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Partial Optimal Transport on the Circle for All Transported Masses in O(N log N)

Soheil Kolouri

arXiv:2608.23910v1cs.LGcs.DS

TL;DR

Partial transport needs a full transported-mass profile, but circular geometry introduces an optimized cut that makes the naive exact method O(N^2 log N). The paper develops PAWC, whose free-gap invariant yields exact profiles, nested plans, and one simultaneous optimal cut in O(N log N) time; it also supports spherical slicing.

  • Problem

    The full partial-transport profile is useful when the transported mass is unknown, but circular geometry prevents direct reuse of efficient line algorithms and makes the exact cut-envelope approach O(N^2 log N).

  • Method

    PAWC uses nested active sets, cyclic-neighbour candidate cells, constant-time local marginals, and a free-gap invariant to construct a cut valid for every cardinality.

  • Results

    PAWC returns exact costs, optimal active sets, plans, and a gap valid for all cardinalities in O(N log N) time and O(N) memory.

  • Takeaways & Limitations

    The complete circular partial-transport profile can be computed in sorting-scale time, and the same construction extends through great-circle slicing to spherical data.

  • Takeaways & Limitations

    The guarantees assume equal atomic weights, W1 geodesic cost, distinct supports, and deterministic handling of degeneracies; transported fraction is set by a rule rather than inferred.

Abstract

from arXiv · show

Partial optimal transport compares two measures while leaving part of the mass unmatched, which is what makes it robust to outliers, occlusion, and clutter. The quantity of interest is usually the whole profile - the optimal cost at every transported cardinality - because the right amount to transport is rarely known in advance, and on the real line the PAWL algorithm returns that profile in $O(N\log N)$. Much data is periodic rather than linear: angles, phases, orientations, time of day, hue, and every direction obtained by projecting onto a great circle. On the circle the same problem acquires a global circulation, or equivalently an optimized cut, which the naive exact method handles by running the line algorithm once per support gap, at $O(N^{2}\log N)$. We show that this factor $N$ is unnecessary. The line structure survives in cut-free form, and a free-gap invariant supplies, at every step, a cut at which all previous local updates remain valid line updates. This yields PAWC: an exact $O(N\log N)$ time, $O(N)$ memory algorithm returning all $K+1$ costs, nested active sets and plans in one run, together with a single gap that is simultaneously optimal for every cardinality. Slicing over great circles extends it to $\mathbb{S}^{d-1}$. Empirically the whole profile costs $0.56$ms at $N=4096$ against $1.5$s for a single transported fraction from a general solver; on occluded, cluttered mpeg-7 shapes, holding the descriptor fixed and varying only the cost, it retains $66\%$ of the clean-data retrieval score against $16\%$ for balanced circular OT, and on $\mathbb{S}^{2}$ it halves the fitting error of spherical sliced Wasserstein against contaminated targets, synthetic and real. Code is available at https://github.com/mint-vu/Partial_Wasserstein_on_Circles.

1 INTRODUCTION

Partial transport leaves unmatched mass out of the correspondence, making it suitable for outliers and occlusion. This paper targets the full cost profile on periodic domains, where circular geometry turns efficient line algorithms into a cut-optimization problem.

  • Motivation: The full profile k 7→C◦_k records the minimum cost for every transported cardinality, exposing the tradeoff between coverage and geometric fidelity.It allows transported mass to be selected after examining the profile rather than fixed before solving.
  • Periodic geometry: Angles, phases, orientations, time of day, hue, and great-circle projections are naturally represented on the circle.The circle therefore arises in both periodic variables and sliced directional data.
  • The circular challenge: Circular geometry has no canonical boundary, introducing a free circulation or an optimized cut absent from the line.Consequently, no single unwrapping recovers the full partial-transport profile.
  • The circular challenge: The exact cut-envelope method runs PAWL separately over O(N) support gaps, costing O(N^2 log N).The paper’s central challenge is avoiding this repeated computation while retaining exactness.
  • Contribution: PAWC uses cut-free line structure and a free-gap invariant to return all costs, nested active sets, and plans in one O(N log N) run.It also constructs one gap that is optimal for every cardinality and extends through great-circle slicing to S^{d-1}.

