UniPROT: Uniform Prototype Selection

• UniPROT is a framework that selects a small set of source examples with uniformly equal weights to accurately summarize a target distribution.
• It reformulates the prototype selection problem using partial optimal transport so that the originally super-additive objective becomes a submodular one amenable to greedy optimization with a (1-1/e) guarantee.
• Empirical results demonstrate that UniPROT improves minority-class accuracy and maintains robust performance in varied settings, including imbalanced image benchmarks and large language model training.

UniPROT is a subset selection framework for choosing a small set of representative source examples that summarize a target distribution while enforcing that every selected prototype carries equal weight. It formulates prototype selection through optimal transport (OT), but replaces the standard unequal-weight regime with a uniformly weighted prototype distribution and then uses a partial optimal transport reformulation to obtain a monotone submodular objective with greedy optimization guarantees. The method is introduced as “Uniform Prototype Selection via Partial Optimal Transport with Submodular Guarantees” and is positioned against prototype-selection procedures whose implicit weighting tends to favor majority classes in imbalanced data (Chanda et al., 13 Apr 2026).

1. Problem setting and motivation

UniPROT studies prototype selection from a source set

$S=\{x_i\}_{i=1}^m$

to represent a target set

$T=\{y_j\}_{j=1}^n.$

Their empirical distributions are written as

$$p=\sum_{i=1}^m p_i \delta_{x_i}, \qquad v=\sum_{j=1}^n v_j \delta_{y_j},$$

with $p\in \Delta^m$ and $v\in \Delta^n$ (Chanda et al., 13 Apr 2026). The generic task is to choose a subset $P\subseteq S$ with $|P|\le k$ that best represents the target distribution.

The central motivation is class imbalance. Existing subset-selection methods such as k-medoids and facility-location-style objectives do not directly enforce equal importance among selected points; instead, the effective prototype weights emerge implicitly from the transport plan or objective. The paper argues that these learned weights often concentrate on majority classes, so minority-class examples may either be excluded or selected with low effective contribution. UniPROT addresses this by requiring the selected prototypes themselves to form a uniformly weighted distribution (Chanda et al., 13 Apr 2026).

This design targets settings in which the prototype set is expected to function as a faithful summary rather than merely a sparse support for a reweighted approximation. A plausible implication is that the method is especially suited to regimes where representational fairness across source subpopulations matters as much as aggregate approximation quality.

2. Uniform-weighted optimal transport formulation

For a selected subset $P\subseteq S$, UniPROT imposes the uniform prototype measure

$\mu_P = \frac{1}{|P|}\mathbf{1}_P,$

where $\mathbf{1}_P$ is the indicator vector of the subset (Chanda et al., 13 Apr 2026). The method then seeks a subset whose uniformly weighted empirical measure is close to the target distribution under OT.

Using a similarity matrix $T=\{y_j\}_{j=1}^n.$0 defined by

$T=\{y_j\}_{j=1}^n.$1

the OT objective is written as

$T=\{y_j\}_{j=1}^n.$2

with coupling set

$T=\{y_j\}_{j=1}^n.$3

UniPROT poses the prototype-selection problem as

$T=\{y_j\}_{j=1}^n.$4

with $T=\{y_j\}_{j=1}^n.$5 (Chanda et al., 13 Apr 2026).

This formulation is conceptually direct: each selected prototype is assigned equal mass, and the subset is judged by how well that uniform measure can be transported to the target. In contrast to weighted-prototype schemes, the equality of source-side contributions is not emergent but hard-coded into the definition of the selected representation.

