Spatial Pairformer

• The paper introduces Spatial Pairformer, advancing geometry-aware modeling by aligning spatial features with learned Riemannian metrics.
• It employs pullback and quotient metrics to ensure that latent space interpolation respects data fidelity and underlying spatial structures.
• The method integrates SPD manifold techniques and optimization preconditioning to enhance performance on complex spatial data tasks.

Riemannian metric matching denotes a family of constructions in which a Riemannian metric is chosen, learned, recovered, or optimized so that the geometry it induces agrees with the structure one wants to model. In the literature considered here, that agreement appears in several distinct forms: pullback metrics that make latent-space infinitesimal lengths match observation-space infinitesimal lengths in probabilistic generative models, quotient metrics that identify shapes modulo reparameterization, geometry-aware discrepancies for covariance matrices on the manifold of Hermitian or symmetric positive definite matrices, sample-based recovery of intrinsic geometry through diffusion operators or random graphs, and metrics designed as preconditioners for Riemannian optimization (Tosi et al., 2014, Brigant et al., 2024, Cohen et al., 12 May 2026, Bamberger et al., 12 Jun 2026, Mishra et al., 2014).

1. Scope and mathematical viewpoint

In the reviewed literature, “matching” rarely means only a direct tensor-to-tensor comparison $g \leftrightarrow \tilde g$. More often, the matched objects are induced quantities: geodesic lengths, local inner products, covariance structure, reparameterization classes, volume elements, or transport paths. A concise abstract formulation appears in the review literature as

$$\operatorname*{argmin}_{g\in \mathcal G_\theta\subset \mathcal G_M} L(\mathcal F(g),X), \qquad \mathcal F(g)=(d_g,\operatorname{Exp}_g,T_g,\mathrm{vol}_g),$$

so the learned or selected metric is judged through the geometry it induces rather than through a standalone tensor discrepancy (Gruffaz et al., 7 Mar 2025).

This viewpoint is already implicit in standard Riemannian geometry. A metric $g_p$ is a smoothly varying inner product on each tangent space $T_pM$; it defines curve length

$$L(\gamma)=\int_0^1 \sqrt{g_{\gamma(t)}(\dot\gamma(t),\dot\gamma(t))}\,dt,$$

geodesic distance

$$d_g(x,y)=\inf_{\gamma(0)=x,\ \gamma(1)=y} L(\gamma),$$

and volume through $\sqrt{\det g}$ in local coordinates (Gruffaz et al., 7 Mar 2025). Consequently, metric matching can be understood as enforcing agreement between these induced geometric objects and data, constraints, or task structure.

The literature considered here also shows that there is no single canonical domain for the problem. The relevant manifold may be a latent-variable manifold, a quotient shape space, the manifold of SPD or HPD matrices, a fixed-connectivity mesh manifold, or an ambient Euclidean space endowed with a data-dependent metric (Tosi et al., 2014, Bauer et al., 2018, Lin, 2019, Herzog et al., 2020, Kapuśniak et al., 2024). This suggests that the unifying principle is not a specific optimization template, but the systematic replacement of Euclidean geometry by a geometry whose local quadratic form is better aligned with the structure being modeled.

2. Pullback metrics, uncertainty, and data-manifold interpolation

A foundational instance of metric matching appears in probabilistic generative latent-variable models. For a smooth generative map $f$, the correct latent metric is the pullback of the observation-space Euclidean metric: $G = J^\top J,$ with Jacobian $J$, and the induced inner product

$$\operatorname*{argmin}_{g\in \mathcal G_\theta\subset \mathcal G_M} L(\mathcal F(g),X), \qquad \mathcal F(g)=(d_g,\operatorname{Exp}_g,T_g,\mathrm{vol}_g),$$0

This makes latent-space infinitesimal lengths match observation-space infinitesimal lengths under the map (Tosi et al., 2014).

In the probabilistic setting of the GP-LVM, the Jacobian is random, and so is the local metric. The key formula is

$$\operatorname*{argmin}_{g\in \mathcal G_\theta\subset \mathcal G_M} L(\mathcal F(g),X), \qquad \mathcal F(g)=(d_g,\operatorname{Exp}_g,T_g,\mathrm{vol}_g),$$1

The first term is the metric induced by the posterior mean map, while the second is an uncertainty correction. The paper explicitly interprets this geometrically: increasing uncertainty expands the metric tensor, increases curve lengths, and makes geodesics avoid uncertain regions (Tosi et al., 2014). Operationally, the expected metric is inserted into the usual Riemannian length functional,

$$\operatorname*{argmin}_{g\in \mathcal G_\theta\subset \mathcal G_M} L(\mathcal F(g),X), \qquad \mathcal F(g)=(d_g,\operatorname{Exp}_g,T_g,\mathrm{vol}_g),$$2

and geodesics are obtained by solving the geodesic boundary value problem numerically. In this setting, metric matching means that interpolation in latent space is forced to respect both manifold distortion and model confidence.

