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Target-Constrained Pooling for Trajectory Extrapolation

Ning Wang, Claire Cardie, Sainyam Galhotra

arXiv:2610.13034v1cs.LG

TL;DR

Partial target trajectories permit many possible continuations, and related completed trajectories offer information about the unresolved endpoint. TCP pools two endpoint signals, constrains their estimates, and calibrates their corrections; across 25,245 targets, it reduces mean RMSE for every target-only forecaster and improves all 100 comparisons.

  • Problem

    Observed data can be consistent with multiple underlying functions that predict differently outside the observed range, making additional information or structural assumptions necessary for reliable extrapolation.

    Source: §1 · p. 1

  • Method

    TCP separates the finite-horizon endpoint from continuation shape, estimates the endpoint through uniform and matched-increment pooling, and calibrates target-constrained corrections to the free forecast.

    Source: §1 · p. 2, §4.1 · p. 3, §4.5 · p. 5

  • Results

    35.8% to 42.1% reductions in mean RMSE across 25,245 targets improved all 100 forecaster, dataset, and direction comparisons for all ten target-only forecasters.

    Source: §1 · p. 2

  • Takeaways & Limitations

    Uniform and matched-increment endpoint estimates consistently outperform either signal alone, and their combination also beats random donor endpoints or donor-level averaging.

    Source: §6 · p. 9

    The complementary endpoint result assumes matched source responses follow a source-specific level-offset-plus-noise model.

    Source: §4.2 · p. 4

Abstract

from arXiv · show

Trajectory extrapolation from a partial target trajectory is underdetermined, but completed related trajectories provide information about the unresolved continuation. We propose target-constrained pooling (TCP), which separates the finite-horizon endpoint from continuation shape and estimates that endpoint from two complementary source signals. Uniform pooling captures final response levels, while matched-increment pooling captures the remaining change after aligning source and target states. TCP selects representative sources, projects both endpoint estimates onto a target-consistent region, converts them into corrections of the free forecast, and controls their influence through strength calibration and degradation-aware weighting. Across 25,245 targets from five domains and both forecasting directions, TCP reduces mean RMSE by 35.8% to 42.1% for all ten target-only forecasters and improves all 100 forecaster, dataset, and direction comparisons. The two endpoint signals consistently outperform either alone, while representative selection remains within 0.64% to 1.02% of the full donor pool using 12.7x fewer donor inputs and 138.8x fewer matching distance computations.

1 Introduction

Partial trajectories can support many incompatible continuations, so extrapolation needs additional information or structural assumptions. The introduction motivates pooling related trajectories and presents TCP as an endpoint-transfer approach, reporting gains across forecasters, datasets, and directions.

1 Introduction

  • Extrapolation is underdetermined because functions that fit observed data equally well can produce very different predictions beyond the observed range.Reliable extrapolation therefore requires additional information or structural assumptions that constrain possible continuations.Source: §1 · p. 1
  • Related units with longer observed trajectories can inform unresolved target continuations, but differences in scale, limiting behavior, and shape make poorly matched pooling a source of bias.Prior approaches regulate pooling under heterogeneity, while the introduction identifies how to pool related trajectories for extrapolation as unresolved.Source: §1 · p. 1
  • TCP estimates the unresolved finite-horizon endpoint using uniform and matched-increment pooling, then projects endpoint estimates and converts them into calibrated corrections to the free forecast.Representative-source selection and degradation-aware weighting control the transferred information.Source: §1 · p. 2
  • Within-cohort median RMS continuation-shape discrepancy was 23.7% lower, with 12.2% to 31.4% reductions across datasets, while the median within-cohort endpoint gap was 24.8% of the observed-prefix response range.This separates the remaining response change from the shape of its continuation.Source: §1 · pp. 1–2
  • 35.8% to 42.1% mean RMSE reductions across 25,245 targets improved all 100 forecaster, dataset, and direction comparisons for all ten target-only forecasters.TCP also remained effective when related trajectories entered the base fit.Source: §1 · p. 2

2 Related Work

Related work studies trajectory extrapolation, pooling across heterogeneous units, and structural constraints; TCP combines these ideas by transferring source information through endpoint pooling.

