Take It or Leave It: Intent-Controlled Partial Optimal Transport

This paper introduces Intent-Controlled Partial Optimal Transport (IC-POT), a novel framework that generalizes partial optimal transport by replacing global mass rejection with pointwise, intent-driven rejection costs based on side information, thereby enabling more structured and reliable matching in applications like positive-unlabeled learning and multi-modal satellite data analysis.

The Gist

Imagine you are trying to match two different groups of people for a dance. One group is the "Source" (let's say, dancers from New York) and the other is the "Target" (dancers from London).

The Old Way (Standard Optimal Transport):
Traditionally, the rule was strict: Every single dancer must find a partner. Even if a New York dancer is wearing a clown nose and a London dancer is wearing a tutu, the algorithm forces them to pair up just to make the numbers match. This often leads to silly, forced matches that don't make sense.

The "Partial" Way (Previous Solutions):
Later, researchers said, "Okay, we can leave some people unmatched." But they suffered from a "one-rule-fits-all" problem. Imagine a manager who says, "We can leave 10% of the dancers on the sidelines," but they can only rank everyone by a single metric, like "dance skill." If the 10% worst dancers are all French, they get kicked out. The system has no way to say, "Kick out the worst 10%, but if two dancers are equally bad, please keep the French one." It cannot handle a secondary preference or a "tie-breaker" rule. It forces a single, rigid ranking that ignores nuance.

The New Way (IC-POT - "Take It or Leave It"):
This paper introduces Intent-Controlled Partial Optimal Transport (IC-POT). Instead of a single ranking rule, it gives every single dancer a personal "rejection price tag."

Think of it like a bouncer at a club, but the bouncer is different for every person:

• The "Take It" Rule: If a dancer is reliable, well-dressed, and fits the vibe, their "rejection price" is high. The algorithm thinks, "It costs too much to kick this person out, so we must try to find them a partner."
• The "Leave It" Rule: If a dancer is clearly out of place (maybe they are a clown in a formal ball, or their data is noisy), their "rejection price" is low. The algorithm thinks, "It's cheap to leave this person on the sidelines, so we will."

Why This Matters:
These individual price tags allow you to encode secondary criteria that the old "one-rule" systems couldn't handle. You can still say, "Drop about 10% of the dancers," but now you can add, "Among the borderline cases, favor the French dancers a bit." By adjusting the price tag for French dancers to be slightly higher, the system automatically keeps them over non-French dancers with similar skill levels. Old partial OT couldn't do this; ICPOT can.

How It Works in Real Life (The Paper's Examples)

The authors show this works in three specific scenarios:

1. The "Guessing Game" (Positive-Unlabeled Learning)
Imagine you are trying to find all the cats in a photo, but you only have a few labeled cat photos and a huge pile of unlabeled photos (some cats, some dogs).

• The Problem: Some cats are hidden in the shadows (hard to see), while others are bright and clear. A standard "partial" method might throw away the shadowy cats because it's trying to be efficient.
• The IC-POT Fix: The system knows that "shadowy" areas are just hard to see, not necessarily "not cats." It puts a high price tag on rejecting shadowy cats. It keeps them in the match. It puts a low price tag on the obvious dogs. The result? It finds more cats without getting confused by dogs.

2. The "Language Barrier" (Open-Partial Domain Adaptation)
Imagine teaching a computer to recognize objects in photos from a new country. Some objects exist in both countries (cars, trees), but some only exist in the new country (unique local animals).

• The Problem: The computer might try to force a match between a local animal and a car because it's desperate to pair everyone up.
• The IC-POT Fix: The system looks at the "confidence" of the match. If a local animal is very confident in its own identity but has no match in the old country, the system gives it a low rejection price. It says, "Leave this animal unmatched; it doesn't belong to the old list." But if a car is clearly a car, the price to reject it is high, so it gets matched.

3. The "Ocean View" (Geophysical Data)
This is the most visual example. The authors compared two different satellite cameras looking at ocean waves.

