Support-Preserving Manifold Assimilation

• SPMA is a geometric assimilation concept that preserves support structure, manifold constraints, and distribution morphology rather than relying solely on Euclidean error minimization.
• It encompasses various methodologies—such as Wasserstein barycenter optimization, tangent-space projection on constraint manifolds, DTW-based feature alignment, and latent atlas continuation—that address different challenges in data assimilation.
• The approach improves performance in non-Gaussian, biased, or discontinuous settings, though it requires significant computational resources and careful parameter tuning.

Support-Preserving Manifold Assimilation (SPMA) denotes a support-preserving, geometry-aware view of assimilation in which updates are constructed to respect support location, manifold structure, and distribution morphology rather than only minimizing coordinatewise discrepancy in a flat Euclidean state space. In the cited literature, this idea appears in several operational forms: as a Wasserstein-space analogue for ensemble data assimilation, as constraint-preserving particle flow on nonlinear manifolds, as feature-aligned transport for discontinuous compressible-flow states, and as shared-manifold continuation in continual learning under compatible shift (Tamang et al., 2020, Subrahmanya et al., 2024, Subrahmanya et al., 1 May 2025, Kobs, 20 Mar 2026).

1. Conceptual basis

A recurring SPMA principle is that assimilation should preserve the morphology of distributions or states instead of relying on pointwise averaging or Euclidean penalization alone. In the Wasserstein formulation, the central contrast is between Eulerian penalization of state-vector error and Lagrangian transport of probability mass; in the feature-preserving filtering formulation, the contrast is between naive convex averaging and interpolation along feature-matched characteristics; in the constrained particle-flow formulation, the contrast is between unconstrained posterior updates and updates confined to a nonlinear admissible manifold; in the continual-learning formulation, the contrast is between preserving parameters or logits and preserving the old latent support itself (Tamang et al., 2020, Subrahmanya et al., 2024, Subrahmanya et al., 1 May 2025, Kobs, 20 Mar 2026).

Within this perspective, “support-preserving” has several concrete meanings. It can mean that mass is transported across disjoint supports rather than compared at fixed coordinates, that sharp structures such as shocks and contact discontinuities are aligned before mixing, that stochastic particle trajectories remain on a constraint manifold $\mathcal{M}=\{x\in\mathbb{R}^s:\ g(x)=0\}$, or that old latent anchors remain on an atlas fitted to frozen teacher features. “Manifold” is likewise used in more than one sense: a Riemannian manifold of probability measures equipped with the Wasserstein metric, a nonlinear physical constraint manifold, a feature manifold traced by matched coherent structures, or a latent atlas of local charts under compatible distribution shift.

This usage makes SPMA a family of geometric assimilation principles rather than a single update rule. A plausible implication is that the unifying object is not a particular optimizer or filter, but the requirement that posterior states remain faithful to the admissible support on which the relevant dynamics, structures, or representations live.

2. Wasserstein-space realization and distribution morphology

"Ensemble Riemannian Data Assimilation over the Wasserstein Space" formulates a concrete realization of the SPMA idea by replacing Euclidean error penalization with a Fréchet barycenter problem over the Wasserstein space of probability histograms (Tamang et al., 2020). In standard 3D-Var, the analysis minimizes

$$J(\mathbf{x}) \propto \frac{1}{2}(\mathbf{x}-\mathbf{x}_b)^\top \mathbf{B}^{-1}(\mathbf{x}-\mathbf{x}_b) +\frac{1}{2}(\mathbf{y}-\mathcal{H}(\mathbf{x}))^\top \mathbf{R}^{-1}(\mathbf{y}-\mathcal{H}(\mathbf{x})),$$

where the penalty is a weighted Euclidean norm in state space. The EnRDA formulation instead defines the analysis distribution as

$$p(\mathbf{x}_a) = \arg\min_{\mathbf{p}_x} \left\{ \eta\, d_{\mathcal W}^2(\mathbf{p}_x,\mathbf{p}_{x_b}) + (1-\eta)\, d_{\mathcal W}^2\!\left(\mathbf{p}_x,\;|\det[\mathcal H'(\mathbf{x})]|\,\mathbf{p}_y\right) \right\},$$

so that the optimizer is a distribution lying between background and observations along a geodesic in probability space rather than an additive state correction.