2 BACKGROUND

The paper formulates circular partial 1-Wasserstein transport for uniformly weighted empirical measures and studies its entire transported-mass profile. It extends line-based partial transport into a setting where prior circular work addressed balanced transport but not prescribed-mass partial transport.

  • Problem formulation: The circular ground space is L = R/LZ with geodesic distance dS1(x, y) = min{|x − y|, L − |x − y|}.The empirical measures use common atom weight w, with N = n + m and K = min(n, m).
  • Problem formulation: PW◦(s) is the minimum cost of transporting exactly s units of mass, with at most one atom’s full mass w participating per atom.At integer mass s = kw, the feasible set scales to cardinality-k matching.
  • Problem formulation: The computed object is the profile k 7→C◦_k, not a single transported-mass cost.The active set A(M) contains atoms incident to a matching.
  • Related work: PAWL is the first line algorithm returning the entire partial profile in O(n log n), and PAWC extends its line structure to the circle.Earlier partial-transport solvers were cubic, quadratic, or lacked prescribed transported mass.
  • Related work: Prior circular methods solve balanced W1 through cut or circulation optimization, while prescribed-mass partial transport on the circle remained unaddressed.The exact circular reference method evaluates all support-gap cuts, producing O(N^2 log N) complexity.

3 STRUCTURE OF CIRCULAR PARTIAL TRANSPORT

Circular partial transport retains an incremental structure: optimal matchings can be nested, and each extension activates cyclic-neighbour atoms through a local candidate cell. A free-gap invariant makes these local updates compatible with a common cut.

  • Nested active sets: Every optimal cardinality-k matching has an optimal cardinality-(k + 1) extension that adds one previously inactive source atom and one target atom.Thus the profile can be built incrementally without revisiting earlier activation decisions.
  • Cyclic neighbours: The two atoms added at each extension are consecutive in the cyclic ordering of the inactive atoms.Only adjacent opposite-type inactive pairs need to be considered as candidate cells.
  • Local updates: Activating a candidate cell revokes the sorted interior matching, replaces it with the sorted closed-cell matching, and leaves outside matches untouched.The exact increment m(C) = cR(C) − cR(C◦) depends only on atoms in the cell.
  • Free-gap invariant: Each current cell contains a free original support gap, meaning one not used by any previously selected cell.When cells merge, the new cell inherits a free gap from a parent, preserving the invariant at constant cost.
  • Greedy choice: The minimum current-cell marginal is the exact greedy increment, and activating any minimizer yields a circularly optimal next active set.The common-cut argument makes the local line updates mutually comparable.
  • Simultaneous cut: PAWC constructs one original support gap that is optimal for every cardinality, then recovers each plan by sorting active atoms in the unwrapped order.This avoids searching separately over cuts.

4 THE PAWC ALGORITHM: O(1) MARGINALS, O(N log N) OVERALL

PAWC evaluates circular candidate marginals in constant time after preprocessing and processes all activations with a lazy-deletion heap. It returns the exact full profile and a simultaneous optimal cut in O(N log N) time and O(N) memory, with spherical slicing as an extension.

  • Constant-time marginals: A doubled cyclic sequence and prefix arrays reduce each balanced candidate marginal to an O(1) table lookup.For interval [a, b], m([a, b]) = (Q_b − Q_a−1) − (Q_b−1 − Q_a), with the second term zero for empty interiors.
  • Complexity and outputs: Under Assumption 1, PAWC returns the exact C◦_k, an optimal active set for every k, and a gap valid for all k in O(N log N) time and O(N) memory.The heap contains O(N) entries, each popped once, while sorting dominates the complexity.
  • Spherical extension: Spherical slicing projects atoms onto great circles and runs PAWC once per slice, returning the complete profile for each slice.With M slices, the total cost is O(M(N log N + dN)).

5 SLICING TO THE SPHERE

PAWC extends exact partial W1 transport from the circle to spheres by slicing directional data onto great circles. Each slice returns the full transported-mass profile, with uniform concentration guarantees over that profile.

