Factorized Conformal Prediction Via Learned Residual-Law Retrieval
Local uncertainty. Global calibration. FacCP builds prediction sets that adapt to query-dependent uncertainty in high dimensions. It learns a representation that brings examples with similar residual distributions together, then retrieves relevant residuals to estimate each query’s uncertainty. This local residual law determines the set’s shape; a single global conformal quantile calibrates its inflation, preserving finite-sample marginal coverage under exchangeability while using the entire calibration sample. The same learned representation supports ellipsoids, hyperboxes, zonotopes, and density-based sets without retraining.
FacCP connects the quality of local uncertainty geometry to local coverage: under a geometry-approximation condition, better geometry bounds how much the score distribution varies across queries. Experiments show why local calibration alone cannot repair a misspecified shape. Across high-dimensional robotic reachability and multimodal autonomous driving, FacCP produces tighter calibrated sets, with particularly strong data and parameter efficiency when training data or model capacity is limited.
Estimating a separate calibration threshold near each query can leave too few relevant examples, especially in high dimensions. Changing the threshold also cannot recover a shape that the chosen score cannot express. FacCP uses local residuals, the predictor’s errors, to build the shape, then uses the entire calibration sample to choose one global inflation threshold.
PIT-CP and Rectified CP adjust a chosen scalar score to improve local coverage. But once a score has discarded direction or multiple possible outcomes, adjusting that number cannot bring them back. Rescaling a distance-based ball, for example, cannot turn it into a tilted ellipsoid. FacCP learns the residual distribution and constructs the geometry before reducing it to a score.
SLCP adapts a chosen scalar-score distribution using nearby examples, but its neighborhoods depend on the chosen features and kernel. Irrelevant features can obscure which examples share the same uncertainty, while scalar scores cannot recover geometry they discard. FacCP explicitly learns a space for retrieving similar multivariate errors, adapting the set’s shape before reducing it to a score.
Learning geometry directly can tie the model to one set family and be data- and parameter-inefficient for high-dimensional structured outputs. FacCP retrieves a residual distribution instead: one learned representation supports ellipsoids, hyperboxes, zonotopes, and density-based sets without retraining, with separate conformal calibration for each constructor. This reuse is especially useful when training data or model capacity is limited.
References: LCP — Guan (2023); RLCP — Hore & Barber (2025); PIT-CP — Laplante (2026); Rectified CP — Plassier et al. (2025a); SLCP — Han et al. (2022); CP-SLS-MPC — Srinivasan et al. (2026).
Prediction sets should adapt to how prediction errors vary with the query. Computing retrieval weights directly from distances in the input X can be unreliable: irrelevant features affect those distances without providing information about uncertainty. In high dimensions, this makes relevant reference examples harder to identify and residual distributions harder to estimate. The localization metric must therefore reflect changes in uncertainty, rather than all variation in the input.
FacCP learns a lower-dimensional representation for computing retrieval weights. The representation is trained with an energy-score objective that evaluates the residual distribution retrieved for each query. Its target is the full conditional residual law, rather than a particular covariance or set shape. By retaining uncertainty-relevant information and suppressing irrelevant variation, the learned space can make localization more statistically efficient.
For a new query, FacCP locates reference examples nearby in the learned representation. Proximity is trained to reflect similarity in conditional residual distributions, so examples can be informative even when their original inputs differ substantially. Only the queries are embedded: the associated residuals remain in their original space. The representation determines which residuals to emphasize without compressing their multivariate structure.
FacCP assigns each reference residual a normalized weight that decreases with distance from the query in the learned space. A temperature parameter controls how local the weighting is. The weighted collection of observed residuals estimates the query’s conditional residual law. This distribution is the shared intermediate object used to construct local geometry, separating uncertainty estimation from the choice of prediction-set family.
A geometry constructor converts the retrieved residual law into a family of nested sets. For ellipsoids, the weighted residual mean and covariance determine the center, orientation, and relative axis lengths; an inflation parameter controls overall size. The same learned representation also supports hyperboxes, zonotopes, and density-based sets without retraining. Each constructor is then calibrated separately.
