Coordinate-Preserving Transport

• Coordinate-preserving transport is a class of methods that strictly maintain inherent structures—such as coordinate axes, mesh topology, or measures—while executing transformation tasks.
• Techniques range from coordinate-wise monotone optimal transport in latent generative models to volume-preserving flows in fluid mechanics and signed-permutation corrections in neural networks.
• These approaches ensure minimal distortion by enforcing invariances, offering robust solutions in optimal transport theory, geometric mechanics, and numerical discretizations.

Searching arXiv for recent and relevant papers on coordinate-preserving transport and related structure-preserving transport formulations. Coordinate-preserving transport denotes a class of transport constructions in which the transformation is constrained to preserve an explicit coordinate, index, geometric, or measure structure while still accomplishing a transport objective. Across the literature, this phrase refers to several closely related regimes: coordinate-wise monotone optimal transport in latent spaces of generative models, volume- and measure-preserving Lagrangian transport in incompressible flows, signed-permutation transport of residual-stream coordinates in RMSNorm Transformers, operator- and geometry-preserving transport between metric-measure spaces, and structure-preserving numerical transport schemes for PDEs (Agustsson et al., 2017). A common theme is that transport is not treated as an arbitrary map between distributions or states, but as a map restricted by a native structure—such as coordinate axes, latent priors, mesh topology, Wasserstein tangent coordinates, or an intrinsic geometric operator—which must remain meaningful after transport.

1. Coordinate preservation as a transport principle

In its most literal form, coordinate-preserving transport modifies a prescribed transformation as little as possible while enforcing a target invariance. In latent-variable generative modeling, this appears when coordinate-wise linear operations such as interpolation, vicinity sampling, and vector arithmetic are corrected by optimal transport maps so that the resulting latent variables are again distributed exactly according to the prior distribution on which the generator was trained (Agustsson et al., 2017). The original operations are defined coordinate-wise in $\mathbb{R}^d$, and the correction is likewise chosen coordinate-wise; the preserved structure is therefore the latent coordinate system itself.

A distinct but related meaning arises in incompressible fluid transport. There, the flow map $\varphi_0^t$ is volume-preserving, with $\det D\varphi_0^t=1$, and the transport formula is rewritten in Lagrangian coordinates so that the transport of a conserved scalar through a surface becomes a count of trajectory crossings weighted by the initial density (Karrasch, 2015). In that setting, coordinate preservation is expressed as invariance of Lebesgue measure under the flow, together with a coordinate-free interpretation via degree and donating regions.

In RMSNorm Transformers, the relevant coordinates are residual-stream indices. The native gauge of the residual stream is not fully fixed by the function represented by the model: for RMSNorm with generic per-channel gain, residual coordinates are defined only up to the signed-permutation group $B_d = S_d \ltimes \{\pm 1\}^d$ (Sweeney, 30 Jun 2026). Coordinate-preserving transport therefore means identifying and applying the correct signed permutation when moving steering vectors, sparse autoencoders, top-$k$ neuron sets, optimizer state, or merge alignments across checkpoints.

A broader generalization appears in optimal transport on metric-measure spaces. Convex Distance Operator Transport aligns distributions while preserving intrinsic geometric structure through a regularization term based on distance operators and conditional expectation operators, rather than only point correspondences (Chung et al., 1 Jun 2026). Here the preserved structure is not a Euclidean coordinate axis but an operator-level encoding of geometry.

This diversity suggests that “coordinate-preserving transport” is best understood as an umbrella term for transport procedures in which admissible transformations are restricted by a native representation. A plausible implication is that the relevant notion of “coordinate” is architecture- or problem-dependent: Euclidean coordinates in latent spaces, spatial coordinates on meshes, residual indices in neural networks, or operator coordinates in geometry-aware transport.