The support-preserving content is explicit in optimal transport. In the discrete setting, the optimal transport plan solves

$\mathbf{U}^a = \arg\min_{\mathbf{U}\ge 0} \langle \mathbf{C},\mathbf{U}\rangle \quad \text{s.t.} \quad \mathbf{U}\mathbbm{1}_N=\mathbf{p}_x,\; \mathbf{U}^\top \mathbbm{1}_M=\mathbf{p}_y,$

with ground cost $c_{ij}=\|\mathbf{x}_i-\mathbf{y}_j\|_q^q$. For $q=2$, the minimum cost defines the squared 2-Wasserstein distance. The paper emphasizes that this metric transports mass and therefore represents translations, shape changes, and matching of disjoint supports in a way that Euclidean distances do not. It also notes the decomposition

$$d_{\mathcal W}^2(\mathbf{p}_x,\mathbf{p}_y) = d_{\mathcal W}^2(\overline{\mathbf{p}_x},\overline{\mathbf{p}_y}) + \|\boldsymbol{\mu}_x-\boldsymbol{\mu}_y\|_2^2,$$

which separates centered-shape mismatch from translation.

The ensemble formulation writes the background as

$$p(\mathbf{x}_b)=\frac{1}{M}\sum_{i=1}^M \delta_{\mathbf{x}_i},$$

and computes the background-observation discrepancy through an optimal transport coupling rather than a covariance-weighted innovation. Computational feasibility is obtained through entropic regularization and Sinkhorn scaling,

$$\mathbf{U}^a=\operatorname{diag}(\mathbf{v})\,\mathbf{K}\,\operatorname{diag}(\mathbf{w}),$$

so the analysis update is a transport plan whose marginals match the background and observation histograms.

The reported strengths are specific. EnRDA is designed for biased, non-Gaussian, dissipative, and chaotic dynamics; in advection-diffusion experiments it preserved the shape of bimodal fields and reduced bias and uBRMSE better than 3D-Var under systematic model error; in Lorenz-63 it was more robust than particle filtering when observations fell outside particle support and improved over EnKF under bias, though the improvement over EnKF was more modest. The limitations are equally explicit: the method is computationally expensive, requires empirical tuning of $\eta$ and $$J(\mathbf{x}) \propto \frac{1}{2}(\mathbf{x}-\mathbf{x}_b)^\top \mathbf{B}^{-1}(\mathbf{x}-\mathbf{x}_b) +\frac{1}{2}(\mathbf{y}-\mathcal{H}(\mathbf{x}))^\top \mathbf{R}^{-1}(\mathbf{y}-\mathcal{H}(\mathbf{x})),$$0, assumes a smooth bijective observation operator, and does not naturally propagate information from observed to unobserved variables in partially observed systems. The paper is also careful that EnRDA does not universally outperform minimum mean-squared estimators in unbiased approximately Gaussian settings.

3. Nonlinear constraint manifolds and stochastic flow preservation

"Preserving Nonlinear Constraints in Variational Flow Filtering Data Assimilation" extends the variational Fokker-Planck particle flow filter to what it explicitly identifies as support-preserving manifold assimilation, namely data assimilation in which the posterior ensemble must remain on a nonlinear constraint manifold (Subrahmanya et al., 2024). The admissible states are defined by a nonlinear constraint function $$J(\mathbf{x}) \propto \frac{1}{2}(\mathbf{x}-\mathbf{x}_b)^\top \mathbf{B}^{-1}(\mathbf{x}-\mathbf{x}_b) +\frac{1}{2}(\mathbf{y}-\mathcal{H}(\mathbf{x}))^\top \mathbf{R}^{-1}(\mathbf{y}-\mathcal{H}(\mathbf{x})),$$1 and manifold

$$J(\mathbf{x}) \propto \frac{1}{2}(\mathbf{x}-\mathbf{x}_b)^\top \mathbf{B}^{-1}(\mathbf{x}-\mathbf{x}_b) +\frac{1}{2}(\mathbf{y}-\mathcal{H}(\mathbf{x}))^\top \mathbf{R}^{-1}(\mathbf{y}-\mathcal{H}(\mathbf{x})),$$2

