Multi-Domain Clustering via Measure Quantization

Rafael Pereira Eufrazio; Eduardo Fernandes Montesuma; Charles Casimiro Cavalcante

cs.LGSep 18, 2026

Multi-Domain Clustering via Measure Quantization

Authors: Rafael Pereira Eufrazio, Eduardo Fernandes Montesuma, Charles Casimiro Cavalcante

Organizations: Instituto Federal de Educac¸˜ao, Ciˆencia e Tecnologia do Cear´a, Canind´e-CE, Brazil · Federal University of Ceara, Fortaleza-CE, Brazil · Sigma Nova Science, Paris, France

Abstract

Clustering is a fundamental task in data analysis, typically addressed through centroid-based methods such as K-means. In this work, we present a general framework for multi-domain clustering via measure quantization: given samples from multiple domains, we learn a shared set of cluster prototypes by minimizing a probability metric, such as the Sinkhorn divergence or the Maximum Mean Discrepancy, between each domain's probability measure and the measure of prototypes. Data points are then assigned to clusters either via nearest centroid, or via optimal transport, a collaborative strategy that couples all samples within a domain. A mini-batch optimization strategy makes both fitting and assignment scalable, reducing memory and computational cost while preserving clustering performance. Experimental results on 5 multi-domain benchmarks spanning image, audio and sensor data show that our Sinkhorn-based method consistently outperforms classical and multi-domain clustering baselines, and that this advantage persists when scaling to hundreds of thousands of samples.

Figures & tables

Figure 1: Overview of multi-domain clustering via measure quantization. (a) Toy example with

K{=}3

domains and the fitted Sinkhorn prototypes

\nu^{\star}

. (b)-(c) Loss landscape

z_{c}\mapsto\frac{1}{K}\sum_{k}\mathbb{D}(\mu_{k},\nu)

and its gradient flow

-\nabla_{z_{c}}\frac{1}{K}\sum_{k}\mathbb{D}(\mu_{k},\nu)

(streamlines) for one prototype, while others are fixed (white stars), under

\mathbb{W}_{p,\epsilon}

and

\mathbb{MMD}_{\kappa}

Dataset	Modality	# Features	# Classes	# Domains	# Samples
Office 31	Image	2048	31	3	4,110
Caltech-Office 10	Image	4096	10	4	2,533
Office-Home	Image	2048	65	4	15,500
DomainNet	Image	2048	345	6	586,575
TAU Urban Scenes	Audio	768	10	10	20,800
Tennessee Eastman Process	Sensor data	64	29	6	17,289

Table 1: Summary of datasets used in our experiments.

Tier	Method	OH				O31				C10				TAU				TEP				Avg. Rank
Tier	Method	Acc	NMI	ARI	GM	Acc	NMI	ARI	GM	Acc	NMI	ARI	GM	Acc	NMI	ARI	GM	Acc	NMI	ARI	GM	Avg. Rank
Pooled	KMeans	0.59	0.70	0.41	0.56	0.81	0.87	0.71	0.79	0.65	0.66	0.43	0.57	0.42	0.38	0.24	0.34	0.24	0.44	0.05	0.18	4.6
	Spectral	0.58	0.70	0.33	0.51	0.73	0.84	0.55	0.70	0.62	0.69	0.38	0.55	0.40	0.41	0.22	0.33	0.30	0.60	0.16	0.31	5.4
	Ward	0.60	0.70	0.39	0.55	0.85	0.89	0.75	0.83	0.67	0.72	0.50	0.62	0.40	0.35	0.22	0.32	0.27	0.48	0.06	0.19	4.2
	BIRCH	0.59	0.70	0.41	0.56	0.84	0.89	0.73	0.82	0.66	0.69	0.50	0.61	0.40	0.35	0.22	0.32	0.21	0.40	0.04	0.15	4.6
Multi-domain	MWMS	0.59	0.69	0.46	0.57	0.79	0.84	0.70	0.77	0.72	0.71	0.51	0.64	0.42	0.38	0.24	0.34	0.22	0.50	0.08	0.21	3.4

Table 2: Per-domain clustering performance (within-domain Hungarian matching, averaged over domains) on the five multi-domain benchmarks. Per dataset: Hungarian Acc. / NMI / ARI and their geometric mean (GM); last column: average rank across datasets by GM (lower is better). Sinkhorn uses

p{=}1

, euclidean cost. Best GM per dataset and best avg. rank in bold.

Figure 2: Clustering methods over the Caltech-Office 10 dataset. Overall, using the Sinkhorn divergence as a criterion yields the most consistent clustering of the data.

Figure 3: Ablation of hyper-parameters

(p,m,\epsilon,\kappa)

of our methods.

Figure 4: Ablation of assignment mechanism of our methods.

Figure 5: Scaling experiment on DomainNet.

Clustering

Clustering groups data points by shared structure, developing methods to discover, validate, and interpret groups across diverse data and constraints.

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Gromov-Wasserstein Quantization and Clustering: Structure, Rates, and Algorithms

Florian Beier, Stephan Eckstein

Department of Mathematics, University of Tübingen, Germany

Clustering Vector Quantization Optimal Transport Wasserstein Distance

Open paper AskPDF arXiv

Clustering is a fundamental class of data analysis techniques with the most important representatives being centroid-based methods like

k

-means. Such methods are strongly connected to quantization problems, which aim to approximate general probability measures with discrete ones. For example,

k

-means corresponds to quantization with respect to the Wasserstein distance. While Wasserstein quantization clusters points within a fixed space, this paper studies Gromov-Wasserstein (GW) quantization, which additionally aims at clustering the ambient geometry of the space. We show existence of solutions to the GW quantization problem and give a characterization that justifies an analogue to the

k

-means algorithm (Lloyd's algorithm) to approximate them numerically. We further calculate the quantization rate for usual Euclidean geometries that are used in the GW context, and relate it to standard Wasserstein quantization rates. Finally, numerical experiments show that GW quantization opens up many modeling possibilities beyond normal clustering methods (e.g., for geodesic distances of 3D shapes or structured pruning of neural networks) and that the introduced algorithm leads to useful numerical solutions with approximation quality often in line with theoretically optimal rates.

Abstract

Clustering

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