Trajectory extrapolation

  • Prior trajectory extrapolation uses parametric models, Bayesian predictors, or regularities across related curves; TCP instead separates the finite-horizon endpoint from continuation shape and transfers information through the endpoint.Examples include learning-curve forecasting, latent Kronecker Gaussian processes [Lin et al., 2024, 2025], and NSL-PFN [Lee et al., 2025].Source: §2 · p. 2

Pooling across related units

  • Empirical-Bayes, historical-borrowing, and transfer-learning methods regulate sharing under heterogeneity; TCP applies pooling at the endpoint level using final-level and matched-change signals with representative-source selection.The cited approaches include empirical-Bayes methods and historical borrowing or transfer learning (Hanneke and Kpotufe, 2019).Source: §2 · p. 2

Structural constraints and calibrated transfer

  • Bounds and monotonicity restrict extrapolations consistent with observed data (Manski, 1990, 1997); TCP combines such target-side constraints with source-side endpoint information and calibrated corrections.The observed boundary and response direction constrain transferred endpoints, while calibration controls correction strength and combination.Source: §2 · p. 2

3 Problem Setup

The forecasting task is to estimate a target’s unobserved continuation from its observed prefix and related trajectories, with inputs ordered toward the forecast horizon.

3 Problem Setup

  • Inputs are ordered toward the forecasting horizon, with physical input order reversed for reverse evaluation before applying the same notation.Source: §3 · p. 2
  • The last observed input τ is the observation boundary, and T > τ is the finite forecasting horizon for the unobserved target continuation.The continuation comprises responses for τ < d ≤ T.Source: §3 · p. 3
  • The target prefix and related source trajectories are the available information; the target continuation remains unobserved and must be estimated through the finite horizon.Sources are observed through the same forecasting horizon.Source: §3 · p. 3

4 Method

TCP separates the unresolved finite-horizon endpoint from continuation shape, estimates that endpoint from related trajectories, and uses it to correct a free forecast. Separate strength calibration and degradation-aware weighting control how the two endpoint-based corrections affect the forecast.

4.1 Continuation Decomposition

  • The continuation is decomposed into the observed boundary, remaining response change, and normalized shape, so related trajectories estimate the endpoint while the target forecaster supplies the shape.The target’s observed response direction defines a consistent endpoint region; projection onto it cannot increase endpoint error when the true endpoint lies within that region.Source: §4.1 · p. 3

4.2 Endpoint Estimation from Source Levels and Matched Changes

  • TCP selects a representative subset of permitted sources to reduce source and matching work, then projects both raw endpoint estimates onto the target-consistent region.The selection criterion balances subset size against discrepancy from the full source pool; projection does not increase endpoint error when the true endpoint is in the region.Source: §4.2 · pp. 3–4
  • Uniform pooling uses source endpoint levels, while matched-increment pooling transfers remaining changes after aligning source stages to the target; their errors trade off source-level offsets against matching-stage error.TCP retains both endpoint estimates and calibrates their contributions separately.Source: §4.2 · p. 4

4.3 Endpoint-Guided Forecast Correction

  • For each projected endpoint, TCP distributes its difference from the free forecast’s endpoint across the continuation, then projects the result to preserve the boundary, reach the endpoint, and follow its response direction.The residual bridge minimizes squared changes between successive residuals under endpoint conditions.Source: §4.3 · p. 4

4.4 Calibrated Endpoint Corrections

  • TCP calibrates the two endpoint-guided correction strengths separately on pseudo-target trajectories, choosing strengths whose calibration error is no greater than either the free forecast or the full correction.The strengths are selected by minimizing calibration error over [0, 1].Source: §4.4 · pp. 4–5

4.5 Degradation-Aware Combination

  • TCP mixes the calibrated branches to minimize positive error degradation relative to the free forecast, subject to mean error no greater than equal averaging.The selected weight moves from equal averaging toward the degradation-minimizing weight only as far as the accuracy constraint permits.Source: §4.5 · p. 5

5 Experiments

Across 25,245 targets, TCP lowers mean RMSE for all ten target-only forecasters and improves all 100 forecaster, dataset, and direction comparisons. Experiments examine source access, endpoint construction, and how transferred information should modify the free forecast.