  • Slicing to the sphere: Great-circle projections map spherical measures to circle instances of the partial 1-Wasserstein problem.The projection uses a Stiefel-manifold frame and reads each projected measure in the great circle’s own coordinate.
  • Slicing to the sphere: One pass over M slices computes the estimate at every transported mass in O(M(N log N + dN)) time.This follows because PAWC returns the whole profile for each slice.
  • Slicing to the sphere: M ≥ (π^2K^2w^2/2ε^2) log(2(K + 1)/δ) slices suffice for ε-accuracy on the entire profile with probability 1 − δ.The bound is stated for uniform concentration over the profile.
  • Slicing to the sphere: At full transported mass, the spherical construction reduces to spherical sliced Wasserstein, making the partial method a strict generalisation.The full-mass endpoint corresponds to the balanced case.

6 EXPERIMENTS

Experiments validate PAWC’s exactness and scaling, then show that partial transport improves robustness to clutter, occlusion, and contaminated spherical targets. The evaluations cover synthetic adversarial instances, MPEG-7 shape matching, and spherical fitting.

  • 6.1 EXACTNESS AND SCALING: Across 504 adversarial instances, PAWC matches an LP oracle to 4.7 × 10^-15 and satisfies every tested structural invariant.The checked invariants include nestedness, free gaps, cell balance, feasibility, and validity of θ⋆.
  • 6.1 EXACTNESS AND SCALING: PAWC’s complete-profile scaling has fitted slopes 1.03–1.29, versus 2.00 for the cut-envelope baseline.Heap operation counts are exactly linear, and the excess above one is attributed to cache effects.
  • 6.1 EXACTNESS AND SCALING: 0.56 ms computes all transported fractions at N = 2048, versus 1.54 s for one fraction from POT’s general partial solver.At N = 256, the corresponding times are 0.09 ms and 4.4 ms.
  • 6.2 PARTIAL MATCHING OF SHAPE DESCRIPTORS: At 20% clutter and full visibility, PAWC retains 66% of the clean retrieval ceiling, while balanced circular OT retains 16%.The reported bullseye values are 0.302 for PAWC and 0.071 for balanced OT, with a factor-of-4.3 advantage.
  • 6.2 PARTIAL MATCHING OF SHAPE DESCRIPTORS: PAWC remains at orientation success 1.00 at 40% clutter, while balanced circular OT falls to 0.45.A fixed ρ = 0.8 also beats both baselines wherever clutter is present.
  • 6.3 ROBUST FITTING ON S2: With contamination from ε = 0.1 upward, the prescribed spherical trimming rule reduces energy distance by 45–68% on the synthetic mixture and 53–70% on the earthquake catalogue.The ε = 0 row is the clean-data control where SSW is best.
  • 6.3 ROBUST FITTING ON S2: The catalogue exception at ε = 0.4 favors trimming less than nominal contamination because some uniform noise lands on fault lines.This is the one reported cell where the rule underperforms.

7 DISCUSSION

The discussion positions PAWC as a single-sweep exact solution for complete circular partial-transport profiles, while identifying mass, cost, and spherical-sampling boundaries. The reported slice-count bound is substantially more conservative than the tested empirical requirement.

  • 7 DISCUSSION: PAWC computes the complete circular profile in O(N log N), avoiding cut search and returning one cut valid for every cardinality.The method’s memory usage is O(N).
  • 7 DISCUSSION: Equal atomic weights make each activation add exactly one source and one target; non-uniform masses require event-driven handling of saturating capacities.This is an explicit scope boundary of the current result.
  • 7 DISCUSSION: The constant-time chain oracle relies on the W1 geodesic cost, while p > 1 and general Monge costs require separate analysis.The supplied discussion identifies these costs as outside the established guarantee.

A A WORKED EXAMPLE, STEP BY STEP

The worked example traces PAWC’s cyclic cell and heap operations from initialization through a simultaneous optimal cut. It demonstrates how wrap-around matches, merges, lazy skips, and free-gap inheritance produce the full profile in one sweep.

