---
title: Intrinsic Convex Riemannian Proximal Gradient
url: https://www.emergentmind.com/topics/intrinsic-convex-riemannian-proximal-gradient-crpg
type: topic
---

# Intrinsic Convex Riemannian Proximal Gradient

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 [2507.16055]. Related work places CRPG within a broader lineage that includes the proximal point algorithm in CAT(0) spaces [1206.7074], retraction-based Riemannian proximal gradient methods [1909.06065], nonconvex intrinsic manifold proximal-gradient methods [2506.09775], and manifold-identification-based proximal-gradient acceleration on active smooth manifolds in nonsmooth optimization [2012.12936].

## 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 [2507.16055]. The principal setting in which the method is developed is a Hadamard manifold, that is, a complete, simply connected manifold with nonpositive sectional curvature [2507.16055]. This geometric assumption ensures that squared distance is geodesically convex and that the manifold proximal map is well posed [2507.16055][1206.7074].

The method is “intrinsic” in the precise sense that it uses manifold objects only: geodesic distance \(d\), exponential map \(\operatorname{Exp}\), logarithm map \(\log\), Riemannian gradient, and manifold subdifferentials [2507.16055]. This distinguishes it from retraction-based and embedding-based variants that either solve tangent-space subproblems or invoke ambient Euclidean structure [1909.06065][2506.09775].

For a geodesically convex open set \(U\subset \mathcal M\), geodesic convexity is expressed via the subdifferential relation
\[
f(q)\ge f(p)+\langle X_p,\log_p(q)\rangle
\]
for all \(p,q\in U\) and all \(X_p\in \partial f(p)\), while \(\mu\)-strong geodesic convexity strengthens this to
\[
f(q)\ge f(p)+\langle X_p,\log_p(q)\rangle+\frac{\mu}{2}d^2(p,q).
\tag{1}
\]
When \(f\) is differentiable, \(L_f\)-smoothness is defined by
\[
f(q)\le f(p)+\langle \operatorname{grad} f(p),\log_p(q)\rangle+\frac{L_f}{2}d^2(p,q).
\tag{2}
\]
These assumptions are the manifold analogues of Euclidean smoothness and convexity used in proximal-gradient theory [2507.16055].

The optimality condition for the composite problem is
\[
0_p\in \operatorname{grad} g(p)+\partial h(p).
\tag{3}
\]
This is the stationarity notion that CRPG targets, and in the geodesically convex setting it characterizes minimizers [2507.16055][2506.09775].

## 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 \(r\colon \mathcal M\to \mathbb R\cup\{+\infty\}\) and \(\lambda>0\),
\[
\operatorname{prox}^M_{\lambda r}(x)
=
\operatorname*{arg\,min}_{y\in\mathcal M}
\left\{
r(y)+\frac{1}{2\lambda}d(x,y)^2
\right\}.
\tag{4}
\]
On a Hadamard manifold, \(d(\cdot,\cdot)^2\) is \(1\)-strongly convex along geodesics, so the proximal objective is strictly convex and admits a unique minimizer [2507.16055]. 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 [1206.7074].

The first-order optimality condition for \(x^+=\operatorname{prox}^M_{\lambda r}(x)\) is
\[
0\in \partial r(x^+) + \frac{1}{\lambda}\log_{x^+}(x).
\tag{5}
\]

CRPG combines this proximal operator with a geodesic gradient step. At iteration \(k\),
\[
y_k=\operatorname{Exp}_{x_k}\bigl(-\alpha_k\,\operatorname{grad} g(x_k)\bigr),
\tag{6}
\]
followed by
\[
x_{k+1}=\operatorname{prox}^M_{\alpha_k h}(y_k).
\tag{7}
\]
Equivalently, with \(I_\lambda(p)=\operatorname{Exp}_p(-\lambda\operatorname{grad} g(p))\), one can write
\[
x_{k+1}=\operatorname{prox}^M_{\alpha_k h}(I_{\alpha_k}(x_k)).
\]
The intrinsic optimality condition of the composite step is
\[
0\in \partial h(x_{k+1})+\frac{1}{\alpha_k}\log_{x_{k+1}}(y_k).
\tag{8}
\]
In Euclidean space these expressions reduce exactly to standard proximal gradient,
\[
y_k=x_k-\alpha_k\nabla g(x_k),\qquad
x_{k+1}=\operatorname{prox}_{\alpha_k h}(y_k),
\]
so CRPG is a direct manifold generalization rather than a qualitatively different method [2507.16055].

