Filtered Conformal Ellipsoids

• Filtered Conformal Ellipsoids are geometric and statistical constructions that blend conformal mapping with filtering techniques to reduce area distortion and adapt prediction sets.
• The methodology integrates spherical conformal maps, quasi-conformal deformations, and axis optimization or state-space filtering to tailor ellipsoidal shapes to underlying data or surface geometry.
• Applications include texture mapping, medical imaging, and multivariate time series forecasting, yielding improved accuracy by efficiently managing cross-coordinate dependencies.

Searching arXiv for the cited works and closely related usage of the term. Tool call: arxiv_search({"query":"all:\"Filtered Conformal Ellipsoids\" OR id:(Choi, 2023) OR id:(Limmer, 15 Jun 2026) OR id:(Henderson et al., 2024)","max_results":10,"sort_by":"submittedDate","sort_order":"descending"}) Found relevant arXiv entries matching the topic and cited ids, including:

• (Limmer, 15 Jun 2026) — "Filtered Conformal Ellipsoids for Graph-Native Time Series"
• (Henderson et al., 2024) — "Adaptive inference with random ellipsoids through Conformal Conditional Linear Expectation"
• (Choi, 2023) — "Fast ellipsoidal conformal and quasi-conformal parameterization of genus-0 closed surfaces" Filtered conformal ellipsoids denote an ellipsoidal construction that has been developed in two distinct technical settings. In geometry processing, the term refers to constructing low-distortion, bijective parameterizations of genus-0 closed surfaces on tri-axial ellipsoids by combining a spherical conformal map with quasi-conformal filtering and axis optimization (Choi, 2023). In conformal prediction for multivariate time series, it refers to a frozen state-space filter that emits a one-step predictive mean and covariance, followed by split-conformal calibration of Mahalanobis scores to produce a single joint prediction ellipsoid (Limmer, 15 Jun 2026). A closely related regression framework, Conformal Conditional Linear Expectation (CCLE), constructs explicit random ellipsoids whose center and shape adapt to covariates through residual covariance analysis (Henderson et al., 2024). This suggests a shared organizing idea: the ellipsoid shape is chosen by structure in the data or geometry, while a separate procedure controls distortion or coverage.

1. Conceptual scope and motivating problem

In the surface-parameterization setting, the starting point is that a genus-0 closed surface $M$ is topologically equivalent to a sphere, so there exists a conformal diffeomorphism $f_{\mathrm{conf}}: M \to S^2$. The practical problem is that mapping an elongated or flattened surface to the rotationally symmetric sphere often entails large area distortion even when angles are preserved. The geometric mismatch between the input surface’s aspect ratios and the sphere drives significant variation in the Jacobian determinant, which complicates area-sensitive texture mapping, remeshing, and shape analysis. The proposed remedy is an ellipsoidal conformal parameterization onto

$$E=\{(x,y,z)\in\mathbb{R}^3:x^2/a^2+y^2/b^2+z^2/c^2=1\},$$

with radii $(a,b,c)>0$, so that tuning $(a,b,c)$ can filter out area distortion that is otherwise unavoidable on $S^2$ (Choi, 2023).

In the time-series setting, the motivating problem is different but structurally analogous. Joint prediction sets for multivariate time series should control a single event while adapting to cross-coordinate dependence. A filtered conformal ellipsoid uses a learned, frozen state-space filter to emit one-step predictive Gaussian laws with mean and covariance, and then uses split-conformal calibration to set a single scalar radius. At time $t$, the filter emits $(\mu_t,\Sigma_t)$, the conformal score is the Mahalanobis distance of the realized outcome $y_t$ to $(\mu_t,\Sigma_t)$, and the prediction set is

$f_{\mathrm{conf}}: M \to S^2$0

This achieves target marginal coverage of $f_{\mathrm{conf}}: M \to S^2$1 without relying on Gaussian tail probabilities, while $f_{\mathrm{conf}}: M \to S^2$2 controls the shape of the set (Limmer, 15 Jun 2026).

The CCLE framework addresses multivariate regression rather than graph-native time series, but it also produces explicit ellipsoids. It introduces two new conformity scores based on a covariance analysis of residuals and input points, and the resulting prediction sets are ellipsoids whose geometry adapts to the covariates. The paper studies asymptotic properties of these ellipsoids and shows that their volume is reduced compared to that of classic balls, under ellipticity assumptions (Henderson et al., 2024).

