---
title: Affine-Invariant Log-Det Metric
url: https://www.emergentmind.com/topics/affine-invariant-log-det-metric
type: topic
---

# Affine-Invariant Log-Det Metric

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 [1906.01349]. 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 [1412.7146]. 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 [1501.02393].

## 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 [1501.02393].

| 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 [1906.01349]. 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 [1412.7146].

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 [2604.01163]. 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 \(O_n\), and the quotient identification is
\[
\mathrm{GL}_n/O_n \simeq \mathrm{SPD}_n,
\qquad [A]\mapsto AA^\top.
\]
A metric \(g\) is affine-invariant exactly when each congruence map is an isometry:
\[
g_{A\Sigma A^\top}(AVA^\top,AWA^\top)=g_\Sigma(V,W).
\]
These are the paper’s formal affine-invariance conditions on SPD matrices [1906.01349].

At the identity, every orthogonally invariant scalar product on \(T_{I_n}\mathrm{SPD}_n\simeq \mathrm{Sym}_n\) has the form
\[
g^1_{I_n}(V_1,W_1)=\alpha\,\mathrm{tr}(V_1W_1)+\beta\,\mathrm{tr}(V_1)\mathrm{tr}(W_1),
\]
with
\[
\alpha>0,\qquad \beta>-\frac{\alpha}{n}.
\]
Transporting tangent vectors by \(\Sigma^{-1/2}\) yields the full affine-invariant family
\[
g^1_\Sigma(V,W)=\alpha\,\mathrm{tr}(\Sigma^{-1}V\Sigma^{-1}W)+\beta\,\mathrm{tr}(\Sigma^{-1}V)\mathrm{tr}(\Sigma^{-1}W).
\]
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 [1906.01349].

The same two-parameter family is also written as
\[
G_\Sigma^{\alpha,\beta}(V,V)=\alpha\,\operatorname{tr}(\Sigma^{-1}V\Sigma^{-1}V)+\beta\,\operatorname{tr}(\Sigma^{-1}V)^2,
\]
with the same parameter constraints. The classical choice is \((\alpha,\beta)=(1,0)\) [2103.04621]. In the \(O(n)\)-invariant classification of SPD metrics, this family is also described as the affine-invariant family
\[
g^{A(\alpha,\beta)}_\Sigma(X,X)=\alpha\,tr\big((\Sigma^{-1}X)^2\big)+\beta\,tr(\Sigma^{-1}X)^2,
\]
and the standard \(\beta=0\) case is the usual affine-invariant metric, sometimes normalized by \(\alpha=1\) or \(\alpha=\tfrac12\) depending on convention [2109.05768].

## 3. Geodesics, logarithms, distance, and curvature

For the affine-invariant case \(f=\mathrm{Id}\), the geodesic through \(\Sigma\) with initial tangent \(V\) is
\[
\gamma^1_{(\Sigma,V)}(t)=\Sigma^{1/2}\exp\!\big(t\,\Sigma^{-1/2}V\Sigma^{-1/2}\big)\Sigma^{1/2},
\]
and the logarithm map is
\[
\mathrm{Log}^1_\Sigma(\Lambda)=\Sigma^{1/2}\log\!\big(\Sigma^{-1/2}\Lambda\Sigma^{-1/2}\big)\Sigma^{1/2}.
\]
The distance is written in the cited source as
\[
d_1(\Sigma,\Lambda)=\sum_{k=1}^n (\log \lambda_k)^2,
\]
where \(\lambda_k\) are the eigenvalues of
\[
\Sigma^{-1/2}\Lambda\Sigma^{-1/2}.
\]
The same source notes that conventionally this is usually the **squared** distance, while many texts write the actual distance as
\[
d_1(\Sigma,\Lambda)=\Big(\sum_{k=1}^n(\log\lambda_k)^2\Big)^{1/2}.
\]
The essential point is the standard log-spectrum formula [1906.01349].

