Curvature-Aware Riemannian Consensus

• The paper introduces a consensus update that replaces linear differences with logarithmic maps and retraction via exponential maps, ensuring intrinsic operations on manifolds.
• Curvature bounds directly influence admissible step sizes, guaranteeing valid operation within the injectivity radius and controlling convergence through Hessian and cosine-law comparisons.
• Variants extend the basic protocol to include Lie-group translations, gradient tracking, and stochastic diffusion, enhancing decentralized optimization across diverse manifold settings.

Curvature-aware Riemannian consensus step denotes an intrinsic update used in distributed consensus and decentralized optimization when agent states lie on a Riemannian manifold rather than in a Euclidean vector space. Its defining operation is to replace linear differences by logarithmic maps, perform mixing in the tangent space at the current point, and return to the manifold by the exponential map. In the foundational formulation, each node follows the negative gradient of a pairwise squared-distance energy, while later variants use weighted Fréchet-type objectives, Lie-group left translations, gradient tracking, or stochastic diffusion. Across these formulations, curvature is not merely a background geometric feature: it enters the admissible step size, the domain on which the logarithm map is single-valued, the Hessian or cosine-law comparison bounds used in the proofs, and the resulting contraction or regret constants (Tron et al., 2012, Sahinoglu et al., 9 Sep 2025).

1. Intrinsic form of the consensus update

The basic intrinsic consensus objective on a connected undirected graph $G=(V,E)$ is

$$\varphi(\mathbf{x})=\frac12\sum_{\{i,j\}\in E} d^2(x_i,x_j), \qquad \mathbf{x}=(x_1,\dots,x_N)\in M^N,$$

with $d(\cdot,\cdot)$ the geodesic distance on $M$. Using

$\frac12 \grad_x d^2(x,y) = -\log_x(y),$

the gradient with respect to the $i$-th component is

$\grad_{x_i}\varphi(\mathbf{x}) = -\sum_{j\in N_i}\log_{x_i}(x_j).$

The resulting Riemannian consensus protocol is

$$x_i(l+1) = \exp_{x_i(l)}\!\left( \varepsilon \sum_{j\in N_i}\log_{x_i(l)}(x_j(l)) \right),$$

equivalently, a gradient step on $\varphi$ executed intrinsically on the manifold (Tron et al., 2012).

This construction is the direct manifold analogue of Euclidean averaging consensus. In Euclidean space, subtraction $x_j-x_i$ is available globally and addition $$\varphi(\mathbf{x})=\frac12\sum_{\{i,j\}\in E} d^2(x_i,x_j), \qquad \mathbf{x}=(x_1,\dots,x_N)\in M^N,$$0 stays in the state space. On a manifold, subtraction is replaced by $$\varphi(\mathbf{x})=\frac12\sum_{\{i,j\}\in E} d^2(x_i,x_j), \qquad \mathbf{x}=(x_1,\dots,x_N)\in M^N,$$1, the tangent vector at $$\varphi(\mathbf{x})=\frac12\sum_{\{i,j\}\in E} d^2(x_i,x_j), \qquad \mathbf{x}=(x_1,\dots,x_N)\in M^N,$$2 pointing toward $$\varphi(\mathbf{x})=\frac12\sum_{\{i,j\}\in E} d^2(x_i,x_j), \qquad \mathbf{x}=(x_1,\dots,x_N)\in M^N,$$3, and addition is replaced by $$\varphi(\mathbf{x})=\frac12\sum_{\{i,j\}\in E} d^2(x_i,x_j), \qquad \mathbf{x}=(x_1,\dots,x_N)\in M^N,$$4, which follows the geodesic starting at $$\varphi(\mathbf{x})=\frac12\sum_{\{i,j\}\in E} d^2(x_i,x_j), \qquad \mathbf{x}=(x_1,\dots,x_N)\in M^N,$$5 with initial velocity $$\varphi(\mathbf{x})=\frac12\sum_{\{i,j\}\in E} d^2(x_i,x_j), \qquad \mathbf{x}=(x_1,\dots,x_N)\in M^N,$$6. When $$\varphi(\mathbf{x})=\frac12\sum_{\{i,j\}\in E} d^2(x_i,x_j), \qquad \mathbf{x}=(x_1,\dots,x_N)\in M^N,$$7, $$\varphi(\mathbf{x})=\frac12\sum_{\{i,j\}\in E} d^2(x_i,x_j), \qquad \mathbf{x}=(x_1,\dots,x_N)\in M^N,$$8 and $$\varphi(\mathbf{x})=\frac12\sum_{\{i,j\}\in E} d^2(x_i,x_j), \qquad \mathbf{x}=(x_1,\dots,x_N)\in M^N,$$9, so the Riemannian update reduces exactly to the classical linear rule (Tron et al., 2012).