3. Super-additivity and the intractability of the naive objective

To analyze the uniform-weighted OT objective, the paper introduces

$T=\{y_j\}_{j=1}^n.$6

with

$T=\{y_j\}_{j=1}^n.$7

as the associated proxy (Chanda et al., 13 Apr 2026). Three properties are established:

$T=\{y_j\}_{j=1}^n.$8

$T=\{y_j\}_{j=1}^n.$9

and, for disjoint sets,

$$p=\sum_{i=1}^m p_i \delta_{x_i}, \qquad v=\sum_{j=1}^n v_j \delta_{y_j},$$0

The last property is the key obstruction: $$p=\sum_{i=1}^m p_i \delta_{x_i}, \qquad v=\sum_{j=1}^n v_j \delta_{y_j},$$1 is super-additive rather than submodular (Chanda et al., 13 Apr 2026). That places the problem outside the standard regime in which greedy subset selection admits classical approximation results. Greedy algorithms are well aligned with diminishing returns, but the naive uniform-OT objective exhibits increasing returns across disjoint additions.

The significance of this point is methodological. UniPROT does not merely propose a new scoring function; it identifies that the most natural uniform-weight OT formulation has the wrong combinatorial curvature for efficient approximation. The method’s main technical contribution is therefore not the initial OT objective itself, but the reformulation that restores tractability without abandoning the equal-weight principle.

4. Partial optimal transport reformulation and submodularity

UniPROT replaces the exact OT marginal constraints with a partial optimal transport (POT) construction. The reformulated objective is

$$p=\sum_{i=1}^m p_i \delta_{x_i}, \qquad v=\sum_{j=1}^n v_j \delta_{y_j},$$2

where

$$p=\sum_{i=1}^m p_i \delta_{x_i}, \qquad v=\sum_{j=1}^n v_j \delta_{y_j},$$3

(Chanda et al., 13 Apr 2026).

Under this reformulation, the source marginal remains exact, but the target marginal is relaxed to an inequality constraint. The paper proves that $$p=\sum_{i=1}^m p_i \delta_{x_i}, \qquad v=\sum_{j=1}^n v_j \delta_{y_j},$$4 is non-negative, monotone, and submodular under the cardinality constraint $$p=\sum_{i=1}^m p_i \delta_{x_i}, \qquad v=\sum_{j=1}^n v_j \delta_{y_j},$$5 (Chanda et al., 13 Apr 2026). That single result changes the computational status of the problem: the objective now falls into the classical monotone-submodular maximization regime.

The paper further states an equivalence at cardinality $$p=\sum_{i=1}^m p_i \delta_{x_i}, \qquad v=\sum_{j=1}^n v_j \delta_{y_j},$$6: if $$p=\sum_{i=1}^m p_i \delta_{x_i}, \qquad v=\sum_{j=1}^n v_j \delta_{y_j},$$7 is an optimal solution to the original problem with $$p=\sum_{i=1}^m p_i \delta_{x_i}, \qquad v=\sum_{j=1}^n v_j \delta_{y_j},$$8, then $$p=\sum_{i=1}^m p_i \delta_{x_i}, \qquad v=\sum_{j=1}^n v_j \delta_{y_j},$$9 is also optimal for the POT-based reformulation, and conversely (Chanda et al., 13 Apr 2026). Thus the POT relaxation is not merely heuristic at the target budget. It preserves the optimum among size-$p\in \Delta^m$0 solutions while exposing submodular structure.

This suggests that the paper’s core conceptual move is to trade exactness of marginal matching away from the optimization path while preserving exactness at the final selection budget. That is the technical basis for its claim of “submodular guarantees.”

5. Greedy optimization and approximation guarantees

Because $p\in \Delta^m$1 is non-negative, monotone, and submodular, UniPROT can be optimized by the standard greedy procedure: start from the empty set, repeatedly add the element with largest marginal gain, and stop after $p\in \Delta^m$2 selections (Chanda et al., 13 Apr 2026). The paper gives the approximation guarantee

$p\in \Delta^m$3

where $p\in \Delta^m$4 and $p\in \Delta^m$5 is the optimal size-$p\in \Delta^m$6 solution (Chanda et al., 13 Apr 2026).

