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Frobenius norm

The Frobenius norm is a measure of the magnitude or size of a matrix, calculated as the square root of the sum of the absolute squares of all its elements. It represents the direct extension of the standard Euclidean vector norm to matrices, effectively treating the matrix entries as coordinates of a single flat vector. Mathematically, the Frobenius norm can also be expressed as the square root of the trace of the matrix multiplied by its conjugate transpose, or equivalently as the square root of the sum of the squares of its singular values. Because it is unitarily invariant, continuous, and computationally straightforward, it is widely utilized in numerical linear algebra, data analysis, optimization, and machine learning to measure matrix distances, quantify approximation errors, and formulate regularization penalties.

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Co-regularized Multi-view Spectral Clustering

Co-regularized Multi-view Spectral Clustering

Abhishek Kumar, Piyush Rai, Hal Daumé

OrganizationsUniversity of MarylandUniversity of Utah

Why you should read this

Proposes a multi-view spectral clustering framework that co-regularizes graph Laplacians across different data representations to find consistent cluster assignments across diverse views.

In many clustering problems, we have access to multiple views of the data each of which could be individually used for clustering. Exploiting information from multiple views, one can hope to find a clustering that is more accurate than the ones obtained using the individual views. Often these different views admit same underlying clustering of the data, so we can approach this problem by looking for clusterings that are consistent across the views, i.e., corresponding data points in each view should have same cluster membership. We propose a spectral clustering framework that achieves this goal by co-regularizing the clustering hypotheses, and propose two co-regularization schemes to accomplish this. Experimental comparisons with a number of baselines on two synthetic and three real-world datasets establish the efficacy of our proposed approaches.

Added

2026-09-25