2. Geometric filtered conformal ellipsoids on genus-0 surfaces

The geometric construction composes a conformal map to the sphere with a carefully designed quasi-conformal deformation from the sphere to the ellipsoid:

$f_{\mathrm{conf}}: M \to S^2$3

where $f_{\mathrm{conf}}: M \to S^2$4 is a spherical conformal parameterization and $f_{\mathrm{conf}}: M \to S^2$5 is a quasi-conformal map. A concrete fast instantiation is

$f_{\mathrm{conf}}: M \to S^2$6

where $f_{\mathrm{conf}}: M \to S^2$7 is an initial spherical conformal map, $f_{\mathrm{conf}}: M \to S^2$8 is the north-pole stereographic projection, $f_{\mathrm{conf}}: M \to S^2$9 is a Möbius transformation aligning poles and orientation, $$E=\{(x,y,z)\in\mathbb{R}^3:x^2/a^2+y^2/b^2+z^2/c^2=1\},$$0 is a scalar balancing factor that filters area distortion near poles, $$E=\{(x,y,z)\in\mathbb{R}^3:x^2/a^2+y^2/b^2+z^2/c^2=1\},$$1 is a quasi-conformal planar map whose $$E=\{(x,y,z)\in\mathbb{R}^3:x^2/a^2+y^2/b^2+z^2/c^2=1\},$$2 matches the inverse ellipsoidal projection’s $$E=\{(x,y,z)\in\mathbb{R}^3:x^2/a^2+y^2/b^2+z^2/c^2=1\},$$3, and $$E=\{(x,y,z)\in\mathbb{R}^3:x^2/a^2+y^2/b^2+z^2/c^2=1\},$$4 is the inverse ellipsoidal stereographic projection (Choi, 2023).

Axis selection is part of the filtering mechanism. The framework gives several initializations for $$E=\{(x,y,z)\in\mathbb{R}^3:x^2/a^2+y^2/b^2+z^2/c^2=1\},$$5. A PCA-based choice computes the covariance of the zero-centered vertex cloud, diagonalizes $$E=\{(x,y,z)\in\mathbb{R}^3:x^2/a^2+y^2/b^2+z^2/c^2=1\},$$6, and sets $$E=\{(x,y,z)\in\mathbb{R}^3:x^2/a^2+y^2/b^2+z^2/c^2=1\},$$7 proportional to $$E=\{(x,y,z)\in\mathbb{R}^3:x^2/a^2+y^2/b^2+z^2/c^2=1\},$$8. A normalized bounding-box variant is

$$E=\{(x,y,z)\in\mathbb{R}^3:x^2/a^2+y^2/b^2+z^2/c^2=1\},$$9

with

$(a,b,c)>0$0

A least-squares best-fit ellipsoid minimizing

$(a,b,c)>0$1

subject to $(a,b,c)>0$2 is another initialization. After an initial ellipsoidal parameterization $(a,b,c)>0$3, the axes can be filtered further by minimizing area distortion in terms of the logged area ratio $(a,b,c)>0$4 by gradient descent on $(a,b,c)>0$5 (Choi, 2023).

The conformal stage uses standard discretizations. For a triangle mesh $(a,b,c)>0$6, the discrete Laplace–Beltrami operator with cotan weights is

$(a,b,c)>0$7

Several standard algorithms produce $(a,b,c)>0$8: harmonic energy minimization, discrete Ricci flow, and the FLASH approach. Stereographic maps are

$(a,b,c)>0$9

and the compositions satisfy

$(a,b,c)$0

To align desired poles and orientation on $(a,b,c)$1, the Möbius map is

$(a,b,c)$2

which sends $(a,b,c)$3 and $(a,b,c)$4 (Choi, 2023).

3. Quasi-conformal filtering, cancellation, and distortion control

The filtering stage is formulated in quasi-conformal theory. A map $(a,b,c)$5 is quasi-conformal if it satisfies the Beltrami equation

$(a,b,c)$6

Its pointwise dilatation is

$(a,b,c)$7

and the Jacobian determinant is

$(a,b,c)$8

so $(a,b,c)$9 whenever $S^2$0, guaranteeing local injectivity; global bijectivity follows under appropriate normalization and absence of fold-overs. Given $S^2$1 per face on a planar triangulation, the Linear Beltrami solver reconstructs $S^2$2 by solving

$S^2$3

with the stated SPD coefficient matrix $S^2$4, yielding sparse SPD linear systems solved efficiently by conjugate gradient or multigrid (Choi, 2023).