The underlying VFP flow evolves particles in pseudo-time $$J(\mathbf{x}) \propto \frac{1}{2}(\mathbf{x}-\mathbf{x}_b)^\top \mathbf{B}^{-1}(\mathbf{x}-\mathbf{x}_b) +\frac{1}{2}(\mathbf{y}-\mathcal{H}(\mathbf{x}))^\top \mathbf{R}^{-1}(\mathbf{y}-\mathcal{H}(\mathbf{x})),$$3 through a McKean–Vlasov Itô SDE,

$$J(\mathbf{x}) \propto \frac{1}{2}(\mathbf{x}-\mathbf{x}_b)^\top \mathbf{B}^{-1}(\mathbf{x}-\mathbf{x}_b) +\frac{1}{2}(\mathbf{y}-\mathcal{H}(\mathbf{x}))^\top \mathbf{R}^{-1}(\mathbf{y}-\mathcal{H}(\mathbf{x})),$$4

with a corresponding Fokker–Planck equation for the particle density.

Two constraint-preserving modifications are proposed. VFPSTAB adds a stabilizing drift toward the manifold,

$$J(\mathbf{x}) \propto \frac{1}{2}(\mathbf{x}-\mathbf{x}_b)^\top \mathbf{B}^{-1}(\mathbf{x}-\mathbf{x}_b) +\frac{1}{2}(\mathbf{y}-\mathcal{H}(\mathbf{x}))^\top \mathbf{R}^{-1}(\mathbf{y}-\mathcal{H}(\mathbf{x})),$$5

where $$J(\mathbf{x}) \propto \frac{1}{2}(\mathbf{x}-\mathbf{x}_b)^\top \mathbf{B}^{-1}(\mathbf{x}-\mathbf{x}_b) +\frac{1}{2}(\mathbf{y}-\mathcal{H}(\mathbf{x}))^\top \mathbf{R}^{-1}(\mathbf{y}-\mathcal{H}(\mathbf{x})),$$6. This reduces constraint violation but does not enforce $$J(\mathbf{x}) \propto \frac{1}{2}(\mathbf{x}-\mathbf{x}_b)^\top \mathbf{B}^{-1}(\mathbf{x}-\mathbf{x}_b) +\frac{1}{2}(\mathbf{y}-\mathcal{H}(\mathbf{x}))^\top \mathbf{R}^{-1}(\mathbf{y}-\mathcal{H}(\mathbf{x})),$$7 exactly. VFPDAE instead rewrites the flow as a stochastic differential-algebraic equation and eliminates the algebraic variable to obtain the tangent-space-projected dynamics

$$J(\mathbf{x}) \propto \frac{1}{2}(\mathbf{x}-\mathbf{x}_b)^\top \mathbf{B}^{-1}(\mathbf{x}-\mathbf{x}_b) +\frac{1}{2}(\mathbf{y}-\mathcal{H}(\mathbf{x}))^\top \mathbf{R}^{-1}(\mathbf{y}-\mathcal{H}(\mathbf{x})),$$8

The projection matrix

$$J(\mathbf{x}) \propto \frac{1}{2}(\mathbf{x}-\mathbf{x}_b)^\top \mathbf{B}^{-1}(\mathbf{x}-\mathbf{x}_b) +\frac{1}{2}(\mathbf{y}-\mathcal{H}(\mathbf{x}))^\top \mathbf{R}^{-1}(\mathbf{y}-\mathcal{H}(\mathbf{x})),$$9

removes the component normal to the manifold, so exact preservation is enforced up to numerical solve tolerance.

The paper also develops an IMEX-style “evolve then project” integrator. An unconstrained Euler–Maruyama step

$$p(\mathbf{x}_a) = \arg\min_{\mathbf{p}_x} \left\{ \eta\, d_{\mathcal W}^2(\mathbf{p}_x,\mathbf{p}_{x_b}) + (1-\eta)\, d_{\mathcal W}^2\!\left(\mathbf{p}_x,\;|\det[\mathcal H'(\mathbf{x})]|\,\mathbf{p}_y\right) \right\},$$0

is followed by a low-dimensional root solve for the projection variable so that the updated state satisfies the constraint. This is motivated by the scarcity of numerical methods for index-2 SDAEs.