A closely related but distinct construction appears in Metric Flow Matching. There the ambient space $$\operatorname*{argmin}_{g\in \mathcal G_\theta\subset \mathcal G_M} L(\mathcal F(g),X), \qquad \mathcal F(g)=(d_g,\operatorname{Exp}_g,T_g,\mathrm{vol}_g),$$3 is equipped with a data-dependent Riemannian metric

$$\operatorname*{argmin}_{g\in \mathcal G_\theta\subset \mathcal G_M} L(\mathcal F(g),X), \qquad \mathcal F(g)=(d_g,\operatorname{Exp}_g,T_g,\mathrm{vol}_g),$$4

and Euclidean straight-line conditional interpolants are replaced by approximate geodesics learned by minimizing the metric kinetic energy

$$\operatorname*{argmin}_{g\in \mathcal G_\theta\subset \mathcal G_M} L(\mathcal F(g),X), \qquad \mathcal F(g)=(d_g,\operatorname{Exp}_g,T_g,\mathrm{vol}_g),$$5

The subsequent flow-matching objective becomes

$$\operatorname*{argmin}_{g\in \mathcal G_\theta\subset \mathcal G_M} L(\mathcal F(g),X), \qquad \mathcal F(g)=(d_g,\operatorname{Exp}_g,T_g,\mathrm{vol}_g),$$6

Here the metric enters twice: it shapes the interpolating path and it defines the norm in which vector fields are matched (Kapuśniak et al., 2024). In both the GP-LVM and MFM settings, the metric is the device that aligns interpolation geometry with the support and uncertainty structure of the data manifold.

3. Quotient geometry and shape matching for curves and surfaces

A second major meaning of Riemannian metric matching concerns shapes modulo reparameterization. For manifold-valued open curves,

$$\operatorname*{argmin}_{g\in \mathcal G_\theta\subset \mathcal G_M} L(\mathcal F(g),X), \qquad \mathcal F(g)=(d_g,\operatorname{Exp}_g,T_g,\mathrm{vol}_g),$$7

the reparameterization group $$\operatorname*{argmin}_{g\in \mathcal G_\theta\subset \mathcal G_M} L(\mathcal F(g),X), \qquad \mathcal F(g)=(d_g,\operatorname{Exp}_g,T_g,\mathrm{vol}_g),$$8 acts by right composition, and the shape space is the quotient

$$\operatorname*{argmin}_{g\in \mathcal G_\theta\subset \mathcal G_M} L(\mathcal F(g),X), \qquad \mathcal F(g)=(d_g,\operatorname{Exp}_g,T_g,\mathrm{vol}_g),$$9

If a metric $g_p$0 on $g_p$1 is reparameterization-invariant, then the quotient inherits a metric and the distance between two shapes is

$g_p$2

The vertical space consists of infinitesimal reparameterizations, the horizontal space is its $g_p$3-orthogonal complement, and horizontal geodesics in the total space project to geodesics in shape space (Brigant et al., 2024). In this framework, metric matching means choosing or computing a metric so that distances compare shapes rather than parameterizations.

For elastic metrics, the quotient problem becomes algorithmically explicit. The paper on optimal matching between manifold-valued curves derives a canonical decomposition of any path into a horizontal part and a path of reparameterizations, proves that horizontalization decreases length, and uses this to iteratively approximate quotient geodesics (Brigant et al., 2024). The broader survey on metric registration of curves and surfaces places this within a common submersion framework and shows how exact endpoint constraints can be relaxed through reparameterization-invariant chordal penalties such as oriented varifold distances (Bauer et al., 2018). The relaxed problem has the form

$g_p$4

where $g_p$5 vanishes when the endpoint belongs to the target quotient fiber.

Intrinsic Sobolev metrics on parametrized surfaces give a complementary perspective. For regular surfaces $g_p$6, one can define an inner $g_p$7-type metric directly on surface deformation fields,

$g_p$8

and perform geodesic shooting in the manifold of parametrized surfaces (Bauer et al., 2011). This matches deformation cost to the evolving intrinsic geometry of the surface rather than to an ambient diffeomorphic flow.

A finite-dimensional analogue arises for planar triangular meshes with fixed connectivity. There the admissible mesh space is an open manifold, and a geodesically complete metric is constructed by a barrier term: $g_p$9 The function $T_pM$0 penalizes small triangle heights, small boundary vertex–edge distances, and large deviation from a reference mesh, so geodesics preserve connectivity and prevent degeneration (Herzog et al., 2020). Across these shape settings, the matched object is the geometry of shape change after quotienting out nuisance parameterization or preserving admissibility constraints.