The 2025 intrinsic nonconvex generalization retains the same forward–backward structure and defines the exact geodesic intrinsic step as
\[
y_k=\operatorname{Exp}_{x_k}(-t_k\operatorname{grad} g(x_k)),\qquad
x_{k+1}=\operatorname{prox}_{t_k h}^M(y_k),
\]
with a gradient mapping
\[
G_{g,h}(t;p):=-(1/t)\Log_p(I_{g,h}(t;p)),
\]
and the equivalence \(G(t;p^*)=0\) if and only if \(p^*\) is stationary for \(f=g+h\) [2506.09775]. This places convex CRPG as a specialization of a more general intrinsic Riemannian proximal-gradient framework [2506.09775].

## 3. Step-size rules and fundamental inequalities

CRPG admits both fixed and backtracking step-size strategies. The fixed choice is
\[
\alpha_k=1/L_g.
\]
The backtracking strategy starts with \(s\in (0,2/L_g)\), maintains a warm start \(\lambda_k=\min\{s,\theta\lambda_{k-1}\}\) with \(\theta\ge 1\), and repeatedly shrinks \(\lambda_k\leftarrow \eta\lambda_k\), \(\eta\in(0,1)\), until the smoothness acceptance condition
\[
g\bigl(I_{\lambda_k}(x_k)\bigr)
\le
g(x_k)+
\langle \operatorname{grad} g(x_k),\log_{x_k}I_{\lambda_k}(x_k)\rangle
+\frac{1}{2\lambda_k}d^2\bigl(x_k,I_{\lambda_k}(x_k)\bigr)
\tag{9}
\]
holds [2507.16055]. Under this rule,
\[
\frac{\beta}{L_g}\le \alpha_k\le \frac{\alpha}{L_g},
\tag{17}
\]
with the constants \(\alpha,\beta\) determined by whether one uses the fixed or backtracking regime [2507.16055].