Evaluation is reported on the double pendulum, Korteweg–de Vries, and incompressible Navier–Stokes equations, using spatio-temporal RMSE and a constraint RMSE (CRMSE). The main findings are method-specific rather than universal. In the double pendulum, ETKF variants achieved slightly lower RMSE than VFP variants, but VFPDAE and ETKFP preserved constraints to machine precision; asymptotic CRMSE values were reported as ETKF $$p(\mathbf{x}_a) = \arg\min_{\mathbf{p}_x} \left\{ \eta\, d_{\mathcal W}^2(\mathbf{p}_x,\mathbf{p}_{x_b}) + (1-\eta)\, d_{\mathcal W}^2\!\left(\mathbf{p}_x,\;|\det[\mathcal H'(\mathbf{x})]|\,\mathbf{p}_y\right) \right\},$$1, ETKFA $$p(\mathbf{x}_a) = \arg\min_{\mathbf{p}_x} \left\{ \eta\, d_{\mathcal W}^2(\mathbf{p}_x,\mathbf{p}_{x_b}) + (1-\eta)\, d_{\mathcal W}^2\!\left(\mathbf{p}_x,\;|\det[\mathcal H'(\mathbf{x})]|\,\mathbf{p}_y\right) \right\},$$2, VFP $$p(\mathbf{x}_a) = \arg\min_{\mathbf{p}_x} \left\{ \eta\, d_{\mathcal W}^2(\mathbf{p}_x,\mathbf{p}_{x_b}) + (1-\eta)\, d_{\mathcal W}^2\!\left(\mathbf{p}_x,\;|\det[\mathcal H'(\mathbf{x})]|\,\mathbf{p}_y\right) \right\},$$3, VFPSTAB $$p(\mathbf{x}_a) = \arg\min_{\mathbf{p}_x} \left\{ \eta\, d_{\mathcal W}^2(\mathbf{p}_x,\mathbf{p}_{x_b}) + (1-\eta)\, d_{\mathcal W}^2\!\left(\mathbf{p}_x,\;|\det[\mathcal H'(\mathbf{x})]|\,\mathbf{p}_y\right) \right\},$$4, and VFPDAE and ETKFP at machine precision. In KdV, VFPDAE outperformed all ETKF-based methods in RMSE and preserved constraints to machine precision. In incompressible Navier–Stokes, VFPDAE performed better than ETKF and ETKFP but worse than the best localized methods, while ETKFP, LETKFP, and VFPDAE preserved divergence to numerical precision and VFPDAE and ETKFP preserved enstrophy to the numerical limit. The paper also notes that projection-based methods can fail in difficult cases, as illustrated by a LETKFP spike in enstrophy CRMSE when projection failed at one step.

4. Feature-aligned transport for discontinuous flow states

"Feature preserving data assimilation via feature alignment" presents a feature-preserving ensemble transform particle filter for compressible-flow systems whose states contain sharp structures such as shocks, contact discontinuities, and rarefaction fronts (Subrahmanya et al., 1 May 2025). Its SPMA-like contribution is transport-based assimilation that respects manifold or support structure by aligning sharp features before convex combination. The underlying motivation is that linear averaging across misaligned discontinuities produces physically meaningless intermediate states and repeated averaging erodes features over multiple assimilation cycles.

The baseline ETPF writes the equally weighted analysis particles as

$$p(\mathbf{x}_a) = \arg\min_{\mathbf{p}_x} \left\{ \eta\, d_{\mathcal W}^2(\mathbf{p}_x,\mathbf{p}_{x_b}) + (1-\eta)\, d_{\mathcal W}^2\!\left(\mathbf{p}_x,\;|\det[\mathcal H'(\mathbf{x})]|\,\mathbf{p}_y\right) \right\},$$5