The key obstruction is that the simple anisotropic scaling

$S^2$5

from $S^2$6 to $S^2$7 is not conformal unless $S^2$8. The framework instead designs the correction on the plane via the inverse ellipsoidal stereographic projection

$S^2$9

Its planar pullback metric yields a Beltrami coefficient

$t$0

where $t$1 are the coefficients of the first fundamental form induced by $t$2 (Choi, 2023).

The cancellation step reconstructs $t$3 with

$t$4

via LBS. Then the composition rule for Beltrami coefficients implies that

$t$5

so $t$6 is conformal. This corrected inverse projection is the key filter: it cancels the distortion introduced by the nonconformal ellipsoidal projection, producing a conformal map from $t$7 to $t$8 (Choi, 2023).

Distortion is evaluated face-wise. If $t$9 is the $(\mu_t,\Sigma_t)$0 Jacobian on a face $(\mu_t,\Sigma_t)$1 and $(\mu_t,\Sigma_t)$2 are its singular values, then conformal distortion is

$(\mu_t,\Sigma_t)$3

while area distortion uses

$(\mu_t,\Sigma_t)$4

The framework also regularizes $(\mu_t,\Sigma_t)$5 face-wise by minimizing

$(\mu_t,\Sigma_t)$6

which suppresses local spikes and improves stability (Choi, 2023).

A further filtering device is the perimeter-based scalar $(\mu_t,\Sigma_t)$7. Let $(\mu_t,\Sigma_t)$8 be the outermost triangle and $(\mu_t,\Sigma_t)$9 the innermost triangle around the origin in the planar parameterization $y_t$0. Since

$y_t$1

the product of perimeters is invariant under scaling, and the choice

$y_t$2

equalizes the perimeters of the North/South pole triangles after inverse ellipsoidal projection. This filters uneven sampling near poles that otherwise arises from stereographic maps (Choi, 2023).

4. Filtered conformal ellipsoids for graph-native time series

In the time-series formulation, the filter emits

$y_t$3

and the Mahalanobis conformity score is

$y_t$4

Given calibration scores $y_t$5, the split-conformal ellipsoid is

$y_t$6

where $y_t$7 is the $y_t$8-th order statistic with

$y_t$9

The Gaussian negative log-likelihood is

$(\mu_t,\Sigma_t)$0

and the ellipsoid volume is

$(\mu_t,\Sigma_t)$1

The filter is used to choose the ellipsoid shape; conformal calibration chooses the scalar radius (Limmer, 15 Jun 2026).

The rationale for a single joint prediction set is explicit. For multivariate time series, the event of interest is joint correctness, and controlling a single joint miscoverage event accounts for cross-coordinate dependence, avoiding the overly conservative Bonferroni-style inflation that arises if each coordinate is treated independently. Ellipsoids adapt to the learned cross-sectional correlations via $(\mu_t,\Sigma_t)$2 so that the shape aligns with principal directions of uncertainty, yielding sharper joint sets than axis-aligned rectangles. The framework therefore contrasts with standard conformal prediction that often uses axis-aligned sets or exchangeability-based finite-sample guarantees in IID settings (Limmer, 15 Jun 2026).

The central difficulty is dependence. Filtered scores $(\mu_t,\Sigma_t)$3 are serially dependent because $(\mu_t,\Sigma_t)$4 depend on the past, and learned recurrent filters need not contract in raw hidden-state space. The analysis therefore introduces an observable predictive-law quotient: two hidden states are equivalent if, over a fixed horizon $(\mu_t,\Sigma_t)$5, they emit the same sequence of Gaussian predictive laws. Distances are measured between sequences of emitted laws using a statistical metric such as Fisher–Rao or Kullback–Leibler. The assumptions linking small excess NLL to contraction are the Stable Bayes Gaussian-projection filter, Covariance bounds

$(\mu_t,\Sigma_t)$6

and the Finite-horizon observability Fisher condition (Limmer, 15 Jun 2026).

Under these assumptions, the main contraction statement is that there exist constants $(\mu_t,\Sigma_t)$7 such that

$(\mu_t,\Sigma_t)$8

and

$(\mu_t,\Sigma_t)$9

In particular, if $f_{\mathrm{conf}}: M \to S^2$00 and $f_{\mathrm{conf}}: M \to S^2$01, then the emitted-law sequence contracts in expectation at rate $f_{\mathrm{conf}}: M \to S^2$02 with an additive perturbation $f_{\mathrm{conf}}: M \to S^2$03 (Limmer, 15 Jun 2026).

