A Riemannian Framework for Low-Rank Structured Elliptical Models

Bouchard, Florent; Breloy, Arnaud; