Metric Extrapolation Problem

• Metric Extrapolation Problem is a framework that extends data beyond observed domains by leveraging optimization in structured metric spaces, such as Wasserstein geometry.
• It incorporates variational principles and metric projections to regularize extrapolation in diverse settings, including probability measures and heavy-tailed predictions.
• Applications span constructing conditional Wasserstein barycenters, variable-metric optimization methods, and certified off-domain error bounds in function extrapolation.

The Metric Extrapolation Problem denotes a class of extrapolation problems in which extension beyond observed data, known measures, or feasible iterates is posed through an explicit metric, a metric-induced variational principle, or a metric projection. In the literature, the phrase appears in several technically distinct settings: extending minimizing geodesics in Wasserstein space, predicting heavy-tailed random variables through excursion-metric projections, constructing conditional Wasserstein barycenters for distributional extrapolation, accelerating composite optimization by combining extrapolation with variable metrics, and deriving deterministic or statistical bounds for off-domain function evaluation (Gallouët et al., 2024, Makogin et al., 2022, Fan et al., 2021, Bonettini et al., 2015, Cao et al., 2022, Pfister et al., 2024, Brown et al., 2024).

1. Conceptual scope

In the surveyed works, the word metric has several meanings. It may refer to a distance on a space of probability measures, such as the quadratic Wasserstein distance $W_2$; to a probability-based discrepancy between random variables, such as the excursion metric; to a Hilbertian norm on an extrapolation region $\Xi$; or to an iteration-dependent positive definite matrix that changes the geometry of an optimization algorithm. This suggests that the topic is not a single formal problem with one canonical definition, but a recurring construction in which extrapolation is constrained or regularized by geometry rather than by pointwise continuation alone.

Setting	Metric or geometry	Extrapolated object
Wasserstein geometry	$W_2$ and generalized geodesics	probability measures
Excursion-based prediction	$\mathcal{E}_m$ or $\mathcal{E}_{H_\alpha}$	random variables or random fields
Variable-metric optimization	SPD matrices $M_k$ or $U_n$	iterates of splitting schemes
Function certification	$\|\cdot\|_\Xi$, $L^2$, relative $L^2$	deterministic functions

A common structural feature is that extrapolation is not treated as arbitrary continuation. Instead, one solves an optimization or feasibility problem in a metric space. Representative examples include the negative-weight barycenter formulation in Wasserstein space, metric projections onto excursion-metric feasible predictors, projection of a baseline function onto anchor-defined feasible sets in $\Xi$0, and projection-safeguarded inertial steps in forward–backward algorithms (Gallouët et al., 2024, Makogin et al., 2022, Hay et al., 10 Mar 2026, Bonettini et al., 2015).

2. Variational extrapolation in Wasserstein space

The most explicit use of the term occurs in Wasserstein geometry. For $\Xi$1 and $\Xi$2, the metric extrapolation operator is defined as the unique minimizer of

$\Xi$3

This problem extends minimizing geodesics “for all times” by allowing negative coefficients in the classical variational characterization of Wasserstein barycenters. The formulation admits two equivalent convex descriptions: a Toland-duality-based dual problem and a barycentric optimal transport problem. Existence and uniqueness follow from a lower bound and strong convexity along generalized geodesics, and the paper develops an entropic regularization together with a Sinkhorn-type scheme for computation (Gallouët et al., 2024).

A central point is that metric extrapolation is not identical to naïvely continuing McCann’s displacement interpolation. The continuation of a minimizing geodesic beyond $\Xi$4 may fail because particle trajectories can cross at $\Xi$5. The variational problem bypasses this by reordering mass through convex order when necessary. This is also the bridge to recent stability theory. A later analysis relates metric extrapolation to backward and forward Wasserstein projections in convex order, proving that backward projection is non-expansive in $\Xi$6 with Lipschitz constant $\Xi$7, that metric extrapolation is Lipschitz in $\Xi$8 with constant $\Xi$9, and that forward projection enjoys Hölder continuity under regularity assumptions. In one dimension, the extrapolated measure has an explicit quantile-space form,

$W_2$0

where $W_2$1 is the $W_2$2-projection onto nondecreasing functions (Kim et al., 25 Jul 2025).

A related but distinct distributional extrapolation framework uses conditional Wasserstein barycenters. If response measures lie on a unique $W_2$3-geodesic and predictors lie on a Euclidean geodesic, then the global conditional barycenter model recovers the same geodesic for interpolation and, under extendability, also for extrapolation. The global weights

$W_2$4

may become negative outside the convex hull, which is essential for linear extrapolation. Sample computation is performed with entropic regularization and Sinkhorn barycenters, and asymptotic rates are established for both global and local models (Fan et al., 2021).