  • STEP 1 — PRECOMPUTATION ON THE DOUBLED SEQUENCE: Doubling the sorted cyclic sequence enables PAWL-style prefix ranks, predecessors, and costs to be precomputed on a line.The resulting Q-table supplies constant-time marginal calculations for balanced intervals.
  • STEP 2 — CELLS, FREE GAPS, AND THE INITIAL HEAP: Initially, inactive atoms partition the circle into cells, each carrying a free original gap and a representative.Opposite-type endpoint cells are candidates, while same-type gaps still retain free-gap representatives.
  • STEP 3 — ITERATION k = 1: At k = 1, selecting (x3, y2) with marginal 0.5 merges three cells and creates candidate [x2, y3] with marginal 1.3.The selected cell’s gap becomes used, while the merged cell inherits a representative from an unselected neighbor.
  • STEP 4 — ITERATION k = 2: At k = 2, the wrap-around cell (y4, x1) is selected with marginal 1.0, producing a match that crosses 0 and cannot be found by a cut near 0.The merged candidate [x4, y1] has marginal 3.6.
  • STEP 5 — ITERATION k = 3: At k = 3, lazy deletion skips a stale heap entry, then [x2, y3] activates and rematches x2 → y2 and x3 → y3, revoking x3 → y2.The cumulative cost becomes 2.8, and the merged candidate [y1, x4] has marginal 3.8.
  • STEP 6 — ITERATION k = 4: At k = 4, the cheaper directed endgame cell [x4, y1] is selected, shifting the circulation and completing the active set.The two remaining directed cells have marginals 3.6 and 3.8.
  • STEP 6 — ITERATION k = 4: The maintained representative θ⋆ = (y1, x2) is a cut where plain PAWL reproduces every circular cost and plan.The example shows that per-k freeness is insufficient: a gap can work for some cardinalities but fail for another.
  • STEP 6 — ITERATION k = 4: The marginal sequence 0.5, 1.0, 1.3, 3.6 is non-decreasing, with an elbow at k = 3.The next sample after k = 3 costs nearly three times more to transport.

C.1 THE LINE STRUCTURE INHERITED FROM PAWL

The line structure inherited from PAWL makes each circular activation local: neighboring inactive atoms define balanced intervals whose cost changes can be evaluated as line updates. A free-gap invariant then provides a common cut, enabling an exact reference profile across all cardinalities.

  • On the line, optimal active sets grow by activating one source and one target, with newly activated opposite-type atoms adjacent among inactive atoms.
  • For neighboring endpoints, the intervening active atoms form a balanced contiguous chain, so rematching changes only within the endpoints’ interval.
  • Distinct support locations make non-neighbor extensions strictly more expensive, upgrading the neighbor result from existential to universal for optimal nested extensions.
  • On the circle, a balanced active set has a free circulation level chosen at a weighted median, yielding at least one support gap with zero optimal flow.
  • A common gap remains flow-free when a neighboring pair is activated, so both circular costs reduce simultaneously to line costs.
  • Running PAWL over all support gaps is exact but costs O(N^2 log N), motivating a method that avoids recomputing the profile for every cut.

D.1 NESTED OPTIMAL ACTIVE SETS

Optimal circular matchings can be organized into nested active sets, while each extension activates cyclically neighboring inactive atoms. Together with a common optimal cut and local cell updates, this supplies the structural basis for efficient incremental computation.

  • D.1 NESTED OPTIMAL ACTIVE SETS: Nested optimality preserves every previously active atom while adding exactly one previously inactive source and one target at each cardinality increase.
  • D.1 NESTED OPTIMAL ACTIVE SETS: Minimum-cost flow augmentation explains nestedness: internal path atoms remain matched, while only the unmatched source and target endpoints become newly active.
  • D.1 NESTED OPTIMAL ACTIVE SETS: Nested active sets do not imply nested matching edges, because augmentation may revoke several old edges and replace them while preserving all active atoms.
  • D.2 A PAIRWISE COMMON-CUT LEMMA: A common flow-free support gap lets the active set and its extension share one line representation, reducing candidate extensions to local circular cells.
  • D.3 CYCLIC NEIGHBOR ADDITION: For any circularly optimal cardinality-k matching, an optimal extension can activate source and target atoms consecutive in the cyclic inactive ordering.
  • D.2 A PAIRWISE COMMON-CUT LEMMA: The local marginal uses ordinary distance inside the unwrapped cell rather than geodesic distance, preventing matches from crossing the complementary arc.

F FREE GAPS, THE EXACT GREEDY THEOREM AND THE SIMULTANEOUS CUT

The free-gap invariant preserves a cut outside all selected cells, enabling exact greedy activation on the circle. This yields nested optimal active sets and one cut that is simultaneously optimal for every transported cardinality.