4. SPD and HPD manifolds: covariance fitting, learned geodesics, and Cholesky geometry

For covariance-like data, the relevant manifold is $T_pM$1, the manifold of HPD or SPD matrices. The array-processing paper on Riemannian covariance matching formulates the matching problem as

$T_pM$2

where $T_pM$3 is the sample covariance and $T_pM$4 is the model covariance implied by a spatial power spectrum (Cohen et al., 12 May 2026). The central claim is that covariance matrices lie on the manifold of HPD matrices, so Euclidean discrepancies are geometrically inappropriate. The method SERCOM therefore replaces Euclidean covariance matching by the Jensen–Bregman LogDet divergence

$T_pM$5

SERCOM does not optimize the exact affine-invariant geodesic distance; it optimizes a geometry-aware divergence surrogate that is closely linked to affine-invariant geometry but computable through log-determinants and matrix inverses rather than eigendecompositions (Cohen et al., 12 May 2026).

Supervised Riemannian metric learning on SPD matrices takes a different route. Under the log-Euclidean framework, learning a Riemannian metric reduces to learning a Mahalanobis metric on $T_pM$6, which induces the geodesic

$T_pM$7

Here matching means that pairs labeled similar or dissimilar in the task are made close or far under a learned geodesic distance that remains valid in the log-Euclidean Riemannian sense (Vemulapalli et al., 2015).

A third SPD construction is the Log-Cholesky metric. Cholesky space $T_pM$8 is endowed with a commutative Lie group law

$T_pM$9

and a bi-invariant metric

$$L(\gamma)=\int_0^1 \sqrt{g_{\gamma(t)}(\dot\gamma(t),\dot\gamma(t))}\,dt,$$0

which is then pushed forward to $$L(\gamma)=\int_0^1 \sqrt{g_{\gamma(t)}(\dot\gamma(t),\dot\gamma(t))}\,dt,$$1 via $$L(\gamma)=\int_0^1 \sqrt{g_{\gamma(t)}(\dot\gamma(t),\dot\gamma(t))}\,dt,$$2. The resulting distance is

$$L(\gamma)=\int_0^1 \sqrt{g_{\gamma(t)}(\dot\gamma(t),\dot\gamma(t))}\,dt,$$3

and the Fréchet average has a closed form with determinant

$$L(\gamma)=\int_0^1 \sqrt{g_{\gamma(t)}(\dot\gamma(t),\dot\gamma(t))}\,dt,$$4

so the determinant of the average lies between the minimum and maximum determinants of the inputs (Lin, 2019).

Setting	Matching object	Representative formulation
HPD covariance fitting	Model covariance to sample covariance	$$L(\gamma)=\int_0^1 \sqrt{g_{\gamma(t)}(\dot\gamma(t),\dot\gamma(t))}\,dt,$$5 with JBLD (Cohen et al., 12 May 2026)
Supervised SPD metric learning	Geodesic distance to pairwise labels	Learned log-Euclidean distance $$L(\gamma)=\int_0^1 \sqrt{g_{\gamma(t)}(\dot\gamma(t),\dot\gamma(t))}\,dt,$$6 (Vemulapalli et al., 2015)
Log-Cholesky geometry	Interpolation, averaging, transport on SPD	Flat bi-invariant metric via Cholesky factors (Lin, 2019)

Taken together, these constructions show that Riemannian metric matching on matrix manifolds may target covariance fidelity, discriminative comparison, or stable interpolation and averaging.

5. Recovering intrinsic geometry from operators, data, and graphs

Another major strand treats metric matching as geometry recovery. On a fixed triangle mesh, the cotangent discrete Laplace–Beltrami operator determines the Euclidean polyhedral metric uniquely up to a global scaling, and conversely the metric determines the operator. The discrete metric is represented by edge lengths, the Laplacian by cotangent weights, and the proof constructs a convex energy whose gradient is the edge-weight vector (Gu et al., 2010). In this setting, matching Laplace–Beltrami data is equivalent to matching the discrete metric up to scale.

A spectral smooth-manifold analogue is given by Riemannian Bergman metrics. Starting from the first $$L(\gamma)=\int_0^1 \sqrt{g_{\gamma(t)}(\dot\gamma(t),\dot\gamma(t))}\,dt,$$7 eigenspaces of the Laplace–Beltrami operator of a reference metric $$L(\gamma)=\int_0^1 \sqrt{g_{\gamma(t)}(\dot\gamma(t),\dot\gamma(t))}\,dt,$$8, one obtains a finite-dimensional family of pullback Euclidean metrics. The resulting Bergman metrics approximate any given Riemannian metric asymptotically, and the union of these finite-dimensional spaces is $$L(\gamma)=\int_0^1 \sqrt{g_{\gamma(t)}(\dot\gamma(t),\dot\gamma(t))}\,dt,$$9-dense in $$d_g(x,y)=\inf_{\gamma(0)=x,\ \gamma(1)=y} L(\gamma),$$0 (Potash, 2013). This provides a finite-dimensional surrogate for metric matching on the infinite-dimensional manifold of smooth metrics.