This deterministic transport avoids weight degeneracy, but because the analysis particles are convex combinations of forecast particles, it inherits feature smearing. The proposed modification inserts two additional steps: feature extraction and feature alignment. In one dimension, for density $$p(\mathbf{x}_a) = \arg\min_{\mathbf{p}_x} \left\{ \eta\, d_{\mathcal W}^2(\mathbf{p}_x,\mathbf{p}_{x_b}) + (1-\eta)\, d_{\mathcal W}^2\!\left(\mathbf{p}_x,\;|\det[\mathcal H'(\mathbf{x})]|\,\mathbf{p}_y\right) \right\},$$6 on a uniform grid, the feature vector is taken as a backward finite difference,

$$p(\mathbf{x}_a) = \arg\min_{\mathbf{p}_x} \left\{ \eta\, d_{\mathcal W}^2(\mathbf{p}_x,\mathbf{p}_{x_b}) + (1-\eta)\, d_{\mathcal W}^2\!\left(\mathbf{p}_x,\;|\det[\mathcal H'(\mathbf{x})]|\,\mathbf{p}_y\right) \right\},$$7

In two dimensions, a mixed finite difference is used to approximate $$p(\mathbf{x}_a) = \arg\min_{\mathbf{p}_x} \left\{ \eta\, d_{\mathcal W}^2(\mathbf{p}_x,\mathbf{p}_{x_b}) + (1-\eta)\, d_{\mathcal W}^2\!\left(\mathbf{p}_x,\;|\det[\mathcal H'(\mathbf{x})]|\,\mathbf{p}_y\right) \right\},$$8.

Alignment is performed by dynamic time warping. For two sequences, DTW chooses an alignment path $$p(\mathbf{x}_a) = \arg\min_{\mathbf{p}_x} \left\{ \eta\, d_{\mathcal W}^2(\mathbf{p}_x,\mathbf{p}_{x_b}) + (1-\eta)\, d_{\mathcal W}^2\!\left(\mathbf{p}_x,\;|\det[\mathcal H'(\mathbf{x})]|\,\mathbf{p}_y\right) \right\},$$9 that minimizes the aligned $\mathbf{U}^a = \arg\min_{\mathbf{U}\ge 0} \langle \mathbf{C},\mathbf{U}\rangle \quad \text{s.t.} \quad \mathbf{U}\mathbbm{1}_N=\mathbf{p}_x,\; \mathbf{U}^\top \mathbbm{1}_M=\mathbf{p}_y,$0-norm cost. With the alignment path in hand, the method defines aligned convex combinations. For two one-dimensional density fields $\mathbf{U}^a = \arg\min_{\mathbf{U}\ge 0} \langle \mathbf{C},\mathbf{U}\rangle \quad \text{s.t.} \quad \mathbf{U}\mathbbm{1}_N=\mathbf{p}_x,\; \mathbf{U}^\top \mathbbm{1}_M=\mathbf{p}_y,$1 and $\mathbf{U}^a = \arg\min_{\mathbf{U}\ge 0} \langle \mathbf{C},\mathbf{U}\rangle \quad \text{s.t.} \quad \mathbf{U}\mathbbm{1}_N=\mathbf{p}_x,\; \mathbf{U}^\top \mathbbm{1}_M=\mathbf{p}_y,$2, and mixing parameter $\mathbf{U}^a = \arg\min_{\mathbf{U}\ge 0} \langle \mathbf{C},\mathbf{U}\rangle \quad \text{s.t.} \quad \mathbf{U}\mathbbm{1}_N=\mathbf{p}_x,\; \mathbf{U}^\top \mathbbm{1}_M=\mathbf{p}_y,$3, an interpolant $\mathbf{U}^a = \arg\min_{\mathbf{U}\ge 0} \langle \mathbf{C},\mathbf{U}\rangle \quad \text{s.t.} \quad \mathbf{U}\mathbbm{1}_N=\mathbf{p}_x,\; \mathbf{U}^\top \mathbbm{1}_M=\mathbf{p}_y,$4 is constructed so that

