Riemann-Oracle: Matrix Nearness Framework

• Riemann-Oracle is a framework that decomposes matrix nearness problems into an inner least-squares subproblem and an outer Riemannian manifold optimization.
• It employs Tikhonov regularization to ensure numerical stability and overcome discontinuities in the outer objective function.
• The unified approach applies to diverse challenges including structured singular matrices, matrix polynomials, distance to instability, and approximate GCD problems.

Riemann-Oracle is a general-purpose framework for matrix nearness problems, possibly with additional linear constraints, in which the original minimization is split into two nested problems: an inner problem that is solved exactly or cheaply, and an outer problem that is recast as unconstrained optimization over a smooth real Riemannian manifold. In this framework, the “oracle” is an auxiliary parameter $\theta$ encoding extra information about a minimizer—such as an eigenvector, a set of eigenvectors, or a null vector—while the “Riemann” component is the manifold structure of the outer search space. The framework is developed for a broad class of matrix and matrix-polynomial problems, including nearest structured singular matrices, nearest singular matrix polynomials, approximate greatest common divisor problems, structured distance to instability, and nearest matrices with prescribed nullity (Gnazzo et al., 2024).

1. General formulation

The basic problem treated by Riemann-Oracle is the classical matrix nearness task

$\min_{X\in \mathcal Q}\|A-X\|,$

with emphasis on the Frobenius norm and with an extension to matrix polynomials $A(x)=\sum_{i=0}^d A_i x^i$, where distance is measured by the coefficientwise Frobenius norm. The key structural step is to decompose the admissible set as

$\mathcal Q=\bigcup_\theta \mathcal Q_\theta,$

and, for fixed $\theta$, solve the stiffened subproblem

$$f(\theta)=\min \|A-X\| \quad \text{s.t. } X\in\mathcal Q_\theta.$$

The original problem is then reduced to

$\min_\theta f(\theta).$

In the examples studied, $\theta$ is often naturally constrained to lie on a manifold such as a unit sphere, a Stiefel manifold, or a Grassmann manifold, and the outer problem is therefore treated as optimization on a smooth real embedded Riemannian manifold (Gnazzo et al., 2024).

This formulation is intended to expose a common nested structure across problems that are usually presented separately. The paper’s central claim is that many matrix nearness problems of practical interest are equivalent, in this sense, to a Riemannian optimization problem. A notable feature is that the outer objective need not be everywhere well behaved: the paper shows that it can be discontinuous, so the reduction is not merely formal and requires regularization to become computationally reliable (Gnazzo et al., 2024).

2. Core model: nearest structured singular matrix

The framework is developed most explicitly for the nearest structured singular matrix problem. Let $A\in\mathbb F^{m\times n}$, $m\ge n$, and let admissible perturbations lie in the linear space

$\min_{X\in \mathcal Q}\|A-X\|,$0

with linearly independent basis matrices $\min_{X\in \mathcal Q}\|A-X\|,$1. Under the standing normalization

$\min_{X\in \mathcal Q}\|A-X\|,$2

one has $\min_{X\in \mathcal Q}\|A-X\|,$3.

Singularity of $\min_{X\in \mathcal Q}\|A-X\|,$4 is encoded by the existence of a unit vector $\min_{X\in \mathcal Q}\|A-X\|,$5 such that

$\min_{X\in \mathcal Q}\|A-X\|,$6

For fixed $\min_{X\in \mathcal Q}\|A-X\|,$7, the paper defines

$\min_{X\in \mathcal Q}\|A-X\|,$8

so that the constraint becomes

$\min_{X\in \mathcal Q}\|A-X\|,$9

The inner problem is therefore

$A(x)=\sum_{i=0}^d A_i x^i$0

equivalently

$A(x)=\sum_{i=0}^d A_i x^i$1

When $A(x)=\sum_{i=0}^d A_i x^i$2, the minimum is attained at

$A(x)=\sum_{i=0}^d A_i x^i$3

If $A(x)=\sum_{i=0}^d A_i x^i$4 has full row rank, this simplifies to

$A(x)=\sum_{i=0}^d A_i x^i$5

The outer problem then becomes

$A(x)=\sum_{i=0}^d A_i x^i$6

with the unit sphere as the outer manifold. Even in the complex case, the sphere is treated as a smooth real embedded Riemannian manifold. This model is the template from which the paper derives its other specializations (Gnazzo et al., 2024).

3. Discontinuity and regularization

A central technical point is that the unregularized outer objective is not automatically smooth. It is defined by

$A(x)=\sum_{i=0}^d A_i x^i$7

The paper shows that $A(x)=\sum_{i=0}^d A_i x^i$8 can have jump discontinuities, even at a global minimizer. In the structured singular matrix setting, discontinuity is tied to feasibility and rank changes of $A(x)=\sum_{i=0}^d A_i x^i$9. For polynomial problems, additional mechanisms appear: the degree of a kernel vector $\mathcal Q=\bigcup_\theta \mathcal Q_\theta,$0 may drop, or $\mathcal Q=\bigcup_\theta \mathcal Q_\theta,$1 may cease to be a minimal basis vector. These phenomena make direct application of smooth manifold methods unreliable (Gnazzo et al., 2024).

