A Novel Theoretical Analysis for Clustering Heteroscedastic Gaussian Data without Knowledge of the Number of Clusters

• URL: http://arxiv.org/abs/2604.01943v1
• Date: Thu, 02 Apr 2026 12:06:21 GMT
• Title: A Novel Theoretical Analysis for Clustering Heteroscedastic Gaussian Data without Knowledge of the Number of Clusters
• Abstract summary: We introduce a novel cost function to estimate the fixed-points of a certain function.<n>The zeros of the gradient of this cost function turn out to be the fixed-points of a certain function.<n>We derive a new clustering algorithm called CENTRE-X that works by estimating the fixed-points of the identified function.
• Score: 0.9519647470165687
• License: http://creativecommons.org/licenses/by/4.0/
• Abstract: This paper addresses the problem of clustering measurement vectors that are heteroscedastic in that they can have different covariance matrices. From the assumption that the measurement vectors within a given cluster are Gaussian distributed with possibly different and unknown covariant matrices around the cluster centroid, we introduce a novel cost function to estimate the centroids. The zeros of the gradient of this cost function turn out to be the fixed-points of a certain function. As such, the approach generalizes the methodology employed to derive the existing Mean-Shift algorithm. But as a main and novel theoretical result compared to Mean-Shift, this paper shows that the sole fixed-points of the identified function tend to be the cluster centroids if both the number of measurements per cluster and the distances between centroids are large enough. As a second contribution, this paper introduces the Wald kernel for clustering. This kernel is defined as the p-value of the Wald hypothesis test for testing the mean of a Gaussian. As such, the Wald kernel measures the plausibility that a measurement vector belongs to a given cluster and it scales better with the dimension of the measurement vectors than the usual Gaussian kernel. Finally, the proposed theoretical framework allows us to derive a new clustering algorithm called CENTRE-X that works by estimating the fixed-points of the identified function. As Mean-Shift, CENTRE-X requires no prior knowledge of the number of clusters. It relies on a Wald hypothesis test to significantly reduce the number of fixed points to calculate compared to the Mean-Shift algorithm, thus resulting in a clear gain in complexity. Simulation results on synthetic and real data sets show that CENTRE-X has comparable or better performance than standard clustering algorithms K-means and Mean-Shift, even when the covariance matrices are not perfectly known.

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