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arxiv: 2604.22469 · v1 · submitted 2026-04-24 · 📡 eess.SP

The manifold of unitary and symmetric matrices: characterization, Riemannian optimization and application to BD-RIS design

Ignacio Santamaria , Carlos Beltr\'an , Eduard Jorswieck , Mohammad Soleymani , Jesus Guti\'errez This is my paper

Pith reviewed 2026-05-08 10:24 UTC · model grok-4.3

classification 📡 eess.SP
keywords Riemannian optimizationunitary symmetric matricesBD-RISMIMOscattering matrix optimizationreconfigurable intelligent surfaceslow-rank matrices
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The pith

Riemannian optimization on unitary symmetric matrices improves BD-RIS MIMO performance.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

This paper characterizes the geometry of the manifold of unitary and symmetric matrices, called U_s, which models the scattering matrices of passive reciprocal devices like beyond-diagonal reconfigurable intelligent surfaces. It derives the tangent space, a simple retraction, and closed-form geodesics to enable two new Riemannian manifold optimization algorithms: a line-search scheme and a phase-optimization update. These are applied to BD-RIS MIMO problems including sum-gain and rate maximization, where they outperform prior methods, and it is shown that the optimal matrix is low-rank when BD-RIS elements exceed the antenna count.

Core claim

The geometry of the set of unitary symmetric matrices admits a Riemannian structure with explicit tangent space, retraction, and geodesics, which supports the design of efficient optimization algorithms for BD-RIS scattering matrices in MIMO systems that achieve superior sum-gain and rates compared to existing techniques while exploiting low-rank structure when the RIS is large.

What carries the argument

The manifold of unitary symmetric matrices U_s, with its derived tangent space, simple retraction, and closed-form geodesics that enable line-search and geodesic phase-optimization algorithms.

Load-bearing premise

The proposed retraction and closed-form geodesics remain computationally tractable and numerically stable for practical BD-RIS MIMO dimensions, and local optima found by the algorithms are close enough to the global optimum for the non-convex objectives.

What would settle it

Running the Riemannian line-search and phase-optimization algorithms on a BD-RIS MIMO setup with more elements than antennas and verifying if the optimized matrix has rank equal to the antenna count, or comparing the achieved performance metrics against baseline methods to see if they are higher.

Figures

Figures reproduced from arXiv: 2604.22469 by Carlos Beltr\'an, Eduard Jorswieck, Ignacio Santamaria, Jesus Guti\'errez, Mohammad Soleymani.

Figure 1
Figure 1. Figure 1: Structure of the low-rank symmetric matrix, Θlr, in Theorem 2. The inner block, Θ˜, is an r × r symmetric matrix with r = Nt + Nr. FΘlrGH and hence Heq(Θ) = Heq(Θlr), and ii) ΘlrΘH lr ⪯ IM. The structure of the low-rank matrix Θlr is depicted in view at source ↗
Figure 2
Figure 2. Figure 2: Convergence curves for the MO algorithms with line search (LS) and view at source ↗
Figure 3
Figure 3. Figure 3: Run time in secs of the MO algorithms with line search and phase view at source ↗
Figure 4
Figure 4. Figure 4: Rate vs. number of BD-RIS elements achieved with different view at source ↗
Figure 5
Figure 5. Figure 5: MSE vs. number of BD-RIS elements for MO algorithms optimizing view at source ↗
read the original abstract

This paper proposes and analyzes Riemannian optimization algorithms on the manifold of unitary and symmetric matrices, denoted ${\cal {U}}_s$, which naturally models the scattering matrices of passive and reciprocal devices such as beyond-diagonal reconfigurable intelligent surfaces (BD-RISs). Despite its relevance, the geometry of ${\cal {U}}_s$ has remained largely unexplored, and existing BD-RIS optimization methods either ignore the symmetry constraint or rely on costly Takagi-based parameterizations. We first provide a rigorous geometric characterization of ${\cal {U}}_s$, deriving its tangent space, a simple retraction, and closed-form expressions for geodesics. Building on these results, we develop two Riemannian manifold optimization (MO) algorithms tailored to ${\cal {U}}_s$: a line-search (LS) based scheme and a phase-optimization (PO) update along geodesics. We then apply the proposed framework to BD-RIS-assisted multiple-input multiple-output (MIMO) links, addressing sum-gain maximization, rate maximization, and minimum mean-square error problems, where they outperform existing approaches. Furthermore, we show that when the number of BD-RIS elements exceeds the total number of antennas, the optimal scattering matrix is low-rank, which motivates and enables efficient low-rank variants of the proposed algorithms.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

3 major / 2 minor

Summary. The paper claims to rigorously characterize the manifold of unitary and symmetric matrices (denoted U_s) by deriving its tangent space, a simple retraction, and closed-form geodesic expressions. Building on this, it develops two Riemannian manifold optimization algorithms (a line-search scheme and a phase-optimization update along geodesics) and applies them to BD-RIS-assisted MIMO systems for sum-gain maximization, rate maximization, and MMSE problems, reporting outperformance over existing methods. It further shows that when the number of BD-RIS elements exceeds the total number of antennas, the optimal scattering matrix is low-rank, motivating efficient low-rank algorithm variants.