Because of the size-$p\in \Delta^m$7 equivalence between the original super-additive objective and the POT-based reformulation, this $p\in \Delta^m$8 result also applies relative to the original uniform-prototype selection problem. This is the main theoretical guarantee associated with UniPROT.

The paper also considers approximate marginal gains. Writing the exact gain as

$p\in \Delta^m$9

it derives a closed-form approximation from the current transport solution; in practice this reduces to sorting the similarity row $v\in \Delta^n$0, giving an $v\in \Delta^n$1 estimate per candidate (Chanda et al., 13 Apr 2026). Lemma 5 then provides a weaker but explicit guarantee for the approximate-greedy variant,

$v\in \Delta^n$2

for a constant $v\in \Delta^n$3 defined from row-wise similarity statistics (Chanda et al., 13 Apr 2026).

Algorithmically, UniPROT therefore operates by alternating between POT solves for the current subset and efficient evaluations of candidate additions. The paper characterizes this as scalable and comparable in complexity to classical greedy k-medoids-style selection (Chanda et al., 13 Apr 2026).

6. Empirical behavior under imbalance

The empirical argument for UniPROT is that enforcing uniform prototype weights improves minority-class representation without damaging majority-class performance. On long-tailed image benchmarks including CIFAR10-LT and synthetic imbalanced MNIST, the paper reports consistent improvements in minority-class accuracy over weighted prototype methods such as k-medoids, while maintaining majority-class accuracy (Chanda et al., 13 Apr 2026).

The paper also reports robustness to the prototype budget: as fewer prototypes are selected, UniPROT degrades less sharply than competing methods (Chanda et al., 13 Apr 2026). This is presented as evidence that the uniform-weight constraint remains useful even when the summary budget is severely limited.

Beyond classical imbalanced classification, the method is evaluated in LLM training. In fine-tuning experiments on MATHINSTRUCT and SUPERGLUE, UniPROT is used for batch selection under domain imbalance and is reported to outperform GREATS, CoLM, GradNorm, and SBERT on both in-domain and out-of-domain evaluation sets (Chanda et al., 13 Apr 2026). In pretraining on OpenWebText with LLaMA-3 500M and 60M models, it is reported to outperform full-batch training and other subset-selection baselines, with lower validation perplexity and more stable training (Chanda et al., 13 Apr 2026).

These experiments extend the method’s scope beyond prototype visualization or coreset construction. They present UniPROT as a general subset-selection mechanism for imbalance-sensitive training pipelines, including settings where source domains rather than class labels define the skew.

7. Interpretation and position within subset selection research

UniPROT occupies a specific position within OT-based and combinatorial subset selection. Its distinctive feature is not the use of OT alone, nor greedy optimization alone, but the combination of three elements: uniform prototype weights, a POT reformulation of the source–target matching problem, and a submodular analysis that restores approximation guarantees (Chanda et al., 13 Apr 2026).

The paper’s practical message is that subset selection should not merely identify “important” examples and then allow implicit weights to absorb imbalance. Instead, if the goal is a fair or balanced summary of a target distribution, the selected representatives themselves should contribute uniformly. UniPROT turns that principle into an explicit optimization objective and then into a tractable greedy algorithm with theoretical guarantees (Chanda et al., 13 Apr 2026).

A recurrent misconception would be to treat the method as a generic OT selector differing from k-medoids only in distance geometry. The paper argues that the decisive difference is the enforced equality of prototype contributions. In that sense, UniPROT is better understood as a uniform-measure selection framework than as a minor OT variant. Its theoretical contribution is the observation that this uniformity requirement initially yields a super-additive objective, and that partial optimal transport provides a principled route from that difficult formulation to a submodular one.

The broader implication is that imbalance-aware subset selection can be reframed as a measure-design problem. UniPROT’s answer is that the chosen subset should define a uniformly weighted source distribution, and that enforcing this constraint can improve minority representation across imbalanced classification, fine-tuning, and pretraining without compromising performance on dominant modes (Chanda et al., 13 Apr 2026).