Additive Noise, Shift Recovery, and Signed Signals in the Cumulative Distribution Transform

Abstract: The cumulative distribution transform (CDT) is a quantile-based transport representation that exactly linearizes one-dimensional translations of positive densities. We study how this structure behaves under additive perturbations and how it can be exploited for shift recovery. Under a local nondegeneracy condition, we derive a first-order expansion showing that additive noise in physical space induces a nonlocal perturbation in CDT space through the primitive of the noise, weighted by the reciprocal density. This yields an explicit description of transform-domain sensitivity and shows, in particular, that perturbations are amplified in low-density regions. When the physical-space perturbation is modeled as a centered Gaussian random field, the induced first-order CDT perturbation is again Gaussian, with an explicit covariance kernel. We then use this structure to study recovery in CDT coordinates. In the known-template setting, the transport shift is obtained by projection onto the constant mode, giving an explicit estimator together with exactness in the noiseless case and a stability bound under perturbations. In the unknown-template setting, multiple observations permit joint recovery of the shifts and a common template up to the natural constant-mode gauge, leading to a simple de-shift--and--average procedure. We also consider a signed-signal analogue based on the signed cumulative distribution transform (SCDT), where shifts are estimated numerically by feature matching and unknown templates are recovered by alternating alignment and averaging. Numerical experiments validate the perturbation analysis and illustrate effective recovery for both density-valued and signed signals.

• The paper derives a first-order CDT noise expansion showing that additive perturbations propagate nonlocally through integrated noise and are amplified in low-density regions, with validated quadratic residual decay.
• The paper uses the CDT’s constant translation mode to recover shifts and estimate unknown templates through weighted-mean projection and de-shifted averaging, achieving robust results down to 10 dB SNR.
• The paper extends alignment to signed signals with the SCDT through grid-based shift search and alternating averaging, while identifying limitations involving tails, approximate centering, grid resolution, and higher-dimensional generalization.

Overview and setting

This paper studies the cumulative distribution transform (CDT) — the quantile map of a positive density relative to a fixed reference density — as a transport-based representation in which translations become exact additive constants. The authors, Antil, Khatri, and Saxena, address two questions that have received comparatively little rigorous treatment: how additive perturbations propagate through the CDT, and how the resulting transform-domain structure can be exploited for shift recovery and template estimation. The analysis is confined to one spatial dimension, where the CDT admits closed-form manipulations, and is complemented by numerical analogues for signed signals via the signed cumulative distribution transform (SCDT).

The paper's central structural observation is well known but worth restating: if $w_s(x) = w(x-s)$, then $\widehat{w}_s(\alpha) = \widehat{w}(\alpha) + s$, so a translated family $M_g = \{g(\cdot - s)\}$ maps to an affine segment in CDT space. This has direct consequences for reduced-order modeling: for a translated Gaussian-mixture family, the first three singular values of the mean-centered snapshot matrix drop from $1.434\times 10^1$, $1.071\times 10^1$, and $6.029$ in physical space to $2.227\times 10^2$, $3.042\times 10^{-3}$, and $7.367\times 10^{-6}$ in CDT space. The transformed family is effectively one-dimensional, which is the geometric basis for everything that follows.

First-order noise propagation

The main analytical result is a first-order expansion for the CDT under zero-mass additive perturbations $u_\delta = u + \delta\eta$. Under the local nondegeneracy condition $\widehat{w}_s(\alpha) = \widehat{w}(\alpha) + s$0 at the relevant quantile location, the paper proves

$\widehat{w}_s(\alpha) = \widehat{w}(\alpha) + s$1

where $\widehat{w}_s(\alpha) = \widehat{w}(\alpha) + s$2 is the primitive of the noise. Three features of this formula deserve emphasis. First, the CDT acts nonlocally on additive perturbations: the effect at coordinate $\widehat{w}_s(\alpha) = \widehat{w}(\alpha) + s$3 depends on the integrated noise up to the quantile location $\widehat{w}_s(\alpha) = \widehat{w}(\alpha) + s$4, not on pointwise values. Second, the reciprocal density factor acts as a signal-dependent gain, so perturbations are amplified in low-density regions; the numerics report a tail-to-center gain ratio of approximately 14.4 for a translated Gaussian. Third, when $\widehat{w}_s(\alpha) = \widehat{w}(\alpha) + s$5 is a centered Gaussian random field with covariance kernel $\widehat{w}_s(\alpha) = \widehat{w}(\alpha) + s$6, the linearized CDT noise $\widehat{w}_s(\alpha) = \widehat{w}(\alpha) + s$7 is again centered Gaussian, with covariance

$\widehat{w}_s(\alpha) = \widehat{w}(\alpha) + s$8

Consequently, even spatially uncorrelated (white) noise becomes correlated in CDT space through the integration step, and the observed CDT covariance scales as $\widehat{w}_s(\alpha) = \widehat{w}(\alpha) + s$9 times the inverse-square density product to leading order.

Semi-analytic experiments validate the expansion sharply: the residual $M_g = \{g(\cdot - s)\}$0 decays quadratically in both $M_g = \{g(\cdot - s)\}$1 and $M_g = \{g(\cdot - s)\}$2, with measured slopes 2.056 and 2.070 respectively, and the quotient error decays linearly with slope 1.056. These are strong confirmations of the asymptotic theory rather than merely qualitative agreement.

