Clustering and Pairing Strategy

• Clustering and Pairing Strategy is a framework that uses pairwise comparisons to dynamically partition data, improving scalability and accuracy.
• It integrates methods such as binary classification, constraint-based clustering, and probabilistic models to refine cluster assignments.
• These strategies are applied in diverse areas like unsupervised learning, federated optimization, and network analysis to drive robust performance.

Clustering and Pairing Strategy

Clustering and pairing strategies span a spectrum of methodological innovations that utilize pairwise relationships—whether observed, inferred, or externally imposed—to guide the partitioning of data. These frameworks play essential roles in unsupervised and semi-supervised learning, network analysis, constraint-based learning, federated optimization, large-scale evaluation, and beyond. The technical landscape detailed below synthesizes core approaches and recent developments from modern arXiv literature, unified by a rigorous focus on pairwise abstractions and algorithmic realizations.

1. Reformulation of Clustering as Pairwise Binary Classification

A key direction exemplified by "Learn to Cluster Faces via Pairwise Classification" (Liu et al., 2022) foregoes traditional global affinity graphs in favor of a paradigm treating clustering as a binary classification problem over pairs. Given a dataset of deep embeddings $$F = [\mathbf f_1,\ldots,\mathbf f_N] \in \mathbb R^{N \times D}$$, the cluster assignment is implicitly modeled by a learned classifier $C(\mathbf f_i, \mathbf f_j) \in \{0,1\}$ that directly predicts whether two points belong to the same cluster. The classifier is trained via balanced positive/negative sampling and binary cross-entropy minimization: $$\mathcal L = -\sum_{(i,j)}\left[\delta_{ij} \log p_{ij} + (1-\delta_{ij}) \log (1-p_{ij}) \right],$$ where $p_{ij} = C(\mathbf f_i, \mathbf f_j)$ and $\delta_{ij} = [y_i = y_j]$.

To sidestep $O(N^2)$ pairwise computation at test time, a rank-weighted density criterion selects only $O(N)$ "high-value" pairs. For each point $x_i$, its $k$-nearest neighbors $\{x_{ij}\}$ are used to compute a local density

$$d'_i = \sum_{j=1}^k f(j) s_{ij}, \quad f(j) = (k-j)^p,$$

where $s_{ij}$ is the similarity and $p>0$ is a hyperparameter. Only pairs connecting each $x_i$ to its nearest neighbor with a higher density, per $d'_{i^*} > d'_i$, are considered for classification and subsequent merging via connected components. Empirically, this approach yields state-of-the-art pairwise and BCubed $F$-scores at substantially reduced runtime and memory compared to graph-based baselines (Liu et al., 2022).

2. Pairwise Constraints and General Constraint-Based Clustering

Stochastic, soft, and hard pairwise constraints have become integral tools for encoding domain knowledge and fairness into clustering objectives. The framework in (Brubach et al., 2021) encapsulates these as Stochastic Pairwise Constraints (SPC): for sets $P_q$ of pairs, each comes with a constraint to ensure that no more than a $\psi_q$-fraction are separated by the randomized clustering solution $\mathcal D$: $$\sum_{e \in P_q} z_e \leq \psi_q |P_q|, \quad z_e = \Pr_{\phi \sim \mathcal D}\left[\phi(j) \neq \phi(j')\right].$$ This abstraction unifies traditional must-link, cannot-link, and probabilistic/fairness constraints, and can be imposed on $k$-center, median, or means formulations. The generic algorithmic template involves: (1) standard (or approximate) clustering on the unconstrained objective to determine centers, (2) an LP assignment step incorporating SPC, and (3) randomized rounding to achieve bounded violations and radius/cost guarantees. For pure must-link constraints, improved $2$-approximation bounds are achievable (Brubach et al., 2021).