Later formulations preserve the same geometric pattern while changing the local objective. On Lie groups with a bi-invariant metric, the consensus force can be written as

$d(\cdot,\cdot)$0

which is the Lie-group form of the gradient of a squared-distance consensus potential. The identities

$d(\cdot,\cdot)$1

make the update explicit in terms of left translation and Lie-algebra logarithms (Kraisler et al., 2023).

A weighted local version appears in decentralized online optimization beyond Hadamard geometry. Given local points $d(\cdot,\cdot)$2, agent $d(\cdot,\cdot)$3 updates by

$d(\cdot,\cdot)$4

or, equivalently,

$d(\cdot,\cdot)$5

This is an intrinsic gradient step on a weighted Fréchet-type consensus objective (Sahinoglu et al., 9 Sep 2025).

2. Curvature, injectivity, and step-size design

The descriptor “curvature-aware” refers to an explicit dependence of the update analysis on sectional curvature bounds, injectivity, and domain diameter. In the bounded-curvature framework, sectional curvature satisfies

$d(\cdot,\cdot)$6

and the analysis proceeds through bounds on second derivatives of squared distance. For the pairwise cost, the Hessian bound is expressed through

$d(\cdot,\cdot)$7

leading to the admissible range

$d(\cdot,\cdot)$8

with $d(\cdot,\cdot)$9 the maximum graph degree. The role of curvature is therefore quantitative: it controls how large a gradient step can be while maintaining descent (Tron et al., 2012).

In Hadamard manifolds, curvature enters through comparison inequalities and through the factor

$M$0

which appears in regret constants and in the analysis of simplified consensus steps. More negative curvature worsens constants through $M$1, although nonpositive curvature still guarantees unique geodesics and unique weighted Fréchet means (Chen et al., 2024).

Beyond Hadamard manifolds, curvature-sensitive upper and lower cosine-law replacements are written as

$M$2

$M$3

with

$M$4

These constants encode triangle distortion over a geodesically convex set $M$5 of diameter $M$6. The consensus step size is then chosen as

$M$7

which optimizes the quadratic tradeoff $M$8 appearing in the contraction proof (Sahinoglu et al., 9 Sep 2025).

The same step-size choice reappears in intrinsic decentralized stochastic optimization on manifolds with bounded sectional curvature. There the assumptions

$M$9

and, if $\frac12 \grad_x d^2(x,y) = -\log_x(y),$0,

$\frac12 \grad_x d^2(x,y) = -\log_x(y),$1

ensure that the logarithm and exponential maps remain valid along the update. The paper further proves that both the stochastic gradient step and the consensus correction stay inside the injectivity radius when $\frac12 \grad_x d^2(x,y) = -\log_x(y),$2 and $\frac12 \grad_x d^2(x,y) = -\log_x(y),$3 are chosen as prescribed (Nguyen et al., 17 Mar 2026).

A related bounded-curvature control appears in intrinsic consensus-based optimization. There the consensus direction is localized by cutoffs and built from weighted logarithmic vectors, while bounded sectional curvature and a positive injectivity radius guarantee that $\frac12 \grad_x d^2(x,y) = -\log_x(y),$4 is single-valued and smooth on the active region. The estimate

$\frac12 \grad_x d^2(x,y) = -\log_x(y),$5

is a representative curvature-dependent Lipschitz bound (Huang et al., 12 Jun 2026).

