n$4 yields an abelian Lie group law $SPD_n$5, and the map

$SPD_n$6

is simultaneously a group isomorphism and a Riemannian isometry (Khare et al., 31 Jul 2025). This identifies the Log-Cholesky geometry as globally flat, not merely locally Euclidean.

The flatness has immediate consequences for Fréchet means. Because $SPD_n$7 is geodesically flat and simply-connected, any Fréchet mean is unique (Lin, 2019). For SPD matrices $SPD_n$8 with $SPD_n$9, the finite-sample Log-Cholesky average is

$L\mapsto X=LL^{\top}$0

Equivalently, off-diagonal entries average arithmetically while diagonal entries average geometrically (Lin, 2019, Khare et al., 31 Jul 2025).

The determinant of this average satisfies

$L\mapsto X=LL^{\top}$1

and therefore

$L\mapsto X=LL^{\top}$2

Lin describes this as fully circumventing swelling effect (Lin, 2019). In this metric, the determinant of the Fréchet average lies between the minimum and the maximum of the determinants of the original SPD matrices.

5. Relation to affine-invariant and Log-Euclidean geometries

The Log-Cholesky metric is often compared with the classical affine-invariant metric

$L\mapsto X=LL^{\top}$3

whose geodesics are

$L\mapsto X=LL^{\top}$4

and whose distance is

$L\mapsto X=LL^{\top}$5

Khare and Vishwakarma emphasize that the affine-invariant metric has nonpositive sectional curvature and is not flat, whereas the Log-Cholesky metric is flat, admits a global additive coordinate $L\mapsto X=LL^{\top}$6, and reduces geodesics, distance, and means to Euclidean ones in $L\mapsto X=LL^{\top}$7-space. They state that the two metrics coincide only infinitesimally at the identity, but globally they are quite different (Khare et al., 31 Jul 2025).

The distinction from the Log-Euclidean metric is equally operational. Lin’s comparison states that the affine-invariant metric needs two symmetric square roots, one inverse, and one matrix logarithm, while the Log-Euclidean metric needs two full matrix logarithms and subtraction. By contrast, the Log-Cholesky metric requires only Cholesky decompositions, vector exponentials, and triangular solves (Lin, 2019). This suggests that the term “log” in Log-Cholesky refers to the diagonal of the Cholesky factor, not to a full matrix logarithm on the SPD matrix itself.

The numerical consequences are explicit in the literature. Lin states that the presented metric is simpler, more computationally efficient, and numerically stabler than existing metrics such as the affine-invariant metric and Log-Euclidean metric (Lin, 2019). In the implementation-oriented summary, all routines are described as numerically stable for very ill-conditioned $L\mapsto X=LL^{\top}$8 and experimentally stable to condition $L\mapsto X=LL^{\top}$9, whereas full log/exp become inaccurate (Lin, 2019). The 2024 product-geometry paper makes the same point in a comparative way: the existing Cholesky metric is computationally efficient and numerically stable, and the new diagonal variants are proposed as numerically stabler than the existing Cholesky metric (Chen et al., 2024).

A common misconception is that flatness makes the metric extrinsic or non-Riemannian in any substantive sense. The cited papers state the opposite: the Log-Cholesky construction is a genuine Riemannian geometry, with its own Levi-Civita connection, geodesics, exponential and logarithm maps, and parallel transport, all inherited through a global Riemannian isometry from Cholesky space (Lin, 2019).

6. Generalizations, product geometries, and applications

Subsequent work places the Log-Cholesky metric inside a broader family of Cholesky-based geometries. The 2024 paper “Product Geometries on Cholesky Manifolds with Applications to SPD Manifolds” first unveils that the existing popular Riemannian metric on the Cholesky manifold can be generally characterized as the product metric of a Euclidean metric and a Riemannian metric on the space of $L$00-dimensional positive vectors. On this basis it proposes two new metrics on Cholesky manifolds, the Diagonal Power Euclidean Metric and the Diagonal Generalized Bures-Wasserstein Metric, and studies gyro structures and deformed metrics associated with them (Chen et al., 2024). In that presentation, the Log-Cholesky metric functions as the canonical baseline from which new diagonal geometries are derived.

Khare and Vishwakarma extend the picture beyond positive definiteness. The union

$L$01

is a dense cone in all symmetric matrices, and $L$02 is exactly the identity-component subgroup, of index $L$03, inside a larger non-connected abelian group. The log-Cholesky metric on $L$04 is recovered as the restriction of a compatible bi-invariant metric on this larger object (Khare et al., 31 Jul 2025). Their abstract further states that the cones $L$05 and $L$06 admit Wishart densities with signed Bartlett decompositions. A plausible implication is that the Log-Cholesky formalism is linked not only to deterministic geometry but also to probability theory on structured cones.

The metric has also been used as a modeling ingredient in multiway covariance analysis. In “Enhancing the Tensor Normal via Geometrically Parameterized Cholesky Factors,” the geometric expansion of the multi-way covariance’s Cholesky factor is explicitly inspired by the Fréchet mean under the log-Cholesky metric (Simonis et al., 14 Apr 2025). In that framework, the closed-form mean

$L$07

and its decomposition into arithmetic averaging of strictly lower-triangular parts and geometric averaging of diagonals are lifted to Kronecker-structured Cholesky factors (Simonis et al., 14 Apr 2025). The paper states that, by parameterizing vector normal covariances through such a Cholesky factor representation, one of the structural components in the covariance of the vectorized data can be eliminated without compromising the analytical tractability of the likelihood.

In the two-way tensor-normal setting, the same paper writes

$L$08

with gradient

$L$09

and reports the use of Hamiltonian Monte Carlo in a fixed-mean setting for two-way covariance relevancy detection of components, as well as in a seasonally-varying covariance process regime (Simonis et al., 14 Apr 2025). Within this application domain, the Log-Cholesky metric is not only a geometric device for SPD matrices but also a coordinate system for statistically tractable covariance parameterizations.