• The Problem: One camera (SWIM) sees waves clearly but gets "static" (noise) in certain directions. The other camera (SAR) sees waves well but gets "blurred" in other directions due to physics.
• The IC-POT Fix: The system uses physics knowledge as the price tag.
  • If a wave is blurry in Camera A but clear in Camera B, the system says, "This is a real wave, but Camera A is just having a bad day. Don't reject it." (High price to reject).
  • If a wave is clear in Camera A but looks like "static" in Camera B, the system says, "Camera B is just seeing noise. Reject this match." (Low price to reject).
  • Result: They get a perfect map of the waves by ignoring the specific "glitches" of each camera, rather than trying to force a match between a real wave and a glitch.

The Big Takeaway

The paper argues that not all mismatches are created equal.

• Old Method: Uses a one-rule-fits-all approach, ranking everyone by a single metric and kicking out the bottom 10% regardless of other important factors.
• IC-POT: Uses per-item, multi-criterion-aware rejection prices. It looks at each piece of data individually, allowing you to balance the need to drop data with specific preferences (like favoring certain groups or trusting specific sensors) for every single decision.

It turns the decision of "what to throw away" from a blunt, single-metric instrument into a precise, intelligent tool.

Technical Summary

Technical Summary: Intent-Controlled Partial Optimal Transport (IC-POT)

Problem Statement

Classical Optimal Transport (OT) enforces a rigid constraint where all source mass must be transported and all target mass must be explained. This "full-participation" assumption often leads to artificial correspondences or negative transfer when comparing distributions where only a subset of mass is relevant or reliable.

While Partial Optimal Transport (POT) relaxes this by allowing mass to remain unmatched, existing formulations typically rely on global control mechanisms. These include a scalar transported-mass budget, a uniform scalar rebate, or global marginal penalties. These mechanisms control how much mass is rejected but not which specific points should be protected or discarded. Consequently, they fail to address applications where the decision to leave mass unmatched depends on side-specific reliability, support geometry, or external information (e.g., sampling bias in Positive-Unlabeled learning, confidence in Domain Adaptation, or sensor-specific artifacts in geophysics).

Methodology: IC-POT

The authors introduce Intent-Controlled Partial Optimal Transport (IC-POT), a targeted generalization of POT that replaces the global rejection paradigm with pointwise rejection costs on both the source and target measures.

Formulation

Given discrete supports $X = \{x_i\}$ and $Y = \{y_j\}$ with masses $\mu$ and $\nu$, and a transport cost matrix $C$, IC-POT introduces slack variables $u$ (unmatched source mass) and $v$ (unmatched target mass). The optimization problem is:

$$\min_{P, u, v} \langle C, P \rangle + \langle c_s, u \rangle + \langle c_t, v \rangle$$
subject to:
$$P\mathbf{1} + u = \mu, \quad P^\top\mathbf{1} + v = \nu, \quad P, u, v \geq 0$$

Here, $c_s \in \mathbb{R}^n_+$ and $c_t \in \mathbb{R}^m_+$ are pointwise unmatched costs. Unlike global rebates, these costs price the local alternative of leaving specific mass unmatched directly on the original supports.

Structural Properties

The paper establishes several key theoretical properties:

1. Reduced Lagrangian Form: The problem is equivalent to minimizing $\sum_{i,j} (C_{ij} - c_s(i) - c_t(j))P_{ij}$ over sub-couplings, effectively replacing the scalar rebate of classical POT with a separable, pointwise rebate.
2. Dual Interpretation: The dual formulation reveals that $c_s(i)$ and $c_t(j)$ act as local acceptance thresholds (caps) for the dual variables. A point is rejected if its dual variable hits this cap.
3. Admissibility and Sparsity: An edge $(i, j)$ can only be active in an optimal transport plan if $C_{ij} \leq c_s(i) + c_t(j)$. This provides an exact, pre-computation rule for pruning the transport graph, ensuring sparsity based on the specific rejection costs.
4. Augmented-Support Equivalence: IC-POT can be recast as a standard balanced Kantorovich OT problem on an augmented support (adding a dummy point to each marginal), proving well-posedness within the discrete OT framework.

