Local Spectral Conformal Inference

• Local Spectral Conformal Inference (LSCI) is a framework that adapts spectral conformal prediction for function-valued operator models, ensuring valid, localized uncertainty quantification.
• It employs projection-based depth scoring and a similarity kernel to weight calibration residuals, producing prediction sets that capture complex, local variability in high-dimensional outputs.
• LSCI provides finite-sample marginal coverage guarantees and has demonstrated empirical success in applications like air quality forecasting and energy demand prediction.

Local Spectral Conformal Inference (LSCI) comprises a framework for locally adaptive, distribution-free uncertainty quantification in function-valued regression settings—particularly for operator models such as deep neural operators that map input functions to output functions. The key innovation enables statistically valid, data-driven prediction sets at the level of entire functions, exploiting localized spectral properties and projection-based data depth. LSCI targets regimes where the residual distribution depends sensitively on the test input, and assumes only local exchangeability: residuals associated with spectrally similar inputs are nearly exchangeable, allowing for locally adaptive conformal inference.

1. Core Principles and Motivation

LSCI is motivated by the need for rigorous uncertainty quantification in operator models, which learn mappings between infinite-dimensional or high-dimensional function spaces. While deep neural operators and related regression frameworks achieve high accuracy, they do not provide predictive sets or intervals with statistical guarantees. Classical conformal prediction methods assume full exchangeability among residuals, an assumption that is not realistic when function-valued data exhibit local heteroskedasticity or nonstationarity. LSCI addresses this by restricting attention to local regions—defined spectral-geometrically—where exchangeability holds approximately, and builds prediction sets using a combination of projection-based depth scoring and localized conformal inference (Harris et al., 28 Jul 2025).

2. Methodological Framework

LSCI comprises two principal methodological components:

A. Projection-Based Depth Scoring

For a calibration dataset $\{(f_t, g_t)\}_{t=1}^n$ (with $f_t$ input functions, $g_t$ output functions), and a parametric model $\Gamma_\theta$, LSCI computes residual functions $r_t = g_t - \Gamma_\theta(f_t)$. To quantify the conformity of a residual $r$, LSCI defines a depth function against a local empirical measure. The depth is constructed by projecting $r$ onto a family of continuous linear functionals $\Phi$ (e.g., Fourier, wavelet, principal components):

$$D^\Phi(h \mid P) = \inf_{\phi \in \Phi} D(\phi(h) \mid \phi(P)),$$

where $D(\cdot \mid \cdot)$ is a univariate data depth (e.g., Tukey depth), and $P$ is an empirical residual distribution weighted for locality.

B. Localization and Weighted Conformal Inference

For a new input $f_{n+1}$, a similarity kernel $H(f_t, f_{n+1}) = \exp(-\lambda \|f_t - f_{n+1}\|)$ is used to assign weights $w_t$ to calibration residuals $r_t$. The localized, weighted empirical distribution then defines the central region:

$$D_\alpha^\Phi(f_{n+1}) = \{ r \in \mathcal{G} : D^\Phi(r \mid G_{n+1}) \geq q_{1-\alpha}(f_{n+1}) \}$$

where $q_{1-\alpha}(f_{n+1})$ is the $(1-\alpha)$ quantile of depth scores under the localized empirical distribution $$G_{n+1} = \sum_t w_t \delta_{r_t} + w_{n+1} \delta_\infty$$.

The corresponding prediction set for the test output is given by

$$C_\alpha(f_{n+1}) = \Gamma_\theta(f_{n+1}) + D_\alpha^\Phi(f_{n+1}),$$

which is a set of functions (or, after pointwise collapse, a band) achieving local coverage.

3. Theoretical Guarantees

LSCI provides approximate finite-sample marginal coverage under local exchangeability. This statistical property requires that for any pair of inputs $f_t, f_{n+1}$, the distributions of associated residuals differ in total variation by at most $d_{TV}(R, R^t)$. Under this condition, and for the localization weighting induced by $H$, the true coverage gap is bounded:

$$\sum_t w_t d_{TV}(R, R^t) \leq \frac{\sum_t \exp(-\lambda d_t) d_t}{\sum_t \exp(-\lambda d_t)},$$

where $d_t$ is the spectral pre-metric between $f_t$ and $f_{n+1}$. This ensures that as localization is tightened and the modeling assumption is accurate in a local neighborhood, the coverage of $C_\alpha(f_{n+1})$ approaches the target rate $1-\alpha$.

