Affine-Invariant Log-Det Metric

• The topic is characterized by its formulation on SPD matrices, where the affine-invariant Riemannian metric is expressed through matrix logarithms and extended via log-det divergence limits.
• It leverages a two-parameter family that captures both standard geodesic distances and quotient-affine structures, providing a unified framework for diverse geometric constructions.
• Applications span finite-dimensional SPD matrices and infinite-dimensional operators, clarifying determinant-based metrics in both classical and quotient settings.

Searching arXiv for the cited papers and closely related terminology on affine-invariant SPD geometry and log-det divergences. The expression affine-invariant log-det metric does not denote a single uniformly defined object across the cited literature. On the manifold of symmetric positive definite matrices, the central object is the classical affine-invariant Riemannian metric, whose geodesic distance is written through matrix logarithms and, in trace-extended forms, may include an explicit log-determinant term (Thanwerdas et al., 2019). In the log-det divergence literature, the Affine Invariant Riemannian Metric (AIRM) appears as a limiting or special case of broader Alpha–Beta Log-Det divergence families rather than as an independently introduced log-det divergence (Cichocki et al., 2014). Several nearby constructions—log-Euclidean metrics, Jensen–Bregman LogDet divergences, quotient-affine metrics on correlation matrices, and equiaffine-covariant affine-normal directions driven by a log-determinant curvature quantity—are related but not identical (Vemulapalli et al., 2015).

1. Terminology and interpretive scope

In the cited sources, the phrase affine-invariant log-det metric is best treated as terminologically ambiguous. The papers distinguish between the affine-invariant metric on the SPD cone, log-Euclidean geometry, and log-det divergences such as Stein or Jensen–Bregman LogDet quantities; they do not conflate them (Vemulapalli et al., 2015).

Interpretation	Object	Status in cited literature
Affine-invariant metric	Riemannian metric on $\mathrm{SPD}_n$	Explicitly defined
Log-Euclidean metric	Different SPD metric based on $\log$	Explicitly defined
Log-det divergence	Determinant-based divergence family	Explicitly defined
“Affine-invariant log-det metric”	Hybrid phrase	Not explicitly standardized

A precise reading therefore depends on context. In the SPD-matrix literature, the nearest canonical object is the affine-invariant metric

$$g^1_\Sigma(V,W) = \alpha\,\mathrm{tr}(\Sigma^{-1}V\Sigma^{-1}W) + \beta\,\mathrm{tr}(\Sigma^{-1}V)\mathrm{tr}(\Sigma^{-1}W), \qquad \alpha>0,\ \beta>-\frac{\alpha}{n},$$

together with its geodesic distance expressed through logarithms of generalized eigenvalues (Thanwerdas et al., 2019). In the Alpha–Beta Log-Det literature, the nearest corresponding statement is that the AIRM is recovered as the $(\alpha,\beta)=(0,0)$ limit, or equivalently as a symmetric logarithmic limit of a broader log-det divergence family (Cichocki et al., 2014).

The ambiguity is reinforced by adjacent but distinct usages. One paper on affine normal directions uses the gradient of $\log\det(H_T)$ to reorganize an affine-differential-geometric direction, but explicitly states that it does not define a Riemannian metric tensor under that name (Niu et al., 1 Apr 2026). A plausible implication is that “affine-invariant log-det metric” should be reserved, when used at all, for the SPD-manifold setting rather than for every affine-geometric construction involving $\log\det$.

2. Affine-invariant geometry on the SPD cone

The basic manifold is

$M=\mathrm{SPD}_n,$

the set of $n\times n$ symmetric positive definite matrices. Its defining affine symmetry is the congruence action of $\mathrm{GL}_n$: $\eta_A^1(\Sigma)=A\Sigma A^\top.$ This action is transitive, the stabilizer at the identity is $\log$0, and the quotient identification is

$\log$1

A metric $\log$2 is affine-invariant exactly when each congruence map is an isometry: $\log$3 These are the paper’s formal affine-invariance conditions on SPD matrices (Thanwerdas et al., 2019).