The PCCC algorithm (Baumann et al., 2022) advances this further by supporting mixed hard/soft must-link and cannot-link constraints, each with user-specified confidence weights $w_{ij}$. Through integer programming over assignment variables, iterative LM contraction, and q-nearest center reduction heuristics, it efficiently scales to tens of thousands of datapoints and millions of pairwise constraints. The objective encompasses both within-cluster compactness and pairwise penalty terms: $$\min_{x,y,z} \sum_{i=1}^n\sum_{l=1}^k d_{il} x_{il} + P\left(\sum_{\mathrm{SCL}} w_{ij} y_{ij} + \sum_{\mathrm{SML}} w_{ij} z_{ij}\right)$$ with tight constraint satisfaction or penalty-based relaxation as appropriate (Baumann et al., 2022).

3. Probabilistic, PAC-Bayesian, and Generative Pairwise Models

A probabilistic reinterpretation of clustering treats the labels (and constraints) as jointly distributed latent variables, providing formal generalization guarantees and optimization principles. In the generative approach (Yu et al., 2018), must-link and cannot-link pairs are incorporated directly into the joint likelihood of a mixture model: $$J(X, Z \mid M, C, \theta) = L(X, Z \mid \theta) \cdot S(X, Z, M \mid \theta) \cdot D(X, Z, C \mid \theta),$$ where $S$ and $D$ encode the enforcement or renormalized likelihood of pairwise constraints. The EM algorithm is adapted so that responsibilities account for both the observed data and the connectivity information, enabling globally consistent probabilistic updates without ad-hoc penalties (Yu et al., 2018).

From a generalization perspective, PAC-Bayesian analyses (Seldin, 2010, Hardoon et al., 2010) provide explicit trade-offs between the empirical fit and the information retained by learned cluster assignments, leading to risk bounds of the form: $$kl(\widehat L(\mathcal Q) \| L(\mathcal Q)) \leq \frac{|\mathcal{X}| \overline{I}(X;C) + |C| \ln|\mathcal{X}| + |C|^2 \ln|W| + \frac{1}{2} \ln(4N) - \ln\delta}{N},$$ where $\widehat{L}$ is empirical loss, $L$ generalization loss, and $\overline{I}(X;C)$ mutual information regularizer. Alternating projection and block-coordinate algorithms optimize the tradeoff, guiding hyperparameter ($|C|$, weighting) and sample complexity selection (Seldin, 2010).

For paired multi-view data, PWCA (Hardoon et al., 2010) extends the PAC-Bayes machinery—enforcing empirical consistency between paired views by minimizing the Hilbert norm subject to the constraint that the cluster predictions $\{F(X_i) = G(Y_i)\}$ agree for all samples. Solutions adopt kernel CCA-type algorithms, yielding paired latent representations with provable generalization bounds (Hardoon et al., 2010).

4. Pairing Strategies in Evaluation, Matching, and Graph Algorithms

Efficiently pairing clusters or data items is foundational not only to clustering procedures but to the evaluation of clustering quality at scale. The Stable Matching Based Pairing (SMBP) algorithm (Karbasian et al., 22 Sep 2024) leverages Gale–Shapley stable matching to find an approximately optimal one-to-one mapping between clusters of different clustering solutions, using the overlap matrix as weights. With a computational cost of $O(N^2)$ versus the $O(N^3)$ of maximum-weight matching (Hungarian), SMBP achieves near-optimal accuracy (within $1\%$) even for large numbers of clusters, supporting scalable external validation (Karbasian et al., 22 Sep 2024).

In agglomerative graph clustering, node pair sampling is used to induce a "distance" $d(a,b) = p(a)p(b)/p(a,b)$ between clusters, where $p(a,b)$ is the empirical edge sampling probability. This reducible, ratio-based criterion permits near-linear nearest-neighbor-chain agglomeration, yielding a regular dendrogram that represents multi-scale community structure without parameter tuning (Bonald et al., 2018). Hierarchical and pair-based sampling principles thus guide both the clustering process and its computational tractability.