A major theoretical contribution of intrinsic CRPG is a set of manifold prox-gradient inequalities generalizing the Euclidean three-point and descent inequalities. Let
\[
\mathcal L_g(p;q):=
g(p)-g(q)-\langle \operatorname{grad} g(q),\log_q(p)\rangle.
\tag{13}
\]
Then, under the smoothness acceptance condition, the intrinsic prox-gradient step satisfies curvature-dependent inequalities involving \(\mathcal L_g(p;q)\), \(d^2(p,\operatorname{prox}_{\lambda h}(I_\lambda(q)))\), and geometric factors \(\zeta_{1,\kappa_{\min}}\), \(\zeta_{2,\kappa_{\max}}\) that encode sectional curvature bounds [2507.16055]. On Hadamard manifolds, these reduce to the fundamental inequality
\[
\begin{aligned}
f(p)-f(\operatorname{prox}_{\lambda h}(I_\lambda(q)))
&\ge
\mathcal L_g(p;q)
+
\frac{1}{2\lambda}d^2\bigl(p,\operatorname{prox}_{\lambda h}(I_\lambda(q))\bigr) \\
&\quad
-
\frac{\zeta_{1,\kappa_{\min}(d(q,I_\lambda(q))) }+1}{4\lambda}d^2(p,q)
-
\frac{\zeta_{1,\kappa_{\min}(d(p,q)) }-1}{4\lambda}
d^2\bigl(q,I_\lambda(q)\bigr).
\end{aligned}
\tag{15}
\]
A key corollary is the sufficient decrease relation
\[
f(p)-f(\operatorname{prox}_{\lambda h}(I_\lambda(p)))
\ge
\frac{1}{2\lambda}
d^2\bigl(p,\operatorname{prox}_{\lambda h}(I_\lambda(p))\bigr),
\tag{16}
\]
provided the smoothness acceptance holds [2507.16055]. 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 \(\zeta_{1,\kappa_1}\), \(\zeta_{2,\kappa_2}\), \(\sigma_{\kappa_1,\kappa_2}\), and step-size conditions of the form
\[
\lambda\in (0,\min\{\lambda_\delta,\zeta_\delta/L_g\}),
\]
yielding
\[
f(p)-f(I(\lambda;p))
\ge
(\lambda\zeta_\delta-\lambda^2L_g)\|G(\lambda;p)\|^2/2
\]
for the exact geodesic variant [2506.09775]. 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 \(O(1/k)\) worst-case convergence rate in function values [2507.16055]. Writing \(\Delta_k=f(x_k)-f_{\mathrm opt}\), with \(S=\arg\min f\neq\emptyset\), and defining \(R=\operatorname{diam}(L_{x_0})\) and the gradient-step distance \(D_k=d(x_k,I_{\alpha_k}(x_k))\), the analysis proves a two-regime contraction. Either
\[
\frac{\alpha_{k-1}\Delta_{k-1}}{\zeta_1(D_{k-1})d^2(x_{k-1},p^*)}\ge 1
\quad\Rightarrow\quad
\Delta_k\le \tfrac12 \Delta_{k-1},
\tag{19}
\]
or otherwise
\[
\Delta_k
\le
\left(
1-\frac{\beta\,\Delta_{k-1}}{2L_g\,\zeta_1(R_\alpha)\,R^2}
\right)\Delta_{k-1}.
\tag{20}
\]
From this one obtains the iteration bound
\[
k\ge
\max\left\{
\left\lceil \log_2 \frac{\Delta_0}{\varepsilon}\right\rceil,
\left\lceil \frac{\Delta_0-\varepsilon}{\delta\,\varepsilon\,\Delta_0}\right\rceil
\right\},
\tag{21}
\]
with
\[
\delta:=
\min\left\{
\frac{\beta}{2L_g\,\zeta_1(R_\alpha)\,R^2},
\frac{1}{\Delta_0}
\right\},
\]
to guarantee \(\Delta_k\le \varepsilon\) [2507.16055]. The explicit appearance of \(\zeta_1(\cdot)\) reflects curvature dependence.

For \(\varepsilon\)-stationarity in the convex setting, the complexity improves from the nonconvex \(O(1/\varepsilon^2)\) baseline to \(O(1/\varepsilon)\). The theory proves that there exists \(X_{x_k}\in \partial f(x_k)\) with \(\|X_{x_k}\|\le \varepsilon\) after at most
\[
k\le
\left\lceil
\frac{2(L_g+\sigma_{\kappa_{\min}})}{\varepsilon}
\sqrt{\frac{8\alpha}{\delta\,L_g(2-\alpha)}}
\right\rceil,
\tag{22}
\]
where \(G^+\) bounds \(\|\operatorname{grad} g(p)\|\) on the initial sublevel set and \(\sigma_{\kappa_{\min}}\) is a curvature-dependent constant [2507.16055].

Under \(\mu\)-strong geodesic convexity of \(f=g+h\), CRPG achieves a global linear rate:
\[
\Delta_{k+1}
\le
\left[
1-
\min\left\{
\frac{\beta\mu}{4L_g\,\zeta_{1,\kappa_{\min}(R_\alpha)}},
\frac12
\right\}
\right]
\Delta_k,
\tag{23}
\]
hence
\[
k\ge
\left\lceil
\max\left\{
\frac{4L_g\,\zeta_{1,\kappa_{\min}(R_\alpha)}}{\beta\mu},
2
\right\}
\log \frac{\Delta_0}{\varepsilon}
\right\rceil
\tag{24}
\]
suffices for \(\Delta_k\le \varepsilon\) [2507.16055]. Distance convergence to the unique minimizer \(p^*\) is also linear:
\[
\frac{\mu}{2}d^2(x_k,p^*)
\le
\left[
1-
\min\left\{
\frac{\beta\mu}{4L_g\,\zeta_{1,\kappa_{\min}(R_\alpha)}},
\frac12
\right\}
\right]^k
\Delta_0.
\tag{25}
\]

