Diversity-Aware Dataset Partitioning

• Diversity-aware dataset partitioning is the process of splitting datasets into subsets that optimize for fairness, homogeneity, or diversity by balancing key attributes.
• It leverages methodologies from clustering, submodular optimization, integer programming, and information theory to ensure subgroup robustness and improved generalization.
• Practical implementations use feature engineering, constrained sampling, and empirical benchmarking to measure diversity indices and optimize model performance.

Diversity-aware dataset partitioning refers to the structured decomposition of datasets into subsets that maximize intra-part subset homogeneity or diversity, balance key attributes, or preserve relevant statistical properties. Motivated by constraints in machine learning, data summarization, and fairness, these partitioning regimes serve objectives including generalization, bias mitigation, subgroup-level robustness, and downstream efficiency. Approaches synthesize algorithmic ideas from clustering, submodular optimization, integer programming, and information theory, often under domain-specific or cross-domain diversity indices. Methodologies are informed by theoretical analyses (e.g., scaling laws, geometric interpretations) as well as empirical benchmarking in vision, language, tabular, and biological contexts.

1. Diversity Metrics and Formal Problem Statements

Formulating diversity-aware partitioning requires the specification of metrics appropriate for the domain and downstream task.

Combinatorial Diversity.

In multi-attribute datasets, the Shannon entropy-based fairness index quantifies the representation of protected groups in a subset: $$D(S) = -\sum_{i=1}^p s_i \log s_i, \quad s_i = \frac{|S \cap X_i|}{|S|}$$ where $X_i$ is the set of items with sensitive attribute $i$ (Celis et al., 2016). This is maximized when $S$ is equitably balanced across all attributes.

Geometric Diversity.

Spread in feature space can be captured via the determinant of the kernel matrix of selected items: $G(S) = \det(K_{S,S})$ where $K(x, y)$ is typically the dot-product or RBF kernel of feature vectors (Celis et al., 2016).

Simpson's Index (Population Partitioning).

Simpson diversity for a count vector $x \in \mathbb{Z}_+^r$ (for $r$ types) is

$D(x) = \frac{\|x\|_1^2}{\|x\|_2^2}$

maximized when all types are equally represented (Perez-Salazar et al., 2021).

Information-Theoretic Measures.

Hill numbers, entropy, and similarity-sensitive diversity measures (Leinster–Cobbold) generalize classical indices, allowing partitioning over abundance matrices and pairwise similarity $Z$ (Reeve et al., 2014).

2. Partitioning Algorithms and Optimization Formulations

Diversity-aware partitioning algorithms fall into the following principal categories:

Clustering-Based Partitioning.

Prototype-based agglomerative clustering assigns samples to clusters by minimizing pairwise feature distance, adaptively splitting the dataset according to diversity ($\theta$ threshold) (Huang et al., 2023). Each subset is then assigned its own prompt/vector, reducing adaptation difficulty for transfer.

Constrained Sampling (P-DPP).

The $P$-DPP framework samples subsets subject to exact quotas for each protected attribute, maximizing geometric diversity: $$\max_{S \subseteq X} \det(K_{S,S}) \quad \text{s.t.} \quad |S \cap X_i| = k_i ~ \forall i$$ Sampling is performed via MCMC with combinatorial constraints, balancing fairness and diversity simultaneously (Celis et al., 2016).

Integer Linear Programming (ILP).

Partitioning into $k$ groups to maximize intra-group diversity can be modeled as a quadratic objective under cardinality and fairness constraints: $$\max_x \sum_{p=1}^k \sum_{i < j} d_{ij} x_{i,p} x_{j,p}$$ with $d_{ij}$ the pairwise skill dissimilarity and $x_{i,p}$ binary group assignments. Bounds are enforced on group sizes and protected subgroup prevalence (Jenkins et al., 2023).

Submodular Partitioning.

Greedy algorithms partition classes across workers under resource and cardinality constraints, maximizing submodular diversity scores per block (facility-location, graph-cut) and enforcing near-IID feature distributions for distributed learning (He et al., 2022).

Decision-Rule Guided Partitioning (DATE).

Rule-based partitioning identifies distribution-guiding rules (DGRs) via top-down decision tree splits, enforcing within-subset quality constraints and minimal pairwise overlap in rules (Tang et al., 26 Dec 2025). Diversity-quality tradeoff is resolved by Multi-Armed Bandit optimization.

Partitioning Theory and Algorithms for Population Diversity.

For $r$ types and $k$ subgroups, perfect diversity partitioning (Simpson index preserved) is feasible iff $k \leq \gcd(b)$, otherwise geometric optimization recovers maximal min-diversity splits for $r=2$ via extended Euclid and piecewise monotonic transformations (Perez-Salazar et al., 2021).

3. Practical Implementation and Empirical Evaluation

Implementation details depend on domain constraints and computational scale.

Feature Engineering.

Frozen backbone features (vision) (Huang et al., 2023), SIFT histograms (images) (Celis et al., 2016), Laplacian eigenmaps (skills) (Jenkins et al., 2023), and autoencoder embeddings (He et al., 2022) serve as input spaces for clustering and distance computation.

Algorithmic Complexity.