$\mathbf{U}^a = \arg\min_{\mathbf{U}\ge 0} \langle \mathbf{C},\mathbf{U}\rangle \quad \text{s.t.} \quad \mathbf{U}\mathbbm{1}_N=\mathbf{p}_x,\; \mathbf{U}^\top \mathbbm{1}_M=\mathbf{p}_y,$5

after which the aligned mixture is sampled. The same procedure is applied to velocity and energy. In practice, the $\mathbf{U}^a = \arg\min_{\mathbf{U}\ge 0} \langle \mathbf{C},\mathbf{U}\rangle \quad \text{s.t.} \quad \mathbf{U}\mathbbm{1}_N=\mathbf{p}_x,\; \mathbf{U}^\top \mathbbm{1}_M=\mathbf{p}_y,$6-way convex combination from ETPF is rewritten as a sequence of pairwise convex combinations, and each pairwise Euclidean mixture is replaced by an aligned one. The paper recommends retaining the ordinary $\mathbf{U}^a = \arg\min_{\mathbf{U}\ge 0} \langle \mathbf{C},\mathbf{U}\rangle \quad \text{s.t.} \quad \mathbf{U}\mathbbm{1}_N=\mathbf{p}_x,\; \mathbf{U}^\top \mathbbm{1}_M=\mathbf{p}_y,$7 distance in the ETPF transport cost matrix, because DTW is more expensive and is not a true metric; replacing the cost with DTW did not provide noticeable benefit in experiments.

The experiments use the compressible Euler equations with WENO5 spatial discretization, a Lax–Friedrichs flux limiter, and third-order TVD Runge–Kutta time stepping. Observations are sparse pressure measurements only, with 9 sensors in 1D and $\mathbf{U}^a = \arg\min_{\mathbf{U}\ge 0} \langle \mathbf{C},\mathbf{U}\rangle \quad \text{s.t.} \quad \mathbf{U}\mathbbm{1}_N=\mathbf{p}_x,\; \mathbf{U}^\top \mathbbm{1}_M=\mathbf{p}_y,$8 sensors in 2D. Across Sod’s shock tube, Toro’s shock tube, Shu–Osher shock-entropy interaction, and a 2D blast wave, the standard ETPF tracks observations but smears structures, whereas the feature-preserving ETPF maintains sharp interfaces. The relative average ensemble error is reported to remain comparable to standard ETPF, with no systematic degradation in the usual error metric; in Sod’s case the aligned method eventually does better in error as well, while in Toro and Shu–Osher the errors are broadly similar.

The assumptions are narrow and explicit. The method assumes that the same solution features are present in all particles at a given assimilation time. It can fail when particles contain qualitatively different wave sets, as noted for reflective boundaries, large initial uncertainty, differing numbers of waves, and a failing Shu–Osher alignment case. The 2D alignment strategy is acknowledged as a weakness because it aligns columns and rows separately, which can introduce small-scale distortions, and turbulence-related cases tended to get averaged out.

5. Shared-manifold continuation in continual learning

"Continual Learning as Shared-Manifold Continuation Under Compatible Shift" reframes SPMA as a representation-level theory of continual learning in the regime where old and new data should remain on the same latent support (Kobs, 20 Mar 2026). The central claim is intentionally narrower than generic continual learning: under compatible shift, the appropriate inductive bias is to continue the old manifold through the new task with as little topological or geometric damage as possible. The target is therefore not merely old outputs or important parameters, but preservation of the old latent support, local geometry, and chart assignment while allowing local deformation for the new distribution.

The evaluated instantiation is SPMA-OG. It begins by building a chart memory from frozen teacher features $\mathbf{U}^a = \arg\min_{\mathbf{U}\ge 0} \langle \mathbf{C},\mathbf{U}\rangle \quad \text{s.t.} \quad \mathbf{U}\mathbbm{1}_N=\mathbf{p}_x,\; \mathbf{U}^\top \mathbbm{1}_M=\mathbf{p}_y,$9, clustering these features into $c_{ij}=\|\mathbf{x}_i-\mathbf{y}_j\|_q^q$0 local components, and fitting a low-rank Gaussian factor model per cluster,

$c_{ij}=\|\mathbf{x}_i-\mathbf{y}_j\|_q^q$1

with covariance

$c_{ij}=\|\mathbf{x}_i-\mathbf{y}_j\|_q^q$2

Soft chart assignments are then defined from chart scores $c_{ij}=\|\mathbf{x}_i-\mathbf{y}_j\|_q^q$3, and the fine-tuning objective combines new-task cross-entropy, sparse anchor supervised replay, output distillation, global relational geometry preservation, local smoothing over teacher $c_{ij}=\|\mathbf{x}_i-\mathbf{y}_j\|_q^q$4-nearest-neighbor pairs, chart-assignment preservation, and an $c_{ij}=\|\mathbf{x}_i-\mathbf{y}_j\|_q^q$5 parameter-drift penalty.