3. Excursion metrics and prediction without moments

A second major line of work formulates extrapolation as metric projection in spaces of random variables. For a finite nonnegative measure $W_2$5 on $W_2$6, the excursion metric between random variables $W_2$7 is

$W_2$8

It is equivalently the expected measure of the symmetric difference of excursion events. When the excursion levels are chosen from the marginal law, the metric becomes distribution-free and reduces to the Gini metric. Prediction is then posed as minimization of a target functional involving $W_2$9, with unconstrained, law-preserving, and penalized law-preserving variants. Existence of minimizers and weak consistency are proved, and the framework applies even when no moments exist (Makogin et al., 2022).

This moment-free viewpoint is particularly important for heavy-tailed max-stable fields with $\mathcal{E}_m$0-Fréchet marginals. In that setting, the predictor is a max-linear combination

$\mathcal{E}_m$1

and the law-preserving constrained problem becomes

$\mathcal{E}_m$2

The induced pseudometric on max-linear weights depends only on the tail dependence function, and in the simple bivariate case the excursion metric is a monotone transform of the Davis–Resnick distance:

$\mathcal{E}_m$3

The paper proves existence of predictors, demonstrates non-uniqueness in several symmetric cases, and introduces a penalized formulation based on the squared $\mathcal{E}_m$4-Wasserstein distance to enforce approximate marginal exactness. The computational procedure uses learning samples formed by shifts, empirical objectives, and stochastic optimization with SGD or Adam (Makogin et al., 17 Apr 2026).

A recurring misconception in this literature is that extrapolation of extremes can be reduced to covariance-based kriging. The cited work explicitly rejects that for heavy tails: when $\mathcal{E}_m$5, second moments may be infinite, so second-order methods can be inappropriate or unstable. The excursion metric replaces moment assumptions by level-set probabilities and dependence geometry (Makogin et al., 17 Apr 2026).

4. Variable metrics and extrapolation in optimization algorithms

In optimization, metric extrapolation usually refers to combining inertial or optimistic extrapolation with a changing inner product. For composite convex minimization

$\mathcal{E}_m$6

a scaled inertial forward–backward method uses the safeguarded extrapolation

$\mathcal{E}_m$7

where $\mathcal{E}_m$8 is symmetric positive definite and may change at each iteration. The projection of the extrapolated point onto $\mathcal{E}_m$9 is the key device allowing domains strictly smaller than the full space. Under mild metric variation and backtracking, the method achieves the objective-value rate $\mathcal{E}_{H_\alpha}$0, and under stronger summability assumptions on the metric changes with $\mathcal{E}_{H_\alpha}$1, the iterates converge to a minimizer. Numerical experiments cover image deblurring with Poisson noise, compressed sensing, and probability density estimation (Bonettini et al., 2015).

A different family combines Tseng’s forward–backward–forward method with “extrapolation from the past,” variable metrics, and error terms for monotone inclusions. The core iteration evaluates the Lipschitz operator at the previous extrapolated point,

$\mathcal{E}_{H_\alpha}$2

followed by a corrected forward step and an update of $\mathcal{E}_{H_\alpha}$3. In the identity-metric, error-free case, the scheme reduces to an OGDA-type recurrence. The analysis yields summability of residuals and weak convergence to a zero of $\mathcal{E}_{H_\alpha}$4, and the paper also develops primal–dual variants for problems with linear operators and applications to image deblurring (Tongnoi, 2022).

For nonconvex sparse optimization, the variable metric extrapolation proximal iterative hard thresholding method addresses

$\mathcal{E}_{H_\alpha}$5

The $\mathcal{E}_{H_\alpha}$6 proximal step remains Euclidean and closed-form through hard thresholding, while the extrapolation step is a projected quasi-Newton move on the currently identified support,

$\mathcal{E}_{H_\alpha}$7

After finite support identification, the method admits linear convergence under eigenvalue safeguards and superlinear convergence under a Dennis–Moré-type condition. Experiments on compressive sensing and CT reconstruction compare it to nmAPG, niAPG, and nPIHT (Zhang et al., 2021).

5. Error bounds, feasibility, and impossibility results for function extrapolation

Several works treat metric extrapolation as a question of certified off-domain error. For multivariate linear interpolation and extrapolation, if $\mathcal{E}_{H_\alpha}$8 has $\mathcal{E}_{H_\alpha}$9-Lipschitz gradient and $M_k$0 is the affine interpolant at a query point $M_k$1, then

$M_k$2

where $M_k$3 are barycentric coefficients and

$M_k$4

Inside the simplex this recovers a sharp bound, while outside the simplex sharper geometry-dependent estimates can be expressed through the tensor $M_k$5 and an extremal Hessian $M_k$6 (Cao et al., 2022).