The diffusion-geometry paper on scalable geometric modeling of distributions takes the carré du champ operator as the primary object. For a diffusion generator $$d_g(x,y)=\inf_{\gamma(0)=x,\ \gamma(1)=y} L(\gamma),$$1,

$$d_g(x,y)=\inf_{\gamma(0)=x,\ \gamma(1)=y} L(\gamma),$$2

and on a Riemannian manifold with $$d_g(x,y)=\inf_{\gamma(0)=x,\ \gamma(1)=y} L(\gamma),$$3, one has $$d_g(x,y)=\inf_{\gamma(0)=x,\ \gamma(1)=y} L(\gamma),$$4. The paper rewrites the finite-bandwidth kernel CDC as a conditional expectation under Gaussian corruption and trains a network with

$$d_g(x,y)=\inf_{\gamma(0)=x,\ \gamma(1)=y} L(\gamma),$$5

Thus the learned object is a local co-metric or tangent projector in ambient coordinates, obtained without explicit graph construction (Bamberger et al., 12 Jun 2026).

A graph-theoretic recovery result appears in random geometric graphs on a compact Riemannian manifold. When edges are generated with probability $$d_g(x,y)=\inf_{\gamma(0)=x,\ \gamma(1)=y} L(\gamma),$$6, there is an $$d_g(x,y)=\inf_{\gamma(0)=x,\ \gamma(1)=y} L(\gamma),$$7-time algorithm that reconstructs all pairwise intrinsic distances with uniform additive error

$$d_g(x,y)=\inf_{\gamma(0)=x,\ \gamma(1)=y} L(\gamma),$$8

under the paper’s regularity assumptions (Huang et al., 7 Nov 2025). Here the matched object is the sampled intrinsic metric-space structure itself.

A density-based alternative is inverse-volume metric learning. Given a family $$d_g(x,y)=\inf_{\gamma(0)=x,\ \gamma(1)=y} L(\gamma),$$9, one maximizes

$\sqrt{\det g}$0

so that metrics with small local volume at the observed sample points are preferred (Lebanon, 2012). In this literature, matching means aligning the metric with the empirical distribution so that paths through data-dense regions become relatively short.

6. Metric selection for optimization and recurring limitations

In optimization, metric matching is explicitly formulated as preconditioning. For equality-constrained problems, the key proposal is to choose the Riemannian metric from the second-order structure of the Lagrangian,

$\sqrt{\det g}$1

with metric candidate

$\sqrt{\det g}$2

On quotient manifolds, the Lagrangian Hessian is degenerate in vertical directions but captures the second-order information in horizontal directions, so the metric should be built from the quotient-relevant curvature (Mishra et al., 2014). The product-manifold framework sharpens this by defining

$\sqrt{\det g}$3

where $\sqrt{\det g}$4 approximates the diagonal blocks of the Riemannian Hessian (Gao et al., 2023). In both papers, matching means making the metric reflect curvature well enough to reduce the condition number of the Riemannian Hessian and accelerate first-order methods.

A recurring misconception is that Riemannian metric matching must always mean minimizing a geodesic distance between two metric tensors. In the literature considered here, the matched objects are more often model and sample covariances, shapes modulo reparameterization, local diffusion operators, graph-induced geodesic distances, or the second-order structure of an optimization problem (Cohen et al., 12 May 2026, Brigant et al., 2024, Bamberger et al., 12 Jun 2026, Mishra et al., 2014). This suggests that induced-geometry matching is currently more operational than direct metric-to-metric matching on the manifold of all Riemannian metrics.

The same literature also makes the main limitations explicit. Pullback and probabilistic latent-space constructions require differentiable maps, and in the GP-LVM case the practical algorithms use only the expected metric rather than the full random metric (Tosi et al., 2014). SERCOM is on-grid, generally nonconvex, and assumes known noise variance (Cohen et al., 12 May 2026). The curve-shape algorithm is formulated for open immersed curves and emphasizes constructive computation rather than broad convergence theorems (Brigant et al., 2024). The Laplace–Beltrami identifiability theorem assumes fixed mesh combinatorics and Euclidean polyhedral structure (Gu et al., 2010). Random-graph reconstruction assumes a known connection function and geometric regularity constants (Huang et al., 7 Nov 2025). The complete mesh metric applies to planar meshes with fixed connectivity rather than to correspondence-free shape spaces (Herzog et al., 2020).

Across these settings, Riemannian metric matching is best understood as the systematic selection of a metric so that the resulting notions of length, distance, interpolation, transport, averaging, or optimization are faithful to the geometry that the problem actually presents.