The geometry terms are specific. Global geometry preservation penalizes changes in normalized pairwise distances among old anchors, local smoothing emphasizes the teacher neighborhood structure, and chart-assignment preservation uses a KL penalty between teacher and student soft chart probabilities. The broader conceptual SPMA family is stated to include additional continuation and support terms acting directly on new samples, whereas SPMA-OG is the old-geometry-preserving variant actually evaluated.

The experimental setting is also constrained. The paper studies compatible shift rather than generic class-incremental learning. Benchmarks are CIFAR10 compatible shift, Tiny-ImageNet compatible shift, and a synthetic atlas-manifold benchmark. The shift family is fixed: rotation by $c_{ij}=\|\mathbf{x}_i-\mathbf{y}_j\|_q^q$6, translation by $c_{ij}=\|\mathbf{x}_i-\mathbf{y}_j\|_q^q$7 pixels, scale $c_{ij}=\|\mathbf{x}_i-\mathbf{y}_j\|_q^q$8, additive noise with standard deviation $c_{ij}=\|\mathbf{x}_i-\mathbf{y}_j\|_q^q$9, and blur kernel size $q=2$0. Representative runs use seed 7, 10 base epochs, and 5 fine-tuning epochs. Anchor memories are built once from frozen teacher features; there are 64 buffered anchors per class; ER-512 and SPMA-OG both replay 512 anchors; Anchor CE replays 256.

The reported results emphasize both task retention and representation preservation. On CIFAR10 compatible shift, SPMA-OG achieved old-task accuracy $q=2$1 versus $q=2$2 for ER-512, new-task accuracy $q=2$3 versus $q=2$4, CKA $q=2$5 versus $q=2$6, and pairwise-distance correlation $q=2$7 versus $q=2$8. On Tiny-ImageNet compatible shift, SPMA-OG achieved old-task accuracy $q=2$9 versus $$d_{\mathcal W}^2(\mathbf{p}_x,\mathbf{p}_y) = d_{\mathcal W}^2(\overline{\mathbf{p}_x},\overline{\mathbf{p}_y}) + \|\boldsymbol{\mu}_x-\boldsymbol{\mu}_y\|_2^2,$$0, new-task accuracy $$d_{\mathcal W}^2(\mathbf{p}_x,\mathbf{p}_y) = d_{\mathcal W}^2(\overline{\mathbf{p}_x},\overline{\mathbf{p}_y}) + \|\boldsymbol{\mu}_x-\boldsymbol{\mu}_y\|_2^2,$$1 versus $$d_{\mathcal W}^2(\mathbf{p}_x,\mathbf{p}_y) = d_{\mathcal W}^2(\overline{\mathbf{p}_x},\overline{\mathbf{p}_y}) + \|\boldsymbol{\mu}_x-\boldsymbol{\mu}_y\|_2^2,$$2, CKA $$d_{\mathcal W}^2(\mathbf{p}_x,\mathbf{p}_y) = d_{\mathcal W}^2(\overline{\mathbf{p}_x},\overline{\mathbf{p}_y}) + \|\boldsymbol{\mu}_x-\boldsymbol{\mu}_y\|_2^2,$$3 versus $$d_{\mathcal W}^2(\mathbf{p}_x,\mathbf{p}_y) = d_{\mathcal W}^2(\overline{\mathbf{p}_x},\overline{\mathbf{p}_y}) + \|\boldsymbol{\mu}_x-\boldsymbol{\mu}_y\|_2^2,$$4, and pairwise-distance correlation $$d_{\mathcal W}^2(\mathbf{p}_x,\mathbf{p}_y) = d_{\mathcal W}^2(\overline{\mathbf{p}_x},\overline{\mathbf{p}_y}) + \|\boldsymbol{\mu}_x-\boldsymbol{\mu}_y\|_2^2,$$5 versus $$d_{\mathcal W}^2(\mathbf{p}_x,\mathbf{p}_y) = d_{\mathcal W}^2(\overline{\mathbf{p}_x},\overline{\mathbf{p}_y}) + \|\boldsymbol{\mu}_x-\boldsymbol{\mu}_y\|_2^2,$$6. On the synthetic atlas-manifold benchmark, SPMA-OG versus replay-anchor gave old-task accuracy $$d_{\mathcal W}^2(\mathbf{p}_x,\mathbf{p}_y) = d_{\mathcal W}^2(\overline{\mathbf{p}_x},\overline{\mathbf{p}_y}) + \|\boldsymbol{\mu}_x-\boldsymbol{\mu}_y\|_2^2,$$7 versus $$d_{\mathcal W}^2(\mathbf{p}_x,\mathbf{p}_y) = d_{\mathcal W}^2(\overline{\mathbf{p}_x},\overline{\mathbf{p}_y}) + \|\boldsymbol{\mu}_x-\boldsymbol{\mu}_y\|_2^2,$$8, new-task accuracy $$d_{\mathcal W}^2(\mathbf{p}_x,\mathbf{p}_y) = d_{\mathcal W}^2(\overline{\mathbf{p}_x},\overline{\mathbf{p}_y}) + \|\boldsymbol{\mu}_x-\boldsymbol{\mu}_y\|_2^2,$$9 versus $$p(\mathbf{x}_b)=\frac{1}{M}\sum_{i=1}^M \delta_{\mathbf{x}_i},$$0, anchor CKA $$p(\mathbf{x}_b)=\frac{1}{M}\sum_{i=1}^M \delta_{\mathbf{x}_i},$$1 versus $$p(\mathbf{x}_b)=\frac{1}{M}\sum_{i=1}^M \delta_{\mathbf{x}_i},$$2, and anchor pairwise-distance correlation $$p(\mathbf{x}_b)=\frac{1}{M}\sum_{i=1}^M \delta_{\mathbf{x}_i},$$3 versus $$p(\mathbf{x}_b)=\frac{1}{M}\sum_{i=1}^M \delta_{\mathbf{x}_i},$$4. The paper is explicit that this is not a claim of generic state-of-the-art continual learning and not a validation for generic class-incremental learning.