Two caveats attach to these results. The nondegeneracy bound need not be uniform in $M_g = \{g(\cdot - s)\}$3: for densities such as Gaussians it holds pointwise but degrades in the far tails, so the amplification statement is a local one. Moreover, the admissibility requirement that $M_g = \{g(\cdot - s)\}$4 remain a positive normalized density forces positivity clipping in the strongest-noise numerical regimes, which the authors themselves flag as a stress test of the density model rather than an unconstrained perturbation regime.

Shift estimation with a known template

When the template is known, the observation model in CDT space is $M_g = \{g(\cdot - s)\}$5. Because the shift lies in the constant mode $M_g = \{g(\cdot - s)\}$6 while the centered residual lies in its orthogonal complement, the least-squares problem $M_g = \{g(\cdot - s)\}$7 admits the explicit solution

$M_g = \{g(\cdot - s)\}$8

the difference of $M_g = \{g(\cdot - s)\}$9-weighted means. The estimator is exact in the noiseless case and satisfies the stability bound $1.434\times 10^1$0 under perturbations. After de-shifting, the residual $1.434\times 10^1$1 has identically zero weighted mean, so transport and shape variability are cleanly separated by projection onto a single mode. Numerically, shift RMSE remains small down to 10 dB SNR, and averaging after CDT alignment produces visibly sharper physical-space reconstructions than direct averaging of shifted noisy observations, which blurs the template.

An important limitation stated plainly in the paper: from a single observation with known template, one recovers only $1.434\times 10^1$2; the pair $1.434\times 10^1$3 cannot be separated, and an unknown template cannot be identified without additional observations.

Joint recovery of template and shifts

With multiple observations sharing a common unknown template, the model $1.434\times 10^1$4 becomes identifiable once two conventions are imposed: centered residuals ($1.434\times 10^1$5) and the gauge $1.434\times 10^1$6. Under these assumptions the shifts are read off exactly as the constant modes $1.434\times 10^1$7, and the template is recovered by de-shift-and-average:

$1.434\times 10^1$8

with error equal to the average residual $1.434\times 10^1$9. The resulting algorithm requires no optimization beyond means and averages, and the gauge ambiguity between template and shifts is resolved computationally by centering the estimate and compensating the shifts. Experiments show exact reconstruction in the noiseless case and moderate shift RMSE and template $1.071\times 10^1$0 error down to 10 dB, with degradation concentrated in the clipped high-noise regime.

One dependency worth noting: all weighted means and orthogonality conditions are defined relative to the chosen reference density $1.071\times 10^1$1, so the numerical values of estimated shifts and recovered templates depend on that choice. The paper recommends selecting $1.071\times 10^1$2 to capture the support of the observed family but offers no systematic selection criterion.

SCDT procedures for signed signals

For signed signals $1.071\times 10^1$3, the SCDT applies transport coordinates to the positive and negative parts together with their masses. Crucially, no closed-form analogue of the constant-mode projection exists in this setting — translated signals are organized in SCDT feature space but not exactly linearized. The paper therefore proposes numerical substitutes: shifts are estimated by minimizing SCDT feature mismatch over a discrete candidate grid $1.071\times 10^1$4, and unknown templates are recovered by alternating alignment (in physical space) and averaging. Recovery quality thus depends explicitly on the width and resolution of $1.071\times 10^1$5, a practical limitation the authors acknowledge. Experiments on Gabor, sawtooth, and square-wave signals with shifts in $1.071\times 10^1$6 show accurate recovery of the dominant translation structure over a broad SNR range, with the discontinuous square signal the most difficult case, particularly at low SNR. Unlike the CDT algorithms, the SCDT outputs live directly in signal space and require no inverse-transform post-processing.

Limitations and open questions

The paper's results carry several qualifications. All perturbation guarantees are first-order and local: they require strict positivity of the density at each quantile location, exclude flat or vanishing regions, and provide no uniform-in-$1.071\times 10^1$7 bounds, so behavior in tails and near zeros of the density remains uncontrolled. The joint recovery proposition assumes exactly centered residuals, whereas in practice centering holds only approximately, and the template error inherits the average residual without a quantitative statistical rate. For the SCDT, the absence of an exact translation identity leaves convergence of the alternating align-and-average iteration unanalyzed, and the accuracy of the grid-search shift estimator is not bounded theoretically. Finally, the entire framework is strictly one-dimensional; extension to higher-dimensional transport coordinates, where no comparable quantile representation exists, is left entirely open.

Conclusion

The paper provides a precise perturbation theory for the CDT under additive zero-mass noise, showing that physical-space noise enters transform space nonlocally through the noise primitive scaled by reciprocal density, with an explicit Gaussian covariance structure at leading order. It converts this structure into simple, provably exact-in-the-noiseless-case estimators for shift recovery and joint template estimation via projection onto the constant mode, and supplies practical grid-based SCDT analogues for signed signals. The numerical evidence — quadratic decay of the perturbation residual, singular-value collapse in CDT space, and robust recovery down to moderate SNR — supports the theoretical claims within their stated assumptions, while leaving uniformity in the tails, statistical rates under approximate centering, and higher-dimensional extensions as open problems.