5. Calibration under dependence and learned covariance structure

Coverage analysis under dependence replaces exchangeability with dependence-aware concentration of the empirical calibration quantile. For thresholded indicators

$f_{\mathrm{conf}}: M \to S^2$04

the threshold-autocovariance envelope is defined by

$f_{\mathrm{conf}}: M \to S^2$05

and

$f_{\mathrm{conf}}: M \to S^2$06

Then

$f_{\mathrm{conf}}: M \to S^2$07

where $f_{\mathrm{conf}}: M \to S^2$08. If the score CDF $f_{\mathrm{conf}}: M \to S^2$09 is continuous at $f_{\mathrm{conf}}: M \to S^2$10 and has density bounded below by $f_{\mathrm{conf}}: M \to S^2$11 near $f_{\mathrm{conf}}: M \to S^2$12, then with an appropriate choice of $f_{\mathrm{conf}}: M \to S^2$13 the realized coverage satisfies

$f_{\mathrm{conf}}: M \to S^2$14

A sharper Bernstein-type bound is obtained under geometric $f_{\mathrm{conf}}: M \to S^2$15-mixing, where $f_{\mathrm{conf}}: M \to S^2$16 (Limmer, 15 Jun 2026).

The implementation is instantiated with a GCN-GRU filter. The architecture combines a Graph Convolutional Network, a Gated Recurrent Unit, and an emission head mapping hidden states to $f_{\mathrm{conf}}: M \to S^2$17. For scalability, the covariance is parameterized as

$f_{\mathrm{conf}}: M \to S^2$18

where $f_{\mathrm{conf}}: M \to S^2$19 with $f_{\mathrm{conf}}: M \to S^2$20 and $f_{\mathrm{conf}}: M \to S^2$21. The Woodbury identity gives

$f_{\mathrm{conf}}: M \to S^2$22

and

$f_{\mathrm{conf}}: M \to S^2$23

Training minimizes average Gaussian NLL on a training split, after which the parameters are frozen and calibration proceeds by computing the order statistic of the calibration Mahalanobis scores (Limmer, 15 Jun 2026).

Empirically, the learned filter gives sharper at-target ellipsoids than static-covariance and non-filter baselines on moderate-size graph-native traffic benchmarks, specifically METRLA-$f_{\mathrm{conf}}: M \to S^2$24 and PEMSBAY-$f_{\mathrm{conf}}: M \to S^2$25. The abstract states that, at full-graph scale and on non-graph-native datasets, factor and copula baselines can be stronger. Practical guidance in the paper emphasizes contiguous calibration blocks, optional block-resampling or block-quantiles, rolling recalibration, and monitoring the stability of $f_{\mathrm{conf}}: M \to S^2$26 over time (Limmer, 15 Jun 2026).

6. CCLE, random ellipsoids, and related adaptive constructions

CCLE develops two conformity scores in a general multivariate regression framework with inputs $f_{\mathrm{conf}}: M \to S^2$27, outputs $f_{\mathrm{conf}}: M \to S^2$28, predictor $f_{\mathrm{conf}}: M \to S^2$29, and residuals $f_{\mathrm{conf}}: M \to S^2$30. The first score constructs an augmented sample using

$f_{\mathrm{conf}}: M \to S^2$31

centers it through the Helmert projector, forms the ridge empirical covariance

$f_{\mathrm{conf}}: M \to S^2$32

and defines the score matrix

$f_{\mathrm{conf}}: M \to S^2$33

The exact conformal set

$f_{\mathrm{conf}}: M \to S^2$34

is expensive because the calibration scores also depend on $f_{\mathrm{conf}}: M \to S^2$35, so the paper introduces the conservative $f_{\mathrm{conf}}: M \to S^2$36-independent approximation

$f_{\mathrm{conf}}: M \to S^2$37

with coverage at least $f_{\mathrm{conf}}: M \to S^2$38 (Henderson et al., 2024).

The resulting conservative joint-Mahalanobis set is exactly an ellipsoid

$f_{\mathrm{conf}}: M \to S^2$39

where

$f_{\mathrm{conf}}: M \to S^2$40

$f_{\mathrm{conf}}: M \to S^2$41

and

$f_{\mathrm{conf}}: M \to S^2$42

The adjusted score $f_{\mathrm{conf}}: M \to S^2$43 yields a second ellipsoid

$f_{\mathrm{conf}}: M \to S^2$44

with the same $f_{\mathrm{conf}}: M \to S^2$45 and $f_{\mathrm{conf}}: M \to S^2$46 but a different radius. The paper states that $f_{\mathrm{conf}}: M \to S^2$47 is possible, while $f_{\mathrm{conf}}: M \to S^2$48 is never empty; conversely, $f_{\mathrm{conf}}: M \to S^2$49 can be full space but $f_{\mathrm{conf}}: M \to S^2$50 as $f_{\mathrm{conf}}: M \to S^2$51 (Henderson et al., 2024).

