PASS: Ambiguity Guided Subsets for Scalable Classical and Quantum Constrained Clustering

TL;DR

PASS addresses scalable pairwise-constrained clustering by collapsing must-link components into pseudo-points and focusing optimization on a small, ambiguity- and violation-driven subset, preserving feasibility while enabling efficient SSE minimization. It introduces constraint-aware margin and information-geometric Fisher–Rao selectors to construct the working set $S$, with a budget that scales as $m = \min(\alpha|\mathcal{S}|, |V| + \beta k \log |\mathcal{S}|)$, and solves a reduced 0-1 ILP on $S$ (size $O(mG)$). The framework extends to quantum clustering through a reduced QUBO of size $|S|\times k$, enabling a quantum refinement (q-PCKMeans) that improves feasibility and scales to larger problems than full encodings would allow. Across diverse benchmarks, PASS matches or exceeds state-of-the-art SSE with substantially lower runtime, and its quantum extension demonstrates a viable path toward quantum-hybrid acceleration for dense constraint graphs and large datasets.

Abstract

Pairwise-constrained clustering augments unsupervised partitioning with side information by enforcing must-link (ML) and cannot-link (CL) constraints between specific samples, yielding labelings that respect known affinities and separations. However, ML and CL constraints add an extra layer of complexity to the clustering problem, with current methods struggling in data scalability, especially in niche applications like quantum or quantum-hybrid clustering. We propose PASS, a pairwise-constraints and ambiguity-driven subset selection framework that preserves ML and CL constraints satisfaction while allowing scalable, high-quality clustering solution. PASS collapses ML constraints into pseudo-points and offers two selectors: a constraint-aware margin rule that collects near-boundary points and all detected CL violations, and an information-geometric rule that scores points via a Fisher-Rao distance derived from soft assignment posteriors, then selects the highest-information subset under a simple budget. Across diverse benchmarks, PASS attains competitive SSE at substantially lower cost than exact or penalty-based methods, and remains effective in regimes where prior approaches fail.

Figures (7)

• Figure 1: Subset selection with $k = 3$. Left: original data with ambiguous points in red and cannot-links as dashed red edges. Right: induced subproblem on $S$ with the constraint graph.
• Figure 2: Ambiguity-based subset selection using Fisher-Rao geometry. (a) Fisher-Rao ambiguity scores $J_i$ for each data point (red = ambiguous, blue = certain). Centroids (black squares) and decision boundaries (gray contours) illustrate the clustering structure. Cannot-link constraints are represented by solid black lines, with a constraint violation shown as a dashed black line. (b) Selected subset $S^*$ ($|S^*|=6$, $|V|=1$) including the constraint violation and the most ambiguous points.
• Figure 3: Total runtime vs. $n$ (log--log) shows near-linear scaling. Runtime grows $\sim$201$\times$ for a 1,633$\times$ dataset size increase.
• Figure 4: Maximum ILP binaries vs. $n$ (log--log). Across the range, the maximum binaries increase $\sim$884$\times$.
• Figure 5: One-hot preserving mixer for $k=3$: apply the three two-qubit blocks $e^{-i\beta(X_{i,g}X_{i,h}+Y_{i,g}Y_{i,h})}$ for the pairs $(g,h)=(1,2),(1,3),(2,3)$. No single-qubit $R_X$ rotations are used.

Theorems & Definitions (19)

• Definition 1: Ambiguity and Constraint Sets
• Definition 2: Reduced Working Subset
• Proposition 3: Feasibility Preservation
• proof
• Definition 4: Fisher--Rao Ambiguity Score
• Definition 5: Budgeted Information-Geometric Subset
• Proposition 6: Greedy Optimality under a Modular Objective
• Definition 7: Point-wise XY Mixer
• Lemma 8: Invariance of the One-Hot Subspace
• proof