2. Distribution-preserving coordinate transport in latent spaces

The paper "Optimal transport maps for distribution preserving operations on latent spaces of Generative Models" formalizes a particularly concrete version of coordinate-preserving transport (Agustsson et al., 2017). A generator

$G:\mathbb{R}^d\to\mathbb{R}^{d'}$

is trained with a fixed latent prior $p_z$, typically either i.i.d. uniform on $[-1,1]^d$ or i.i.d. Gaussian $\mathcal{N}(0,1)^d$. After training, one commonly applies coordinate-wise linear latent operations: $$y_t = t z_1 + (1-t) z_2,\qquad y = \sum_{i=1}^n t_i z_i,\qquad y_j = z_1 + \epsilon u_j,\qquad y = z_3 + (z_2-z_1).$$

The central observation is that these operations do not preserve the latent prior distribution. For two-point interpolation,

$\varphi_0^t$0

one has

$\varphi_0^t$1

which differs from $\varphi_0^t$2 unless $\varphi_0^t$3. The same variance mismatch appears for $\varphi_0^t$4-point interpolation, vicinity sampling, and analogies. In high dimension, the mismatch becomes a geometric shell effect: midpoint samples lie on a norm scale different from that of the prior, so they are outliers relative to the region on which the generator was trained.

The remedy is posed as a Monge optimal transport problem. If $\varphi_0^t$5 has distribution $\varphi_0^t$6, the corrected operation is

$\varphi_0^t$7

where $\varphi_0^t$8 is chosen so that $\varphi_0^t$9 and minimizes

$\det D\varphi_0^t=1$0

With additive cost

$\det D\varphi_0^t=1$1

i.i.d. coordinates, and coordinate-wise latent operations, the $\det D\varphi_0^t=1$2-dimensional transport problem decomposes into $\det D\varphi_0^t=1$3 one-dimensional optimal transport problems. The resulting map is applied coordinate-wise: $\det D\varphi_0^t=1$4

For convex one-dimensional costs, the optimal map is the monotone rearrangement

$\det D\varphi_0^t=1$5

Hence the corrected latent operation is

$\det D\varphi_0^t=1$6

This preserves order along each coordinate axis, introduces no cross-coordinate mixing, and matches the prior exactly.

For Gaussian priors, the map reduces to a global scalar rescaling. For two-point interpolation,

$\det D\varphi_0^t=1$7

and analogous formulas hold for $\det D\varphi_0^t=1$8-point interpolation, vicinity sampling, and analogies. For uniform priors, the correction is nonlinear but still scalar and monotone, obtained from the one-dimensional CDFs of the distorted marginals.

This construction exemplifies coordinate-preserving transport in a strong sense: the corrected map does not rotate or mix latent coordinates, but only performs the minimal monotone deformation required to restore the training prior.

3. Measure-preserving and coordinate-free transport in continuum mechanics

A second major interpretation comes from Lagrangian transport through surfaces in incompressible flows (Karrasch, 2015). Let $\det D\varphi_0^t=1$9 be a smooth, time-dependent, volume-preserving velocity field on $B_d = S_d \ltimes \{\pm 1\}^d$0, with flow map $B_d = S_d \ltimes \{\pm 1\}^d$1 satisfying

$B_d = S_d \ltimes \{\pm 1\}^d$2

For a conserved scalar $B_d = S_d \ltimes \{\pm 1\}^d$3 obeying

$B_d = S_d \ltimes \{\pm 1\}^d$4

transport through a surface is classically measured by the Eulerian flux integral. The paper reformulates this in Lagrangian coordinates by introducing

$B_d = S_d \ltimes \{\pm 1\}^d$5

For a stationary section $B_d = S_d \ltimes \{\pm 1\}^d$6, the Eulerian flux

$B_d = S_d \ltimes \{\pm 1\}^d$7

is shown to equal

$B_d = S_d \ltimes \{\pm 1\}^d$8

where $B_d = S_d \ltimes \{\pm 1\}^d$9 is the set of initial points whose trajectories cross the section with net crossing number $k$0. The key topological quantity is the degree

$k$1

which counts signed crossings of trajectories.