At the identity, every orthogonally invariant scalar product on $\log$4 has the form

$\log$5

with

$\log$6

Transporting tangent vectors by $\log$7 yields the full affine-invariant family

$\log$8

An important point made explicitly in the cited literature is that there is not one unique affine-invariant metric, but already a one-parameter family up to overall scale (Thanwerdas et al., 2019).

The same two-parameter family is also written as

$\log$9

with the same parameter constraints. The classical choice is $$g^1_\Sigma(V,W) = \alpha\,\mathrm{tr}(\Sigma^{-1}V\Sigma^{-1}W) + \beta\,\mathrm{tr}(\Sigma^{-1}V)\mathrm{tr}(\Sigma^{-1}W), \qquad \alpha>0,\ \beta>-\frac{\alpha}{n},$$0 (Thanwerdas et al., 2021). In the $$g^1_\Sigma(V,W) = \alpha\,\mathrm{tr}(\Sigma^{-1}V\Sigma^{-1}W) + \beta\,\mathrm{tr}(\Sigma^{-1}V)\mathrm{tr}(\Sigma^{-1}W), \qquad \alpha>0,\ \beta>-\frac{\alpha}{n},$$1-invariant classification of SPD metrics, this family is also described as the affine-invariant family

$$g^1_\Sigma(V,W) = \alpha\,\mathrm{tr}(\Sigma^{-1}V\Sigma^{-1}W) + \beta\,\mathrm{tr}(\Sigma^{-1}V)\mathrm{tr}(\Sigma^{-1}W), \qquad \alpha>0,\ \beta>-\frac{\alpha}{n},$$2

and the standard $$g^1_\Sigma(V,W) = \alpha\,\mathrm{tr}(\Sigma^{-1}V\Sigma^{-1}W) + \beta\,\mathrm{tr}(\Sigma^{-1}V)\mathrm{tr}(\Sigma^{-1}W), \qquad \alpha>0,\ \beta>-\frac{\alpha}{n},$$3 case is the usual affine-invariant metric, sometimes normalized by $$g^1_\Sigma(V,W) = \alpha\,\mathrm{tr}(\Sigma^{-1}V\Sigma^{-1}W) + \beta\,\mathrm{tr}(\Sigma^{-1}V)\mathrm{tr}(\Sigma^{-1}W), \qquad \alpha>0,\ \beta>-\frac{\alpha}{n},$$4 or $$g^1_\Sigma(V,W) = \alpha\,\mathrm{tr}(\Sigma^{-1}V\Sigma^{-1}W) + \beta\,\mathrm{tr}(\Sigma^{-1}V)\mathrm{tr}(\Sigma^{-1}W), \qquad \alpha>0,\ \beta>-\frac{\alpha}{n},$$5 depending on convention (Thanwerdas et al., 2021).

3. Geodesics, logarithms, distance, and curvature

For the affine-invariant case $$g^1_\Sigma(V,W) = \alpha\,\mathrm{tr}(\Sigma^{-1}V\Sigma^{-1}W) + \beta\,\mathrm{tr}(\Sigma^{-1}V)\mathrm{tr}(\Sigma^{-1}W), \qquad \alpha>0,\ \beta>-\frac{\alpha}{n},$$6, the geodesic through $$g^1_\Sigma(V,W) = \alpha\,\mathrm{tr}(\Sigma^{-1}V\Sigma^{-1}W) + \beta\,\mathrm{tr}(\Sigma^{-1}V)\mathrm{tr}(\Sigma^{-1}W), \qquad \alpha>0,\ \beta>-\frac{\alpha}{n},$$7 with initial tangent $$g^1_\Sigma(V,W) = \alpha\,\mathrm{tr}(\Sigma^{-1}V\Sigma^{-1}W) + \beta\,\mathrm{tr}(\Sigma^{-1}V)\mathrm{tr}(\Sigma^{-1}W), \qquad \alpha>0,\ \beta>-\frac{\alpha}{n},$$8 is

