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Splitting Methods for Convex Clustering
Eric C. Chi, Kenneth Lange
TL;DR
Clustering includes the vexing issue of determining the number of clusters. The paper introduces two splitting algorithms, and analysis and simulations indicate that accelerated AMA is preferred over ADMM.
Problem
Determining the number of clusters is a vexing issue in clustering.
Method
The paper introduces two splitting algorithms for convex clustering and uses a splitting perspective that permits centroid penalties with an arbitrary norm.
Results
Complexity analysis and simulations show that accelerated AMA appears to be the algorithm of choice, while ADMM does not fare as well.
Takeaways & Limitations
The splitting perspective provides a framework that can accommodate arbitrary-norm centroid penalties and potentially improve solutions when few features drive clustering.
Takeaways & Limitations
Both algorithms deserve further investigation, including whether alternative ADMM variants can close the observed gap.
Abstract
from arXiv · showhide
Clustering is a fundamental problem in many scientific applications. Standard methods such as $k$-means, Gaussian mixture models, and hierarchical clustering, however, are beset by local minima, which are sometimes drastically suboptimal. Recently introduced convex relaxations of $k$-means and hierarchical clustering shrink cluster centroids toward one another and ensure a unique global minimizer. In this work we present two splitting methods for solving the convex clustering problem. The first is an instance of the alternating direction method of multipliers (ADMM); the second is an instance of the alternating minimization algorithm (AMA). In contrast to previously considered algorithms, our ADMM and AMA formulations provide simple and unified frameworks for solving the convex clustering problem under the previously studied norms and open the door to potentially novel norms. We demonstrate the performance of our algorithm on both simulated and real data examples. While the differences between the two algorithms appear to be minor on the surface, complexity analysis and numerical experiments show AMA to be significantly more efficient.
1 Introduction
Convex clustering replaces difficult, locally trapped clustering procedures with a globally optimized convex formulation whose solution path spans cluster counts. The paper introduces unified ADMM and AMA splitting frameworks that support arbitrary norms, with AMA shown to be more efficient.
- Methods: The paper introduces two variable-splitting frameworks—ADMM and AMA—for minimizing the convex clustering objective under an arbitrary norm.Both methods rely on computing the proximal map of the norm, so they apply when that map can be computed efficiently.
- Convex clustering: The convex relaxation shrinks centroids together as γ increases, producing a continuous, globally optimal path that can reduce the need to preselect the number of clusters.At γ = 0 each point forms its own cluster, while sufficiently large γ coalesces all points.
- Theory and complexity: The algorithms are theoretically characterized, including convergence to the unique global minimizer and complexity scaling with graph connectivity.The analysis also identifies an AMA step size linked to the graph Laplacian.
- Motivation: Standard clustering methods can be NP-hard, become trapped in suboptimal local minima, or require the number of clusters to be determined separately.The convex formulation addresses these challenges through a continuous clustering path.
- Comparison: Complexity analysis and simulations indicate that AMA is more efficient than ADMM, although both methods provide generic splitting formulations.The frameworks extend beyond the previously studied norms and support potentially novel norms.
- Scalability: K-nearest-neighbor weights make storage and computation requirements grow linearly with problem size while enhancing solution quality.The paper also provides guidance for choosing weights wij.
2 Properties of the solution path
The convex clustering solution path exists continuously in the regularization parameter and weights, reaches a common mean for sufficiently large γ on connected graphs, and is not generally agglomerative.
- Path continuity: The solution path U(γ) exists and depends continuously on both γ and the weight matrix w.This continuity motivates warm starts when solving problems over a grid of γ values.
- Large-γ behavior: For a connected graph, sufficiently large γ makes every centroid equal the overall sample average.Thus the path eventually ends in a single common cluster.
- Path structure: Clustering paths are not guaranteed to be agglomerative because centroids may fuse and later unfuse as γ increases.The algorithms allow such fission events to preserve the true solution path and global minimization.
3 Algorithms to Compute the Clustering Path
The paper reformulates convex clustering with variable splitting and develops ADMM and AMA updates based on linear systems and proximal maps. The formulations support common norms while simplifying the penalty terms.
- Variable splitting: ADMM and AMA split each centroid difference into an auxiliary variable constrained to equal the difference between paired centroids.The edge set contains centroid pairs with nonzero weights.
- ADMM: ADMM minimizes the augmented Lagrangian one variable block at a time, producing linear-transform and proximal-map updates.The multiplier update uses the constraint residual.
- AMA: AMA updates the first block using the ordinary Lagrangian, corresponding to setting the augmentation parameter ν to 0.AMA shares the V and Λ updates with ADMM and separates the minimization across each u_i.
- ADMM updates: For fully connected weights, ADMM can compute the U update analytically by expressing its matrix as diagonal plus rank one and applying Sherman-Morrison.This explicit update increases the number of auxiliary and multiplier variables, creating a storage trade-off.
- AMA updates: AMA tracks only one dual variable per nonzero weight, so sparse weights keep the variables introduced by splitting from becoming prohibitive.The paper also reports that sparse weights often produce better-quality clusterings.
4 Convergence
Both algorithms converge under stated conditions, but ADMM has broader guarantees while AMA requires a bounded step parameter. ADMM iterates converge to the unique global minimizer of the convex clustering criterion.
- General convergence: ADMM converges for any ν > 0, whereas AMA convergence is guaranteed only when ν is not too large.For AMA, the sufficient condition is ν < 2α/ρ(A^tA).