• P-DPP samplers are polynomial-time in constant $p$ (protected attributes), otherwise brute-force or MCMC is used.
• Greedy submodular partitioning runs in $O(n^2)$, significantly accelerated for high $L$ (classes).
• Decision-rule discovery (DATE) leverages priority queues, model-sharing, and bandit loops for efficient rule selection.

Empirical Metrics.

Experiments measure not only diversity indices ($D(S)$, $G(S)$, Simpson, Hill) but also ML outcomes (generalization error, subgroup risk, accuracy), convergence speed, and resource allocation efficacy (Huang et al., 2023, He et al., 2022, Rolf et al., 2021, Tang et al., 26 Dec 2025). Partition quality is functionally linked with downstream learning curves, fairness, and robustness.

Guidelines and Limitations.

Optimal partition numbers are dictated by diversity and size tradeoff: excessive environments lead to sparsity and diminished benefit, while too few reduces OOD generalization. Partition quality—measured via Rand index, overlap, min/max diversity—correlates with empirical gains (Teney et al., 2020, Tang et al., 26 Dec 2025). Sampling and partitioning strategies must be tailored to data modality and application constraints.

4. Theoretical Analyses and Design Considerations

Scaling Laws for Subgroup Performance.

Empirical and theoretical evidence supports power-law scaling of subgroup risk: $$R_g(h_p) \approx \sigma_g^2 (np_g)^{-\alpha} + \tau_g^2 n^{-\beta} + \delta_g$$ Optimal sample allocations derived by convex optimization yield closed-form solutions for balancing population vs. subgroup risk (Rolf et al., 2021).

Partition Geometry.

Simpson diversity admits geometric reinterpretations as quadratic cones: feasible partitions preserve the angle to the all-ones vector, leading to efficient algorithms for $r=2$ types and pseudo-polynomial strategies for higher dimension (Perez-Salazar et al., 2021).

Unified Diversity Partitioning.

Information-theoretic frameworks (Hill numbers, Leinster–Cobbold similarity diversities) support algebraic partitioning of subcommunity and metacommunity $\alpha$, $\beta$, $\gamma$ using power means and similarity matrices, generalizable across ecological, genetic, or functional data (Reeve et al., 2014).

5. Applications, Extensions, and Future Directions

Diversity-aware partitioning is applied across domains:

Transfer and Prompt Learning.

DAM-VP shows that fine-grained clustering and meta-prompt initialization jointly accelerate adaptation and boost performance when handling visually heterogeneous datasets under frozen backbone constraints (Huang et al., 2023).

Fair Summarization and Training.

Fair subsampling (P-DPP, constrained k-DPPs) delivers image sets with maximal attribute balance and geometric spread, crucial for unbiased summarization and downstream training set construction (Celis et al., 2016).

Distributed Learning.

Submodular partitioning ensures resource-balanced, near-IID splits for heterogeneous model training and achieves superior accuracy and convergence times over naive random partitions or class-balanced splits (He et al., 2022).

Tabular Data Generation.

DATE demonstrates that jointly optimizing partition rule distinctness and within-part quality, with Multi-Armed Bandit selection, produces fewer, higher-quality synthetic samples, improving model accuracy and reasoning ability on tabular benchmarks (Tang et al., 26 Dec 2025).

Ecological and Population Studies.

Frameworks for diversity partitioning enable ecological insights into subcommunity redundancy, distinctive reservoirs, and spatio-temporal diversity dynamics, with R-package support for rapid implementation (Reeve et al., 2014).

Limitations and Open Challenges.

Scalability to large numbers of protected attribute combinations, efficient discovery of environments (rather than externally imposed splits), theoretical mixing-time analysis for constrained samplers, and integration of richer multivariate or individual-fairness indices remain open questions. Extension to active learning, streaming data, and hierarchical selection mechanisms are active areas for future research (Tang et al., 26 Dec 2025, Perez-Salazar et al., 2021).

6. Summary Table: Diversity-Aware Partitioning Techniques

Approach	Diversity Metric	Optimization/Algorithm
DAM-VP (Huang et al., 2023)	Feature clustering, Euclidean/cosine	Diversity-adaptive clustering, meta-prompt bootstrapping
P-DPP (Celis et al., 2016)	Shannon entropy, determinant (kernel)	MCMC sampling, characteristic polynomial exact sampling
Constrained ILP (Jenkins et al., 2023)	Skill dissimilarity (embedding dist.)	Integer linear/quadratic programming, MILP solvers
Submodular (He et al., 2022)	Similarity, facility-location	Greedy submodular maximization (“balanced round robin”)
DATE (Tang et al., 26 Dec 2025)	Decision-rule overlap	Top-down decision tree splits, Multi-Armed Bandit selection
Scaling law (Rolf et al., 2021)	Subgroup risk/reduction	Convex allocation optimization, pilot sampling, risk modeling
Partition theory (Perez-Salazar et al., 2021, Reeve et al., 2014)	Simpson index, Hill numbers, similarity-sensitive	Geometric, information-theoretic, power mean aggregation

Each methodology reflects distinct trade-offs in partition granularity, diversity indices, scalability, and downstream effect, and should be chosen to match dataset heterogeneity and task-specific constraints.