3. Consensus sets, critical points, and contraction

For the pairwise energy $\frac12 \grad_x d^2(x,y) = -\log_x(y),$6, the global minimizers are exactly the consensus manifold

$\frac12 \grad_x d^2(x,y) = -\log_x(y),$7

provided the communication graph is connected. In a tubular neighborhood $\frac12 \grad_x d^2(x,y) = -\log_x(y),$8 determined by the convexity radius, the only critical points of $\frac12 \grad_x d^2(x,y) = -\log_x(y),$9 are points of $i$0. Under an admissible fixed step size, $i$1 decreases strictly unless the iterate is stationary, every cluster point is a critical point, and sufficiently clustered initial conditions imply convergence to $i$2. Two global cases are singled out: convergence from arbitrary initialization when $i$3, and convergence on a tree with linear topology when adjacent initial states satisfy $i$4. For manifolds of constant non-negative curvature, the iterates remain in the convex hull of the initial data and all nodes converge to a single consensus point $i$5 (Tron et al., 2012).

In decentralized online Riemannian optimization on Hadamard manifolds, the consensus step is the exact weighted Fréchet mean

$i$6

and it satisfies the linear variance reduction property

$i$7

where

$i$8

This is the manifold counterpart of Euclidean spectral contraction, with rate controlled by the second largest singular value of the mixing matrix (Chen et al., 2024).

Beyond Hadamard geometry, the one-step intrinsic update still admits geometric decay of disagreement. For

$i$9

the contraction theorem states that with $\grad_{x_i}\varphi(\mathbf{x}) = -\sum_{j\in N_i}\log_{x_i}(x_j).$0,

$\grad_{x_i}\varphi(\mathbf{x}) = -\sum_{j\in N_i}\log_{x_i}(x_j).$1

where

$\grad_{x_i}\varphi(\mathbf{x}) = -\sum_{j\in N_i}\log_{x_i}(x_j).$2

The dependence on $\grad_{x_i}\varphi(\mathbf{x}) = -\sum_{j\in N_i}\log_{x_i}(x_j).$3, $\grad_{x_i}\varphi(\mathbf{x}) = -\sum_{j\in N_i}\log_{x_i}(x_j).$4, and curvature constants $\grad_{x_i}\varphi(\mathbf{x}) = -\sum_{j\in N_i}\log_{x_i}(x_j).$5 makes explicit how geometry and network topology jointly determine the speed of consensus (Sahinoglu et al., 9 Sep 2025).

In the stochastic diffusion setting, the same curvature-aware contraction yields a non-asymptotic network consensus error bound of order $\grad_{x_i}\varphi(\mathbf{x}) = -\sum_{j\in N_i}\log_{x_i}(x_j).$6: $\grad_{x_i}\varphi(\mathbf{x}) = -\sum_{j\in N_i}\log_{x_i}(x_j).$7 This estimate replaces the steady-state neighborhood typical of fixed-step decentralized Riemannian schemes by a decaying consensus error under diminishing step sizes (Nguyen et al., 17 Mar 2026).

4. Consensus versus Riemannian center of mass

A recurrent distinction in the literature is the difference between reaching agreement and reaching the Riemannian center of mass. On a Lie group $\grad_{x_i}\varphi(\mathbf{x}) = -\sum_{j\in N_i}\log_{x_i}(x_j).$8 with a bi-invariant metric, the Riemannian center of mass of data $\grad_{x_i}\varphi(\mathbf{x}) = -\sum_{j\in N_i}\log_{x_i}(x_j).$9 is defined as a minimizer of

$$x_i(l+1) = \exp_{x_i(l)}\!\left( \varepsilon \sum_{j\in N_i}\log_{x_i(l)}(x_j(l)) \right),$$0

and, inside a geodesically convex ball, it is characterized by the Karcher equation

$$x_i(l+1) = \exp_{x_i(l)}\!\left( \varepsilon \sum_{j\in N_i}\log_{x_i(l)}(x_j(l)) \right),$$1