The equality between Eulerian and Lagrangian forms relies decisively on volume preservation. Because $k$2, no additional Jacobian weighting appears when changing from Eulerian to Lagrangian coordinates. The transport is therefore a pure rearrangement of Lebesgue measure, and the flux becomes a measure-preserving count of trajectory crossings. The formalism is coordinate-free in the sense that it depends on the geometry of trajectories and surfaces, not on a particular parameterization.

This paper also gives an algorithmic interpretation. One computes $k$3 by integrating trajectories backward, identifies donating regions $k$4, and evaluates

$k$5

or its weighted analogue for nonconstant $k$6. In two dimensions, the degree is computed through winding numbers of the image of the boundary.

A plausible connection to coordinate-preserving transport more broadly is that measure preservation plays the role of the invariant coordinate structure. Rather than preserving Euclidean axes, the flow preserves the reference measure and thereby permits a Lagrangian transport description with no coordinate-dependent distortion term.

4. Gauge-correct coordinate transport in RMSNorm Transformers

In RMSNorm Transformers, coordinate-preserving transport is not about physical space but about residual-stream indices and their native symmetry group (Sweeney, 30 Jun 2026). Residual activations are row-vectors

$k$7

and a gauge acts on the right: $k$8

The paper identifies an architecture-dependent discrete gauge. For LayerNorm, the residual-stream gauge is $k$9. For RMSNorm with generic per-channel gain $G:\mathbb{R}^d\to\mathbb{R}^{d'}$0, the maximal native discrete gauge is the signed-permutation group

$G:\mathbb{R}^d\to\mathbb{R}^{d'}$1

Thus permutation-only alignment is symmetry-incomplete for RMSNorm models.

Given paired probe activations $G:\mathbb{R}^d\to\mathbb{R}^{d'}$2, define cross-covariance

$G:\mathbb{R}^d\to\mathbb{R}^{d'}$3

If source coordinate $G:\mathbb{R}^d\to\mathbb{R}^{d'}$4 matches target coordinate $G:\mathbb{R}^d\to\mathbb{R}^{d'}$5 with sign $G:\mathbb{R}^d\to\mathbb{R}^{d'}$6, then the matching problem is

$G:\mathbb{R}^d\to\mathbb{R}^{d'}$7

For fixed permutation, the optimal sign is

$G:\mathbb{R}^d\to\mathbb{R}^{d'}$8

so the optimal permutation is found by

$G:\mathbb{R}^d\to\mathbb{R}^{d'}$9

This is solved by Hungarian or Jonker–Volgenant assignment on cost matrix $p_z$0, followed by sign extraction from the matched entries. The resulting forward gauge is

$p_z$1

The paper proves a structural failure mode for naive signed-correlation matching without sign marginalization: with decorrelated coordinates, asymptotic permutation accuracy is capped at the positive-sign fraction of the true gauge, which is approximately $p_z$2 under i.i.d. uniform signs. Sign-marginalized matching removes this ceiling.

The practical relevance is that many coordinate-indexed objects transform covariantly only under the full $p_z$3 gauge. The paper reports that composing local $p_z$4 gauges along same-base fine-tuning trajectories recovers 91.1% of cross-run coordinates at 1500 steps versus 60.3% for endpoint matching, and that the gain is not explained by merely routing through the base. It also reports that TinyLlama SAE reconstruction has NMSE 0.004 under $p_z$5 versus 1.08 under $p_z$6, that Qwen sentiment steering preserves 95.8% of its effect versus 17.2%, that refusal steering reverses sign under $p_z$7, and that signed transport of AdamW state preserves the resumed trajectory while permutation-only state follows a different one.

These results establish a highly specific sense of coordinate-preserving transport: the transported object must preserve index-level identity under the model’s native gauge, not merely the function realized by the checkpoint. This suggests that “coordinate preservation” in neural networks is meaningful only relative to an explicit gauge.

