Riemannian KL Property Analysis

• The Riemannian KL property is a regularity condition on geodesically complete manifolds that guarantees a gradient–suboptimality differential inequality near critical points.
• It underpins convergence theorems for manifold-based algorithms, including proximal and steepest descent methods, by providing explicit linear or sublinear rate estimates.
• This property extends classical KL results to nonconvex, nonsmooth functions, enabling robust and predictable convergence analysis across diverse optimization problems.

The Riemannian Kurdyka–Łojasiewicz (KL) property extends the key regularity phenomenon underlying sharp global and local convergence of descent-type algorithms from the classical Euclidean to the Riemannian manifold context. It guarantees, for a large class of nonconvex and possibly nonsmooth functions on geodesically complete Riemannian manifolds, a powerful gradient–suboptimality differential inequality near critical points, governing the interplay between residuals and step sizes within iterative optimization schemes. The Riemannian KL property now underpins a host of abstract and algorithmic convergence theorems for inexact proximal, projected, and steepest descent methods on manifolds, and, crucially, provides explicit rates that depend on the KL exponent.

1. Subdifferentials, Gradients, and Quasi-Distances on Riemannian Manifolds

Let $(M,g)$ denote an $n$-dimensional complete Riemannian manifold with metric $g$, distance $d(\cdot,\cdot)$, and geodesic structure. For $f:M\to\mathbb{R}\cup\{+\infty\}$, proper and lower semicontinuous (lsc), the relevant first-order objects are:

• Fréchet subdifferential at $x\in\operatorname{dom}f$:

$$\hat\partial f(x) =\left\{dh_x: h\in C^1(M),\ f-h \text{ minimal at } x\right\}$$

• Limiting subdifferential:

$$\partial f(x) = \left\{v\in T_xM: \exists (x_k,v_k)\to(x,v),\ v_k\in\hat\partial f(x_k),\ f(x_k)\to f(x)\right\}$$

• Critical points are $x$ such that $0\in\partial f(x)$.

A quasi-distance $n$0 satisfies

• $n$1,
• $n$2, continuity in the second argument.

The Riemannian gradient $n$3 at $n$4 satisfies $n$5 for all $n$6.

2. Formal Statement of the Riemannian KL Property

Given a proper lsc $n$7 and a critical point $n$8, $n$9 is said to have the Kurdyka–Łojasiewicz property at $g$0 if there exist:

• $g$1,
• geodesically convex neighborhood $g$2 of $g$3,
• $g$4, continuous, concave, $g$5 on $g$6 with $g$7 and $g$8,

such that for all $g$9 with $d(\cdot,\cdot)$0,

$d(\cdot,\cdot)$1

or, in the subdifferential form,

$d(\cdot,\cdot)$2

A global KL function satisfies this property at every point of $d(\cdot,\cdot)$3 (Bento et al., 2011, Yu et al., 10 Feb 2025).

3. Classes of Functions and Manifold Requirements

The Riemannian KL property is guaranteed for key function classes:

• Analytic functions on analytic manifolds (classical Łojasiewicz setup).
• Definable functions in an o-minimal structure (semi-algebraic, subanalytic, restricted analytic, etc.).
• Morse functions ($d(\cdot,\cdot)$4 with nondegenerate critical points): KL with exponent $d(\cdot,\cdot)$5.
• Nonsmooth, lower-semicontinuous definable functions on definable submanifolds.

For algorithmic convergence, manifold requirements depend on method:

• Proximal-point: $d(\cdot,\cdot)$6 must be complete (no curvature sign constraint), Hopf–Rinow applies.
• Steepest descent: $d(\cdot,\cdot)$7 assumed Hadamard (complete, simply connected, nonpositive sectional curvature; unique minimizing geodesics and global exponential map) (Bento et al., 2011, Yu et al., 10 Feb 2025).

4. KL Property, Exponent, and Desingularizing Functions

A typical desingularizing function is $d(\cdot,\cdot)$8 with $d(\cdot,\cdot)$9, $f:M\to\mathbb{R}\cup\{+\infty\}$0. The key inequality becomes:

$f:M\to\mathbb{R}\cup\{+\infty\}$1

for some $f:M\to\mathbb{R}\cup\{+\infty\}$2. The exponent $f:M\to\mathbb{R}\cup\{+\infty\}$3 quantifies critical point sharpness:

• $f:M\to\mathbb{R}\cup\{+\infty\}$4: “sharp” case—finite-step convergence.
• $f:M\to\mathbb{R}\cup\{+\infty\}$5: local linear convergence rate for descent methods.
• $f:M\to\mathbb{R}\cup\{+\infty\}$6: local sublinear rate, specifically $f:M\to\mathbb{R}\cup\{+\infty\}$7.