The distributed reformulation lifts the problem to $$x_i(l+1) = \exp_{x_i(l)}\!\left( \varepsilon \sum_{j\in N_i}\log_{x_i(l)}(x_j(l)) \right),$$2: $$x_i(l+1) = \exp_{x_i(l)}\!\left( \varepsilon \sum_{j\in N_i}\log_{x_i(l)}(x_j(l)) \right),$$3 where

$$x_i(l+1) = \exp_{x_i(l)}\!\left( \varepsilon \sum_{j\in N_i}\log_{x_i(l)}(x_j(l)) \right),$$4

is the agreement submanifold. Consensus is monitored by

$$x_i(l+1) = \exp_{x_i(l)}\!\left( \varepsilon \sum_{j\in N_i}\log_{x_i(l)}(x_j(l)) \right),$$5

On a connected graph, driving $$x_i(l+1) = \exp_{x_i(l)}\!\left( \varepsilon \sum_{j\in N_i}\log_{x_i(l)}(x_j(l)) \right),$$6 is equivalent to enforcing $$x_i(l+1) = \exp_{x_i(l)}\!\left( \varepsilon \sum_{j\in N_i}\log_{x_i(l)}(x_j(l)) \right),$$7 (Kraisler et al., 2023).

A distributed gradient flow

$$x_i(l+1) = \exp_{x_i(l)}\!\left( \varepsilon \sum_{j\in N_i}\log_{x_i(l)}(x_j(l)) \right),$$8

is not sufficient, because it may admit stationary points away from the agreement submanifold. To remove that defect, the algorithm introduces gradient tracking on the Lie algebra: $$x_i(l+1) = \exp_{x_i(l)}\!\left( \varepsilon \sum_{j\in N_i}\log_{x_i(l)}(x_j(l)) \right),$$9 with the final coupled dynamics

$\varphi$0

In local form,

$\varphi$1

$\varphi$2

The limit-point characterization is exact: if the trajectory converges inside the convexity region, then

$\varphi$3

for every $\varphi$4, where $\varphi$5 is the Riemannian center of mass. In $\varphi$6, the same dynamics are globally exponentially convergent from arbitrary initial $\varphi$7 when $\varphi$8 (Kraisler et al., 2023).

This distinction corrects a common misconception. Consensus-only schemes synchronize agent states, but they do not necessarily converge to the Fréchet or Karcher mean of the data. The center-of-mass problem requires an additional mechanism, here supplied by gradient tracking, that forces the consensus point to satisfy the mean equation (Kraisler et al., 2023).

5. Online, stochastic, and collective extensions

In decentralized online Riemannian optimization on Hadamard manifolds, each round combines a projected local gradient step with a consensus step based on a weighted Fréchet mean. The exact consensus operator is

$\varphi$9

and the resulting dynamic regret scales as

$x_j-x_i$0

where $x_j-x_i$1 is the path variation and $x_j-x_i$2 measures network connectivity. To reduce computational cost, the paper also replaces the exact Fréchet mean by a single Riemannian gradient step in the tangent space and proves that this simplified consensus still achieves the same dynamic regret order, albeit with a weaker contraction factor than $x_j-x_i$3 (Chen et al., 2024).

Intrinsic decentralized stochastic Riemannian optimization on manifolds with bounded sectional curvature uses a two-stage diffusion update,

$x_j-x_i$4

with $x_j-x_i$5 and $x_j-x_i$6. The diminishing step-size regime yields an $x_j-x_i$7 network consensus error and an $x_j-x_i$8 ergodic bound for the global optimality gap, while allowing larger initial gradient steps than fixed-step baselines (Nguyen et al., 17 Mar 2026).

A further extension replaces consensus toward neighbors by consensus toward a base-point-dependent weighted barycentric field. In intrinsic consensus-based optimization on bounded-curvature manifolds, the Euclidean drift $x_j-x_i$9 is replaced by

$$\varphi(\mathbf{x})=\frac12\sum_{\{i,j\}\in E} d^2(x_i,x_j), \qquad \mathbf{x}=(x_1,\dots,x_N)\in M^N,$$00

and the associated “consensus point” is

$$\varphi(\mathbf{x})=\frac12\sum_{\{i,j\}\in E} d^2(x_i,x_j), \qquad \mathbf{x}=(x_1,\dots,x_N)\in M^N,$$01

Here the consensus object is no longer a global point independent of the base state; it is a vector field on $$\varphi(\mathbf{x})=\frac12\sum_{\{i,j\}\in E} d^2(x_i,x_j), \qquad \mathbf{x}=(x_1,\dots,x_N)\in M^N,$$02. The paper proves global well-posedness of the particle system and of the McKean–Vlasov dynamics, together with convergence of the mean-field equation toward a global minimizer under suitable conditions (Huang et al., 12 Jun 2026).