- AMA condition: For convex clustering, the strong convexity modulus in the AMA condition is α = 1.The condition depends on the largest eigenvalue of A^tA.
- AMA condition: The matrix A^tA equals L⊗I, so ρ(A^tA) equals the largest eigenvalue of the associated graph Laplacian.The Laplacian spectral radius is bounded by n, with equality for a fully connected positive-weight graph.
- Practical step sizes: A fixed step size ν < 2/n works in practice for dense graphs with fewer than 1000 data points.Sparse bounded-degree graphs admit a sharper bound, and the relevant quantities can be computed quickly in O(n + ε) operations.
- ADMM guarantee: The ADMM iterates converge to the unique global minimizer U* of the convex clustering criterion Fγ(U).Uniqueness follows from the properties established for Fγ(U).
5 Acceleration
Both AMA and ADMM can be accelerated at little additional computational cost. In particular, AMA can use Nesterov acceleration because it is a proximal gradient algorithm.
- AMA acceleration: AMA is a proximal gradient algorithm and can be accelerated using Nesterov’s method.The paper presents an accelerated AMA algorithm.
- Comparison: Both algorithms therefore admit acceleration without substantial additional computational cost.This statement is the paper’s general motivation for considering accelerated variants.
- ADMM acceleration: Acceleration methods also exist for ADMM, although the paper does not consider those methods in detail.The cited ADMM acceleration methods are presented as related work.
6 Computational Complexity
AMA has lower complexity on sparse graphs, while explicit ADMM updates retain quadratic dependence on the number of data points. Restricting connectivity to k nearest neighbors makes AMA’s storage and operation counts linear in n.
- AMA complexity: A single AMA iteration costs O(εp) for ℓ1 and ℓ2 norms and O(εp log p) for the ℓ∞ norm.These costs include duality-gap checking under the assumption n = O(ε).
- AMA complexity: The best known convergence bounds for AMA and its accelerated variant are sublinear because f*(A^tλ) is not strongly convex.The relevant matrix A^t is typically short and fat.
- Sparse graphs: Limiting each node to its k nearest neighbors makes ε = O(kn), yielding linear operation counts and storage O(knp).Without this restriction, the worst-case complexity and storage are quadratic in n.
- ADMM complexity: ADMM storage or computational costs remain quadratic regardless of how the edge set is limited.Its storage requirements in the fully connected case are O(pn^2 + np).
7 Practical Implementation
This section covers weight selection, clustering-path interpretation, and implementation details for recovering assignments. Sparse, locally structured weights can improve clustering quality and computational efficiency, while assignments can be extracted in O(n) operations.
- Weight selection: Weight choice can dramatically affect the quality of the clustering path.
- Weight selection: Sparse nearest-neighbor weights improve both computational efficiency and clustering quality.
- Weight selection: Combining neighborhood indicators with Gaussian kernels increases the clustering path’s sensitivity to local data density.
- Making cluster assignments: Both ADMM and AMA recover clustering assignments in O(n) operations using the differences variable V.
- Making cluster assignments: After obtaining V, breadth-first search finds connected components whose edges join point pairs with zero differences; each component is a cluster.
8 Numerical Experiments
The experiments examine how weight construction affects clustering paths and compare runtime across algorithms on synthetic and real data. Locally density-sensitive sparse weights generally improve clustering quality and speed, while AMA is especially efficient at larger problem sizes.
- Experimental design: The experiments evaluate weight choices on one synthetic and three real data sets, plus runtime comparisons with Hocking et al.’s subgradient algorithm.
- Weight effects: Limiting positive weights to nearest neighbors and using Gaussian kernels improves clustering quality and computational efficiency.
- Qualitative comparisons: For the halfmoons data, k = 10 with φ = 0.5 produces the best clustering path, whereas too many neighbors can misassign difficult points.
- Qualitative comparisons: Locally density-sensitive weights yield more sensible paths for Iris, including some separation between overlapping virginica and versicolor species.
- Qualitative comparisons: For Senate voting, density-sensitive weights produce better clustering with clear party separation and identify an outlying Democrat.
- Qualitative comparisons: For mammal dentition, density-sensitive weights give superior results and can differ sensibly from clustering based on the first two PCA components.
- Timing comparisons: All three algorithms scale quadratically under dense weights, while sparse weights shorten runtimes and make ADMM competitive for small to moderate n.
- Timing comparisons: For 500 points, AMA averages 7 seconds versus 6 to 7 minutes for the subgradient and ADMM methods.
9 Conclusion & Future Work
The paper presents ADMM and AMA splitting algorithms for convex clustering, with analyses of computational complexity and convergence. The conclusion highlights AMA’s efficiency, flexible extensions, and several open questions about clustering paths, sparsity, and optimality.
- The paper introduces two splitting algorithms, ADMM and AMA, for solving the convex clustering problem.
- The splitting perspective supports path following and permits centroid penalties based on arbitrary norms.
- The algorithms’ projection operators enable analysis of their computational complexity and convergence properties.
- AMA is favored by the paper’s complexity analysis and simulations, although alternative ADMM variants may reduce the observed performance gap.
- Future work includes parallel updates, active-set strategies, structured centroid sparsity, and determining when fusion events remain permanent as γ increases.
- Sparse weight graphs can improve cluster quality and computational efficiency, while worst-case AMA storage and computation depend quadratically on the number of graph edges.
- The analysis leaves optimal clustering largely unresolved, and sparse-centroid extensions introduce the additional challenge of choosing the penalization amount.