Kempf-Ness Optimization for GL-Actions

• Kempf–Ness optimization for GL-actions is a geometric-analytic approach that connects invariant theory with convex optimization using moment maps and gradient flows.
• It employs algorithmic strategies such as Riemannian gradient descent and one-parameter subgroup methods to establish numerical stability and unique minimizers.
• This framework facilitates Morse-theoretic stratification and has broad applications in quantum information, tensor invariance, and geometric invariant theory.

Kempf–Ness optimization for general linear group (GL) actions constitutes a geometric–analytic approach to extremal problems arising in the representation theory of real or complex reductive Lie groups. It is foundational in real and complex geometric invariant theory (GIT), connecting orbit structure, moment map theory, convexity, and numerical criteria for stability. At its core, Kempf–Ness optimization seeks canonical representatives in orbit closures through convex geometric functionals, providing both existence and uniqueness theorems for minimizers, algorithmic gradient flows, and stratification for singular settings.

1. Real and Complex Reductive Groups and Their Linear Actions

A real reductive group $G \subset \GL(V)$, for a finite-dimensional inner-product space $V$, is characterized by a Cartan decomposition $\ggo = \kg \oplus \pg$ of its Lie algebra, where $\kg = \ggo \cap \mathfrak{so}(V)$ (skew-symmetric) and $\pg = \ggo \cap \Sym(V)$ (symmetric). The standard example is $\GL(n, \R) = O(n) \exp(\Sym(n))$, admitting the structure $G = K \exp(\pg)$ with $K$ maximal compact. In the complex setting, $\GL(n, \C)$ and its maximal compact $U(n)$ serve as the prototypical objects. These groups admit natural linear actions on Euclidean or Hermitian vector spaces, tensor spaces, and associated projective and flag varieties (Böhm et al., 2017, Hoskins, 2012, Maculan, 2014).

2. Kempf–Ness Functionals and the Moment Map

For a linear $G$-action on $V$ (real or complex), the Kempf–Ness function attaches to each vector $v \in V$ the function $F_v(g) = \|g \cdot v\|^2$. Geometrically, $F_v(g)$ measures the squared distance from the orbit point $g \cdot v$ to the origin. Critical points of $F_v$ along directions in $\pg$ correspond to vanishing of the real moment map

$\mu(v) \in \pg, \quad \langle \mu(v), A \rangle = \frac{1}{\|v\|^2} \langle A \cdot v, v \rangle, \quad \forall A \in \pg.$

This moment map captures the “optimal balance” of the orbit and forms the basis for both variational and stratification results (Böhm et al., 2017, Biliotti, 2019, Biliotti et al., 2016).

3. The Kempf–Ness Theorem and Closed Orbits

The Kempf–Ness theorem establishes a crucial link between critical points of the Kempf–Ness functional and closedness of orbits. For real reductive $G$, the orbit $G \cdot v$ is closed in $V$ if and only if there exists $g \in G$ such that $\mu(g \cdot v) = 0$; such $g \cdot v$ is (up to $K$) the unique minimizer of $F_v$, canonically representing the orbit in the orbit closure (Böhm et al., 2017):

• In the abelian case, this minimal point is uniquely determined by strict convexity.
• For general $G$, the analysis reduces via polar decomposition $G = K \exp(\pg)$ and convexity along geodesics in the symmetric space $G/K$.
• Non-closed orbits “drift to infinity” and have no finite critical point for the moment map.

In the context of GIT, this result realizes the symplectic quotient as the quotient by $K$ of the zero-level set of the moment map and establishes deep correspondence with algebro-geometric quotient constructions (Maculan, 2014, Hoskins, 2012).

4. Stratification: Kirwan–Ness and Morse–Hesselink Theory

Beyond stable orbits, Kempf–Ness optimization provides a Morse-theoretic stratification of the null cone and orbit closures via the negative gradient flow of the “energy” functional $E(v) = \|\mu(v)\|^2$. Critical points of $E$ (solutions to $\mu(v) = \beta \in \pg$) yield smooth, $G$-invariant submanifolds $S_\beta$, stratifying the null cone according to their asymptotic limit under flow (Böhm et al., 2017, Hoskins, 2012):

• Each $S_\beta = \{ v: \lim_{t \to \infty} v(t) \in \Crit(E) \cap \mu^{-1}(K \cdot \beta) \}$ where $v(t)$ solves $\dot{v} = -\nabla E(v)$.
• Hesselink’s 1-PS stratification (by adapted one-parameter subgroups) coincides with the Morse stratification induced by the moment map norm squared (Hoskins, 2012).
• Stratification provides a canonical decomposition of representation spaces and moduli, pivotal for singular orbits and semistable points.