In nonparametric statistics, extrapolation is defined as inference on a conditional function evaluated outside the support of the conditioning variable. The paper introduces the assumption that the conditional function attains its minimal and maximal directional derivative, in each direction, within the observed support. For $M_k$7 and a closed set $M_k$8, this yields lower and upper extrapolation bounds

$M_k$9

constructed by Taylor expansion along the direction from an anchor point $U_n$0 to $U_n$1. Plug-in versions based on differentiable estimators are shown to be consistent, and the framework is applied to conditional expectations, conditional quantiles, prediction intervals, and minimax point prediction outside support (Pfister et al., 2024).

For completely monotone functions, the feasibility question becomes sharply asymmetric. If two completely monotone functions agree on an interval to relative $U_n$2 precision $U_n$3, extrapolation to the left of the interval is impossible: discrepancies can be made arbitrarily large. To the right, however, the maximal discrepancy obeys a power law. On a rescaled interval $U_n$4, the exponent at $U_n$5 is

$U_n$6

and for a general interval $U_n$7 it becomes

$U_n$8

The same work also studies a local problem around a fixed completely monotone function, derives optimality conditions through a Caprini-type dual certificate, and solves the single-exponential case exactly (Brown et al., 2024).

A more recent feasibility-based approach uses anchor functions. An anchor $U_n$9 is any function equipped with a certified radius $\|\cdot\|_\Xi$0 such that $\|\cdot\|_\Xi$1. The corresponding feasible set is the $\|\cdot\|_\Xi$2-ball

$\|\cdot\|_\Xi$3

and for multiple anchors one takes the intersection $\|\cdot\|_\Xi$4. If a baseline approximation $\|\cdot\|_\Xi$5 is projected onto $\|\cdot\|_\Xi$6 in the $\|\cdot\|_\Xi$7-metric, the resulting predictor cannot increase the error on $\|\cdot\|_\Xi$8; for a single ball, the paper proves an explicit lower and upper bound on the improvement in $\|\cdot\|_\Xi$9. The certification constants include a tight spectral condition number $L^2$0 and a probabilistic variant based on random Rayleigh quotients (Hay et al., 10 Mar 2026).

6. Applications, diagnostics, and recurring limitations

The metric-extrapolation viewpoint has been used in domains where extrapolation quality is not well captured by standard in-sample criteria. In image analysis, the time-discrete metamorphosis model defines a discrete exponential map that extrapolates images along a geodesic in a Riemannian metric combining transport cost and intensity variation. A single shooting step can be formulated entirely in the deformation variables, and local existence and uniqueness are proved for sufficiently small time steps. Applications include synthetic ellipses, portraits, human faces, and animal motion (Effland et al., 2017).

In transductive model assessment, interpolation and extrapolation points are encoded by target-specific design and covariance structures. The estimators tAI and Loss(Opt_t) extend Akaike-like prediction-error correction to targets that differ from the training sample, including future times and spatial interpolation grids. The correction depends explicitly on $L^2$1, $L^2$2, $L^2$3, $L^2$4, and $L^2$5, and the paper shows that in-sample criteria such as cAI and mAI can underestimate prediction error at extrapolation points (Rabinowicz et al., 2018).

In neural network interatomic potentials, the failure of standard test metrics is especially explicit. On the 3BPA benchmark, small differences in in-domain energy and force RMSEs do not predict out-of-distribution error or molecular-dynamics stability in the high-accuracy regime. The proposed alternative is loss entropy, a training-data-only metric built from loss-landscape probes around the trained parameters,

$L^2$6

with a combined energy–force version

$L^2$7

Across NequIP and MACE models, larger loss entropy correlates with flatter minima, improved out-of-distribution behavior, longer time-to-failure in high-temperature molecular dynamics, and better learning-curve slopes (Vita et al., 2023).

At a more abstract level, metric extrapolation has also been formulated for operator norms. One constructs a family $L^2$8 over a half-plane, imposes curvature conditions on bundle metrics, and uses plurisubharmonicity or monotonicity to transfer endpoint bounds to finite $L^2$9. In one complex-geometric application, this yields sharp Ohsawa–Takegoshi-type extension estimates by embedding the target norm into a curvature-controlled family of weighted Bergman norms (Lempert, 2015).

Across these literatures, several limitations recur. Extrapolation is often only as reliable as the geometric or probabilistic structure imposed: absolute continuity and unique geodesics in Wasserstein space, stationarity or ergodicity in random-field prediction, convexity and Lipschitz assumptions in splitting methods, derivative-extremum assumptions in nonparametric inference, or positivity and analyticity in completely monotone extrapolation. The literature also repeatedly shows that naïve surrogates can fail: in-sample prediction error may be irrelevant at target points (Rabinowicz et al., 2018), low test RMSE may not predict dynamical stability (Vita et al., 2023), covariance-based second-order methods may be inappropriate for heavy tails (Makogin et al., 17 Apr 2026), and left-sided extrapolation can be impossible even under strong regularity classes (Brown et al., 2024). These results collectively frame the Metric Extrapolation Problem as a problem of geometry, certification, and stability rather than of continuation alone.