6. Extrinsic formulations, special manifolds, and limitations

Not all manifold consensus schemes are fully intrinsic. On compact submanifolds $$\varphi(\mathbf{x})=\frac12\sum_{\{i,j\}\in E} d^2(x_i,x_j), \qquad \mathbf{x}=(x_1,\dots,x_N)\in M^N,$$03, one can minimize the extrinsic disagreement energy

$$\varphi(\mathbf{x})=\frac12\sum_{\{i,j\}\in E} d^2(x_i,x_j), \qquad \mathbf{x}=(x_1,\dots,x_N)\in M^N,$$04

whose Euclidean gradient is

$$\varphi(\mathbf{x})=\frac12\sum_{\{i,j\}\in E} d^2(x_i,x_j), \qquad \mathbf{x}=(x_1,\dots,x_N)\in M^N,$$05

The update is then a Riemannian gradient step obtained by tangent projection and retraction,

$$\varphi(\mathbf{x})=\frac12\sum_{\{i,j\}\in E} d^2(x_i,x_j), \qquad \mathbf{x}=(x_1,\dots,x_N)\in M^N,$$06

Its local linear analysis depends on the proximal smoothness radius $$\varphi(\mathbf{x})=\frac12\sum_{\{i,j\}\in E} d^2(x_i,x_j), \qquad \mathbf{x}=(x_1,\dots,x_N)\in M^N,$$07, local Lipschitz continuity of the projection $$\varphi(\mathbf{x})=\frac12\sum_{\{i,j\}\in E} d^2(x_i,x_j), \qquad \mathbf{x}=(x_1,\dots,x_N)\in M^N,$$08, a normal inequality with constant $$\varphi(\mathbf{x})=\frac12\sum_{\{i,j\}\in E} d^2(x_i,x_j), \qquad \mathbf{x}=(x_1,\dots,x_N)\in M^N,$$09, and retraction approximation constants $$\varphi(\mathbf{x})=\frac12\sum_{\{i,j\}\in E} d^2(x_i,x_j), \qquad \mathbf{x}=(x_1,\dots,x_N)\in M^N,$$10. These geometric quantities replace the global convexity arguments available in Euclidean spaces (Hu et al., 2023).

On the Stiefel manifold, the Distributed Riemannian Consensus on Stiefel manifold (DRCS) takes the form

$$\varphi(\mathbf{x})=\frac12\sum_{\{i,j\}\in E} d^2(x_i,x_j), \qquad \mathbf{x}=(x_1,\dots,x_N)\in M^N,$$11

with polar retraction

$$\varphi(\mathbf{x})=\frac12\sum_{\{i,j\}\in E} d^2(x_i,x_j), \qquad \mathbf{x}=(x_1,\dots,x_N)\in M^N,$$12

The local rate is linear, and asymptotically it matches the Euclidean rate governed by $$\varphi(\mathbf{x})=\frac12\sum_{\{i,j\}\in E} d^2(x_i,x_j), \qquad \mathbf{x}=(x_1,\dots,x_N)\in M^N,$$13. The proof hinges on a Riemannian restricted secant inequality and on local-region invariance conditions that keep iterates within a neighborhood where curvature-induced second-order terms remain controlled (Chen et al., 2021).