This result generalizes classical conformal prediction’s exact marginal coverage under full exchangeability, providing a non-asymptotic, quantifiable bound based on the local geometry and residual distribution.

4. Implementation Strategy

Efficient implementation of LSCI consists of:

• Precomputing projections $\phi(r_t)$ for each $r_t$ using a chosen family $\Phi$ (e.g., low-order Fourier coefficients).
• For each test input $f_{n+1}$, computing weights $w_t$ via $H(f_t, f_{n+1})$.
• Evaluating univariate depth $D(\phi(r) \mid \phi(P))$ efficiently for each $\phi$ and residual $r$.
• Using inverse transform sampling (Algorithm 2 in (Harris et al., 28 Jul 2025)) to generate an ensemble $\{r^{(m)}\}$ from the distribution defined by the localized region $D_\alpha^\Phi(f_{n+1})$.
• Forming the ensemble-based prediction band by aggregating (e.g., pointwise minima/maxima or quantiles) over sampled functions $g^{(m)} = \Gamma_\theta(f_{n+1}) + r^{(m)}$.

Selection of the projection family balances computational efficiency (favoring low-dimensional or fast transforms) with adaptivity; data-driven bases such as functional principal component analysis (FPCA) may be beneficial in high-complexity settings.

5. Empirical Results and Scope

LSCI demonstrates empirical validity and adaptivity across both synthetic and real-world operator learning tasks:

• For Gaussian process regression and time-series autoregressive operator learning, LSCI achieves empirical coverage at the nominal $1-\alpha$ rate while maintaining locally adaptive prediction widths.
• In air quality forecasting, energy demand curve prediction, and global weather data, LSCI adapts spatially and temporally, with band widths reflecting non-homogeneous residual variation (e.g., seasonality or regions of higher volatility).
• Comparisons with traditional conformal prediction for functional data, as well as adaptive scoring rule approaches, indicate that LSCI yields tighter and more informative prediction sets without sacrificing statistical coverage (Harris et al., 28 Jul 2025).
• Practical gains in adaptivity are directly attributable to the localization mechanism and the spectral (projection-based) conformity scoring, which together enable the method to handle high-dimensional outputs and function-valued targets effectively.

6. Relation to Other Spectral and Localized Conformal Methods

The LSCI framework differs from classic spectral clustering and earlier “higher-order” or local spectral conformal clustering methods (1001.1323, Arias-Castro et al., 2013) by its focus on prediction set construction for function-valued operator models within a conformal inference paradigm; nonetheless, the foundational idea of leveraging local spectral structure—either in clustering or in conformity scoring—is shared.

Connections to other localized conformal prediction methods (e.g., kernel-weighted (Guan, 2021), RLCP randomization (Hore et al., 2023), and diagnostic-based partitioning (LeRoy et al., 2021)) are apparent. However, LSCI is distinguished by its focus on function-valued output spaces, its explicit use of spectral projections for depth scoring, and its theoretical guarantees under a formulated local exchangeability assumption tailored to operator regression.

7. Practical Considerations and Future Research

Key considerations for deploying LSCI include the computational cost of repeated projection and depth computations, which may be amortized through precomputed projections or variational approximations. Tuning of localization bandwidth $\lambda$ is critical for balancing local sample adequacy against adaptivity; cross-validation or error-based approaches may be used.

Potential future directions include:

• Amortization of localizer and depth evaluation steps to limit computational overhead.
• Learned or data-driven projection families beyond fixed bases to enhance adaptivity, particularly in highly nonlinear or nonstationary regimes.
• Extension to multivariate and structured outputs, as well as integration with probabilistic neural operator frameworks.
• Application to time-series, spatiotemporal, and high-dimensional data modalities, with spectral localization defined over appropriate domains.

LSCI thus represents a rigorous and general approach for uncertainty quantification in operator learning and other complex regression settings where local spectral structure informs predictive uncertainty and coverage.