Equivalent formulations appear elsewhere as
\[
d_{\mathrm{AI}(P,Q)=\|\log(P^{-1/2}QP^{-1/2})\|_F
\]
or
\[
d_{\mathrm{AI}(\Sigma,\Lambda)=\sqrt{\alpha}\,\|\log(\Sigma^{-1/2}\Lambda\Sigma^{-1/2})\|_F,
\]
with the trace-extended affine family admitting
\[
d(\Sigma,\Lambda)^2=
\alpha\|\log(\Sigma^{-1/2}\Lambda\Sigma^{-1/2})\|^2+
\beta\,\log(\det(\Sigma^{-1}\Lambda))^2.
\]
This last formula is the clearest place where a literal log-determinant contribution enters the affine-invariant family: it is part of the \((\alpha,\beta)\) extension, not a separately named metric [2109.05768].

The Levi-Civita connection for the classical affine-invariant metric is
\[
(\nabla^G_VW)_{|\Sigma}=\partial_VW-\frac{1}{2}(V\Sigma^{-1}W+W\Sigma^{-1}V),
\]
and the curvature formula is the standard nonpositive-curvature expression in SPD geometry [2103.04621]. A later synthesis states that the affine-invariant metric is geodesically complete, that \(\mathrm{SPD}(n)\) is a Riemannian symmetric space for this geometry, and that the sectional curvature is non-positive and bounded [2109.05768].

This logarithmic distance should not be confused with the log-Euclidean metric
\[
\|\log(P_1)-\log(P_2)\|_F,
\]
which the cited literature treats as a different geodesic distance with different invariance properties; in particular, log-Euclidean distance is not affine-invariant [1501.02393].

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

The Alpha–Beta Log-Det divergence on SPD matrices is defined by
\[
D^{(\alpha,\beta)}_{AB}(\mathbf{P}\|\mathbf{Q})=
\frac{1}{\alpha \beta}
\log \det
\frac{\alpha (\mathbf{P}\mathbf{Q}^{-1})^{\beta}+\beta(\mathbf{P}\mathbf{Q}^{-1})^{-\alpha}}{\alpha+\beta},
\qquad
\alpha\neq 0,\ \beta\neq 0,\ \alpha+\beta\neq 0,
\]
with spectral form
\[
D^{(\alpha,\beta)}_{AB}(\mathbf{P}\|\mathbf{Q})=
\frac{1}{\alpha\beta}\sum_{i=1}^n
\log\left(\frac{\alpha\lambda_i^\beta+\beta\lambda_i^{-\alpha}}{\alpha+\beta}\right),
\]
where \(\lambda_i\) are the eigenvalues of \(\mathbf{P}\mathbf{Q}^{-1}\) [1412.7146].

The key limiting statement is
\[
D^{(0,0)}_{AB}(\mathbf{P}\|\mathbf{Q})=
\frac{1}{2}\|\log(\mathbf{Q}^{-1/2}\mathbf{P}\mathbf{Q}^{-1/2})\|_F^2
=
\frac12\sum_{i=1}^n \log^2(\lambda_i),
\]
so that
\[
d_R(\mathbf{P}\|\mathbf{Q})
=
\|\log(\mathbf{Q}^{-1/2}\mathbf{P}\mathbf{Q}^{-1/2})\|_F
=
\sqrt{\sum_{i=1}^n\log^2(\lambda_i)}.
\]
In the paper’s formulation, the Affine Invariant Riemannian Metric is therefore obtained from the AB family at the singular origin \((\alpha,\beta)=(0,0)\), and \(D^{(0,0)}_{AB}\) is one half of the squared AIRM [1412.7146].

The same source also states the local second-order expansion
\[
D^{(\alpha,\beta)}_{AB}(\mathbf{P}+d\mathbf{P}\|\mathbf{P})
=
\frac12\operatorname{tr}(d\mathbf{P}\mathbf{P}^{-1}d\mathbf{P}\mathbf{P}^{-1}),
\]
which means that the full AB family induces the same local Riemannian metric tensor
\[
g_{\mathbf{P}}(d\mathbf{P},d\mathbf{P})
=
\operatorname{tr}(d\mathbf{P}\mathbf{P}^{-1}d\mathbf{P}\mathbf{P}^{-1}).
\]
This places AIRM not only as a limit point of the family but also as its common infinitesimal geometry [1412.7146].

