A Discrete Cumulative Distribution Transform via Optimal Transport

Abstract: This paper develops a fully discrete cumulative distribution transform (CDT) for atomic probability measures on the real line. The transform is defined through monotone quantile maps and admits explicit linear-time algorithms for both forward transformation and inverse reconstruction based solely on cumulative mass matching. Unlike the classical continuous setting, deterministic transport between atomic measures cannot generally split masses, so exact reconstruction may fail at finite resolution. We establish a precise cumulative-mass compatibility criterion for exact finite-resolution recovery and prove weak convergence of reconstructed measures under reference refinement. Several structural properties of the discrete CDT are derived, including translation, composition, and scaling laws, and the framework is extended to a discrete signed cumulative distribution transform with thresholded stabilization near zero crossings. By avoiding continuous interpolation, the proposed framework provides a simple fixed-reference transport representation for discrete data. Numerical examples illustrate translation linearization, compatibility-controlled reconstruction, refinement consistency, and stabilization of the signed transform.

• The paper develops a fully discrete cumulative distribution transform for atomic measures, using cumulative mass matching to compute forward and inverse representations in linear time without interpolation or density estimation.
• The paper proves exact reconstruction occurs if and only if every target cumulative mass level matches a reference level, while refinement with maximum reference mass δ_N yields the uniform bound ||F_μ̃_N − F_μ||∞ ≤ δ_N.
• The paper extends the method to signed signals through separate positive and negative channels, preserving translation linearization but revealing sensitivity near sign changes that dead-zone thresholding can reduce at the cost of information loss.

Overview

The paper develops a fully discrete cumulative distribution transform (CDT) for atomic probability measures on the real line, defined via monotone quantile maps $T_\mu = F_\mu^{-1} \circ F_\sigma$ relative to a fixed discrete reference measure $\sigma = \sum_{j=1}^m q_j \delta_{y_j}$ (2606.12131). The framework requires no interpolation, density estimation, or continuous inverse-CDF approximation: both forward and inverse transforms admit explicit linear-time algorithms based solely on cumulative mass matching. A fixed reference of $m$ atoms maps any target measure with an arbitrary number of atoms to a vector in $\mathbb{R}^m$, yielding a uniform-dimensional transport representation suited to downstream learning tasks.

The central theoretical contribution is a precise characterization of when exact reconstruction is possible at finite resolution. Because deterministic (Monge) transport between atomic measures cannot split mass, the reconstructed pushforward $(T_\mu)_\#\sigma$ generally differs from the target $\mu$. The authors prove that exact recovery holds if and only if every target cumulative mass level is also a reference cumulative level, and they show this obstruction vanishes asymptotically under reference refinement via weak convergence.

Forward transform and algorithms

With cumulative sums $Q_j = \sum_{\ell=1}^j q_\ell$ and $P_i = \sum_{k=1}^i p_k$, the discrete quantile formula gives

$$T_\mu(y_j) = x_i, \qquad i = \min\{k : P_k \ge Q_j\},$$

so each reference atom is matched to the first target support location whose cumulative mass reaches or exceeds the reference cumulative level. The forward algorithm is a non-backtracking two-pointer sweep with complexity $\mathcal{O}(m+n)$; the inverse algorithm aggregates reference masses transported to identical locations in $\sigma = \sum_{j=1}^m q_j \delta_{y_j}$0. The authors emphasize a distinction from standard computational optimal transport: rather than producing a Kantorovich coupling matrix that permits mass splitting, the algorithm constructs a deterministic vector-valued map serving as a fixed-dimensional representation. A floating-point safeguard handles roundoff at the final cumulative level but is inactive in exact arithmetic.

Exact reconstruction and its obstruction

The main structural result states that $\sigma = \sum_{j=1}^m q_j \delta_{y_j}$1 holds if and only if $\sigma = \sum_{j=1}^m q_j \delta_{y_j}$2 (equivalently, it suffices that $\sigma = \sum_{j=1}^m q_j \delta_{y_j}$3 be contained in the reference levels). When the condition holds, each target mass is recovered exactly as the aggregate $\sigma = \sum_{j=1}^m q_j \delta_{y_j}$4 over the block of reference indices transported to $\sigma = \sum_{j=1}^m q_j \delta_{y_j}$5. This criterion delineates precisely where a discrete 1D Monge map exists; when it fails, exact matching requires a mass-splitting Kantorovich plan.

