Cone-Compatible Monge Geometry for High-Dimensional Ordered Optimal Transport

Abstract: High-dimensional optimal transport is seldom available in closed form. The one-dimensional case is exceptional because the order of the real line is compatible with convex transport costs, making monotone rearrangement optimal. This paper studies when an analogous Monge structure can be recovered in higher dimensions from a partial order. We introduce a cone-compatible Monge geometry: a closed convex cone (K) induces the order (x\preceq_K y) whenever (y-x\in K), and is compatible with a cost if ordered pairs satisfy a Monge exchange inequality. For squared Mahalanobis costs (c_M(x,y)=(x-y)^\top M(x-y)), we prove a sharp characterization: compatibility holds exactly when (K) is acute under the (M)-inner product, namely (u^\top Mv\ge0) for all (u,v\in K), equivalently (K\subseteq K_M^*). Under this condition, measures supported on cone chains admit a quantile-type closed-form optimal coupling, yielding exact transport under the original ground cost rather than after projection or metric replacement. We distinguish the resulting cone-chain Wasserstein metric on canonically ordered chain distributions from an extended directed cone transport cost on general measures, and develop feasibility, duality, stability, approximation, Gaussian recovery, statistical, and computational results. The theory is complementary to sliced and tree Wasserstein distances: it is not a universal fast surrogate, but a way to obtain interpretable, direction-valid, original-space monotone transport for ordered high-dimensional data.

• The paper establishes that squared Mahalanobis costs are Monge-compatible exactly when the cone is acute under the M-inner product, with finite-generator tests enabling computational certification and learning penalties.
• Cone-chain measures admit an exact cumulative-overlap coupling that visits at most m+n−1 pairs, reducing equal-size equal-weight transport to a sorting-type formula while preserving the original ground cost.
• The framework distinguishes symmetric cone-chain metrics from asymmetric directed transport, provides soft-relaxation and approximation guarantees, and achieves one-dimensional O(n⁻¹ᐟ²) statistical rates when ordered structure is valid.

Motivation and central question

One-dimensional optimal transport is analytic because the total order of the real line is compatible with convex costs: the no-crossing (Monge) property makes monotone rearrangement optimal, and sorting becomes an optimality certificate. The paper asks whether a high-dimensional partial order can play the same role. Its answer is built on convex cones: a closed convex cone $K \subset \mathbb{R}^d$ induces the partial order $x \preceq_K y \iff y - x \in K$, covering coordinate-wise order, Lorentz-cone progression, and Loewner order as special cases. The core question is when this cone order is compatible with a transport cost in the sense that ordered pairs satisfy the Monge exchange inequality, so that crossing two matched pairs cannot improve the objective.

The authors are explicit that "ordered transport" per se is not new: directional OT [Nutz–Wang], displacement-constrained OT, order-preserving Wasserstein distances for sequences, entrywise order constraints on transport matrices, stochastic-order projections, and causal/adapted OT all use order or direction. The claimed novelty is narrower: cone-cost compatibility as a source of analytic high-dimensional Monge structure under the original ground cost, without projection to slices or replacement by tree metrics.

Sharp compatibility characterization

For squared Mahalanobis costs $c_M(x,y) = (x-y)^\top M(x-y)$, the paper proves a sharp equivalence: $(K, c_M)$ is Monge-compatible if and only if $u^\top M v \ge 0$ for all $u, v \in K$, i.e., $K$ is contained in the dual cone $K_M^*$ under the $M$-inner product. The proof is a direct expansion showing the exchange gap equals exactly $2u^\top Mv$, so the condition is necessary and sufficient for all two-pair exchanges—not merely sufficient. This sharpness is the strongest structural claim of the paper.

A practical corollary is the finite-generator test: for $x \preceq_K y \iff y - x \in K$0, compatibility holds iff $x \preceq_K y \iff y - x \in K$1 is entrywise nonnegative. This yields both a computational certificate and a differentiable penalty $x \preceq_K y \iff y - x \in K$2 for learning cones within neural pipelines. Compatibility extends beyond quadratic costs: for translation costs $x \preceq_K y \iff y - x \in K$3 with $x \preceq_K y \iff y - x \in K$4 convex, the cone-Hessian condition $x \preceq_K y \iff y - x \in K$5 implies compatibility via a submodularity argument.

Closed-form cone-chain transport

The main analytic result states that empirical measures supported on compatible $x \preceq_K y \iff y - x \in K$6-chains admit a quantile-type optimal coupling given by the cumulative-overlap plan

$x \preceq_K y \iff y - x \in K$7

where $x \preceq_K y \iff y - x \in K$8 are cumulative masses. Under strict compatibility and nondegenerate cumulative breakpoints, this plan is the unique optimum. For equal weights and equal support sizes, the cost reduces to the sorting-type formula $x \preceq_K y \iff y - x \in K$9. Computation is a two-pointer sweep visiting at most $c_M(x,y) = (x-y)^\top M(x-y)$0 nonzero pairs, i.e., $c_M(x,y) = (x-y)^\top M(x-y)$1 couplings plus cost evaluation ($c_M(x,y) = (x-y)^\top M(x-y)$2 or $c_M(x,y) = (x-y)^\top M(x-y)$3 per pair depending on $c_M(x,y) = (x-y)^\top M(x-y)$4). Crucially, this solves the original Kantorovich problem under the original Mahalanobis cost—the closed form is exact, not a surrogate.