CCLE is defined through ridge multivariate regression:

$f_{\mathrm{conf}}: M \to S^2$52

which yields

$f_{\mathrm{conf}}: M \to S^2$53

Thus the ellipsoid center is

$f_{\mathrm{conf}}: M \to S^2$54

so CCLE applies an input-dependent correction to the predictor. The paper gives finite-sample marginal coverage under exchangeability for both $f_{\mathrm{conf}}: M \to S^2$55 and $f_{\mathrm{conf}}: M \to S^2$56, asymptotic limits for $f_{\mathrm{conf}}: M \to S^2$57, $f_{\mathrm{conf}}: M \to S^2$58, and the radii, and volume comparisons with classic balls. Under elliptical distributions, the volume of the ellipsoid is reduced compared to that of balls in the stated regimes; under highly anisotropic non-elliptical scenarios, covariance-based ellipsoids can be suboptimal (Henderson et al., 2024).

The paper also gives a time-series remark: set $f_{\mathrm{conf}}: M \to S^2$59 as past residuals $f_{\mathrm{conf}}: M \to S^2$60 and $f_{\mathrm{conf}}: M \to S^2$61 as future residuals $f_{\mathrm{conf}}: M \to S^2$62. CCLE then yields multi-horizon ellipsoids adapting to recent dynamics. The text explicitly notes that the paper does not implement weighted or sliding-window calibration, but that a practitioner can adapt CCLE to filtered settings by using a moving calibration window, covariate-dependent quantiles, and suitable exchangeability or stationarity assumptions (Henderson et al., 2024).

7. Misconceptions, limitations, and applications

A common misconception in the geometric setting is that ellipsoidal parameterization can be obtained by a simple anisotropic scaling of the sphere. The construction explicitly rejects this: $f_{\mathrm{conf}}: M \to S^2$63 is not conformal unless $f_{\mathrm{conf}}: M \to S^2$64, so a nontrivial quasi-conformal correction is required. Another misconception is that ellipsoidal domains are introduced only for aesthetic reasons. The stated objective is to reduce the area distortion induced by the geometric mismatch between an anisotropic genus-0 surface and the sphere, while maintaining bijectivity through quasi-conformal theory (Choi, 2023).

A common misconception in the statistical setting is that an ellipsoidal prediction set must rely on Gaussian tail probabilities. The time-series framework states the opposite: the filter’s covariance controls the shape of the set, while split-conformal calibration chooses the scalar radius, so the construction benefits from a learned predictive covariance without relying on Gaussian tail probabilities for coverage. Another misconception is that joint multivariate coverage can be obtained by coordinate-wise intervals without substantial loss. The framework emphasizes that axis-aligned sets ignore cross-coordinate dependence and can induce overly conservative Bonferroni-style inflation (Limmer, 15 Jun 2026).

The principal limitations are also explicit. For filtered conformal ellipsoids in graph-native time series, assumptions on Gaussian emitted laws and mixing or contraction may be violated in highly nonstationary regimes; sensitivity to filter misspecification affects sharpness and stability; and scalability challenges arise at very large $f_{\mathrm{conf}}: M \to S^2$65 with complex graph structure and large $f_{\mathrm{conf}}: M \to S^2$66. For CCLE, exact finite-sample conditional coverage is impossible in general, covariance-based ellipsoids may be suboptimal under strong non-ellipticity or multimodality, $f_{\mathrm{conf}}: M \to S^2$67-ellipsoids can be full with small $f_{\mathrm{conf}}: M \to S^2$68 and heavy tails, and $f_{\mathrm{conf}}: M \to S^2$69-ellipsoids can be empty when $f_{\mathrm{conf}}: M \to S^2$70 is very atypical (Limmer, 15 Jun 2026, Henderson et al., 2024).

The applications span the two domains. In geometry, the stated uses are texture mapping and UV parameterization on anisotropic targets, remeshing and meshing with near-uniform area elements, and registration and medical imaging for anatomically anisotropic organs such as the hippocampus and skull (Choi, 2023). In time series, the framework is designed for graph-native traffic benchmarks and, more generally, for multivariate settings where a single joint prediction event and learned cross-sectional dependence are central (Limmer, 15 Jun 2026). In regression, CCLE provides covariate-sensitive alternatives to residual-norm balls, and its time-series remark indicates a path to multi-horizon ellipsoids adapting to recent dynamics (Henderson et al., 2024).