  • Free-gap invariant: Every current cell contains an original support gap that no selected cell has used, and this property survives cell merging.Gap labels remain fixed while cells change; a merged cell inherits a free gap from an unselected neighbor.
  • Cut compatibility: A free gap provides a cut outside all previous selections, so earlier circular updates remain valid line-local updates.The cut-compatibility lemma applies inductively because no selected cell crosses the cut.
  • Exact greedy choice: Selecting a minimum-marginal candidate cell produces a circularly optimal active set at the next cardinality.Any minimizing candidate can be activated while preserving optimality, yielding an optimal nested sequence from A0 = ∅ through K.
  • Simultaneous cut: The greedy construction supplies one original gap whose cut realizes the optimal line cost for every cardinality.The same gap is avoided by every selected cell and is therefore simultaneously valid throughout the sequence.
  • Why construction is needed: A cut optimal for full mass need not remain optimal for smaller masses, so reusing such a cut can be unboundedly wrong.The full-mass problem carries no information about small-mass costs, and the paper explicitly distinguishes existence of a simultaneous cut from uniqueness.

G.1 DOUBLED CYCLIC SEQUENCE

The doubled cyclic sequence converts circular cells into bounded intervals and supports constant-time balanced-interval cost queries after linear preprocessing beyond sorting.

  • Cyclic representation: The sorted union support is doubled by appending each point shifted by the circumference, preserving labels across the second copy.Every clockwise circular cell becomes a standard doubled-sequence interval with length less than N.
  • Balanced intervals: Prefix differential ranks and signed coordinate sums identify balanced intervals and their minimal balanced predecessors.The predecessor of an endpoint is the most recent earlier occurrence of the same differential-rank value.
  • Minimal-chain costs: A minimal balanced chain has cost w|S_t − S_p_t|, obtained from the signed coordinate difference between its endpoints.Within such a chain, all increasing-order matched pairs share one orientation, making the absolute displacement sum collapse to this difference.
  • Interval queries: Every balanced interval decomposes uniquely into adjacent minimal chains, whose costs add through prefix differences.The resulting accumulated chain costs provide the interval cost by subtracting two prefix values.
  • Preprocessing: Candidate-cell marginals are evaluated in O(1) time after preprocessing, with all prefix and predecessor values computed in O(N) time after sorting.The preprocessing uses a hash table or rank-indexed array for predecessor lookup.

H CORRECTNESS AND COMPLEXITY

PAWC combines doubled-sequence preprocessing, linked-list cell maintenance, and lazy heap updates to compute the exact circular profile and optimal active sets efficiently. The same circle solver serves spherical sliced transport through great-circle projections.

  • Algorithm: The algorithm maintains inactive atoms in a circular doubly linked list and candidate cells in a min-heap with lazy deletion.Entries remain valid only when endpoints are inactive, adjacent, and of opposite types.
  • Complexity: Sorting costs O(N log N), while preprocessing, initialization, and heap-driven updates use O(N) space and linear-size event structures.At most N initial heap entries and one new candidate insertion per iteration are created, and each entry is popped once.
  • Correctness outputs: The algorithm returns exact costs C◦_k and optimal active sets for every k = 0, ..., K, together with a simultaneous optimal gap.Plans are recovered by cutting at that gap and matching retained atoms in increasing unwrapped order.
  • All transported masses: Intermediate transported masses are obtained by linearly interpolating consecutive optimal plans because the min-cost flow value is convex and piecewise linear.Successive shortest augmentations give nondecreasing marginal costs, so the interpolation is optimal between integer cardinalities.
  • Spherical extension: Spherical slicing projects measures onto great circles, where the circle solver computes each slice’s partial-transport profile without rescaling arc lengths.The projection preserves total mass and maps empirical atoms to atoms; the coordinate circle has circumference 2π.

J.2 SLICED SPHERICAL PARTIAL WASSERSTEIN

The paper extends partial Wasserstein transport from the circle to spheres by averaging circular partial costs over great-circle slices. The resulting profile preserves several structural properties, while its scope and metric interpretation remain limited.