Key Contributions

The paper claims three primary contributions:

1. Explicit Modeling of Unmatched Behavior: It makes the unmatched policy an explicit object in the formulation via slack variables on the original supports, rather than an implicit result of global constraints.
2. Theoretical Characterization: It characterizes the problem as a separable pointwise-rebate generalization of Lagrangian partial transport, establishing dual caps, sparse admissibility rules, and a strict separation from constant-cost partial OT (demonstrated via counterexamples where pointwise costs break symmetries preserved by uniform rules).
3. Empirical Validation: It demonstrates that incorporating pointwise rejection rules driven by side information improves performance in tasks where rejection is structured, specifically in Positive-Unlabeled (PU) learning, Open-Partial Domain Adaptation (OPDA), and geophysical signal comparison.

Experimental Results

1. Positive-Unlabeled (PU) Learning

In PU learning, the goal is to match labeled positives against an unlabeled pool containing both latent positives and negatives.

• Setup: The authors simulate "Selected at Random" (SAR) scenarios where positive samples are under-observed in certain regions (fringes) due to covariate-dependent selection bias.
• Result: A constant-cost partial OT baseline (uniform rejection) fails to protect these under-observed fringe regions, treating them as negatives. IC-POT, using a source-side cost profile that encodes the selection bias (making rejection expensive in low-observation fringes), significantly outperforms the baseline.
• Metrics: In heterogeneous regimes, IC-POT achieved an F1 score of 0.86 compared to 0.52 for the constant-cost baseline.

2. Open-Partial Domain Adaptation (OPDA)

In OPDA, the target domain contains unknown classes that should be rejected.

• Setup: Using a fixed CLIP distillation backbone, the authors modified only the final rejection layer. They compared a uniform partial-W baseline against two IC-POT variants: one using posterior entropy (protecting low-entropy samples) and one using prototype-support (protecting samples with coherent local neighborhood agreement).
• Result: Both IC-POT variants improved upon the uniform baseline across multiple datasets (Office-31, Office-Home, VisDA, DomainNet). The prototype-support variant achieved the highest gains on locally coherent datasets (e.g., 95.12 H-score on Office-31 vs. 94.08 for partial-W).
• Finding: The results suggest that once representation is fixed, performance gains depend on modeling rejection as a structure-dependent policy rather than a uniform scalar rule.

3. Geophysical Case Study: SWIM/SAR Ocean Wave Spectra

This experiment addresses comparing ocean-wave spectra retrieved from two different sensors (SWIM and SAR) with distinct artifacts.

• Context: SAR spectra suffer from "azimuth cutoff" (displacing energy), while SWIM spectra suffer from "speckle" (unreliable directional sectors). The goal is to compare only physically consistent wave systems.
• Method: IC-POT uses side-specific costs derived from physical priors: protecting SAR mass displaced by cutoff (if supported by SWIM) while exposing speckle-dominated or unsupported mass to rejection.
• Result: IC-POT recovered comparable wave energy (0.993) comparable to a high-price global baseline but reduced spurious transport by a factor of 7 (0.031 vs. 0.236).
• Significance: Unlike a scalar rule that forces a trade-off between recovering common systems and rejecting artifacts, IC-POT allows the rejection policy to be defined by the physical nature of the data itself.

Significance and Limitations

The paper argues that IC-POT is significant because it shifts the paradigm of partial transport from "how much to reject" to "what to reject." By making the unmatched policy an explicit, pointwise variable, it allows domain-specific knowledge (sampling bias, confidence, physical priors) to directly inform the transport plan.

Limitations acknowledged by the authors:

• Specification: The unmatched functions ($c_s, c_t$) must be specified by the user based on available side information or diagnostics. The paper does not propose a method for learning these functions automatically from data, though it suggests this as a future direction (e.g., via bilevel optimization).
• Scalability: While the sparse solver is exact, large-scale applications may require further approximations.
• Regularization: The authors note that standard entropic regularization (Sinkhorn) does not directly apply to the augmented-support formulation without altering the objective (introducing a bias on total transported mass) or creating scale mismatches between dummy and real points. Thus, IC-POT is not a drop-in replacement for standard entropic OT solvers.

In conclusion, IC-POT provides a flexible framework for structured rejection in optimal transport, demonstrating that encoding side information into pointwise rejection costs yields superior performance in tasks where the "unmatched" decision is inherently non-uniform.