Signed Cumulative Distribution Transform

• SCDT is an invertible, transport-based representation that decomposes one-dimensional signed signals into normalized positive and negative components.
• It linearizes complex deformations by transforming translations, dilations, and monotone warps into simple additive and multiplicative operations in transform space.
• The framework enhances practical applications in alignment, classification, and parameter estimation while maintaining computational efficiency and explicit inversion.

The Signed Cumulative Distribution Transform (SCDT) is an invertible, transport-based representation for one-dimensional signed signals that extends the cumulative distribution transform (CDT) from nonnegative unit-mass densities to arbitrary finite signed signals. Its defining construction is to decompose a signal into its positive and negative parts, apply CDT to the normalized components relative to a fixed positive reference, and retain the component masses. In this way, the SCDT separates sign, mass, and geometry, while preserving the quantile-based transport structure that makes translations, dilations, and more general monotone deformations tractable in transform space (Aldroubi et al., 2021, Thareja et al., 2022, Antil et al., 9 Jun 2026).

1. Definition and formal construction

For a nonnegative unit-mass signal or density $s$, with cumulative distribution function $F_s$, and a fixed strictly positive reference $s_0$ with cumulative distribution function $F_{s_0}$, the CDT is the monotone transport map

$\mathcal{C}(s)=F_s^{\dagger}\circ F_{s_0},$

where $F^\dagger$ denotes the generalized inverse, or quantile function. In the notation used in several SCDT papers, the transformed signal is the transport map itself; for a uniform reference on $[0,1]$, it reduces to the quantile function $F_s^{-1}$ (Thareja et al., 2022, Rubaiyat et al., 2021).

For a signed signal $s\in L^1(\mathbb{R})$, the SCDT begins with the Jordan decomposition

$$s(x)=s^+(x)-s^-(x),\qquad s^+(x)=\max\{0,s(x)\},\qquad s^-(x)=\max\{0,-s(x)\}.$$

Let

$F_s$0

When $F_s$1, define normalized nonnegative components $F_s$2. The SCDT then stores the CDT of each normalized component together with its mass: $F_s$3 or equivalently,

$F_s$4

depending on the notation adopted in a given paper (Thareja et al., 2022, Antil et al., 9 Jun 2026).

A measure-theoretic formulation places the construction on finite signed measures on the extended real line. If $F_s$5 and $F_s$6 is a non-trivial, atomless, finite positive reference measure, then the SCDT is

$F_s$7

with each $F_s$8 obtained from the generalized inverse of the cumulation of $F_s$9 composed with the reference cumulation. This formulation makes the transform applicable beyond smooth densities and clarifies its status as a bijection between signed measures and pairs of monotone transport maps plus masses (Aldroubi et al., 2021).

A recurring point in the literature is that the SCDT is not a single transport map for a signed signal. Rather, it is a paired representation: one transport map for the positive part, one for the negative part, and mass coordinates for both. This separation is the mechanism by which the transform extends optimal-transport-style geometry to signed data (Gong et al., 2023, Thareja et al., 2022).

2. Inversion, metric structure, and optimal-transport interpretation

The SCDT is invertible. In the density setting, if $s_0$0 and $s_0$1, then the original signed signal is reconstructed from the inverse transport maps and the reference density $s_0$2 by

$s_0$3

followed by

$s_0$4

Equivalent formulas appear in several variants, including the form

$s_0$5

which emphasizes inversion of the component CDTs before rescaling and recombination (Thareja et al., 2022, Antil et al., 9 Jun 2026).

In the measure-theoretic setting, inversion is expressed as a pushforward: $s_0$6 This formulation is important because it does not rely on classical derivatives and therefore extends to finite signed measures and generalized inverse maps (Aldroubi et al., 2021).

The geometric content of the SCDT is inherited from one-dimensional monotone optimal transport. For positive densities, the CDT is the unique monotone rearrangement mapping the reference to the target. For signed signals, the same transport geometry is applied channel-wise to the positive and negative parts. This leads to a Wasserstein-type metric

$s_0$7

which is exactly the squared norm difference of the SCDT representations: $s_0$8 Thus, transport distances between signed signals reduce to Euclidean distances between transform coordinates, with separate contributions from transport geometry and mass mismatch (Thareja et al., 2022, Aldroubi et al., 2021).