5. Geometry-preserving transport beyond Euclidean coordinates

Several recent formulations extend coordinate-preserving transport to settings where the relevant structure is geometric rather than axis-aligned. Convex Distance Operator Transport defines a convex OT discrepancy between attributed compact metric-measure spaces by combining a feature-matching term

$p_z$8

with an operator commutation penalty

$p_z$9

where $[-1,1]^d$0 and $[-1,1]^d$1 are distance operators and $[-1,1]^d$2 is the conditional expectation operator induced by the coupling (Chung et al., 1 Jun 2026). The full objective is

$[-1,1]^d$3

The operator penalty enforces alignment of aggregated distance structures rather than exact pairwise isometries. The resulting discrepancy is a pseudometric on attributed compact metric-measure spaces, and the paper identifies the relation

$[-1,1]^d$4

where $[-1,1]^d$5 is a dispersion term that explains the non-convexity of Gromov–Wasserstein relative to CDOT’s convexity. In this setting, geometry preservation replaces coordinate-axis preservation.

A transport-based reduced-order modeling framework introduces yet another coordinate system: transport signatures derived from OT potentials relative to a fixed reference density $[-1,1]^d$6 (Yu et al., 2 Jul 2026). For each parameter $[-1,1]^d$7, the optimal map from $[-1,1]^d$8 to $[-1,1]^d$9 is

$\mathcal{N}(0,1)^d$0

and the associated signature is

$\mathcal{N}(0,1)^d$1

Because

$\mathcal{N}(0,1)^d$2

signature error controls transport-map error and hence Wasserstein error. The reconstructed solution is always a pushforward of $\mathcal{N}(0,1)^d$3, so mass preservation is built into the model. This is coordinate-preserving in the sense of preserving the OT chart around the reference density.

A more differential-geometric formulation appears in the global-coordinate treatment of mechanics and optimal transport on embedded manifolds (Nguyen, 2023). There, an affine projection $\mathcal{N}(0,1)^d$4 onto $\mathcal{N}(0,1)^d$5 replaces local coordinates and Christoffel symbols by operator-valued formulas in the ambient Euclidean space. Hamilton vector fields, affine and Levi-Civita connections, curvature, and the Kim–McCann cross-curvature are expressed globally in terms of $\mathcal{N}(0,1)^d$6, $\mathcal{N}(0,1)^d$7, and metric operators. This realizes coordinate-preserving transport as transport in a fixed global coordinate system that still exactly respects manifold constraints.

Together, these works indicate that “coordinate preservation” need not mean preservation of literal coordinates. A plausible implication is that the underlying invariant may be any native representation in which transport is semantically or geometrically well posed: distance operators, Wasserstein tangent coordinates, or ambient-coordinate realizations of manifold geometry.

6. Numerical, topological, and structure-preserving discretizations

Coordinate-preserving transport also appears as a numerical design principle for transport PDEs and mesh mappings. In a structure-preserving finite element discretization of the transport equation, the continuous one-dimensional system

$\mathcal{N}(0,1)^d$8

is discretized so that the discrete Hamiltonian

$\mathcal{N}(0,1)^d$9

satisfies exactly the same scattering-energy balance as the continuum: $$y_t = t z_1 + (1-t) z_2,\qquad y = \sum_{i=1}^n t_i z_i,\qquad y_j = z_1 + \epsilon u_j,\qquad y = z_3 + (z_2-z_1).$$0 in one dimension, and the corresponding boundary-flux identity in two dimensions (Toledo-Zucco et al., 2024). For moving meshes with time-dependent basis functions $$y_t = t z_1 + (1-t) z_2,\qquad y = \sum_{i=1}^n t_i z_i,\qquad y_j = z_1 + \epsilon u_j,\qquad y = z_3 + (z_2-z_1).$$1 and spatially uniform $$y_t = t z_1 + (1-t) z_2,\qquad y = \sum_{i=1}^n t_i z_i,\qquad y_j = z_1 + \epsilon u_j,\qquad y = z_3 + (z_2-z_1).$$2, the same balance holds for any time-dependent mesh. In that paper, coordinate preservation means invariance of the scattering-passive structure under a changing spatial chart.

