Manifold Optimization in Communications

• Manifold optimization in communications is a framework leveraging Riemannian methods on curved spaces to address nonconvex, constraint-rich signal and resource allocation problems.
• Techniques like Riemannian gradient descent, conjugate gradient, and trust-region methods enable efficient optimization in applications such as beamforming, RIS, and MIMO systems.
• This approach natively respects geometric constraints, leading to significant gains in performance metrics including sum-rate, SINR, and computational efficiency.

Manifold optimization in communications refers to the rigorous application of Riemannian optimization techniques—optimization directly on curved geometric spaces, or "manifolds"—to solve large-scale, nonconvex, and structurally constrained signal processing and resource allocation problems in advanced wireless systems. These include scenarios featuring massive MIMO, reconfigurable intelligent surfaces (RIS), integrated sensing and communication (ISAC), fluid antenna systems (FAS), distributed antenna arrays, and quantum-inspired architectures. Manifold optimization enables direct navigation of the feasible set without performance-degrading relaxations or penalization, natively respects geometric constraints, and provides algorithmic frameworks with proven convergence and computational advantages over classical Euclidean approaches (Li et al., 9 Feb 2026).

1. Geometric Foundations of Manifold Optimization

A manifold $\mathcal{M}$ in this context is a differentiable set of feasible points, such as the set of unit-norm vectors (sphere), product of complex circles (unit-modulus, also known as the complex circle manifold), or matrices with orthonormal columns (Stiefel manifold). At every point $x\in\mathcal{M}$, the tangent space $T_{x}\mathcal{M}$ encodes allowed infinitesimal variations that respect the constraints locally. The Riemannian metric $\langle \cdot,\cdot \rangle_x$ defines a smoothly varying inner product on each tangent space, inducing a geometry on $\mathcal{M}$ (Li et al., 9 Feb 2026, Tabrizi et al., 10 Aug 2025).

Retraction maps $R_x:T_{x}\mathcal{M}\to\mathcal{M}$ approximate the computationally expensive exponential map (geodesics), serving to return candidate points after tangent-space steps back to the manifold. Vector transport $$\mathcal{T}_\eta:T_{x}\mathcal{M} \to T_{R_x(\eta)}\mathcal{M}$$ carries search directions between different tangent spaces. These ingredients are central for defining Riemannian gradient, conjugate-gradient, and trust-region methods on $\mathcal{M}$ (Li et al., 9 Feb 2026, Tabrizi et al., 10 Aug 2025).

2. Manifold Structures in Communication System Design

A broad spectrum of wireless problems can be reformulated so that their decision variables lie on natural manifolds:

• Complex Sphere/Oblique Manifold: Used for beamforming with per-antenna/unit-modulus constraints or per-BS power normalization. The feasible set is either a sphere (all entries with fixed $\ell_2$ norm) or a product of spheres (columns have unit $\ell_2$ norm, not necessarily orthogonal) (Li et al., 9 Feb 2026, Zhou et al., 2017).
• Complex Circle Manifold (CCM): For phase-shifter vectors in RIS and analog beamforming. Here, $x\in\mathcal{M}$0 with $x\in\mathcal{M}$1 for all $x\in\mathcal{M}$2. The key operations, including tangent space, orthogonal projection, and retraction, are performed entrywise and are supported by rigorous derivations (Tabrizi et al., 10 Aug 2025).
• Stiefel Manifold: For column-orthonormal precoder or beamformer matrices ($x\in\mathcal{M}$3, $x\in\mathcal{M}$4). Tangent space projections and QR-type retractions are fundamental in MIMO/ISAC (Li et al., 9 Feb 2026, Tabrizi et al., 10 Aug 2025, Wang et al., 2021).
• Ellipsoidal Manifold: Arising in distributed antenna/operator beamforming: subspace constraints imposed by channel geometry lead to ellipsoidal normalization, with explicit projection formulas (Zhu et al., 24 Mar 2026).
• Product Manifolds: Many systems, e.g., hybrid beamforming or joint BS-RIS optimization, naturally require product manifold optimization where variables live in coupled manifolds (e.g., a Stiefel for precoding and a circle manifold for RIS phases) (Junior et al., 2024, Zhang et al., 2021).