$$g^1_\Sigma(V,W) = \alpha\,\mathrm{tr}(\Sigma^{-1}V\Sigma^{-1}W) + \beta\,\mathrm{tr}(\Sigma^{-1}V)\mathrm{tr}(\Sigma^{-1}W), \qquad \alpha>0,\ \beta>-\frac{\alpha}{n},$$9

and the logarithm map is

$(\alpha,\beta)=(0,0)$0

The distance is written in the cited source as

$(\alpha,\beta)=(0,0)$1

where $(\alpha,\beta)=(0,0)$2 are the eigenvalues of

$(\alpha,\beta)=(0,0)$3

The same source notes that conventionally this is usually the squared distance, while many texts write the actual distance as

$(\alpha,\beta)=(0,0)$4

The essential point is the standard log-spectrum formula (Thanwerdas et al., 2019).

Equivalent formulations appear elsewhere as

$(\alpha,\beta)=(0,0)$5

or

$(\alpha,\beta)=(0,0)$6

with the trace-extended affine family admitting

$(\alpha,\beta)=(0,0)$7

This last formula is the clearest place where a literal log-determinant contribution enters the affine-invariant family: it is part of the $(\alpha,\beta)=(0,0)$8 extension, not a separately named metric (Thanwerdas et al., 2021).

The Levi-Civita connection for the classical affine-invariant metric is

$(\alpha,\beta)=(0,0)$9

and the curvature formula is the standard nonpositive-curvature expression in SPD geometry (Thanwerdas et al., 2021). A later synthesis states that the affine-invariant metric is geodesically complete, that $\log\det(H_T)$0 is a Riemannian symmetric space for this geometry, and that the sectional curvature is non-positive and bounded (Thanwerdas et al., 2021).

This logarithmic distance should not be confused with the log-Euclidean metric

$\log\det(H_T)$1

which the cited literature treats as a different geodesic distance with different invariance properties; in particular, log-Euclidean distance is not affine-invariant (Vemulapalli et al., 2015).

4. Log-det divergence families and the emergence of AIRM

The Alpha–Beta Log-Det divergence on SPD matrices is defined by

$\log\det(H_T)$2

with spectral form

$\log\det(H_T)$3

where $\log\det(H_T)$4 are the eigenvalues of $\log\det(H_T)$5 (Cichocki et al., 2014).

The key limiting statement is

$\log\det(H_T)$6

so that

$\log\det(H_T)$7

In the paper’s formulation, the Affine Invariant Riemannian Metric is therefore obtained from the AB family at the singular origin $\log\det(H_T)$8, and $\log\det(H_T)$9 is one half of the squared AIRM (Cichocki et al., 2014).

The same source also states the local second-order expansion

$\log\det$0

which means that the full AB family induces the same local Riemannian metric tensor

$\log\det$1

This places AIRM not only as a limit point of the family but also as its common infinitesimal geometry (Cichocki et al., 2014).

The literature simultaneously distinguishes this structure from other determinant-based quantities. Stein’s loss,

$\log\det$2

the S-divergence or Jensen–Bregman LogDet divergence,

$\log\det$3

and the Bhattacharyya or LogDet-zero metric

$\log\det$4

all belong to the same broader determinant-based taxonomy, but they are not identical to the affine-invariant Riemannian metric (Cichocki et al., 2014).

A common misconception addressed indirectly across the cited sources is that any determinant-based SPD dissimilarity is an affine-invariant geodesic metric. The literature is explicit that affine-invariant and log-det divergences are separate categories unless an exact limit or equivalence is stated (Vemulapalli et al., 2015).