This is a genuine departure from the continuous CDT, which is bijective whenever reference and target densities are strictly positive. Here bijectivity breaks down unconditionally unless mass decompositions are compatible. An illustrative example makes the point concrete: with reference $\sigma = \sum_{j=1}^m q_j \delta_{y_j}$6 and target $\sigma = \sum_{j=1}^m q_j \delta_{y_j}$7, reconstruction yields only $\sigma = \sum_{j=1}^m q_j \delta_{y_j}$8; enriching the reference to $\sigma = \sum_{j=1}^m q_j \delta_{y_j}$9 restores exactness. Practically, the criterion enables a priori verification of invertibility given a candidate reference design — a useful design tool, though it means invertibility is resolution-dependent rather than guaranteed.

Structural properties and refinement consistency

Three linearization laws are established, mirroring the continuous theory:

Property	Law
Translation	$m$0
Composition ($m$1 strictly increasing)	$m$2
Scaling ($m$3)	$m$4

Translations, scalings, and monotone deformations thus become additive shifts, compositions, and scalar multiplications in CDT coordinates, exactly as in the continuous setting.

The finite-resolution obstruction is shown to be purely asymptotic. Under refinement with maximal reference mass $m$5, the authors prove the sharp uniform bound

$m$6

which implies weak convergence $m$7. The bound is quantitative and controlled directly by the coarsest reference atom, so reconstruction error degrades gracefully with reference coarseness. Note that convergence is weak (in CDF sup-norm), not in stronger metrics such as Wasserstein distance; the paper does not establish rates beyond the $m$8 bound itself.

Discrete signed CDT

For signed signals $m$9, the discrete SCDT applies the CDT separately to the normalized positive and negative parts, returning $\mathbb{R}^m$0; reconstruction combines the channels as $\mathbb{R}^m$1. Translation linearization carries over channel-wise. The paper identifies a genuinely discrete pathology absent in the continuous signed framework of Aldroubi et al.: near zero crossings, a single sign flip moves an entire atom between channels, so the SCDT can be highly sensitive to small perturbations. The proposed remedy is a dead-zone threshold of width $\mathbb{R}^m$2 that discards coefficients with $\mathbb{R}^m$3 before normalization. This stabilizes the decomposition but introduces a hyperparameter and discards information; the paper offers no principled rule for selecting $\mathbb{R}^m$4 and does not quantify the resulting bias.

Numerical illustrations

Four experiments validate the theory without interpolation or density estimation: (i) empirical measures from Gaussian samples exhibit exact translation linearization in CDT coordinates; (ii) for the target $\mathbb{R}^m$5 under uniform references, exact reconstruction occurs precisely at resolutions where some $\mathbb{R}^m$6 (e.g., $\mathbb{R}^m$7), confirming the compatibility criterion, with approximate reconstruction otherwise; (iii) SCDT components shift additively under signal translation; and (iv) dead-zone thresholding suppresses noise-induced sign oscillations near zero crossings while leaving total positive and negative masses essentially unchanged.

Limitations and open questions

Several restrictions are acknowledged or evident. The theory is confined to one dimension, where monotone rearrangement provides closed-form transport; extension to higher dimensions is listed as future work but is nontrivial since no canonical monotone map exists there. Exact invertibility depends on cumulative-mass compatibility, so the transform is not a bijection on atomic measures in general — a structural limitation the authors treat as intrinsic rather than remedied. The refinement consistency theorem assumes only that maximal reference mass tends to zero and gives no rates beyond $\mathbb{R}^m$8; whether adaptive reference designs can achieve faster convergence or compatibility by construction remains open. For the SCDT, the threshold parameter $\mathbb{R}^m$9 is chosen heuristically, and the interaction between thresholding and downstream task accuracy is unexamined. Finally, all numerical evidence is illustrative rather than comparative; no benchmarks against interpolation-based CDT implementations or alternative discrete transport representations are reported.

Conclusion

The paper provides a rigorous, fully discrete formulation of the CDT for atomic measures, with linear-time forward and inverse algorithms, exact translation/composition/scaling linearization laws, and a sharp if-and-only-if criterion for finite-resolution reconstruction. Its most substantive results are the compatibility characterization of exact invertibility and the uniform error bound $(T_\mu)_\#\sigma$0 establishing asymptotic consistency under refinement. Together with the thresholded signed extension, these results constitute a well-founded transport-based representation for empirical and sampled data whose practical value in classification, estimation, and inverse problems now rests on the higher-dimensional and adaptive-reference extensions the paper leaves open.