Intrinsic Convex Riemannian Proximal Gradient

• Intrinsic Convex Riemannian Proximal Gradient (CRPG) is a manifold optimization framework that solves composite convex problems using intrinsic geodesic steps and proximal maps.
• It alternates a geodesic gradient update on the smooth part with an intrinsic proximal step on the nonsmooth, geodesically convex component, achieving sublinear or linear convergence rates.
• Operating directly on Hadamard manifolds via the exponential and manifold proximal maps, CRPG distinguishes itself from embedding-based or retraction-based methods.

Searching arXiv for the cited CRPG and related Riemannian proximal-gradient papers to ground the article in current literature. Intrinsic Convex Riemannian Proximal Gradient (CRPG) denotes an intrinsic manifold optimization framework for composite objectives of the form $f=g+h$ on Riemannian manifolds, especially Hadamard manifolds, where $g$ is smooth and $h$ is proper, lower semicontinuous, and geodesically convex. Its defining feature is that both the forward step and the backward step are formulated directly on the manifold through the exponential map and the manifold proximal map, rather than through an ambient embedding or a tangent-space surrogate. In the formulation developed for convex optimization, CRPG performs a geodesic gradient step on $g$ followed by an intrinsic proximal step on $h$, and admits a sublinear convergence rate in the convex case and a linear convergence rate in the strongly geodesically convex case (Bergmann et al., 21 Jul 2025). Related work places CRPG within a broader lineage that includes the proximal point algorithm in CAT(0) spaces (Bacak, 2012), retraction-based Riemannian proximal gradient methods (Huang et al., 2019), nonconvex intrinsic manifold proximal-gradient methods (Bergmann et al., 11 Jun 2025), and manifold-identification-based proximal-gradient acceleration on active smooth manifolds in nonsmooth optimization (Bareilles et al., 2020).

1. Definition and problem class

CRPG addresses composite optimization problems on a Riemannian manifold $\mathcal M$ of the form

$\min_{p\in\mathcal M} f(p)=g(p)+h(p),$

where $g$ is smooth and $h$ is possibly nonsmooth but geodesically convex (Bergmann et al., 21 Jul 2025). The principal setting in which the method is developed is a Hadamard manifold, that is, a complete, simply connected manifold with nonpositive sectional curvature (Bergmann et al., 21 Jul 2025). This geometric assumption ensures that squared distance is geodesically convex and that the manifold proximal map is well posed (Bergmann et al., 21 Jul 2025, Bacak, 2012).

The method is “intrinsic” in the precise sense that it uses manifold objects only: geodesic distance $d$, exponential map $g$0, logarithm map $g$1, Riemannian gradient, and manifold subdifferentials (Bergmann et al., 21 Jul 2025). This distinguishes it from retraction-based and embedding-based variants that either solve tangent-space subproblems or invoke ambient Euclidean structure (Huang et al., 2019, Bergmann et al., 11 Jun 2025).

For a geodesically convex open set $g$2, geodesic convexity is expressed via the subdifferential relation

$g$3

for all $g$4 and all $g$5, while $g$6-strong geodesic convexity strengthens this to

$g$7

When $g$8 is differentiable, $g$9-smoothness is defined by

$h$0

These assumptions are the manifold analogues of Euclidean smoothness and convexity used in proximal-gradient theory (Bergmann et al., 21 Jul 2025).

The optimality condition for the composite problem is

$h$1

This is the stationarity notion that CRPG targets, and in the geodesically convex setting it characterizes minimizers (Bergmann et al., 21 Jul 2025, Bergmann et al., 11 Jun 2025).

2. Intrinsic proximal map and algorithmic update

The central geometric primitive underlying CRPG is the manifold proximal map. For a proper, lower semicontinuous, geodesically convex function $h$2 and $h$3,

$h$4

On a Hadamard manifold, $h$5 is $h$6-strongly convex along geodesics, so the proximal objective is strictly convex and admits a unique minimizer (Bergmann et al., 21 Jul 2025). This is consistent with the broader CAT(0) proximal-point theory, where resolvents are uniquely defined because the squared distance term is strictly convex along geodesics (Bacak, 2012).

The first-order optimality condition for $h$7 is

$h$8

CRPG combines this proximal operator with a geodesic gradient step. At iteration $h$9,

$g$0

followed by

$g$1

Equivalently, with $g$2, one can write

$g$3

The intrinsic optimality condition of the composite step is

$g$4

In Euclidean space these expressions reduce exactly to standard proximal gradient,

$g$5

so CRPG is a direct manifold generalization rather than a qualitatively different method (Bergmann et al., 21 Jul 2025).