  • Definition and setup: Sliced spherical partial Wasserstein averages circular partial transport costs over great-circle projections for finite measures with uniformly weighted atoms.The same transported mass s is admissible across all slices because projection preserves mass.
  • Scope and limitations: The construction excludes d = 2 as a non-informative case and does not cover empirical probability measures with different sample counts and therefore nonuniform atom weights.It assumes a common atom weight while allowing the numbers of atoms, and hence total masses, to differ.
  • Inherited properties: The discrepancy is symmetric, nonnegative, and invariant under orthogonal transformations of the sphere.The invariance follows from the symmetry of the circular distance and invariance of the slicing measure.
  • Inherited properties: The spherical profile is nondecreasing, convex, and piecewise linear in transported mass, with breakpoints contained in the admissible atom-mass grid.These properties follow from the corresponding properties of each circular slice profile and survive averaging.
  • Computation and estimation: Monte Carlo sampling over slices provides a uniform accuracy guarantee over the whole profile, rather than only at one transported mass.The concentration argument uses bounded slice costs and a union bound over profile breakpoints.
  • Computation and estimation: One PAWC run per slice returns the entire spherical profile, so transported mass can be selected after computation without recomputing the circular solver.Replacing the solver with cut enumeration would increase the sliced construction's cost from O(N log N) to O(LN^2 log N).

K.4 COMPLEXITY

PAWC computes the complete circular partial-transport profile in near-linear time, while experiments test both its complexity and the reliability of simultaneous-cut claims.

  • Complexity: Baseline cut enumeration follows an O(N^2 log N) slope of 2.00, whereas PAWC fits slopes of 1.03–1.29.The corresponding line solver PAWL fits 1.12–1.25 on the same hardware.
  • Complexity: 0.81N heap deletions and 0.31N insertions remain unchanged across N = 10^4, 10^5, and 10^6, supporting linear total heap work.The measurements attribute excess log–log slope to cache effects rather than a hidden asymptotic term.
  • Simultaneous cut: The deterministic rule is optimal at every k on 86.5% of instances, but its failure rate ranges from 0.1% to 31.5% across families and its excess can reach three orders of magnitude.For balanced n = m instances, its failure rate rises to 27.3%; choosing a different optimal full-mass cut fails on 46.5%.
  • Simultaneous cut: The simultaneously optimal gap set has median size between 0.17N and 0.92N depending on the family, although a uniformly random full-mass-optimal gap can succeed with probability as low as 0.14.The theorem guarantees existence and constructive production, not that every full-mass-optimal gap works.
  • Simultaneous cut: The optimal cut changes with transported cardinality in a majority of instance families, and fixing it at k0 = K/2 fails on 38.9% of instances.This contrasts with the paper’s constructive guarantee of a simultaneously optimal gap.
  • Spherical construction: The spherical estimator’s convergence fit is −0.497 across eighteen configurations, and its bound is never violated across 540 configuration–threshold combinations.The tested piecewise linearity in transported mass also holds exactly.
  • Spherical construction: The effective Hoeffding range is only 0.3%–15% of πs, while the uniform whole-profile bound can exceed the pointwise bound by a factor whose logarithm reaches 10361 at k/K = 0.05.The pointwise bound is recommended for a particular transported mass, while the uniform bound concerns the whole profile.

L.2 WHY NOT UNBALANCED TRANSPORT, AND IS THIS THE SAME OPTIMUM?

The comparison asks whether unbalanced Sinkhorn transport is an adequate substitute for partial transport and whether PAWC returns the same optimum as a general solver. PAWC matches the general solver numerically while retaining a substantial speed advantage and avoids tuning a relaxation weight across visibility conditions.

  • Comparison with unbalanced transport: 0.71 versus 0.85 success at v = 0.35, c = 0.4 shows KL-relaxed Sinkhorn falling behind PAWC as the visible boundary disappears.Sinkhorn matches PAWC for nearly complete queries, but no single relaxation weight is competitive across cells.
  • Comparison with unbalanced transport: 0.61 and 0.44 are the best-cell scores for different Sinkhorn relaxation weights, whereas ρ = 1 − c is a rule rather than a tuned parameter.The rotation sweep costs 0.58 s per query per τ versus 0.029 s for all 360 PAWC solves.
  • Same optimum: 4.3 × 10^-14 is the largest absolute discrepancy between PAWC’s profile value and POT’s partial wasserstein2 at the same cardinality over 864 queries.The comparison uses unit atom weights and therefore evaluates the same partial-transport problem.
  • Same optimum: 13 times faster at n = 128 and 2700 times faster at n = 2048, PAWC returns the same optimum reported by the general partial solver.The speed comparison concerns applications reading the exact profile value at a selected cardinality.
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