3. Algorithmic Frameworks and Key Methods

Manifold optimization in communications leverages variants of the following Riemannian methods:

• Riemannian Gradient Descent (RGD): At iteration $x\in\mathcal{M}$5, update $x\in\mathcal{M}$6, where $x\in\mathcal{M}$7 is the tangent space (Riemannian) gradient obtained by projecting the Euclidean gradient onto $x\in\mathcal{M}$8 (Li et al., 9 Feb 2026, Junior et al., 2024).
• Riemannian Conjugate Gradient (RCG): Uses search directions constructed from current and past gradients using vector transport, allowing for superlinear convergence characteristics (Li et al., 9 Feb 2026, Zhu et al., 24 Mar 2026, Zhang et al., 2021).
• Riemannian Trust-Region and Augmented Lagrangian: Second-order (or constrained) variants such as Riemannian trust-region (RTR) methods and augmented Lagrangian techniques enable handling inequality constraints and fast local convergence. The augmented Lagrangian on the sphere manifold for ISAC beamforming is one such example (Zargari et al., 2024, Li et al., 9 Feb 2026).
• Alternating/Block Optimization on Product Manifolds: For coupled optimization (e.g. BS and RIS), block-wise Riemannian CG or trust-region subproblems are solved for each block variable with others held fixed (Zhang et al., 2021, Wang et al., 2021, Junior et al., 2024).
• Hybrid and Meta-Learning Architectures: Meta-learning algorithms, such as GMML, integrate Riemannian geometry with lightweight neural architectures, operating directly on gradients projected to tangent spaces to enhance spectral efficiency and robustness (Zhu et al., 2024).
• Penalty Approaches for Multiple Constraints: Intersection of manifold-type constraints (e.g., unitarity and symmetry for BD-RIS) is handled by penalty reformulation, optimizing over a main manifold (Stiefel) with deviation from the secondary constraint penalized, followed by post-processing projection (Fidanovski et al., 24 Sep 2025).

4. Application Domains in Advanced Wireless Communications

Manifold optimization underpins a range of next-generation wireless communication problems:

• Beamforming and Precoding: Sum-rate, SINR, secrecy, and fairness objectives under unit-modulus, per-antenna power, and orthonormality are directly optimized using sphere, oblique, or Stiefel manifolds with RGD/RCG/RTR (Li et al., 9 Feb 2026, Zhou et al., 2017, Tabrizi et al., 10 Aug 2025).
• RIS-Aided Communications and Massive MIMO: Joint BS-RIS phase/skew design, energy efficiency, grant-free random access, pilot interference mitigation, and multi-beam configurations rely on CCM and product manifold structures for efficient optimization (Junior et al., 2024, Zhang et al., 2021, Zhu et al., 2024).
• Hybrid Beamforming and Dual-Function Radar-Communication (DFRC): Hybrid analog-digital architectures with constant-modulus and power constraints exploit circle and Stiefel manifolds (or products thereof). Methods include Riemannian ADMM (fully-connected) and product-manifold trust-region (partially-connected) (Wang et al., 2021).
• ISAC Waveform and Resource Allocation: Unimodular (constant-modulus) waveform design for MIMO ISAC, and resource allocation with joint sum-rate and beampattern constraints, are cast as optimization on unit-modulus or sphere manifolds, often with nonsmooth cost functions (Wang et al., 8 Apr 2025, Zargari et al., 2024).
• Distributed Antenna Systems: By exploiting low-dimensional subspace properties, high-dimensional beamformer optimization under per-cluster power constraints reduces to an ellipsoidal product-manifold problem, enabling orders-of-magnitude gains in scalability and runtime (Zhu et al., 24 Mar 2026).
• Quantum Manifold Optimization (QMO): Classical manifold-constrained problems (pilot design, beamforming, RIS phase) are lifted to variational quantum circuits, enabling trace-based gradient updates compatible with quantum devices while maintaining the manifold structure (Rexhepi et al., 13 Apr 2025).
• Beyond-Diagonal RIS and Non-Standard Architectures: Design of symmetric, unitary scattering matrices for reciprocal, beyond-diagonal RIS leverages penalty-manifold approaches with RCG on block-diagonal Stiefel manifolds and post-hoc projection for physical admissibility (Fidanovski et al., 24 Sep 2025).