The 2025 intrinsic nonconvex generalization retains the same forward–backward structure and defines the exact geodesic intrinsic step as

$g$6

with a gradient mapping

$g$7

and the equivalence $g$8 if and only if $g$9 is stationary for $h$0 (Bergmann et al., 11 Jun 2025). This places convex CRPG as a specialization of a more general intrinsic Riemannian proximal-gradient framework (Bergmann et al., 11 Jun 2025).

3. Step-size rules and fundamental inequalities

CRPG admits both fixed and backtracking step-size strategies. The fixed choice is

$h$1

The backtracking strategy starts with $h$2, maintains a warm start $h$3 with $h$4, and repeatedly shrinks $h$5, $h$6, until the smoothness acceptance condition

$h$7

holds (Bergmann et al., 21 Jul 2025). Under this rule,

$h$8

with the constants $h$9 determined by whether one uses the fixed or backtracking regime (Bergmann et al., 21 Jul 2025).

A major theoretical contribution of intrinsic CRPG is a set of manifold prox-gradient inequalities generalizing the Euclidean three-point and descent inequalities. Let

$\mathcal M$0

Then, under the smoothness acceptance condition, the intrinsic prox-gradient step satisfies curvature-dependent inequalities involving $\mathcal M$1, $\mathcal M$2, and geometric factors $\mathcal M$3, $\mathcal M$4 that encode sectional curvature bounds (Bergmann et al., 21 Jul 2025). On Hadamard manifolds, these reduce to the fundamental inequality

$\mathcal M$5

A key corollary is the sufficient decrease relation

$\mathcal M$6

provided the smoothness acceptance holds (Bergmann et al., 21 Jul 2025). This is the direct manifold analogue of the Euclidean one-step decrease estimate.

The broader intrinsic nonconvex method also establishes sufficient decrease, but with curvature-dependent constants $\mathcal M$7, $\mathcal M$8, $\mathcal M$9, and step-size conditions of the form

$\min_{p\in\mathcal M} f(p)=g(p)+h(p),$0

yielding

$\min_{p\in\mathcal M} f(p)=g(p)+h(p),$1

for the exact geodesic variant (Bergmann et al., 11 Jun 2025). This suggests that curvature-sensitive descent control is a structural feature of intrinsic manifold proximal-gradient methods, not only of the convex specialization.

4. Convergence theory

For convex problems, CRPG attains a sublinear $\min_{p\in\mathcal M} f(p)=g(p)+h(p),$2 worst-case convergence rate in function values (Bergmann et al., 21 Jul 2025). Writing $\min_{p\in\mathcal M} f(p)=g(p)+h(p),$3, with $\min_{p\in\mathcal M} f(p)=g(p)+h(p),$4, and defining $\min_{p\in\mathcal M} f(p)=g(p)+h(p),$5 and the gradient-step distance $\min_{p\in\mathcal M} f(p)=g(p)+h(p),$6, the analysis proves a two-regime contraction. Either

$\min_{p\in\mathcal M} f(p)=g(p)+h(p),$7

or otherwise

$\min_{p\in\mathcal M} f(p)=g(p)+h(p),$8

From this one obtains the iteration bound

$\min_{p\in\mathcal M} f(p)=g(p)+h(p),$9

with

$g$0

to guarantee $g$1 (Bergmann et al., 21 Jul 2025). The explicit appearance of $g$2 reflects curvature dependence.

For $g$3-stationarity in the convex setting, the complexity improves from the nonconvex $g$4 baseline to $g$5. The theory proves that there exists $g$6 with $g$7 after at most

$g$8

where $g$9 bounds $h$0 on the initial sublevel set and $h$1 is a curvature-dependent constant (Bergmann et al., 21 Jul 2025).

Under $h$2-strong geodesic convexity of $h$3, CRPG achieves a global linear rate: $h$4 hence

$h$5

suffices for $h$6 (Bergmann et al., 21 Jul 2025). Distance convergence to the unique minimizer $h$7 is also linear: $h$8

These results sharpen earlier intrinsic manifold proximal-gradient analyses. The 2025 nonconvex intrinsic method proves monotonicity, $h$9, $d$0 complexity to $d$1-stationarity, and global stationarity of accumulation points for general nonconvex composite objectives (Bergmann et al., 11 Jun 2025). The 2026 inexact-oracle extension proves that under summable oracle errors $d$2, the search directions vanish, function values converge at accumulation points, and every accumulation point is stationary; under an additional strong inexact oracle and retraction-convexity, function values converge to the optimal value at $d$3 or $d$4, depending on error decay (Huang et al., 24 Jun 2026). This suggests a growing theory in which exact convex CRPG is one particularly well-structured regime within a broader intrinsic proximal-gradient family.