5.2 RQ1. Endpoint Transfer across Target-Only and Source-Pooled Forecasters

  • 35.8% to 42.1% lower mean RMSE for all ten target-only forecasters, with improvements in all 100 forecaster, dataset, and direction comparisons.The comparison covers mean RMSE over held-out continuations.Source: §5.2 · p. 6, Table 1 · p. 7
  • TCP also lowers mean error when related trajectories enter the base fit, with gains of 6.5% to 36.1% using selected sources and 1.6% to 36.1% using all permitted sources.For LKGP, the TCP gain falls from 37.2% in target-only fitting to 6.5% with selected sources and 1.65% with all permitted sources.Source: §5.2 · p. 6

5.3 RQ2. Source Information and Endpoint Construction

  • Own-cohort endpoints add 14.7% to 17.5% improvement beyond alternative-cohort endpoints, and TCP outperforms four other endpoint constructions across all ten forecasters.TCP lowers mean RMSE over U-only by 2.7% to 4.5%, M-only by 1.7% to 2.4%, Random-two by 8.2% to 11.5%, and All-endpoints by 2.4% to 4.6%.Source: §5.3 · p. 7
  • TCP's uniform and matched-increment signals contribute complementary endpoint information, while MMD-based selection stays within 0.64% to 1.02% of the full donor pool using 12.7 times fewer donor inputs and 138.8 times fewer matching distances.The full pool lowers mean RMSE over TCP by 0.64% to 1.02%.Source: §5.3 · p. 8
  • When the direction condition holds, projection lowers mean RMSE by 0.45% to 1.29%, rising to 7.3% to 19.3% when projection is active; inactive differences are near zero.When the endpoint direction condition fails, removing the constraint improves RMSE by 2.1% to 5.7%.Source: §5.3 · p. 8

5.4 RQ3. Calibration and Degradation-Aware Combination

  • Calibration reduces TCP's total degradation by 34.4% to 40.4%, and degradation-aware weighting lowers total degradation by 1.8% to 5.5% relative to equal averaging.For TCP, calibration changes mean RMSE by −0.15% to 2.0%; relative to equal averaging, it lowers mean RMSE by 0.39% to 0.98%.Source: §5.4 · pp. 8–9

6 Conclusion

TCP transfers finite-horizon endpoint information from completed related trajectories into a free forecast. Across 25,245 targets, it improves forecasting across target-only forecasters while combining complementary endpoint signals and using representative donor selection to reduce computation.

6 Conclusion

  • Across 25,245 targets, TCP reduces mean RMSE by 35.8% to 42.1% for all ten target-only forecasters.It improves all 100 forecaster, dataset, and direction comparisons.Source: §6 · p. 9
  • Uniform and matched-increment pooling provide complementary endpoint signals, and their combination outperforms either alone, random donor, and donor-averaging alternatives.MMD-based selection stays within 0.64% to 1.02% of the full donor pool while using 12.7 times fewer donor inputs and 138.8 times fewer matching distances.Source: §6 · p. 9
  • Target-consistent projection improves forecasts most when its direction condition holds, while strength calibration and degradation-aware weighting control how endpoint information changes the free forecast.TCP remains effective when sources also enter the base fit.Source: §6 · p. 9

A Appendix

The appendix identifies Table 2 as defining the reported metrics and states the evaluation and calibration sets used unless otherwise specified.

A Appendix

  • Table 2 is titled “Definitions of the reported metrics,” but the supplied passage does not give the metric definitions.The table contents are not available in the supplied passages.Source: Table 2 · p. 11
  • Unless stated otherwise, E contains all 25,245 evaluation targets and calibration objectives use C.Source: A Appendix · p. 11

A.1 Methodology

The experiments implement TCP as a sequence of stages, from continuation decomposition through degradation-aware combination.

A.1 Methodology

  • The implementation proceeds through continuation decomposition, representative source selection, endpoint construction and projection, common forecast correction, separate strength calibration, and degradation-aware combination.Source: §A.1 · p. 11

A.1.1 Continuation decomposition

The continuation decomposition separates a finite-horizon endpoint from continuation shape; direction constraints alone do not determine either one.