5. Algorithmic and Variational Approaches to Optimization

Kempf–Ness optimization motivates intrinsic algorithmic strategies:

• Gradient Flow: The gradient descent for $g$ minimizing $F_v$ on $G/K$ takes the form

$g^{-1} \dot{g} = -\pr_{\pg} \left( \mu(g \cdot v) \right),$

with discrete time update $g \gets \exp(-\eta \mu(g \cdot v)) g$. This approach converges to the unique $K$-orbit minimizer when the orbit is closed (Böhm et al., 2017, Hoskins, 2012, Maculan, 2014).

• One-Parameter Subgroups: The method of steepest descent along abelian slices ($\exp(\tg)$) provides optimal destabilizing directions, foundational for Hilbert–Mumford theory and numerical criteria (Böhm et al., 2017, Hoskins, 2012).
• Generalizations: For actions on spaces of tensors, such as entanglement polytopes, generalized gradient flows (e.g., $Q$-gradient flows on Hadamard manifolds) minimize convex, invariant cost functionals associated with moment maps, enabling convex optimization over moment polytopes in quantum information and noncommutative invariant theory (Hirai, 15 Nov 2025).

Method	Flow Equation	Key Features
Riemannian Gradient	$g^{-1}\dot{g} = -\pr_\pg[\mu(gv)]$	Converges to closed orbit representative
Subgradient Iteration	$g_{k+1} = \exp(-\tau \mu(g_k v)) g_k$	Discrete, practical for computation
1-PS Descent	Minimize along $\exp(tA)$, $A\in\pg$	Efficient in abelian/diagonalizable settings
Q-Gradient Flow (generalized)	$\dot{x}(t) = -\nabla^Q f(x(t))$	Applies to Hadamard manifolds, moment polytopes

Continuous gradient flows also yield optimality certificates via duality, and discretizations are the basis of algorithms in invariant-theoretic and quantum applications (Hirai, 15 Nov 2025).

6. Generalizations: Tensor Actions and Hadamard Manifolds

For Cartesian actions of $GL_{n_1}(\C) \times \dots \times GL_{n_d}(\C)$ on tensor spaces $\mathcal{V} = \bigotimes_{i=1}^d \C^{n_i}$, the moment map

$$\mu_k(v) H = \frac{\langle v, (I, ..., H, ..., I) v \rangle}{\|v\|^2}$$

captures partial traces (“reduced density matrices”), and the image of the $G$-orbit under the joint spectra defines the entanglement polytope $\Delta(v)$ (Hirai, 15 Nov 2025). The convex geodesic optimization problem minimizes a $K$-invariant convex function $S$ over $\Delta(v)$, with Q-gradient flows providing a theoretical and algorithmic framework. Applications include:

• Quantum functionals: Variational principles for quantum entropy and functionals (Hirai, 15 Nov 2025).
• G-stable rank and Noncommutative rank: Convex programs characterizing rank via moment maps (Hirai, 15 Nov 2025).
• Convex Duality: Inf–sup duality delivers both optimization and certification of minimality in terms of asymptotic cones (flags and weights).

7. Stability Criteria, Convexity, and Hilbert–Mumford Theory

Numerical stability for the action is characterized via the maximal weight function, defined by the asymptotic slope of the Kempf–Ness functional along geodesic rays. For real and complex $GL(n)$, the Hilbert–Mumford criterion and its convex analytic generalizations assert:

• Stability (resp. semistability, polystability) is characterized by positivity (resp. nonnegativity) of maximal weights for all directions in $\pg$ (Biliotti et al., 2016, Biliotti, 2019).
• These conditions reduce in the $GL(n,\R)$ case to explicit inequalities on supports and components, producing combinatorial/polyhedral characterizations for orbits and convex hulls of moment map images (Biliotti, 2019).
• The affine and projective GIT quotients are homeomorphic to the symplectic reductions via the zero loci of the moment map, with Kempf–Ness optimization producing canonical quotient representatives (Hoskins, 2012, Maculan, 2014).

• Real geometric invariant theory and analytic approaches: (Böhm et al., 2017)
• Generalized gradient flows and tensor invariants: (Hirai, 15 Nov 2025)
• Kempf–Ness in both archimedean and non-archimedean settings: (Maculan, 2014)
• Morse and 1-PS stratification, affine Kempf–Ness: (Hoskins, 2012)
• Explicit convexity, real Hilbert–Mumford, gradient maps: (Biliotti, 2019, Biliotti et al., 2016)