To remove this obstruction, the framework introduces Tikhonov regularization: $\mathcal Q=\bigcup_\theta \mathcal Q_\theta,$2 This regularized objective has the closed form

$\mathcal Q=\bigcup_\theta \mathcal Q_\theta,$3

with minimizer

$\mathcal Q=\bigcup_\theta \mathcal Q_\theta,$4

No rank assumptions are required, and the paper proves the pointwise limit

$\mathcal Q=\bigcup_\theta \mathcal Q_\theta,$5

The resulting algorithm solves a sequence of smooth regularized problems with $\mathcal Q=\bigcup_\theta \mathcal Q_\theta,$6, using each solution to initialize the next. The paper embeds this into two classical outer strategies. In the penalty method,

$\mathcal Q=\bigcup_\theta \mathcal Q_\theta,$7

In the augmented Lagrangian formulation, starting from

$\mathcal Q=\bigcup_\theta \mathcal Q_\theta,$8

one introduces $\mathcal Q=\bigcup_\theta \mathcal Q_\theta,$9 and

$\theta$0

The inner minimization remains explicit, yielding

$\theta$1

with multiplier update

$\theta$2

The paper notes that augmented Lagrangian is often faster and more robust than pure penalty, while retaining the same nested structure (Gnazzo et al., 2024).

4. Riemannian optimization and differential structure

After regularization, the outer problem is smooth on the manifold $\theta$3. The implementation described in the paper uses Manopt’s Riemannian trust-region method. For the regularized augmented Lagrangian objective, letting

$\theta$4

the Euclidean gradient is

$\theta$5

The Hessian action on a direction $\theta$6 is

$\theta$7

with

$\theta$8

Manopt then forms the Riemannian gradient by orthogonal projection onto the tangent space and the Riemannian Hessian using the Weingarten correction (Gnazzo et al., 2024).

The paper also records cases where second-order information is less useful. For some problems, only first-order methods are recommended because the dependence of the outer minimizer on additional variables is nonsmooth. This is especially relevant in settings such as distance to instability, where a closed-form parameter minimizer is reused inside the outer Riemannian iteration (Gnazzo et al., 2024).

The convergence discussion is correspondingly local and stationarity-based. The paper cites Riemannian augmented Lagrangian results stating that if the inner solver returns iterates with gradient norm below a tolerance and the outer sequence has a limit point satisfying LICQ, then the limit point satisfies KKT conditions for the original constrained problem. In the structured singular matrix setting, LICQ reduces to a rank condition on

$\theta$9

Because the implementation uses a gradient-based stopping criterion, the strongest guaranteed outcome reported is convergence to stationary points (Gnazzo et al., 2024).

5. Principal specializations

The framework is presented as a common reduction for several nearness problems that are usually treated by separate algorithms. The paper emphasizes that each specialization retains the same architecture: an inner least-squares-type problem in perturbation variables and an outer manifold problem in an oracle parameter (Gnazzo et al., 2024).

Problem class	Outer variable / manifold	Key reduction
Sparse structured singular matrix	$$f(\theta)=\min \|A-X\| \quad \text{s.t. } X\in\mathcal Q_\theta.$$0 on the sphere	$$f(\theta)=\min \|A-X\| \quad \text{s.t. } X\in\mathcal Q_\theta.$$1 becomes diagonal
Singular matrix polynomials	kernel vector via Toeplitz lifting	reduced to structured singular matrix problem
Approximate GCD	cofactor-based oracle	structured singularity of $$f(\theta)=\min \|A-X\| \quad \text{s.t. } X\in\mathcal Q_\theta.$$2
Distance to instability	$$f(\theta)=\min \|A-X\| \quad \text{s.t. } X\in\mathcal Q_\theta.$$3 on the sphere with closed-form $$f(\theta)=\min \|A-X\| \quad \text{s.t. } X\in\mathcal Q_\theta.$$4	nested minimization over $$f(\theta)=\min \|A-X\| \quad \text{s.t. } X\in\mathcal Q_\theta.$$5
Prescribed nullity	orthonormal basis $$f(\theta)=\min \|A-X\| \quad \text{s.t. } X\in\mathcal Q_\theta.$$6 on the Grassmannian	stacked constraints for multiple null vectors

For sparse structured singular matrices, if $$f(\theta)=\min \|A-X\| \quad \text{s.t. } X\in\mathcal Q_\theta.$$7 is a sparsity pattern then $$f(\theta)=\min \|A-X\| \quad \text{s.t. } X\in\mathcal Q_\theta.$$8 and $$f(\theta)=\min \|A-X\| \quad \text{s.t. } X\in\mathcal Q_\theta.$$9 simplify substantially; in fact $\min_\theta f(\theta).$0 becomes diagonal, and the regularized objective reduces to

$\min_\theta f(\theta).$1

with elementwise weights $\min_\theta f(\theta).$2. The paper states that this permits $\min_\theta f(\theta).$3 evaluation (Gnazzo et al., 2024).