6. Scope, misconceptions, and open issues

The literature makes several boundaries of SPMA explicit. It is not a claim that geometry-aware methods universally dominate classical Euclidean or Gaussian estimators. EnRDA is stated not to “magically outperform” minimum mean-squared estimators when the problem is already unbiased and approximately Gaussian; its strength is bias correction and support mismatch rather than universal superiority (Tamang et al., 2020). Likewise, the continual-learning version is presented as appropriate when old and new data are semantically compatible and plausibly lie on the same latent support, not when a genuinely new manifold or representational expansion is required (Kobs, 20 Mar 2026).

A second misconception would be to treat “manifold preservation” as a single mechanism. The cited work shows at least four distinct mechanisms: Wasserstein geodesic interpolation between distributions, tangent-space projection onto nonlinear constraint manifolds, DTW-based alignment of coherent structures before convex combination, and atlas-based anchor regularization in latent space. These mechanisms share a support-preserving objective, but they do not assume the same state representation, observation model, or computational primitive.

The open issues are also domain-specific. For Wasserstein-based assimilation, computational cost, parameter tuning, and partial-observation handling remain central constraints. For constraint-preserving flow filters, exact preservation requires projection or SDAE solves, and projection-based methods can fail in difficult cases. For feature-aligned particle transport, multidimensional alignment and robustness to varying wave sets remain unresolved, and turbulence-related cases tended to get averaged out. For compatible-shift continual learning, the current evidence supports the full bundle of geometric constraints in SPMA-OG but does not isolate chart preservation alone as the sole cause of improvement.

Taken together, these works position SPMA as a precise geometric doctrine: assimilation should preserve admissible support, whether that support is a probability manifold, a nonlinear invariant set, a collection of sharp coherent structures, or a shared latent atlas. This suggests that the main scientific role of SPMA is as an inductive bias for regimes in which support mismatch, constraint violation, feature smearing, or latent-manifold tearing are the dominant failure modes of conventional updates.