A.1.1 Continuation decomposition

  • The finite-horizon endpoint VT = y(T) is the response at the forecast horizon, and the response direction defines the target-consistent region Cb = {v : σ(v − b) ≥ 0}.Source: §A.1.1 · p. 12
  • For any v in Cb and a > 0, the continuation starts at boundary b and reaches v via yv,a(τ + r) = b + (v − b)(r/R)^a.Distinct a values give distinct shapes when v ≠ b; v = b yields a constant continuation.Source: §A.1.1 · p. 12
  • Direction alone determines neither endpoint nor shape; known response bounds restrict endpoints to Cb ∩ [ℓ, u], while additional assumptions can restrict continuation shape.The passage lists rate, curvature, derivative, and parametric assumptions as examples.Source: §A.1.1 · p. 12
  • Projecting a source endpoint outside Cb to boundary b cannot increase its squared error relative to any target endpoint VT in Cb.Source: §A.1.1 · p. 12

A.1.2 Representative calibration and source selection

TCP selects separate calibration and source representatives from permitted out-of-fold trajectories using Gaussian-kernel herding.

A.1.2 Representative calibration and source selection

  • Within each outer fold, calibration candidates are permitted out-of-fold trajectories valid as targets in both directions, while source candidates exclude the selected calibration set.Calibration selection uses five z-scored prefix descriptors; source selection uses full oriented source curves.Source: §A.1.2 · p. 12
  • For each pool, Gaussian-kernel herding selects the smallest prefix maximizing a knee criterion that balances selected-set size against discrepancy from the full pool.The criterion uses the square root of the biased MMD estimate and sets the full-pool discrepancy to zero.Source: §A.1.2 · p. 12
  • Singleton or zero-discrepancy pools retain one member, and when all knee scores are nonpositive TCP retains the first representative.The Gaussian kernel uses median pairwise distance, and herding ties follow pool order.Source: §A.1.2 · p. 12

A.1.3 Endpoint construction and directional projection

TCP estimates the finite-horizon endpoint using uniform and matched-increment pooling, then projects the raw estimates. Matched increments align source and target states to transfer the response change over the target’s continuation length.

A.1.3 Endpoint construction and directional projection

  • Uniform pooling estimates the finite-horizon endpoint in the shared response coordinate using pooled source observations.The supplied derivation states that substituting the source endpoint responses into uniform pooling yields the raw estimate.Source: §A.1.3 · pp. 12–13
  • Matched-increment pooling transfers a source’s response change over the target’s continuation length after matching source and target windows by level and first differences.It compares the target’s last five prefix observations with five-point source windows and resolves distance ties in favor of the later source stage.Source: §A.1.3 · pp. 12–13
  • TCP fits the bandwidth using leave-one-source-out endpoint MAE on completed source trajectories treated as pseudo-targets.It also evaluates a near-uniform candidate and selects it when its MAE is no larger than the finite candidate’s.Source: §A.1.3 · p. 13
  • If the total matched-increment weights are at most 10^-12, TCP uses the median increment.Source: §A.1.3 · p. 13
  • When the true endpoint is in the feasible region, projection cannot increase absolute or squared endpoint error, and feasible raw estimates remain unchanged.Source: §A.1.3 · p. 13

A.1.4 Forecast correction

TCP applies an endpoint-guided operator to each base forecaster’s saved free continuation. It constructs a bridge to the endpoint and projects that bridge onto bounded, monotone sequences anchored at the observed boundary and endpoint.

A.1.4 Forecast correction

  • TCP applies the same endpoint-guided operator to every base forecaster’s saved free continuation.Source: §A.1.4 · p. 13
  • The bridge distributes the endpoint displacement smoothly across the continuation, and the residual bridge uniquely minimizes residual increment energy under the endpoint conditions.Equality holds only when residual increments are equal.Source: §A.1.4 · p. 13
  • TCP projects the bridge onto sequences anchored at the boundary and endpoint, bounded between them, and monotone in the direction of the endpoint displacement.For a nonzero displacement, bounded isotonic regression with clipping gives the solution; when the endpoint equals the boundary, the corrected continuation is constant.Source: §A.1.4 · p. 13

A.1.5 Separate correction-strength calibration

This section examines TCP’s calibration and combination of correction strengths, including degradation-aware weighting under an accuracy constraint. It presents calibration guarantees, computational analysis, and diagnostics of corrections, endpoint-pool comparisons, direction projection, and representative donor selection.