For composite or structured potentials (e.g., in ManPPA), KL exponents propagate: if a primal potential $f:M\to\mathbb{R}\cup\{+\infty\}$8 has exponent $f:M\to\mathbb{R}\cup\{+\infty\}$9, so does an auxiliary lifted potential $x\in\operatorname{dom}f$0, and conversely under additional regularity (Yu et al., 10 Feb 2025).

5. KL Property and Convergence of Descent Methods

The KL property is pivotal for establishing full-sequence convergence in inexact (possibly non-Euclidean) descent algorithms, as shown in (Bento et al., 2011, Yu et al., 10 Feb 2025). The abstract convergence framework employs:

• Proximal-point methods: For a sequence $x\in\operatorname{dom}f$1 generated via

$x\in\operatorname{dom}f$2

$x\in\operatorname{dom}f$3

boundedness and compactness in $x\in\operatorname{dom}f$4 yield convergence of $x\in\operatorname{dom}f$5 to a critical point.

• Steepest descent (smooth, $x\in\operatorname{dom}f$6-Lipschitz gradient): For $x\in\operatorname{dom}f$7,

$x\in\operatorname{dom}f$8

$x\in\operatorname{dom}f$9

With $$\hat\partial f(x) =\left\{dh_x: h\in C^1(M),\ f-h \text{ minimal at } x\right\}$$0 (the Riemannian distance), on Hadamard manifolds, this applies to exponential-mapped iterates with Armijo or fixed-step rules.

Fundamental technical estimates, such as

$$\hat\partial f(x) =\left\{dh_x: h\in C^1(M),\ f-h \text{ minimal at } x\right\}$$1

ensure iterates remain in a KL ball and the steps sum to a finite total, so the full sequence converges (Bento et al., 2011).

6. Application: Manifold Proximal Point and Related Algorithms

For constrained and structured problems, the Riemannian KL property enables convergence analysis of manifold-based algorithms such as the Manifold Proximal Point Algorithm (ManPPA). For instance, on $$\hat\partial f(x) =\left\{dh_x: h\in C^1(M),\ f-h \text{ minimal at } x\right\}$$2 (unit sphere),

$$\hat\partial f(x) =\left\{dh_x: h\in C^1(M),\ f-h \text{ minimal at } x\right\}$$3

with $$\hat\partial f(x) =\left\{dh_x: h\in C^1(M),\ f-h \text{ minimal at } x\right\}$$4 a concave regularizer, KL analysis applies both to the primal objective $$\hat\partial f(x) =\left\{dh_x: h\in C^1(M),\ f-h \text{ minimal at } x\right\}$$5 and to lifted auxiliary potentials $$\hat\partial f(x) =\left\{dh_x: h\in C^1(M),\ f-h \text{ minimal at } x\right\}$$6. Concrete residual-step bounds and descent inequalities facilitate direct application of the abstract KL convergence theorems, guaranteeing:

• Global convergence of the sequence to a critical point.
• Explicit linear or sublinear rates determined by the KL exponent $$\hat\partial f(x) =\left\{dh_x: h\in C^1(M),\ f-h \text{ minimal at } x\right\}$$7.
• Finite-step convergence when $$\hat\partial f(x) =\left\{dh_x: h\in C^1(M),\ f-h \text{ minimal at } x\right\}$$8, such as in the weak-sharp minima setting.

The propagation and equivalence of KL exponents between potential functions are established for key problem classes, making rate conclusions robust to various reformulations (Yu et al., 10 Feb 2025).

7. Examples and Scope of KL Functions on Manifolds

The class of KL functions on manifolds is broad, encompassing:

• Real-analytic functions on analytic manifolds (guaranteed classical Łojasiewicz property).
• Definable functions (semi-algebraic, subanalytic, restricted analytic) on definable submanifolds, ensuring the KL property via o-minimal structure theory.
• Morse functions: Smooth objectives with nondegenerate critical points, with exponent $$\hat\partial f(x) =\left\{dh_x: h\in C^1(M),\ f-h \text{ minimal at } x\right\}$$9.
• Nonsmooth, prox-regular, semialgebraic functions—ubiquitous in composite optimization, statistical estimation, and signal processing on manifolds (Bento et al., 2011, Yu et al., 10 Feb 2025).

The Riemannian KL property thereby serves as the geometric-analytic backbone for convergence analysis, rate guarantees, and regularity theory of nonconvex optimization across a broad swath of smooth and nondifferentiable problems on manifolds.