Correlation Clustering Optimization

• Correlation clustering optimization is a technique that partitions data by balancing pairwise similarity and dissimilarity without predefining the number of clusters.
• It leverages a probabilistic foundation and formulations akin to Potts energy models to align with discrete, non-submodular optimization challenges.
• Scalable algorithms like Expand-and-Explore and Swap-and-Explore enable efficient, automatic cluster number selection, proving effective in vision and high-dimensional applications.

Correlation clustering optimization addresses the problem of partitioning elements, given pairwise similarity (positive affinity) and dissimilarity (negative affinity) scores, so as to maximize agreement within clusters and minimize disagreement across clusters—without specifying the number of clusters a priori. The field has evolved from small-scale, theoretically motivated formulations to large-scale, practical optimization schemes leveraging probabilistic models, connections to discrete energy minimization, and efficient large-scale algorithms. Recent progress includes scalable discrete move-making methods, provably justified objective selection, and application to vision and high-dimensional data.

1. Formalization and Probabilistic Foundations

Correlation clustering is formulated in terms of an affinity matrix $W \in \mathbb{R}^{n \times n}$, accommodating both positive (attraction) and negative (repulsion) entries. Given $U \in \{0,1\}^{n \times k}$, where $U_{ic} = 1$ indicates assignment of item $i$ to cluster $c$, and $\sum_c U_{ic} = 1$ for all $i$, the canonical correlation clustering energy is:

$$\mathcal{E}_{CC}(U) = -\sum_{i,j} W_{ij} \sum_c U_{ic} U_{jc}$$

This energy quantifies the sum of the affinities between points within each cluster; minimizing it balances positive and negative contributions.

A key theoretical insight is the generative probabilistic interpretation: if each pairwise similarity $s_{ij}$ is drawn from $f^+$ when $i$ and $j$ share a cluster, and from $f^-$ otherwise, then setting

$$W_{ij} = \log \left( \frac{f^+(s_{ij})}{f^-(s_{ij})} \right)$$

yields, up to an additive constant, the negative log posterior of the partition $U$ under a uniform prior. The functional intrinsically performs model selection: the induced prior on the number of clusters $k$ discourages degenerate solutions (e.g., $k=1$ or $k=n$) due to the combinatorics encoded in the uniform partition prior (via Stirling numbers of the second kind) (Bagon et al., 2011).

2. Connection to Potts Energy and Conditional Random Fields

Rewriting the clustering assignment in terms of label vectors $L \in \{1,2,\ldots\}^n$, with $l_i$ marking the cluster label of node $i$, the CC energy can be cast as:

$$\mathcal{E}_{CC}(L) = \sum_{i,j} W_{ij} \mathbb{I}[l_i \neq l_j]$$

This is structurally identical to the discrete pairwise Potts (CRF) model, but with essential challenges:

• The energy is generally non-submodular, rendering its global minimization NP-hard.
• There are no unary potentials to guide optimization.
• The number of labels $k$ is not fixed and must be determined as part of optimization.

Recognizing these correspondences enables the adaptation of advanced move-making techniques from graphical models (Bagon et al., 2011).

3. Large-Scale Optimization Algorithms

Addressing optimization in this context requires algorithms resilient to non-submodularity, adaptive label set (k), and high dimensionality. The principal contributions are as follows:

Expand-and-Explore: Extends α-expansion moves to non-submodular CC energies by permitting expansion not only to existing labels (clusters) but also a "new" empty label, dynamically proposing the creation of new clusters. Subproblems in each expansion move are solved with QPBOI, capable of providing optimal solutions (possibly partial) even in non-submodular cases.

Swap-and-Explore: Generalizes αβ-swap to perform label pair (α, β) optimization, including one "new" label per iteration. This allows the solution to adaptively discover a nontrivial number of clusters.

Adaptive-label ICM: Greedily reassigns each variable to the cluster that yields maximal attraction, or if no existing cluster is sufficiently attractive, forms a singleton. This ICM variant enables rapid large-scale inference, especially when the affinity matrix is dense.

All methods optimize directly over the $n$-dimensional label vector, avoiding the memory bottleneck of explicit $n \times n$ adjacency matrix optimization as in earlier methods.

Scalability: These algorithms have been demonstrated on problems with $>100$K variables, which are infeasible for prior convex relaxation or branch-and-cut approaches (Bagon et al., 2011).

4. Model Selection and Automatic Determination of Cluster Number

An essential property of the CC functional, emerging from its generative and combinatorial underpinnings, is its ability to select the number of clusters $k$ automatically during optimization—without explicit regularization or external penalties. The optimization process penalizes both over-fragmentation and trivial solutions. The introduced "explore" steps (i.e., adding a new empty label in move-making iterations) operationalize this capability, yielding recovery of the correct $k$ on synthetic and real data without manual intervention (Bagon et al., 2011).

5. Applications to Computer Vision and Pattern Recognition

Two distinct vision applications exemplify the practical impact of large-scale CC optimization:

Interactive Multi-Object Segmentation: A user provides rough (boundary) scribbles; these define both strong positive and negative affinities at pixel-level granularity. Exploiting the algorithms' scalability, the method segments images into multiple objects, automatically determining the number of regions. Affinity matrices at the 100K$\times$100K scale are handled. No explicit input of $k$ is required.

Unsupervised Face Identification: Given images to be grouped by identity (unknown $k$), a similarity score is learned (e.g., using a Mahalanobis distance and a sigmoid mapping). The affinities are translated into $W_{ij}$ values as above, and the CC solver (e.g., Swap-and-Explore or ICM) both determines $k$ and produces clusters with high purity. On the PUT face dataset, the method outperforms spectral clustering (using the spectral gap) and connected component baselines in both purity and correct $k$ recovery (Bagon et al., 2011).

6. Performance and Comparative Evaluation

Empirical evaluation demonstrates that, on both synthetic and vision datasets:

• Swap-and-Explore and Expand-and-Explore yield lower (better) energy solutions and more accurate $k$ recovery compared to TRW-S and BP (which require fixed $k$).
• Adaptive-label ICM is highly efficient and accurate when $W$ is dense.
• On co-segmentation tasks, the proposed methods match or outperform advanced convex relaxation methods, with the added advantage of scaling to very large graphs.
• The algorithms achieve lower or comparable clustering energy, correct automatic model selection, and typically superior or comparable clustering purity.

The main trade-off among the methods concerns speed versus robustness to affinity matrix sparsity, with ICM excelling on dense problems and move-making methods being more versatile (Bagon et al., 2011).

7. Limitations, Algorithmic Choices, and Future Directions

While the proposed framework overcomes limitations of earlier LP/IP and spectral approaches for large-scale, unconstrained-$k$ clustering, several intrinsic challenges remain:

• Non-submodular optimization barriers limit the guarantee of global optimality.
• Expand/Swap strategies rely on effective QPBOI or related solvers, which can be sensitive to worst-case structure.
• Sparse affinity structures may render move-making algorithms less efficient.

Ongoing research may focus on tighter integration of spectral, convex, or continuous relaxations for initializations in very large or highly unbalanced scenarios. Extensions to structured data (e.g., hierarchical, time-series, or manifold clustering) and hybrid convex-discrete optimization are promising directions.

---

In conclusion, the optimization of correlation clustering via discrete, large-scale algorithms grounded in probabilistic generative models not only enables principled model-selection and recovery of the cluster number but also scales to challenging practical settings, with demonstrated efficacy on classic vision tasks and pattern recognition benchmarks (Bagon et al., 2011).