After fixing the learned geometry, FacCP scores each independent calibration example by the smallest inflation needed to contain its residual. The full calibration sample determines one global conformal threshold, while each query retains its own local shape. Under exchangeability, the resulting prediction set has finite-sample marginal coverage regardless of retrieval accuracy. The same calibrated set also admits the local coverage bounds described next.
For one global threshold to be meaningful locally, the score distributions must be comparable across queries. Under the paper’s geometry-approximation condition, error in local geometry bounds how much these distributions differ, a property called approximate pivotality. The resulting local coverage bound separates geometry error from finite calibration error. Marginal coverage follows from conformal calibration; approximate local coverage additionally requires accurate geometry under the stated assumptions.
One learned representation supports ellipsoids, hyperboxes, zonotopes, and density-based sets without retraining. On planar quadcopter rollouts, FacCP adapts each geometry to the query and produces tighter sets across all four families. Each family is calibrated separately to target 99% simultaneous coverage of the full 100-step trajectory.
The benefit comes from learning which errors are relevant. With 806 input dimensions, raw-input retrieval gives nearly the same ellipsoid log volume as global geometry (−26.66 versus −25.82). Learned retrieval reaches −33.25 with 99.1% measured coverage, showing how a suitable representation can make local adaptation more effective.
FacCP shows its largest gains when training data or model capacity is limited. At roughly 5,000 learned parameters, it produces sets over 1,000× smaller than resCONTRA at comparable coverage; the low-capacity comparison with covariance prediction gives over 250× smaller sets. The capacity sweeps match learned parameter counts and use the same trajectory-level calibration procedure.
With 20% of the training data, sets are 211× smaller than resCONTRA and 46× smaller than covariance prediction while maintaining coverage. Retrieving observed residuals lets the model focus on learning which examples to use. The advantage narrows as direct models receive more data and capacity.
The same construction extends to 200-step 3D quadcopter rollouts: 3,212 input coordinates and a 2,400-dimensional residual trajectory. FacCP reduces mean per-step 12D log volume from −52.52 to −68.99, with measured whole-trajectory coverage of 99.3%, versus 99.4% for global geometry. The animation shows coordinate projections of these reachable sets.
Calibration targets the complete trajectory, rather than individual coordinates or time steps. Retrieval remains particularly effective under limited data and capacity, but the comparison also shows its scope: with enough training data and model capacity, resCONTRA closes the gap and can produce tighter sets.
FacCP adapts uncertainty around six fixed candidate paths from an MTR forecaster, covering a six-second driving horizon. At the 95% target, it reduces mean footprint by 14.9% relative to global SNSA, from 683.8 to 581.9 m²·s. Measured whole-path coverage is 94.46% versus 95.19%, so the smaller footprint comes with a modest empirical coverage difference.
Coverage here requires one candidate tube to contain the entire observed future, without switching candidates along the way. FORCE-OPT produces smaller footprints but reaches 62.29% on this whole-path metric; its calibration targets individual time steps. The plots illustrate why compactness and the coverage event need to be considered together.
Tighter tubes also appear when only the highest-probability forecast is retained. At the 97% target, FacCP reduces mean footprint from 3,356.1 to 2,310.1 m²·s—about 31%—with measured whole-path coverage of 97.53%, versus 97.63% for global SNSA. In the illustrated scene, both cover the observed trajectory, while the local tube uses 76.7% less footprint.
This comparison measures spatial occupancy over time. NLE and MVCS are shown as projections of joint trajectory ellipsoids: a wide projection can still belong to a compact joint set because it captures dependencies across time. The footprint comparison therefore describes spatial tightness, rather than a ranking of joint trajectory volumes.
@misc{FacCP_2026,
title = {Factorized Conformal Prediction via Learned Residual-Law Retrieval},
author = {Nath, Devesh and Rustagi, Rahul and Chou, Glen},
note = {Preprint},
year = {2026}
}