Significance. If the geometric derivations hold and the algorithms prove tractable, this work offers a specialized Riemannian framework for symmetric unitary constraints that arise in reciprocal passive devices. The closed-form primitives and low-rank insight could improve scalability and performance in BD-RIS MIMO optimization compared with Takagi factorizations or unconstrained relaxations, with potential impact on practical system design.

major comments (3)
  1. Abstract and low-rank section: the claim that 'the optimal scattering matrix is low-rank' when the number of BD-RIS elements exceeds the total number of antennas requires precise mathematical clarification. Unitary matrices are full-rank by definition, so the statement must specify whether it refers to a reduced-dimensional submanifold, a low-rank factorization preserving symmetry and unitarity, or the rank of an effective operator; without this, the motivation for the efficient variants is unclear.
  2. Algorithm development and complexity analysis: the retraction and closed-form geodesics rely on matrix functions (exponentials, square roots) whose O(N^3) cost and numerical stability for practical BD-RIS sizes (N=64–256) are not quantified. The manuscript should provide explicit complexity counts, runtime benchmarks, and floating-point error analysis to confirm tractability.
  3. Optimization results sections: for the non-convex sum-gain and rate objectives, the paper should supply evidence that the Riemannian local optima are sufficiently close to global solutions. This could be addressed by reporting statistics over multiple random initializations or comparisons against convex relaxations or global solvers.
minor comments (2)
  1. Notation consistency: the manifold symbol U_s and related objects (tangent space, retraction) should be introduced once with clear definitions and used uniformly.
  2. Experimental tables: performance comparisons should report standard deviations or confidence intervals alongside mean gains to allow assessment of variability.

Simulated Author's Rebuttal

3 responses · 0 unresolved

We thank the referee for their thorough review and valuable suggestions. We have carefully considered each major comment and provide point-by-point responses below, along with our plans for revisions to the manuscript.

read point-by-point responses

  1. Referee: Abstract and low-rank section: the claim that 'the optimal scattering matrix is low-rank' when the number of BD-RIS elements exceeds the total number of antennas requires precise mathematical clarification. Unitary matrices are full-rank by definition, so the statement must specify whether it refers to a reduced-dimensional submanifold, a low-rank factorization preserving symmetry and unitarity, or the rank of an effective operator; without this, the motivation for the efficient variants is unclear.

    Authors: We appreciate the referee's observation regarding the potential ambiguity in our low-rank claim. Upon reflection, the statement in the abstract and low-rank section refers to the fact that, when the number of BD-RIS elements exceeds the total number of antennas, there exists an optimal scattering matrix that admits a low-rank factorization while preserving the required symmetry and unitarity constraints. Specifically, the optimization can be restricted to a reduced-dimensional submanifold parameterized by lower-rank factors. This provides the motivation for developing efficient low-rank variants of our algorithms. We will revise the abstract and the corresponding section to include this precise mathematical clarification. revision: yes

  2. Referee: Algorithm development and complexity analysis: the retraction and closed-form geodesics rely on matrix functions (exponentials, square roots) whose O(N^3) cost and numerical stability for practical BD-RIS sizes (N=64–256) are not quantified. The manuscript should provide explicit complexity counts, runtime benchmarks, and floating-point error analysis to confirm tractability.

    Authors: We agree that quantifying the computational aspects is important for assessing practicality. The proposed retraction and geodesic computations involve matrix square roots and exponentials, which have a complexity of O(N^3) per operation. In the revised manuscript, we will include explicit complexity analyses for the individual primitives and the complete algorithms. Furthermore, we will add runtime benchmarks and floating-point error analyses based on our existing simulations for BD-RIS sizes up to N=256, demonstrating that the methods remain tractable and numerically stable for the considered problem dimensions. revision: yes

  3. Referee: Optimization results sections: for the non-convex sum-gain and rate objectives, the paper should supply evidence that the Riemannian local optima are sufficiently close to global solutions. This could be addressed by reporting statistics over multiple random initializations or comparisons against convex relaxations or global solvers.

    Authors: We recognize the value of providing evidence on the quality of the local optima for these non-convex problems. To address this, we will augment the optimization results sections with statistics (such as average performance and standard deviation) obtained from multiple random initializations of our Riemannian algorithms. This will illustrate the consistency of the solutions. Where computationally feasible, we will also include comparisons with convex relaxations on smaller-scale instances to support that our local solutions are close to global optima. revision: yes

Circularity Check

0 steps flagged

No significant circularity: geometric primitives derived from first principles

full rationale

The paper's core contribution is a direct characterization of the manifold U_s (tangent space, retraction, closed-form geodesics) followed by standard Riemannian algorithms (LS and PO) applied to BD-RIS objectives. These steps do not reduce to fitted parameters renamed as predictions, self-citations as load-bearing premises, or ansatzes smuggled via prior work. The low-rank property when N exceeds antenna count follows from the unitarity/symmetry constraints and dimension counting rather than being presupposed. Performance comparisons are external to the derivation chain. The work is self-contained against external benchmarks with no evidence of the enumerated circular patterns.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The central claim rests on the set of unitary symmetric matrices forming a Riemannian manifold whose standard geometric objects (tangent space, retraction, geodesics) can be derived in closed form and on the assumption that the non-convex wireless optimization problems are well-posed on this manifold. No free parameters, new physical entities, or ad-hoc axioms are introduced in the abstract.

axioms (2)
  • domain assumption The set of unitary symmetric matrices forms a smooth embedded submanifold of the unitary group
    Invoked to justify the application of Riemannian geometry and the derivation of tangent space and geodesics.
  • standard math Standard Riemannian optimization theory (line search, geodesic updates) applies directly once the manifold geometry is known
    Used to construct the LS and PO algorithms without additional derivation.

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