Diversity Maximization (CABS-DM)

• Diversity Maximization (CABS-DM) is a framework that selects a diverse subset of data points using algorithmic techniques under various complex constraints.
• It leverages scalable methods like coresets, streaming algorithms, and convex programming to achieve formal approximation guarantees in combinatorial settings.
• Applications in multimodal pretraining, clustering, and recommendation systems demonstrate its ability to boost performance and ensure fair, representative data selection.

Diversity Maximization (CABS-DM) refers to algorithmic frameworks and methodologies for selecting subsets of items (data points, samples, assignments) that maximize an explicit diversity objective under possibly complex constraints. CABS-DM approaches are of central importance in multimodal data curation, fair coverage, clustering, recommendation, search, and robust learning, and are characterized by the joint emphasis on (1) computational scalability (often via coresets or streaming/distributed routines), (2) formal approximation guarantees, and (3) compatibility with matroid or other combinatorial constraints. The term "CABS-DM" is explicitly used in the context of concept-aware batch sampling for multimodal pretraining (Ghosh et al., 25 Nov 2025), but the design patterns and theoretical techniques summarized under this label appear broadly across literature on diversity maximization under resource, fairness, or independence constraints.

1. Formal Definition and General Framework

Diversity maximization, also referred to as diversified subset selection or max-sum dispersion, considers a ground set $S$ (of $n$ points in a metric or pseudometric space $(S,d)$), an integer $k\leq n$, and potentially a matroid $\mathcal{M} = (S, \mathcal{I})$ encoding independence constraints (e.g., partition, transversal, uniform, or general matroid constraints). The goal is the selection of a feasible $k$-subset $I \in \mathcal{I}$ maximizing a diversity function $D(I)$. Canonical diversity measures include:

• Sum-diversity: $$D_{\mathrm{sum}}(I) = \sum_{1 \leq p < q \leq k} d(i_p, i_q)$$.
• Star-diversity: $$D_{\mathrm{star}}(I) = \max_{c \in I} \sum_{u \in I \setminus \{c\}} d(c, u)$$.
• Tree-, cycle-, bipartition-diversity: Objective defined as the weight of the minimum spanning tree, Hamiltonian cycle, or bipartition cut over $I$ (Ceccarello et al., 2020).
• Remote-clique and other power-diversities: $$f_{\mathrm{clique}}^q(I) = \sum_{1 \leq i < j \leq k} d(i, j)^q$$ for $q \geq 1$ (Cevallos et al., 2018).

The CABS-DM paradigm also extends to streaming, dynamic, and large-scale settings, and includes batch construction for downstream multimodal models (Ghosh et al., 25 Nov 2025). Throughout, the maximization problem is typically NP-hard, warranting scalable approximation strategies.

2. Coreset-Based Approaches and Approximation Schemes

State-of-the-art CABS-DM algorithms rely on coreset constructions: small, representative subsets $T \subseteq S$ with the property that the optimal solution on $T$ approximates the global optimum on $S$ within a specified ratio. In metric spaces of bounded doubling dimension $D$, coreset-based methods achieve approximation schemes for a variety of diversity functions and constraint types (Ceccarello et al., 2020, Ceccarello et al., 2016, Pellizzoni et al., 2023):

• For partition or transversal matroids, if $\epsilon>0$ is a quality parameter, a coreset of size $O(k (k/\epsilon)^D)$ suffices to obtain a $(1-\epsilon)$-approximate solution.
• The generic workflow is as follows:
  1. Partition $S$ into clusters with radius $r \leq (\epsilon/4)\rho_{S,k}$, where $\rho_{S,k}$ is the average farness in the optimal solution.
  2. In each cluster, extract a maximal independent set and aggregate these to obtain $T$.
  3. Solve the diversity maximization problem on $T$. For sum-diversity, a local search achieves a $1/2$-approximation; for other measures, enumeration or exact routines are used (Ceccarello et al., 2020).

In dynamic and distributed models, cover tree data structures or composable coreset constructions further reduce update and query times, supporting streaming and MapReduce implementations (Pellizzoni et al., 2023).

3. Greedy and Convex Programming Algorithms

For matroid-constrained diversity maximization with sum-diversity and negative-type distances, convex programming relaxations provide polynomial-time approximation schemes (PTAS) (Cevallos et al., 2015):

• The max-sum dispersion problem under matroid constraint admits a slice-concave convex quadratic relaxation. Deterministic pairwise-exchange rounding yields an integral basis with objective value at least $(1 - O(\ln k / k))$ times the LP optimum, matching the best possible unless $\text{P} = \text{NP}$.

• For broader classes of constraints and objectives (including monotone submodular utilities and compositional constructs), greedy or pairwise algorithms achieve constant-factor approximations or bicriteria guarantees (Zhang et al., 2020).