The literature also records several instructive special cases. In the intrinsic pairwise framework, $$\varphi(\mathbf{x})=\frac12\sum_{\{i,j\}\in E} d^2(x_i,x_j), \qquad \mathbf{x}=(x_1,\dots,x_N)\in M^N,$$14 has curvature $$\varphi(\mathbf{x})=\frac12\sum_{\{i,j\}\in E} d^2(x_i,x_j), \qquad \mathbf{x}=(x_1,\dots,x_N)\in M^N,$$15 and $$\varphi(\mathbf{x})=\frac12\sum_{\{i,j\}\in E} d^2(x_i,x_j), \qquad \mathbf{x}=(x_1,\dots,x_N)\in M^N,$$16, so the update is exactly standard linear consensus. For $$\varphi(\mathbf{x})=\frac12\sum_{\{i,j\}\in E} d^2(x_i,x_j), \qquad \mathbf{x}=(x_1,\dots,x_N)\in M^N,$$17, the paper lists $$\varphi(\mathbf{x})=\frac12\sum_{\{i,j\}\in E} d^2(x_i,x_j), \qquad \mathbf{x}=(x_1,\dots,x_N)\in M^N,$$18 for $$\varphi(\mathbf{x})=\frac12\sum_{\{i,j\}\in E} d^2(x_i,x_j), \qquad \mathbf{x}=(x_1,\dots,x_N)\in M^N,$$19, $$\varphi(\mathbf{x})=\frac12\sum_{\{i,j\}\in E} d^2(x_i,x_j), \qquad \mathbf{x}=(x_1,\dots,x_N)\in M^N,$$20 for $$\varphi(\mathbf{x})=\frac12\sum_{\{i,j\}\in E} d^2(x_i,x_j), \qquad \mathbf{x}=(x_1,\dots,x_N)\in M^N,$$21, and $$\varphi(\mathbf{x})=\frac12\sum_{\{i,j\}\in E} d^2(x_i,x_j), \qquad \mathbf{x}=(x_1,\dots,x_N)\in M^N,$$22. For $$\varphi(\mathbf{x})=\frac12\sum_{\{i,j\}\in E} d^2(x_i,x_j), \qquad \mathbf{x}=(x_1,\dots,x_N)\in M^N,$$23, it gives $$\varphi(\mathbf{x})=\frac12\sum_{\{i,j\}\in E} d^2(x_i,x_j), \qquad \mathbf{x}=(x_1,\dots,x_N)\in M^N,$$24, $$\varphi(\mathbf{x})=\frac12\sum_{\{i,j\}\in E} d^2(x_i,x_j), \qquad \mathbf{x}=(x_1,\dots,x_N)\in M^N,$$25, and $$\varphi(\mathbf{x})=\frac12\sum_{\{i,j\}\in E} d^2(x_i,x_j), \qquad \mathbf{x}=(x_1,\dots,x_N)\in M^N,$$26. The sphere $$\varphi(\mathbf{x})=\frac12\sum_{\{i,j\}\in E} d^2(x_i,x_j), \qquad \mathbf{x}=(x_1,\dots,x_N)\in M^N,$$27 has constant curvature $$\varphi(\mathbf{x})=\frac12\sum_{\{i,j\}\in E} d^2(x_i,x_j), \qquad \mathbf{x}=(x_1,\dots,x_N)\in M^N,$$28. Hadamard-manifold experiments and analyses further include hyperbolic space and $$\varphi(\mathbf{x})=\frac12\sum_{\{i,j\}\in E} d^2(x_i,x_j), \qquad \mathbf{x}=(x_1,\dots,x_N)\in M^N,$$29 with the affine-invariant metric, while compact-submanifold experiments include Stiefel and Oblique manifolds (Tron et al., 2012, Chen et al., 2024, Hu et al., 2023).

Several limitations recur. The intrinsic pairwise consensus protocol does not preserve the Fréchet mean: even when it converges to consensus, the limit need not be the Fréchet mean of the initial data. Convergence may fail outside the convexity region, and the method may converge to a non-consensus local minimum or fail to reach the desired Fréchet mean when initial states are too spread out. For non-commutative Lie groups, higher-order BCH terms obstruct exact mean preservation in tangent space. These limitations motivated later formulations, especially gradient-tracking-based methods for the Riemannian center of mass and curvature-calibrated consensus steps with explicit variance contraction beyond Hadamard manifolds (Tron et al., 2012).