This Hilbert-space embedding explains why subspace methods, least-squares estimators, and orthogonal projections recur throughout the SCDT literature. The transform replaces a nonlinear comparison problem in signal space with an $s_0$9-type problem on monotone maps and mass coordinates (Gong et al., 2023, Rubaiyat et al., 30 Sep 2025).

3. Transformation laws and linearization properties

The central structural property of the SCDT is its behavior under mass-preserving monotone warps. If $F_{s_0}$0 is a strictly increasing differentiable bijection and

$F_{s_0}$1

then the SCDT satisfies

$F_{s_0}$2

The masses are preserved, while the transport maps are composed with $F_{s_0}$3. This is the precise sense in which domain deformations become simple operations in transform space (Thareja et al., 2022, Aldroubi et al., 2021).

Several important special cases follow immediately. For an affine warp $F_{s_0}$4 with $F_{s_0}$5,

$F_{s_0}$6

Hence translations act additively on each transport component, and dilations act multiplicatively. Amplitude scaling behaves differently: multiplying the signal by a positive scalar leaves the CDT shape coordinates unchanged and scales only the masses (Thareja et al., 2022, Rubaiyat et al., 2023).

For the positive-density CDT, this linearization is exact and especially simple for rigid shifts. If $F_{s_0}$7, then

$F_{s_0}$8

so a translation family becomes an affine line parallel to the constant mode in $F_{s_0}$9. The recent analysis of additive perturbations in CDT coordinates shows that, under a local nondegeneracy condition, additive noise in physical space induces a nonlocal perturbation in transform space through the primitive of the noise weighted by the reciprocal density. In particular, perturbations are amplified in low-density regions. For signed signals, that paper considers an SCDT analogue and uses numerical feature matching for shift estimation rather than a closed-form constant-mode projection (Antil et al., 9 Jun 2026).

A further consequence is convexification of deformation classes. If a signal class is generated from a template by increasing warps $\mathcal{C}(s)=F_s^{\dagger}\circ F_{s_0},$0, then in transform space the class has the form $\mathcal{C}(s)=F_s^{\dagger}\circ F_{s_0},$1. The literature states that the transformed class is convex for every template if and only if the inverse deformation family is convex. In finite-data settings, this often produces “thin” or approximately low-dimensional sets, which motivates nearest-subspace and local-subspace models for classification and regression (Rubaiyat et al., 2021, Rubaiyat et al., 2022).

The transform therefore does not merely encode signals differently; it reorganizes deformation variability into additive, multiplicative, or compositional structure. This suggests why linear estimation procedures in SCDT space can succeed even when the native-domain problem is nonlinear or nonconvex.

4. Discrete, numerical, and algorithmic formulations

Although the classical CDT is stated for continuous densities, recent work has developed a fully discrete CDT and discrete SCDT for atomic measures on the real line. With a fixed atomic reference

$\mathcal{C}(s)=F_s^{\dagger}\circ F_{s_0},$2

and a target probability measure

$\mathcal{C}(s)=F_s^{\dagger}\circ F_{s_0},$3

the discrete CDT is defined by

$\mathcal{C}(s)=F_s^{\dagger}\circ F_{s_0},$4

where $\mathcal{C}(s)=F_s^{\dagger}\circ F_{s_0},$5 are the reference cumulative masses. The inverse reconstruction is the pushforward

$\mathcal{C}(s)=F_s^{\dagger}\circ F_{s_0},$6

In the signed case, if $\mathcal{C}(s)=F_s^{\dagger}\circ F_{s_0},$7, with positive and negative masses $\mathcal{C}(s)=F_s^{\dagger}\circ F_{s_0},$8 and normalized channels $\mathcal{C}(s)=F_s^{\dagger}\circ F_{s_0},$9, then the discrete SCDT is

$F^\dagger$0

and reconstruction is

$F^\dagger$1

This framework exposes a genuine finite-resolution obstruction: deterministic atomic transport cannot split mass, so exact reconstruction holds only under a cumulative-mass compatibility criterion, while weak convergence is recovered under reference refinement (Antil et al., 10 Jun 2026).