A complementary perspective comes from spectral discretization of transport and continuity equations on compact manifolds (Jacobs et al., 2014). By lifting densities to half-densities and discretizing the Lie derivative on a finite-dimensional $$y_t = t z_1 + (1-t) z_2,\qquad y = \sum_{i=1}^n t_i z_i,\qquad y_j = z_1 + \epsilon u_j,\qquad y = z_3 + (z_2-z_1).$$3 subspace, the scheme produces a discrete unitary flow that preserves the $$y_t = t z_1 + (1-t) z_2,\qquad y = \sum_{i=1}^n t_i z_i,\qquad y_j = z_1 + \epsilon u_j,\qquad y = z_3 + (z_2-z_1).$$4 norm of half-densities, exact mass conservation for densities, and the algebraic structure of transported observables. The matrix ODE

$$y_t = t z_1 + (1-t) z_2,\qquad y = \sum_{i=1}^n t_i z_i,\qquad y_j = z_1 + \epsilon u_j,\qquad y = z_3 + (z_2-z_1).$$5

implements the transport of multiplication operators, while the density is reconstructed as the square of the transported half-density. The preserved “coordinates” are the Hilbert-space coordinates in which the flow is unitary.

On triangular meshes, "QC-OT: Optimal Transport with Quasiconformal Mapping" introduces topology structure-preserving OT by relaxing the Delaunay-triangulation updates of semi-discrete OT and then correcting local distortions using quasiconformal maps (Lv et al., 2 Jul 2025). A relaxed feasible set permits empty power cells,

$$y_t = t z_1 + (1-t) z_2,\qquad y = \sum_{i=1}^n t_i z_i,\qquad y_j = z_1 + \epsilon u_j,\qquad y = z_3 + (z_2-z_1).$$6

and a quasiconformal correction step enforces bounded Beltrami coefficient $$y_t = t z_1 + (1-t) z_2,\qquad y = \sum_{i=1}^n t_i z_i,\qquad y_j = z_1 + \epsilon u_j,\qquad y = z_3 + (z_2-z_1).$$7, thereby guaranteeing orientation-preserving diffeomorphic behavior. The result preserves mesh connectivity, avoids triangle flips, and retains a prescribed mass distribution. Here the preserved coordinate structure is the mesh topology and face orientation.

These numerical frameworks reinforce a general pattern: coordinate-preserving transport is often implemented by changing representation so that the desired invariant becomes algebraic—unitarity, energy balance, topological orientation, or a fixed mesh connectivity—and then designing the transport algorithm to preserve that algebraic form exactly.

7. Limitations, ambiguities, and conceptual scope

Despite the recurring terminology, the literature does not define a single universal object called coordinate-preserving transport. In latent OT, the preserved coordinates are i.i.d. scalar coordinates of a latent prior; in RMSNorm Transformers, they are residual indices modulo signed permutations; in incompressible transport, the invariant is Lebesgue measure and the Lagrangian coordinate map; in geometry-aware OT, the preserved object may be a distance operator rather than coordinates in the usual sense (Agustsson et al., 2017).

Several limitations recur. Coordinate-wise latent transport assumes i.i.d. priors and additive costs; for correlated or learned priors, the reduction to scalar monotone rearrangements no longer applies. Signed-permutation transport in Transformers depends on same-base trajectories and probe activations, and no canonical coordinate transport exists for independently specified endpoints if one demands both dependence only on the function and gauge equivariance (Sweeney, 30 Jun 2026). Geometry-preserving convex OT relaxes pairwise distance preservation to aggregated operator alignment, so its zero discrepancy is only a pseudometric notion of equivalence. Topology-preserving OT on meshes relaxes classical convexity guarantees, and the quasiconformal correction is applied after the OT step rather than solved jointly.

At the same time, the accumulated evidence suggests a coherent organizing principle. Coordinate-preserving transport is best understood as transport constrained by a native symmetry or structural representation, together with a correction mechanism that restores the target invariance while minimally disturbing that representation. Depending on the application, the invariant may be prior distribution, coordinate order, signed index identity, volume measure, scattering-energy balance, mesh topology, or operator-level geometry. In that sense, the topic sits at the intersection of optimal transport, geometric mechanics, generative modeling, numerical PDEs, and neural-network gauge theory.