Complete Balanced Metrics

• Complete balanced metrics are defined as Riemannian or Hermitian metrics meeting rigorous invariance, optimal embedding, and duality conditions in various geometric settings.
• They establish deep connections between Kähler geometry, algebraic stability, and information geometry, facilitating precise analytic and geometric characterizations.
• Their applications span from constructing optimal metrics on SPD manifolds to enhancing machine learning classifiers through balanced accuracy in imbalanced datasets.

A complete balanced metric is a Riemannian or Hermitian metric that satisfies rigorous balance properties, often connected with optimal embedding, algebraic stability, or duality on classes of manifolds—such as symmetric positive definite (SPD) matrices, complex projective manifolds, and homogeneous domains. The notion of balance unifies geometric, analytic, and algebraic concepts, establishing critical links to Kähler geometry, information geometry, and the quantitative evaluation of geometric or machine learning structures. Balanced metrics admit precise characterizations, key theorems, and structural results across several mathematical domains.

1. Foundational Definitions and Constructions

A balanced metric, in various settings, is defined in terms of specific invariance, criticality, or duality constraints. In S. Donaldson’s framework for Kähler geometry, a Kähler metric $g$ (with potential $\phi$ and Kähler form $\omega = i\partial\bar\partial\phi$) is balanced if the Rawnsley function

$\varepsilon_\phi(z) = e^{-\phi(z)} K_\phi(z,z)$

is constant on the manifold, where $K_\phi$ is the reproducing kernel for the Hilbert space of $L^2$-integrable holomorphic functions with weight $e^{-\phi}$ (Loi et al., 2010). This condition is equivalent to the fact that the induced projective embedding by an $L^2$-orthonormal basis pulls back the Fubini–Study form to $\omega$ precisely, and it places the embedding at the zero of a moment map for the unitary group action.

For the manifold of SPD matrices, a balanced metric is obtained by interpolating two metrics $g_1$, $g_2$ with canonical (often flat) parallel transports $\Pi^1_{P\to Q}$ and $\Pi^2_{P \to Q}$. Fixing a basepoint $O$ (commonly $I_n$), the balanced bilinear form is

$$g^0_P(X, Y) = \langle \Pi^1_{P\to O} X,\, \Pi^2_{P\to O} Y \rangle_{T_O M}$$

with respect to the Frobenius inner product. If $g^0$ is symmetric and positive-definite at each $P$, it is a new Riemannian metric—termed the balanced metric of $(g_1,g_2)$ (Thanwerdas et al., 2019).

In Hermitian geometry, a metric $h$ is balanced (or semi-Kähler) if the top exterior derivative of the $(n-1)$st wedge of its fundamental form vanishes: $d\left(\omega^{n-1}\right) = 0.$ This is equivalent to the vanishing of the Lee form $\theta = - * d * \omega$ (Giusti et al., 2021).

2. Principal Examples and Metric Families

Balanced metrics arise in several major settings:

• Kähler and Projective Geometry: Balanced metrics on compact Kähler manifolds elucidate optimal projective embeddings and zeros of finite-dimensional moment maps, and their existence is intimately related to the approximation of constant scalar curvature Kähler (cscK) metrics (Loi et al., 2010).
• SPD Manifolds: Canonical metrics include the Euclidean, inverse-Euclidean, affine-invariant, log-Euclidean, and Bogoliubov–Kubo–Mori (BKM) metrics. Affine-invariant and BKM metrics are themselves balanced metrics between pairs of flat structures:
  • $g^A$ (affine-invariant) is the balance of $(g^I, g^E)$;
  • $g^{BKM}$ is the balance of $(g^{LE}, g^E)$ (Thanwerdas et al., 2019).
• Lie Groups: On noncompact real simple Lie groups of inner type and even dimension, there exist invariant balanced Hermitian metrics (with vanishing Chern scalar curvature), while their compact analogs exhibit mutually exclusive properties—either balanced or pluriclosed but not both (Giusti et al., 2021).

New continuous families arise from mixed-power interpolations:

• Mixed-power-Euclidean metrics:

$$g^{E, \theta_1, \theta_2}_\Sigma(X, Y) = \frac{1}{\theta_1 \theta_2} \operatorname{tr}\left(\partial_X \Sigma^{\theta_1} \cdot \partial_Y \Sigma^{\theta_2}\right)$$

These generalize classical metrics, with special cases ($(\theta, \theta)$, $(1,0)$, $(1,-1)$) recovering power-Euclidean, BKM, and affine-invariant metrics, respectively (Thanwerdas et al., 2019).

• Mixed-power-affine metrics: Similarly constructed by pullback of the affine-invariant metric via the power map, yielding a two-parameter family with analogous properties.

Cartan and Cartan–Hartogs domains: Balanced metrics are classified explicitly on irreducible bounded symmetric domains, where multiples $\beta g_B$ of the Bergman metric are balanced iff $\beta > \gamma - 1$ (with $\gamma$ the domain's genus). Only the complex hyperbolic space among Cartan–Hartogs domains admits any balanced multiple (Loi et al., 2010).