The literature simultaneously distinguishes this structure from other determinant-based quantities. Stein’s loss,
\[
\operatorname{tr}(\mathbf{P}\mathbf{Q}^{-1})-\log\det(\mathbf{P}\mathbf{Q}^{-1})-n,
\]
the S-divergence or Jensen–Bregman LogDet divergence,
\[
4\log \frac{\det \frac12(\mathbf{P}+\mathbf{Q})}{\sqrt{\det(\mathbf{P})\det(\mathbf{Q})}},
\]
and the Bhattacharyya or LogDet-zero metric
\[
2\sqrt{\log \frac{\det \frac12(\mathbf{P}+\mathbf{Q})}{\sqrt{\det(\mathbf{P})\det(\mathbf{Q})}}}
\]
all belong to the same broader determinant-based taxonomy, but they are not identical to the affine-invariant Riemannian metric [1412.7146].

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 [1501.02393].

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

For full-rank correlation matrices,
\[
Cor^+(n)=\{\Sigma\in Sym^+(n)\mid Diag(\Sigma)=I_n\},
\]
the relevant construction is not restriction of the SPD affine-invariant metric to the elliptope, but **quotienting out positive diagonal congruences**:
\[
\Sigma_2=D\Sigma_1D,\qquad D\in Diag^+(n).
\]
The quotient map is
\[
\pi(\Sigma)=Diag(\Sigma)^{-1/2}\Sigma Diag(\Sigma)^{-1/2}\in Cor^+(n),
\]
and therefore
\[
Cor^+(n)\simeq Sym^+(n)/Diag^+(n).
\]
The quotient-affine metric is induced from the affine-invariant SPD geometry by this diagonal-scaling quotient construction [2103.04621].

The exact quotient-affine metric formula is
\[
g_C^{\alpha,\beta}(X,X)=
G_C^{\alpha,\beta}(X,X)
-
2\mu^\top[\alpha(I_n+A(C))+2\beta\mathds{1}\mathds{1}^\top]\mu,
\]
where
\[
\mu=(I_n+A(C))^{-1}Diag(C^{-1}X)\mathds{1},
\qquad
A(C)=C\bullet C^{-1}.
\]
Equivalently,
\[
g_C(X,X)=G_C(hor(X),hor(X))
=
G_C(X,X)-G_C(ver(X),ver(X)).
\]
Thus the quotient metric is the ambient affine-invariant energy minus the energy of the vertical component removed by the quotient [2103.04621].

The quotient-geodesic principle states that geodesics of the quotient metric are projections of horizontal geodesics of the ambient SPD manifold, with explicit formula
\[
\gamma_{(C,X)}(t)
=
\pi\!\left(
C^{1/2}\exp\!\big(t\,C^{-1/2}hor(X)C^{-1/2}\big)C^{1/2}
\right).
\]
The quotient-affine metric is geodesically complete, but the logarithm is not available in closed form in general [2103.04621].

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 [2201.06282]. This motivates alternative poly-hyperbolic-Cholesky, Euclidean-Cholesky, and log-Euclidean-Cholesky geometries on \(Cor^+(n)\), which provide Hadamard structures or flat structures with unique logarithms and means [2201.06282].

## 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 \(r=0\) specialization yields
\[
D_0^{(\alpha,\beta)}(X,Y)=\frac{1}{2\alpha\beta}d_{\mathrm{aiHS}}^2(X,Y),
\]
with affine-invariant distance
\[
d_{\mathrm{aiHS}}(X,Y)=
\left\|
\log\!\left(Y^{-1/2}XY^{-1/2}\right)
\right\|_{\mathrm{eHS}}.
\]
This is the exact infinite-dimensional analogue of the finite-dimensional affine-invariant logarithmic distance [1610.08087].

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
\[
R=(B+\mu I)^{-1/2}(A+\gamma I)(B+\mu I)^{-1/2},
\]
and the affine-invariant distance is
\[
d_{\mathrm{aiHS}}(A+\gamma I,B+\mu I)=\|\log R\|_{\mathrm{eHS}}.
\]
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 [1702.03425].

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 [1702.03425]. 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 \(\log\det\) are not metric constructions on the SPD cone. In affine differential geometry of level sets, the identity
\[
\partial_i\log\det(H_T)(z)=\sum_{p,q=1}^n f^{pq}(z)f_{pqi}(z)
\]
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” [2604.01163].

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
\[
\|\log(P^{-1/2}QP^{-1/2})\|_F
\]
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 [1906.01349].

Source: https://www.emergentmind.com/topics/affine-invariant-log-det-metric