In scenarios where greedy element-wise selection fails, pairwise or matching-based selection (e.g., in clustered or partitioned data) yields robust, scalable $O(1)$-approximations even when clusters have heavy overlap (Zhang et al., 2020).

4. Concept-Aware Batch Sampling (CABS-DM) in Multimodal Pretraining

The explicit CABS-DM methodology is central in concept-aware online batch construction for large-scale vision-language pretraining (Ghosh et al., 25 Nov 2025):

• Each super-batch of $B$ examples is filtered to a sub-batch of size $b = (1-f)B$ by greedily selecting samples to maximize per-batch concept coverage.
• Let $\mathcal{C}_i \subseteq \mathcal{V}$ denote the set of annotated concepts for sample $i$, and $\mathcal{F}_c$ the frequency of concept $c$ in the super-batch. Define per-concept targets $t_c$ (typically $b/|\mathcal{V}|$ per $c$).
• The sample selection priority ("diversity gain") is

$$h_{\mathrm{DM}}(i) = \frac{1}{|\mathcal{C}_i|} \sum_{c \in \mathcal{C}_i} \begin{cases} \frac{t_c - n_c}{t_c} + \frac{1}{\mathcal{F}_c} & n_c < t_c \ 0 & \text{otherwise} \end{cases}$$

where $n_c$ is the running count of $c$ in the current sub-batch.

• Empirical results show CABS-DM batch curation yields up to $+7\%$ zero-shot accuracy improvement on long-tailed and rare-class benchmarks relative to uniform IID or offline concept-balancing strategies.
• Implementation is efficient ($\mathcal{O}(B \log B)$ per batch) and compatible with major multimodal model architectures such as CLIP and SigLIP.

5. Theoretical Properties and Complexity

CABS-DM methodologies leverage both geometric and combinatorial properties:

• In metric spaces of doubling dimension $D$, approximation schemes exploit net constructions, strong packing bounds, and cluster-induced proxies to bound the diversity loss due to coresetting (Cevallos et al., 2018, Ceccarello et al., 2020).
• For negative-type distances, slice-concavity and strong union inequalities support deterministic rounding without incurring prohibitive integrality gaps (Cevallos et al., 2015).
• In matroidal or partition-constrained problems, bounds on independence systems, exchange graphs, and submodular monotonicity yield provable $O(1)$-approximation ratios in greedy and batch-construction strategies (Zhang et al., 2020, Aslay et al., 2018).

Several measures remain hard even for restricted inputs: maximizing remote-clique with squared distances ($q=2$) in $\mathbb{R}^3$ is NP-hard (Cevallos et al., 2018); the quadratic-knapsack instance for maximizing diversity of exposure admits no efficient constant-approximation unless P=NP (Matakos et al., 2018).

6. Empirical Evaluation and Applications

Empirical studies across real-world and synthetic datasets confirm the effectiveness of CABS-DM, especially in high-dimensional and large $n$ regimes (Ceccarello et al., 2020, Ghosh et al., 25 Nov 2025):

• Streaming, MapReduce, and online batch-sampling strategies scale to millions of samples with minimal quality loss (typically $<1-2\%$ away from global optima, or at most a known constant-factor) (Ceccarello et al., 2016, Pellizzoni et al., 2023).
• CABS-DM batch selection substantially improves vision-LLM generalization at fixed data budgets, especially on classification tasks where uniform sampling under-represents minority concepts (Ghosh et al., 25 Nov 2025).
• In clustering, recommender systems, and network exposure settings, CABS-DM algorithms outperform local search and element-wise greedy baselines, particularly under strong resource and combinatorial constraints (Zhang et al., 2020, Aslay et al., 2018, Matakos et al., 2018).

7. Principal Limitations and Open Questions

CABS-DM’s practical impact is substantial, but several open directions persist:

• The gap between practical coreset size and provably minimal sufficiency, especially in non-doubling metrics or very high dimensional settings, remains.
• For diversity maximization functions with strong combinatorial dependencies (e.g., quadratic submodular penalties; exposure in social graphs), worst-case hardness precludes general PTAS, but instance-specific upper bounds or SDP relaxations yield high-quality approximations (Matakos et al., 2018).
• Trade-offs between batch diversity and affinity (distributional matching) or downstream task-specific calibration are not fully resolved (Liu et al., 2021, Ghosh et al., 25 Nov 2025).
• Adaptive and dynamic algorithmic techniques for online data curation and federated settings continue to be active research areas.

In summary, CABS-DM—understood as a collection of scalable, provably robust algorithms for diversity maximization under resource and combinatorial constraints—provides a unifying framework with rigorous guarantees and demonstrated empirical effectiveness in large-scale data selection and machine learning contexts (Ghosh et al., 25 Nov 2025, Ceccarello et al., 2020, Cevallos et al., 2015, Pellizzoni et al., 2023, Cevallos et al., 2018, Zhang et al., 2020).