5. Relation to proximal point, retraction-based RPG, and manifold identification

CRPG is best understood as one branch of a larger development in manifold nonsmooth optimization. The deepest historical root is the proximal point algorithm in CAT(0) spaces, where the resolvent

$d$5

is uniquely defined, the proximal sequence is Fejér monotone with respect to the minimizer set, and weak $d$6-convergence to a minimizer follows whenever $d$7 (Bacak, 2012). CRPG inherits this metric-space proximal logic, but supplements it with a forward smooth step on $d$8.

A second line is retraction-based Riemannian proximal gradient. In that formulation, the update is obtained by solving a tangent-space problem

$d$9

then setting $g$00 (Huang et al., 2019). Under retraction-convexity, this framework yields global convergence, $g$01 rates in the convex case, and KL-based convergence of the full sequence to a single stationary point (Huang et al., 2019). When the retraction is the exponential map, retraction convexity reduces to geodesic convexity (Huang et al., 2019). CRPG differs in that it does not formulate the nonsmooth step as a tangent-space surrogate; it uses the manifold proximal map itself (Bergmann et al., 21 Jul 2025).

The intrinsic nonconvex forward–backward formulation of 2025 makes this distinction explicit. It defines

$g$02

and emphasizes that the method “does not require or work in the embedding” (Bergmann et al., 11 Jun 2025). However, for retraction-based variants, that work also states that convergence theory remains open without additional assumptions such as a retraction-specific comparison inequality (Bergmann et al., 11 Jun 2025). This marks a technical divide between exact intrinsic geodesic CRPG and retraction-based surrogates.

A distinct but related viewpoint arises from active-manifold identification in nonsmooth Euclidean composite optimization. The 2020 work on “Newton acceleration on manifolds identified by proximal-gradient methods” shows that proximal-gradient iterates can finitely identify a $g$03 manifold $g$04 of differentiability under partial smoothness, prox-regularity, and qualification conditions (Bareilles et al., 2020). Once $g$05 is identified, the objective restricted to $g$06 becomes smooth, and Riemannian Newton or truncated Newton steps on $g$07 produce superlinear or quadratic convergence under positive-definite Riemannian Hessian assumptions (Bareilles et al., 2020). The paper does not define CRPG explicitly, but it interprets the identified-manifold phase as an intrinsic Riemannian proximal-gradient perspective in which the problem becomes smooth on $g$08 (Bareilles et al., 2020). This suggests a conceptual connection: in convex CRPG, the manifold is given a priori by the problem geometry, whereas in partial-smoothness settings it is revealed adaptively by proximal identification.

6. Geometry, examples, and applications

The theoretical development of intrinsic CRPG is motivated by concrete Hadamard manifolds where manifold proximal maps can be computed either in closed form or through simple inner routines (Bergmann et al., 21 Jul 2025).

Hyperbolic space

In the hyperboloid model of $g$09, points satisfy $g$10 and $g$11, with Minkowski metric $g$12 where $g$13. The distance is

$g$14

and the exponential and logarithm maps are

$g$15

$g$16

For the intrinsic $g$17-regularizer $g$18, the proximal map

$g$19

is computed via a fixed-point iteration $g$20, where $g$21, $g$22 is strictly increasing on $g$23, has a unique fixed point $g$24, and $g$25 monotonically (Bergmann et al., 21 Jul 2025). The proximal solution is $g$26 (Bergmann et al., 21 Jul 2025).

For indicator functions of geodesically convex balls, $g$27, the proximal map reduces to geodesic projection: $g$28 where $g$29 is the unit-speed geodesic (Bergmann et al., 21 Jul 2025).

Symmetric positive definite manifolds

On the manifold $g$30 of symmetric positive definite matrices with the affine-invariant metric,

$g$31

with

$g$32

$g$33

For distance penalties $g$34, the intrinsic prox is a geodesic shrink along the minimal geodesic from $g$35 to $g$36: $g$37 This yields a direct intrinsic implementation of distance-regularized convex optimization on $g$38 (Bergmann et al., 21 Jul 2025).

Numerical problem classes

The convex CRPG paper reports experiments on three representative tasks (Bergmann et al., 21 Jul 2025):

Manifold/problem	Objective structure	Reported comparison
SPD convex example	$g$39, $g$40	Fixed and backtracking CRPG reached the same objective value to within $g$41
Hyperbolic sparse mean	$g$42	CRPG compared to CPPA
Hyperbolic constrained mean	$g$43 with $g$44	CRPG compared to PGA

On the SPD example, both fixed-step and backtracking CRPG reached the same objective value to within $g$45; backtracking used fewer iterations but about twice the runtime, whereas fixed step used more iterations but lower per-iteration cost (Bergmann et al., 21 Jul 2025). On sparse means in hyperbolic space, CRPG with fixed steps had the fewest iterations and lowest runtime among the reported methods, while CPPA generally required more time (Bergmann et al., 21 Jul 2025). On constrained hyperbolic means, CRPG with fixed step had the lowest runtime except in isolated cases; backtracking CRPG was slightly slower than PGA but comparable (Bergmann et al., 21 Jul 2025).