These results sharpen earlier intrinsic manifold proximal-gradient analyses. The 2025 nonconvex intrinsic method proves monotonicity, \(G(\lambda_k;p_k)\to 0\), \(O(1/\sqrt{k})\) complexity to \(\varepsilon\)-stationarity, and global stationarity of accumulation points for general nonconvex composite objectives [2506.09775]. The 2026 inexact-oracle extension proves that under summable oracle errors \(\sum_k\delta_k^{2-q}<\infty\), 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 \(O(1/n)\) or \(O((\ln n)/n)\), depending on error decay [2606.25764]. 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
\[
J_\lambda(x)=\operatorname*{argmin}_{y\in X}
\left\{
f(y)+\frac{1}{2\lambda}d(x,y)^2
\right\}
\]
is uniquely defined, the proximal sequence is Fejér monotone with respect to the minimizer set, and weak \(\Delta\)-convergence to a minimizer follows whenever \(\sum_k\lambda_k=+\infty\) [1206.7074]. CRPG inherits this metric-space proximal logic, but supplements it with a forward smooth step on \(g\).

A second line is retraction-based Riemannian proximal gradient. In that formulation, the update is obtained by solving a tangent-space problem
\[
\ell_k(\eta):=
\langle \operatorname{grad} f(x_k),\eta\rangle_{x_k}
+\frac{L}{2}\|\eta\|_{x_k}^2
+g(R_{x_k}(\eta)),
\]
then setting \(x_{k+1}=R_{x_k}(\eta_k^*)\) [1909.06065]. Under retraction-convexity, this framework yields global convergence, \(O(1/k)\) rates in the convex case, and KL-based convergence of the full sequence to a single stationary point [1909.06065]. When the retraction is the exponential map, retraction convexity reduces to geodesic convexity [1909.06065]. CRPG differs in that it does not formulate the nonsmooth step as a tangent-space surrogate; it uses the manifold proximal map itself [2507.16055].

The intrinsic nonconvex forward–backward formulation of 2025 makes this distinction explicit. It defines
\[
\operatorname{prox}_{t h}^M(x):=
\arg\min_{y\in M} h(y)+\frac{1}{2t}d^2(x,y)
\]
and emphasizes that the method “does not require or work in the embedding” [2506.09775]. However, for retraction-based variants, that work also states that convergence theory remains open without additional assumptions such as a retraction-specific comparison inequality [2506.09775]. 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 \(C^2\) manifold \(\mathcal M\) of differentiability under partial smoothness, prox-regularity, and qualification conditions [2012.12936]. Once \(\mathcal M\) is identified, the objective restricted to \(\mathcal M\) becomes smooth, and Riemannian Newton or truncated Newton steps on \(\mathcal M\) produce superlinear or quadratic convergence under positive-definite Riemannian Hessian assumptions [2012.12936]. 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 \(\mathcal M\) [2012.12936]. 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 [2507.16055].

### Hyperbolic space

In the hyperboloid model of \(\mathcal H^n\), points satisfy \(\langle x,x\rangle_M=-1\) and \(x_{n+1}>0\), with Minkowski metric \(\langle x,y\rangle_M=x^\top J y\) where \(J=\operatorname{diag}(1,\dots,1,-1)\). The distance is
\[
d(x,y)=\operatorname{arcosh}(-\langle x,y\rangle_M),
\]
and the exponential and logarithm maps are
\[
\operatorname{Exp}_x(v)
=
\cosh(\|v\|_M)x
+
\sinh(\|v\|_M)\frac{v}{\|v\|_M},
\]
\[
\log_x(y)
=
\frac{\operatorname{arcosh}(-\langle x,y\rangle_M)}
{\sqrt{\langle x,y\rangle_M^2-1}}
\bigl(y+\langle x,y\rangle_M x\bigr).
\]
For the intrinsic \(\ell_1\)-regularizer \(h(x)=\|x\|_1\), the proximal map
\[
\operatorname{prox}_{\mu\|\cdot\|_1}(x)
=
\arg\min_{y\in\mathcal H^n}
\left\{
\|y\|_1+\frac{1}{2\mu}d^2(x,y)
\right\}
\tag{26}
\]
is computed via a fixed-point iteration \(t_{k+1}=T(t_k)\), where \(T=\sigma_x\circ p_x\), \(T\) is strictly increasing on \([0,x_{\max}]\), has a unique fixed point \(t^*\), and \(t_{k+1}\to t^*\) monotonically [2507.16055]. The proximal solution is \(y^*=p_x(t^*)\) [2507.16055].