The honest complexity accounting matters here: only the within-chain term $c_M(x,y) = (x-y)^\top M(x-y)$5 becomes analytic. If chain order must be discovered, pairwise comparability testing costs $c_M(x,y) = (x-y)^\top M(x-y)$6, and by Dilworth's theorem the number of chains needed to cover the partial order equals its width; when the width approaches $c_M(x,y) = (x-y)^\top M(x-y)$7, the advantage over global OT disappears.

Metric versus directed objects

The paper separates two frequently conflated objects. On canonical $c_M(x,y) = (x-y)^\top M(x-y)$8-chain distribution classes, the quantile-map distance $c_M(x,y) = (x-y)^\top M(x-y)$9 is a genuine metric (via canonical uniqueness and Minkowski's inequality); without the canonical parameterization, multiple representations of the same measure would break well-definedness. Separately, the directed cone OT cost $(K, c_M)$0 constrains displacements to $(K, c_M)$1 almost surely; it is generally asymmetric, possibly infinite, satisfies only an extended directed triangle inequality, and admits Kantorovich duality and existence results for closed $(K, c_M)$2. Feasibility of the directed problem is characterized by a Strassen-type theorem: $(K, c_M)$3 iff $(K, c_M)$4 is dominated by $(K, c_M)$5 in the cone-stochastic order.

A separation proposition establishes that these notions are genuinely distinct: there exist compatible cone-chain problems whose optimal monotone coupling violates the hard directed constraint on every transported pair, and conversely directed-feasible problems whose supports are not chains. Directional feasibility alone therefore does not yield the analytic formula.

Relation to known structures and Gaussian recovery

The framework recovers one-dimensional quantile OT as the case $(K, c_M)$6, and recovers Gaussian $(K, c_M)$7 whenever the Bures map $(K, c_M)$8 is $(K, c_M)$9-isotone ($u^\top M v \ge 0$0), in which case the cone-monotone value equals the standard Bures–Wasserstein value. Notably, Gaussian recovery belongs to the isotone-map formulation, not hard displacement-constrained OT, which is generally too restrictive for full-dimensional Gaussians under pointed cones. For general $u^\top M v \ge 0$1, separable costs with the orthant cone remain compatible, while nonseparable $u^\top M v \ge 0$2 with $u^\top M v \ge 0$3 requires the cone-Hessian condition—explaining why $u^\top M v \ge 0$4 carries the cleanest theory.

Additional structural results include closed-form geodesics and barycenters on the canonical class (convex combinations of canonical quantile maps preserve $u^\top M v \ge 0$5-monotonicity), and dimension-free statistical rates: under bounded support and Lipschitz canonical parameterization, empirical cone-chain distances inherit one-dimensional $u^\top M v \ge 0$6 convergence, since the effective statistical dimension is the chain parameter rather than the ambient dimension.

Soft relaxation and approximation

Strict cone feasibility is relaxed by the soft objective adding $u^\top M v \ge 0$7 to the integrand. The soft values are nondecreasing in $u^\top M v \ge 0$8 and converge upward to the hard directed value, with tight near-minimizer subsequences converging to hard-feasible optima when feasible. Incomparability of real data is handled through a chain-defect quantity $u^\top M v \ge 0$9, giving the approximation bound $u, v \in K$0 for cone-chain approximants—turning lack of order into a measurable error rather than an informal caveat.

Limitations and scope

The paper is unusually candid about boundaries. The solid-angle analysis shows that for approximately uniform difference directions, comparability probability is $u, v \in K$1, and two-way comparability at most $u, v \in K$2; for the orthant cone this is $u, v \in K$3, vanishing exponentially. Cone-compatible OT is therefore a structured-data method, not a general-purpose high-dimensional OT accelerator. Cone selection is task-specific—a modeling hypothesis like a kernel or tree metric—and the compatibility certificate guarantees cost compatibility but not semantic validity of the chosen direction. The framework does not subsume sliced Wasserstein, tree Wasserstein, or directional OT; it complements them by preserving the original ground cost while exploiting original-space order. Supporting propositions quantify the alternatives' weaknesses: finite-slice estimators are non-identifiable for any finite direction set in $u, v \in K$4, and tree-Wasserstein incurs distortion bounded by the ground-metric discrepancy on support pairs. Open questions left by the paper include learnable cones, local cone fields $u, v \in K$5, cone-ordered Schrödinger bridges and flow matching, cone-ordered Gromov-Wasserstein, and statistical learning theory for ordered distribution classes.

Conclusion

This paper identifies cone-cost compatibility as a distinct mechanism for analytic optimal transport: when a convex cone's induced partial order satisfies the Monge exchange inequality under the ground cost—sharply characterized as acuteness under the $u, v \in K$6-inner product—measures supported on compatible chains admit exact, sparse, differentiable quantile-type solutions of the original Kantorovich problem. By carefully separating metrics from directed costs, certifying feasibility, quantifying approximation and statistical behavior, and stating complexity honestly including order-discovery costs, the work positions cone-compatible transport as an interpretable geometry for progression-like ordered data rather than a universal fast surrogate.