The discrete theory also introduces thresholded stabilization near zero crossings. Because a small perturbation can flip a discrete coefficient from the positive channel to the negative channel, a dead-zone rule around zero can suppress unstable channel switching. This is a specifically signed phenomenon and highlights that zero crossings are a numerical singular locus for the split-channel representation (Antil et al., 10 Jun 2026).

For sampled continuous signals, practical SCDT computation is based on positive/negative splitting, normalization, cumulative sums, generalized inverse evaluation, and interpolation. The numerical workload is typically linear in the number of samples for CDF construction and quantile inversion. In the 2026 shift-recovery study, CDT/SCDT computation on a grid is described as $F^\dagger$2 per signal for CDF/quantile construction, with known-template CDT shift estimation also $F^\dagger$3, unknown-template de-shift-and-average $F^\dagger$4, and SCDT numerical matching $F^\dagger$5 per observation per iteration (Antil et al., 9 Jun 2026).

These computational descriptions show a persistent pattern across the literature: the main numerical ingredients are monotone cumulative functions and their inverses. No transport plan matrix is needed in the one-dimensional setting, which is one reason the SCDT remains attractive for large-scale or repeatedly solved problems (Antil et al., 10 Jun 2026, Rubaiyat et al., 30 Sep 2025).

5. Applications in signal analysis, estimation, and learning

One major application is alignment and shift recovery. For positive densities, CDT coordinates permit exact linear recovery of translations through projection onto the constant mode when the template is known, and de-shift-and-average procedures when multiple shifted observations are available. In the signed case, the 2026 study uses the SCDT as a transport feature map, estimates shifts numerically by feature matching over a grid, and recovers unknown templates by alternating alignment and averaging. Its numerical experiments report effective recovery for both density-valued and signed signals, with stronger difficulty for discontinuous square-wave signals under stronger noise (Antil et al., 9 Jun 2026).

A second major application is classification. The nearest-subspace search method in SCDT space represents each class by the span of transformed training samples and classifies by projection residual. On synthetic data generated by polynomial warps of prototype signals, the reported result is that the SCDT-NS method achieves near-perfect accuracy with only 16 training samples per class, while none of the compared CNNs attained perfect accuracy even with 256 training samples per class. On an ECG heartbeat classification problem with three classes, the reported results are: DeepConvNet $F^\dagger$6, F1 $F^\dagger$7; ShallowConvNet $F^\dagger$8, F1 $F^\dagger$9; CompactConvNet $[0,1]$0, F1 $[0,1]$1; SCDT-NS $[0,1]$2, F1 $[0,1]$3 (Rubaiyat et al., 2021).

An end-to-end extension replaces global class subspaces by nearest local subspaces enriched with analytic deformation directions. On ten 1D time-series datasets, the reported method achieves the top accuracy in $[0,1]$4 datasets, matches the average arithmetic ranking of $[0,1]$5NN-DTW at $[0,1]$6, attains mean per class error $[0,1]$7, and is described as more data efficient than several deep-learning baselines. The same study also reports that on a synthetic out-of-distribution setup the proposed method achieved near-perfect accuracy with approximately 16 training samples per class (Rubaiyat et al., 2022).

A third application is parameter estimation. For composition models $[0,1]$8, the SCDT converts Wasserstein-type distance minimization into linear least squares in transform space. For polynomial $[0,1]$9, the composition $F_s^{-1}$0 becomes linear in the unknown coefficients, so the estimation problem admits a global minimizer via linear least squares. The reported examples include time delay and dispersion estimation, where native-domain objectives are described as nonconvex and multimodal while the SCDT-domain objectives become convex with a unique global minimum (Thareja et al., 2022).