5. Quotient-affine geometry on full-rank correlation matrices

For full-rank correlation matrices,

$\log\det$5

the relevant construction is not restriction of the SPD affine-invariant metric to the elliptope, but quotienting out positive diagonal congruences: $\log\det$6 The quotient map is

$\log\det$7

and therefore

$\log\det$8

The quotient-affine metric is induced from the affine-invariant SPD geometry by this diagonal-scaling quotient construction (Thanwerdas et al., 2021).

The exact quotient-affine metric formula is

$\log\det$9

where

$M=\mathrm{SPD}_n,$0

Equivalently,

$M=\mathrm{SPD}_n,$1

Thus the quotient metric is the ambient affine-invariant energy minus the energy of the vertical component removed by the quotient (Thanwerdas et al., 2021).

The quotient-geodesic principle states that geodesics of the quotient metric are projections of horizontal geodesics of the ambient SPD manifold, with explicit formula

$M=\mathrm{SPD}_n,$2

The quotient-affine metric is geodesically complete, but the logarithm is not available in closed form in general (Thanwerdas et al., 2021).

A later treatment sharpens the geometric limitations of this quotient-affine construction. Its sectional curvature takes both negative and positive values, is bounded from below, and is unbounded from above. Consequently, the open elliptope with the quotient-affine metric is not Hadamard, so uniqueness of the Riemannian logarithm and of the Fréchet mean is not ensured (Thanwerdas et al., 2022). This motivates alternative poly-hyperbolic-Cholesky, Euclidean-Cholesky, and log-Euclidean-Cholesky geometries on $M=\mathrm{SPD}_n,$3, which provide Hadamard structures or flat structures with unique logarithms and means (Thanwerdas et al., 2022).

6. Infinite-dimensional extensions and adjacent affine/log-det constructions

The finite-dimensional SPD formulas extend to operator settings in two related infinite-dimensional frameworks. On the cone of positive definite unitized trace-class operators, the Alpha–Beta Log-Det divergence is defined through the extended Fredholm determinant, and its $M=\mathrm{SPD}_n,$4 specialization yields

$M=\mathrm{SPD}_n,$5

with affine-invariant distance

$M=\mathrm{SPD}_n,$6

This is the exact infinite-dimensional analogue of the finite-dimensional affine-invariant logarithmic distance (Quang, 2016).

On the larger cone of positive definite unitized Hilbert–Schmidt operators, the same structural picture is recovered using the extended Hilbert–Carleman determinant. The relative operator is

$M=\mathrm{SPD}_n,$7

and the affine-invariant distance is

$M=\mathrm{SPD}_n,$8

The paper states that the symmetric limit of the infinite-dimensional Alpha–Beta Log-Det family converges to one half the square of this affine-invariant distance (Quang, 2017).

These operator-theoretic constructions preserve the same conceptual distinction seen in finite dimensions: the log-det divergence family is broader, and the affine-invariant metric emerges as its canonical symmetric or logarithmic limit (Quang, 2017). This suggests that the phrase affine-invariant log-det metric is most accurate when used to describe that specific limiting relationship, not as a generic label for every determinant-based SPD dissimilarity.

By contrast, other mathematically proximate uses of $M=\mathrm{SPD}_n,$9 are not metric constructions on the SPD cone. In affine differential geometry of level sets, the identity

$n\times n$0

replaces an explicit third-order contraction in the affine normal direction, but the cited work explicitly states that it does not define a Riemannian metric tensor under the name “affine-invariant log-det metric” (Niu et al., 1 Apr 2026).

The technically correct synthesis is therefore narrow. On SPD matrices and their operator analogues, the core object is the affine-invariant Riemannian metric, with distance

$n\times n$1

or its infinite-dimensional counterpart. Its relation to log-det geometry is twofold: logarithms enter the distance through the log-spectrum, and determinant-based divergence families recover the metric in symmetric or singular limits (Thanwerdas et al., 2019).