The 2025 nonconvex intrinsic paper complements these convex experiments with intrinsically formulated sparse PCA on the oblique manifold, Grassmannian median-type problems, and row-sparse low-rank recovery, showing monotone decrease of the objective, convergence of gradient mappings, and practical advantages of backtracking over constant steps in several settings (Bergmann et al., 11 Jun 2025). This suggests that the intrinsic formalism is not limited to Hadamard manifolds, though the strongest convex theory is presently formulated there.

7. Limitations, misconceptions, and later developments

A common misconception is that any “Riemannian proximal gradient” method is intrinsic. This is not the case. The retraction-based RPG framework solves a tangent-space proximal surrogate and updates through a retraction (Huang et al., 2019), while intrinsic CRPG uses the manifold proximal map itself (Bergmann et al., 21 Jul 2025, Bergmann et al., 11 Jun 2025). The distinction matters both conceptually and technically: the intrinsic method does not require ambient embeddings or tangent-space relaxations, whereas retraction-based methods often do (Bergmann et al., 11 Jun 2025).

Another misconception is that manifold proximal-gradient theory is settled once convexity is assumed. In fact, several limitations remain explicit in the literature. The convex CRPG analysis relies on Hadamard geometry, since nonpositive curvature guarantees global existence and uniqueness of geodesics and strong convexity of squared distance (Bergmann et al., 21 Jul 2025). For positively curved manifolds, one must restrict to geodesically convex subsets of bounded diameter, and a global theory of intrinsic CRPG comparable to the Hadamard case is not provided there (Bergmann et al., 21 Jul 2025). Likewise, the 2025 intrinsic nonconvex paper states that convergence theory for the intrinsic retraction variant remains an open problem without additional assumptions (Bergmann et al., 11 Jun 2025).

A further point of clarification concerns rates. The convex intrinsic CRPG paper proves an $g$46 rate for convex problems and a linear rate for strongly convex problems (Bergmann et al., 21 Jul 2025). A later accelerated development proposes a unified intrinsic Riemannian accelerated proximal-gradient method for geodesically convex and geodesically strongly convex problems under $g$47-retraction-convexity, establishing $g$48 in the convex case and a geometric rate in the strongly convex case, with an intrinsic safeguard mechanism for nonconvex settings (Feng et al., 26 Sep 2025). That work does not explicitly use the acronym CRPG, but it describes its method, under geodesic convexity and intrinsic exponential-map-based updates, as an intrinsic Riemannian proximal-gradient scheme with acceleration (Feng et al., 26 Sep 2025). This suggests that accelerated CRPG is now an active direction rather than a settled endpoint.

The relation between convex intrinsic CRPG and Euclidean active-manifold acceleration is also sometimes overstated. The manifold-identification framework of proximal-gradient methods in nonsmooth nonconvex optimization does show that one can pass from a proximal-gradient phase to Riemannian Newton-type steps on an identified active manifold, with superlinear or quadratic convergence (Bareilles et al., 2020). However, this is not the same theory as convex CRPG on a given manifold. A plausible implication is that both viewpoints are manifestations of a broader principle: proximal maps can expose latent smooth geometry, after which manifold methods become natural (Bareilles et al., 2020). But the formal assumptions and guarantees differ substantially.

Finally, the inexact-oracle literature indicates that intrinsic proximal-gradient methods can tolerate imperfect first-order information. The 2026 RPG-IO analysis proves global convergence under summable oracle errors, KL-based full-sequence convergence, and convex function-value rates under a strong inexact oracle (Huang et al., 24 Jun 2026). This suggests that future CRPG implementations may increasingly emphasize oracle inexactness, approximate proximal subproblem solves, and curvature-adaptive complexity guarantees.

Intrinsic Convex Riemannian Proximal Gradient therefore occupies a precise position in contemporary optimization theory: it is the intrinsic manifold counterpart of Euclidean proximal gradient for composite convex problems, grounded in Hadamard geometry, characterized by manifold-valued forward–backward steps, and supported by curvature-aware descent inequalities and convergence guarantees (Bergmann et al., 21 Jul 2025). Its significance lies both in its direct theoretical lineage from metric proximal methods (Bacak, 2012) and in its role as a reference point for accelerated, nonconvex, inexact, and structure-identifying manifold optimization methods developed subsequently (Bergmann et al., 11 Jun 2025, Feng et al., 26 Sep 2025, Huang et al., 24 Jun 2026, Bareilles et al., 2020).