For indicator functions of geodesically convex balls, \(h=\chi_{B(p_0,r)}\), the proximal map reduces to geodesic projection:
\[
\operatorname{prox}_{\lambda h}(p)=
\begin{cases}
p,& p\in B(p_0,r),\\
\gamma_{p_0\to p}\!\left(\frac{r}{d(p,p_0)}\right),&\text{otherwise},
\end{cases}
\tag{27}
\]
where \(\gamma_{p_0\to p}\) is the unit-speed geodesic [2507.16055].

### Symmetric positive definite manifolds

On the manifold \(\mathcal P(n)\) of symmetric positive definite matrices with the affine-invariant metric,
\[
d(X,Y)=
\left\|
\log(X^{-1/2}YX^{-1/2})
\right\|_F,
\]
with
\[
\operatorname{Exp}_X(V)=
X^{1/2}\exp(X^{-1/2}VX^{-1/2})X^{1/2},
\]
\[
\log_X(Y)=
X^{1/2}\log(X^{-1/2}YX^{-1/2})X^{1/2}.
\]
For distance penalties \(h(X)=\tau d(X,\bar Q)\), the intrinsic prox is a geodesic shrink along the minimal geodesic from \(X\) to \(\bar Q\):
\[
\operatorname{prox}_{\lambda h}(X)=
\gamma_{X\to \bar Q}\!\bigl(\min\{\lambda\tau,d(X,\bar Q)\}\bigr).
\tag{28}
\]
This yields a direct intrinsic implementation of distance-regularized convex optimization on \(\mathcal P(n)\) [2507.16055].

### Numerical problem classes

The convex CRPG paper reports experiments on three representative tasks [2507.16055]:

| Manifold/problem | Objective structure | Reported comparison |
|---|---|---|
| SPD convex example | \(g(X)=\log(\det X)^4\), \(h(X)=\tau d(X,\bar Q)\) | Fixed and backtracking CRPG reached the same objective value to within \(1e{-11}\) |
| Hyperbolic sparse mean | \(\frac{1}{2N}\sum_i d^2(p,q_i)+\mu\|p\|_1\) | CRPG compared to CPPA |
| Hyperbolic constrained mean | \(\frac{1}{2N}\sum_i d^2(p,p_i)\) with \(h=\chi_{B(p_0,r)}\) | CRPG compared to PGA |

On the SPD example, both fixed-step and backtracking CRPG reached the same objective value to within \(1e{-11}\); backtracking used fewer iterations but about twice the runtime, whereas fixed step used more iterations but lower per-iteration cost [2507.16055]. 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 [2507.16055]. 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 [2507.16055].

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 [2506.09775]. 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 [1909.06065], while intrinsic CRPG uses the manifold proximal map itself [2507.16055][2506.09775]. 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 [2506.09775].

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 [2507.16055]. 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 [2507.16055]. Likewise, the 2025 intrinsic nonconvex paper states that convergence theory for the intrinsic retraction variant remains an open problem without additional assumptions [2506.09775].

A further point of clarification concerns rates. The convex intrinsic CRPG paper proves an \(O(1/k)\) rate for convex problems and a linear rate for strongly convex problems [2507.16055]. A later accelerated development proposes a unified intrinsic Riemannian accelerated proximal-gradient method for geodesically convex and geodesically strongly convex problems under \(\rho\)-retraction-convexity, establishing \(O(1/k^2)\) in the convex case and a geometric rate in the strongly convex case, with an intrinsic safeguard mechanism for nonconvex settings [2509.21897]. 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 [2509.21897]. 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 [2012.12936]. 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 [2012.12936]. 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 [2606.25764]. 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 [2507.16055]. Its significance lies both in its direct theoretical lineage from metric proximal methods [1206.7074] and in its role as a reference point for accelerated, nonconvex, inexact, and structure-identifying manifold optimization methods developed subsequently [2506.09775][2509.21897][2606.25764][2012.12936].

Source: https://www.emergentmind.com/topics/intrinsic-convex-riemannian-proximal-gradient-crpg