5. Representative Numerical Results and Performance Impact

Manifold optimization in communications demonstrates consistent empirical advantages:

• Performance: Riemannian methods (e.g., RTR and RGD) substantially outperform traditional MM and SDR benchmarks on secrecy rate, convergence speed, and robustness in FAS-assisted secure beamforming (e.g., 66.6% and 50.3% ASR gain, runtime reduction from 35–161 s to 0.1–1.2 s) (Li et al., 9 Feb 2026).
• RIS/Massive MIMO: Double manifold alternating optimization yields 10–30% sum-rate gains over MRT/ZF in multi-IRS systems, reaching near-ideal performance with modest phase quantization ($x\in\mathcal{M}$9 bits) (Zhang et al., 2021). In RIS-aided massive MIMO, Riemannian schemes achieve up to 20% SINR gain over SDP/AO with roughly half the runtime (Junior et al., 2024).
• Hybrid and DFRC: Trust-region methods on product manifolds for mmWave DFRC achieve algorithmic convergence in $T_{x}\mathcal{M}$0 steps, and match fully digital precoders, outperforming alternating minimization and blocking coordinate descent at lower computational cost (Wang et al., 2021).
• Resource Allocation in ISAC: Augmented Lagrangian manifold optimization delivers 10.1% sum-rate improvement over classic optimization-based benchmarks, with quick convergence ($T_{x}\mathcal{M}$1 outer iterations) and optimal beampattern attainment (Zargari et al., 2024).
• Distributed Beamforming: Ellipsoidal manifold RCG converges to optimal WSR stationary points with much lower complexity than conventional manifold or WMMSE methods, especially as the cluster count increases (Zhu et al., 24 Mar 2026).
• Robustness and Dynamics: Manifold-based meta-learning algorithms for RIS phase/precoder design achieve 7.31% spectral efficiency advantage and 23$T_{x}\mathcal{M}$2 acceleration in convergence, and are robust to channel errors and mobility scenarios (Zhu et al., 2024).
• BD-RIS Scattering Matrix Design: Riemannian CG achieves higher sum-rate under symmetry/unitarity constraints than prior interference-nulling designs, with fast convergence (e.g., 50–2500 iterations, depending on architecture) and scalable runtime (Fidanovski et al., 24 Sep 2025).

6. Algorithmic and Implementation Best Practices

• Projection to tangent space and retraction must be implemented in a manner that preserves constraint satisfaction exactly at each iteration; elementwise and QR-based constructions are respectively preferred for circle/oblique and Stiefel manifolds (Tabrizi et al., 10 Aug 2025, Li et al., 9 Feb 2026).
• For non-smooth or nonsmooth+smooth cost functions (e.g., with $T_{x}\mathcal{M}$3 penalties), subgradient projection and Armijo-type line search are necessary to ensure convergent descent (Wang et al., 8 Apr 2025).
• Proper stepsize selection (adaptive or backtracking) is crucial for robust convergence; naive fixed stepsize can lead to divergence, especially for large-scale problems or ill-conditioned objectives (Tabrizi et al., 10 Aug 2025).
• Algorithm variants should be chosen according to problem structure: RGD for large-scale, Hessian-free scenarios; RCG for moderate scale with improved convergence; RTR for problems requiring superlinear convergence or precise constraint handling (Junior et al., 2024).
• Hybridization with learning (meta-learning) or quantum-variational loops extends manifold optimization to adaptive, real-time, and hardware-constrained regimes, retaining performance advantages (Zhu et al., 2024, Rexhepi et al., 13 Apr 2025).

7. Extensions, Limitations, and Future Directions

• Manifold optimization is extensible to nonconvex, nonsmooth, multi-block, and mixed-integer contexts, and supports distributed and meta-learning-based communication system design.
• Intersection of multiple manifolds (e.g., unitarity and symmetry) is algorithmically tractable via penalty reformulation and post-hoc projections, enabling physical feasibility in constrained hardware (e.g., reciprocal BD-RIS) (Fidanovski et al., 24 Sep 2025).
• Quantum manifold optimization offers a route to quantum-enhanced wireless design, where both classical gradient projection and quantum circuits enforce the underlying Riemannian structure (Rexhepi et al., 13 Apr 2025).
• Scalability is established for large antenna counts, cluster decompositions, and high user-densities, primarily due to problem-specific dimension reduction (e.g., the subspace property for distributed beamforming (Zhu et al., 24 Mar 2026)).
• Open directions include robust manifold optimization under time-varying and uncertain CSI, integration with real-time, learning-driven protocols, and efficient quantum-classical hybrid implementations.

Manifold optimization thus forms the mathematical and algorithmic backbone for a wide range of contemporary and emerging wireless communication problems, bringing geometric insight, computational scalability, and algorithmic rigor to the design of complex, constraint-rich radio systems (Li et al., 9 Feb 2026, Tabrizi et al., 10 Aug 2025, Junior et al., 2024).