This least-squares perspective is developed further for dynamical systems and PDE parameter identification. In that setting, one measures a time signal at a fixed sensor, models it as a warped template induced by the governing PDE, and uses SCDT nearest-local-subspace regression for coarse parameter recovery. Reported results include nonlinearity detection accuracy of $F_s^{-1}$1 for SCDT-NLS versus $F_s^{-1}$2 for FT-SVM and $F_s^{-1}$3 for 1D-VGG, dispersion detection accuracy of $F_s^{-1}$4, and near-perfect damage-level identification on the UNESP-CONCEPT structural health monitoring dataset, including $F_s^{-1}$5 for one sensor and $F_s^{-1}$6 for another (Rubaiyat et al., 2023).

The framework has also been lifted to images through the Radon Signed Cumulative Distribution Transform (RSCDT), which applies the one-dimensional SCDT to Radon projections. Reported results include $F_s^{-1}$7 accuracy on a simulated signed-image task for RSCDT-NS versus $F_s^{-1}$8 for an unsigned RCDT-NS baseline that used absolute values, as well as strong performance on geometric shape and sign-language datasets (Gong et al., 2023).

More recent work applies SCDT to hyperspectral anomaly detection. In that setting, each spectrum is modeled as a signed or preprocessed one-dimensional signal, mapped to SCDT coordinates, and background variability is learned by PCA in transform space. The reported AUC values are $F_s^{-1}$9 on AVIRIS-I, $s\in L^1(\mathbb{R})$0 on AVIRIS-II, $s\in L^1(\mathbb{R})$1 on Urban, $s\in L^1(\mathbb{R})$2 on Pavia, and $s\in L^1(\mathbb{R})$3 on Forest for the FPR range $s\in L^1(\mathbb{R})$4, with superiority on four datasets at low false-positive rates and the highest full-range AUC on all five (Rubaiyat et al., 30 Sep 2025).

6. Assumptions, limitations, and current scope

The SCDT is inherently a one-dimensional construction. Its exact optimal-transport interpretation depends on monotone rearrangement on the line, and most theoretical statements assume a strictly positive reference density or an atomless reference measure. For practical density formulas, the positive and negative components are usually assumed absolutely continuous on their supports, while generalized inverses are used to handle non-strict monotonicity (Aldroubi et al., 2021, Thareja et al., 2022).

The transform is best matched to variability generated by monotone, mass-preserving deformations. Non-monotone warps, severe multipath interference, or signal classes not well described by transport of a template can degrade convexity and subspace structure. One study explicitly notes that gearbox vibration signals in the raw time domain do not satisfy the generative assumptions well, whereas performance improves after transformation to the Fourier domain, where finite-support event structure is more compatible with the model (Rubaiyat et al., 2022).

Noise sensitivity is structured rather than benign. In the CDT perturbation analysis, additive noise in physical space becomes nonlocal in transform space through integration, and perturbations are amplified in low-density regions by the factor $s\in L^1(\mathbb{R})$5. Even white noise becomes correlated after transformation. In density recovery under strong noise, positivity clipping and renormalization may be needed to preserve admissibility, which departs from a purely additive model (Antil et al., 9 Jun 2026).

The signed extension introduces its own limitations. Positive and negative parts are transported independently, so interactions between signs are not represented by a single coupled transport map. Near zero crossings, small perturbations can trigger channel switching, motivating thresholded stabilization in the discrete setting (Antil et al., 10 Jun 2026). In alignment problems, the positive-density CDT admits a closed-form constant-mode projection for translation estimation, whereas the SCDT workflow described in the 2026 shift-recovery study uses numerical feature matching rather than a closed-form projection (Antil et al., 9 Jun 2026).

These constraints delimit the present scope of the SCDT rather than diminish its utility. Within its natural regime—one-dimensional signed signals whose dominant variability is transport-like—the transform provides a rare combination of bijectivity, explicit inversion, metric structure, deformation linearization, and computational simplicity. That combination explains its continuing use across classification, parameter estimation, inverse problems, signed-image analysis, and anomaly detection (Aldroubi et al., 2021, Rubaiyat et al., 30 Sep 2025).