For singular matrix polynomials $\min_\theta f(\theta).$4, Toeplitz lifting is used: $\min_\theta f(\theta).$5 This converts polynomial singularity to an ordinary structured singular matrix problem. In the unstructured polynomial case, the paper uses the Kronecker form

$\min_\theta f(\theta).$6

which is described as a much cheaper implementation (Gnazzo et al., 2024).

For approximate GCD, the criterion is the existence of cofactors $\min_\theta f(\theta).$7 such that

$\min_\theta f(\theta).$8

with prescribed degrees, leading to a scaled Sylvester-type matrix $\min_\theta f(\theta).$9. For distance to instability, the nested problem is

$\theta$0

and, for fixed $\theta$1, the $\theta$2-minimizer has the closed form

$\theta$3

For prescribed nullity, the outer oracle is an orthonormal basis $\theta$4 and the natural manifold is the Grassmannian $\theta$5, with stacked inner constraints across the basis vectors (Gnazzo et al., 2024).

6. Numerical behavior and empirical profile

The numerical study is used to support two claims: first, that the framework is not tied to a single algebraic structure; second, that the regularized manifold formulation can be competitive with or superior to specialized methods on demanding instances (Gnazzo et al., 2024).

For the nearest sparse singular matrix problem, the benchmark orani678 has size $\theta$6 with 90158 nonzeros. The method is reported to converge rapidly despite severe ill-conditioning of $\theta$7 throughout the iterations, reaching a structured distance to singularity of about

$\theta$8

The paper remarks that the unregularized objective can be numerically unreliable near the minimizer, whereas the regularized objective behaves stably. Compared with the method of Sicilia et al., the competing approach is reported to use over 500 seconds per outer iteration, while Riemann-Oracle produced output in under a minute (Gnazzo et al., 2024).

The clearest head-to-head comparison is for singular matrix polynomials against the algorithm of Das–Bora. For random quadratic matrix polynomials, performance improves with size: at $\theta$9, Riemann-Oracle found a better solution in 99 out of 100 trials and achieved an average distance about 16% smaller. When degree is varied with $A\in\mathbb F^{m\times n}$0, it is reported to be slightly better in objective value for all degrees $A\in\mathbb F^{m\times n}$1 through $A\in\mathbb F^{m\times n}$2, and for $A\in\mathbb F^{m\times n}$3 about 3 times faster in median runtime (Gnazzo et al., 2024).

In approximate GCD experiments, the method matched the best known results in most cases and often did so without postprocessing. The paper also records two caveats: slight underperformance in a very challenging, badly scaled case where the objective was near machine precision, and convergence to a local minimum with spurious factors in another case. For prescribed nullity, on a structured $A\in\mathbb F^{m\times n}$4 Grcar matrix, the algorithm computed distances for nullity levels $A\in\mathbb F^{m\times n}$5 under both sparsity-preserving and Toeplitz perturbations, and in every case found a solution of exact nullity $A\in\mathbb F^{m\times n}$6 (Gnazzo et al., 2024).

These results support the paper’s broader interpretation of Riemann-Oracle as a systematic reduction rather than a problem-specific heuristic. Its mathematical novelty is described as the identification of a common oracle-plus-manifold structure across many matrix nearness problems, while its practical strength lies in combining explicit inner solves, regularization, and Riemannian outer optimization in a single implementation strategy (Gnazzo et al., 2024).

7. Terminology and relation to other arXiv usages

The name “Riemann-Oracle” is specific to the matrix-nearness framework above and should not be conflated with other arXiv uses of “Riemann” or “oracle.” In query-based foundations of analysis, a real number can be represented as an oracle that answers yes/no questions about inclusive rational intervals (Taylor, 2023). In complexity theory, oracle real Turing machines extend binary Turing machines by adding a real tape and basic arithmetic, leading to oracle separation results for the real polynomial hierarchy (Hamm et al., 13 Feb 2025). In analytic number theory, recent work on Riemann’s Nachlass studies an entire function $A\in\mathbb F^{m\times n}$7 connected to $A\in\mathbb F^{m\times n}$8 and its Fourier-analytic properties (Reyna, 2024). In Reverse Mathematics, global domination principles for everywhere Riemann integrable functions are shown to be equivalent to $A\in\mathbb F^{m\times n}$9 (Sanders et al., 31 May 2026).

Within the matrix-nearness literature, however, the terminology is narrower. The “oracle” is the auxiliary parameter $m\ge n$0 encoding structural information about a minimizer, and the “Riemann” aspect is the recasting of the outer search as optimization on a smooth real Riemannian manifold. A plausible implication is that the title is intended to emphasize this dual structure: oracle parametrization for the inner constraint geometry, and Riemannian optimization for the outer nonlinear search (Gnazzo et al., 2024).