5. Pairwise Strategies in Semi-Supervised, Federated, and Application-Specific Domains

Sparse pairwise measurements and constraints drive significant algorithmic design in both classical and modern settings. Techniques outlined in (Saade et al., 2016) demonstrate that, under information-theoretic limits, $O(n)$ random pairwise measurements (edges) suffice for partial recovery of planted clusters. Belief propagation, non-backtracking spectral methods, and Bethe Hessian algorithms attain these limits, further confirmed by robust empirical evaluations. The random pairing structure, edge-sparsity, and associated complexity analyses are central to their optimality (Saade et al., 2016).

In federated learning with non-i.i.d. client data, nonconvex pairwise fusion penalties (Yu et al., 2022) enable automatic clustering of devices by inducing fusion/splitting between local models. The optimization problem,

$$\min_{\omega_1,...,\omega_m} \sum_{i=1}^m f_i(\omega_i) + \lambda \sum_{i < j} p(\|\omega_i - \omega_j\|),$$

is solved by distributed ADMM with SCAD-type nonconvex thresholding. This allows the number and composition of clusters to be learned adaptively, with guarantees on convergence and statistical recovery (Yu et al., 2022).

In communication and resource allocation, pairing strategies such as those for UAV-assisted NOMA-VLC systems (Pham et al., 2020) employ channel metrics to form user pairs maximizing channel disparity—critical for NOMA sum-rate gains. Clustering-by-pairing is followed by joint placement and power optimization via Harris Hawk Optimization, with demonstrable superiority in throughput and fairness metrics over non-paired alternatives (Pham et al., 2020).

Pairing in financial applications, such as statistical arbitrage with graph-based multi-pair trading (Korniejczuk et al., 15 Jun 2024), uses cluster analysis on correlation networks to define within-cluster mean-reversion signals, then manages multi-instrument pairings and risk through machine learning classifiers and Kelly-optimal allocations. The clustering structure directly shapes the generation and risk control of pairwise trading signals (Korniejczuk et al., 15 Jun 2024).

6. Theoretical Guarantees, Complexity, and Empirical Performance

Clustering and pairing strategies are subject to rigorous statistical and computational analysis. Several works derive optimal or near-optimal sample complexity, bounds on generalization error, and computational complexity (time and space) for their proposed methods (Saade et al., 2016, Seldin, 2010, Yu et al., 2022, Baumann et al., 2022, Bonald et al., 2018, Brubach et al., 2021). For example, random-agglomerative (SACA) and regularized graph-based (RGCA) clustering algorithms (Pasteris et al., 2017) yield explicit $O((n^2/m)K\log(n^2/m))$ misclassification bounds in terms of sample and side-information size.

Practical performance is confirmed by extensive benchmarking: pairwise-classification clustering (Liu et al., 2022) scales to millions of samples with bounded memory and runtime; SMBP (Karbasian et al., 22 Sep 2024) provides sub-second large-scale matching with negligible accuracy loss; pairwise-constrained clustering (PCCC) (Baumann et al., 2022) achieves near-perfect ARI with a small number of constraints while maintaining competitive computational cost.

7. Synthesis and Domain-Specific Extensions

The pairwise angle—whether realized as classification, sampling, constraint integration, or matching—serves as a powerful abstraction in modern clustering. Approaches cast core challenges (scalability, accuracy, incorporation of prior knowledge, fairness, decentralization) into tractable subproblems with empirical and theoretical support. The flexibility and performance of pairwise strategies have enabled major advances in face identity clustering (Liu et al., 2022), semi-supervised learning (Brubach et al., 2021, Baumann et al., 2022, Yu et al., 2018), federated optimization (Yu et al., 2022), graph theory (Bonald et al., 2018, Saade et al., 2016), and domain applications from communication to finance (Pham et al., 2020, Korniejczuk et al., 15 Jun 2024). The diversity and breadth of these strategies affirm the foundational role of pairwise methods in clustering research.