3. Algebraic and Variational Characterizations

A crucial link exists between balanced metrics and algebraic invariants:

• On polarized projective manifolds $(X,L)$, basis divisors and their log-canonical thresholds, as introduced by Fujita–Odaka, give rise to $\delta_m$ invariants (Rubinstein et al., 2020). The coercivity threshold $\delta_m$ of a quantized Ding functional on the $m$th Bergman space is equivalent to the existence of an $m$th-level balanced metric:

  $$\delta_m(L) > 1 \iff \exists\;\text{balanced metric in }B_m$$

  This threshold coincides analytically and algebraically,

  $\delta_m(L) = \delta_m^{\rm alg}(L),$

  where the algebraic version is the infimum of log-canonical thresholds over basis divisors, and the analytic version is an integral condition involving exponentials of deviations from Bott–Chern functionals. The limit $\delta(L) = \lim_{m\to\infty} \delta_m(L)$ characterizes uniform K-stability and uniform Ding stability (Rubinstein et al., 2020).

• In toric and equivariant settings, explicit combinatorial formulas for $\delta_m$ simplify characterizations. The approach generalizes to coupled and weighted settings, leading to new stability thresholds for Kähler–Ricci solitons and $g$-solitons.
• In SPD geometry, the principle of balanced metrics generates smooth, positive-definite Riemannian structures, with explicit geodesic formulas in coordinate systems adapted to the given powers $(\theta_1,\theta_2)$ (Thanwerdas et al., 2019).

4. Dual Connections and Information Geometry

Balanced metrics in information geometry are tightly linked to dual connections:

• The one-parameter family of $\alpha$–connections $\nabla^{(\alpha)}$ on a statistical manifold are dual relative to a specified metric (e.g., the Fisher information metric $g^A$). Notably,
  • $\nabla^{(-1)}$ (mixture) is dual to $\nabla^{(+1)}$ (exponential), and both are dual for $g^A$,
  • $g^A$ is the balanced metric of $(g^I, g^E)$,
  • $g^{BKM}$ is the balanced metric of $(g^{LE}, g^E)$ (Thanwerdas et al., 2019).
• The duality is formalized:
  • If $g_1, g_2$ are flat metrics and $g^0$ is their balanced form, then $(M, g^0, \nabla^{g_1}, \nabla^{g_2})$ forms a dually-flat manifold.
• In the context of machine learning classification, balanced accuracy and error rate arise as equally weighted special cases of the more general Expected Cost framework, making them default choices for class-imbalanced settings (Ferrer, 2022).

5. Existence, Uniqueness, and Geometric Properties

Balanced metrics possess strong existence and uniqueness properties, conditioned on algebro-geometric or analytic stability:

• On compact polarized manifolds, the existence of balanced metrics as critical points of the quantized Ding functional is equivalent to coercivity conditions on the functional and to stability thresholds above one (Rubinstein et al., 2020).
• On non-compact inner-type real semisimple Lie groups and their compact quotients, invariant complex structures $J$ and Hermitian balanced metrics $h$ with $d(\omega^{n-1})=0$ and vanishing Chern scalar curvature $s_{\rm Ch}=0$ exist, while no pluriclosed metrics are possible in these settings (Giusti et al., 2021).
• Projectively induced metrics are strictly more general than balanced metrics; in non-compact settings, there are Kähler–Einstein metrics that are projectively induced but for which no scalar multiple is balanced (Loi et al., 2010). This establishes a precise gap between geometric embedding and the balance condition.

6. Applications and Statistical Perspectives

Applications of balanced metrics span geometric analysis, signal processing, and statistical learning:

• In medical data analysis, particularly involving SPD matrices (e.g., diffusion-tensor imaging, covariance-based classification in BCI), continuously parametric balanced metrics, such as the mixed-power-Euclidean family, can be tuned (e.g., via cross-validation) to optimize classification or clustering (Thanwerdas et al., 2019).
• The interpretation via dual connections enables the definition of Bregman-type divergence functionals and facilitates computation in dual affine coordinates.
• In classification metrics for machine learning, balanced accuracy and balanced error rate correct for class prior imbalances and are derived as EC specializations with uniform per-class weights. They are preferred in practical setups lacking reliable error costs or suffering from significant class imbalance (Ferrer, 2022).

7. Summary of Canonical Properties and Theorems

Balanced metrics unify analytic criticality, algebraic stability, and geometric duality. Summary of principal results:

Setting	Balanced Condition	Key Result
Kähler geometry (Loi et al., 2010)	Constant Rawnsley function, balanced embedding	Approximates cscK metrics, zeros of moment map
Projective manifold (Rubinstein et al., 2020)	$\delta_m > 1$ (analytic=$\delta_m^{\mathrm{alg}}$ algebraic)	Characterizes existence/uniqueness; linked to K-stability
SPD matrices (Thanwerdas et al., 2019)	Bilinear form via parallel transports of flat metrics	Interpolates classical metrics; positive-definite, dual structure
Lie groups (Giusti et al., 2021)	$d(\omega^{n-1})=0$ in left-invariant Hermitian structure	Chern scalar curvature vanishes, no pluriclosed metric
Classification metrics (Ferrer, 2022)	Per-class uniform weight under Expected Cost	Balanced accuracy/error as EC special cases

The principle of complete balanced metrics provides a comprehensive framework for constructing, analyzing, and applying canonical metric structures with well-understood geometric, algebraic, and statistical properties across a diverse range of mathematical and applied domains.