A.1.5 Separate correction-strength calibration

  • Separate strength calibration has a convex continuous RMSE objective, and the degradation-aware mixture has a convex objective over a nonempty accuracy-feasible interval.The calibration-set guarantees bound mean error and degradation relative to equal averaging; if both strengths are zero, TCP returns the free forecast.Source: §A.1.5 · p. 13, §A.1.6 · p. 14

A.1.6 Degradation-aware combination and its guarantees

  • The terminal TCP value is a convex combination of the free terminal value and projected endpoints, with calibrated strengths controlling endpoint influence and the mixture weight balancing branches.Strength and weight stages are optimized sequentially because their joint objective need not be convex; these guarantees apply on calibration set C.Source: §A.1.6 · pp. 14–15

Using all permitted sources instead gives

  • After representative selection, matching and bandwidth work scale quadratically with the retained donor count, while endpoint corrections scale linearly with horizon.The reported candidate-stage distance elements fall from 331,685.23 for Full pool to 2,389.45 for TCP; the bound separates one-time selection cost Tsel(N).Source: Using all permitted sources instead… · pp. 15–16, §A.1.7 · p. 15

A.2 Case studies

  • In forward examples, TCP raised a nearly flat forecast and lowered an overshoot, reducing full-tail RMSE by 48.4% and 47.0%, respectively.For case (a), full-tail RMSE fell from 0.3735 to 0.1928; for case (b), it fell from 0.0173 to 0.0092. Case (a) averaged fully applied U and M corrections, while case (b) selected the M branch.Source: §A.2.1 · p. 17, §A.2.2 · p. 18
  • In reverse examples, TCP raised a forecast below the data and removed an upward jump, reducing full-tail RMSE by 83.9% and 54.5%, respectively.Case (c) fell from 0.1207 to 0.0194 RMSE; case (d) fell from 0.2093 to 0.0952 as the bounded correction stayed within the observed boundary range.Source: §A.2.3 · p. 18, §A.2.4 · p. 18
  • When both projected endpoints equaled the boundary, TCP flattened the continuation; with an attenuated upward correction, its mixture ended between the free and held-out endpoints.Case (e) reduced full-tail RMSE from 0.0176 to 0.0018, an 89.8% reduction; case (f) reduced it from 0.002007 to 0.001226, a 38.9% reduction.Source: §A.2.5 · p. 18, §A.2.6 · p. 18

A.3.1 Data, folds, and coordinates

  • The benchmark covers 25,245 evaluation targets across five datasets and both directions; the source-access comparison refits strengths and weights on the same calibration set.The endpoint-source intervention compares own-cohort endpoints with alternatives under the frozen policy, while the forecaster setup applies the common correction operator to ten saved free forecasts.Source: §A.3.1 · p. 18, §A.3.3 · p. 19, §A.3.2 · p. 19

A.3.5 Endpoint construction, expanded pools, and workload

  • Full pool lowers pooled mean RMSE by 0.64% to 1.02% for all ten forecasters, while the selected pool wins three of 100 dataset–direction comparisons.Total degradation changes by −1.28% to 1.00%; the authors conclude representative selection preserves most endpoint information available from the full pool.Source: §A.3.5 · p. 23
  • TCP used 138.8× fewer matching-distance elements and 12.7× fewer donor–packet inputs than Full pool across 34,160 packets.TCP used 2,389.45 matching-distance elements and 20.29 donor–packet inputs per packet, versus 331,685.23 and 256.85 for Full pool, respectively.Source: §A.3.5 · p. 23, Table 12 · p. 23, Table 13 · p. 23

A.3.6 Direction-constraint diagnostic

  • On condition-holding targets, TCP had lower mean RMSE than the no-direction arm for all ten forecasters; when the condition failed, the no-direction arm did for all ten.Across the complete benchmark, TCP had lower mean RMSE for eight forecasters; gains were largest when projection was active on condition-holding targets.Source: §A.3.6 · p. 24, Table 15 · p. 24, Table 16 · p. 24

A.3.7 Calibration and weighting controls

  • Calibrated TCP reduced total RMSE degradation by 34.4% to 40.4% versus full-strength TCP, while RMSE-only weighting lowered mean RMSE and TCP reduced degradation by 8.7% to 11.2%.TCP also had lower mean RMSE than U-only and M-only for all ten forecasters; degradation-only weighting lowered degradation while increasing mean RMSE.Source: Table 18 · p. 25, §A.3.7 · p. 26, Table 19 · p. 26
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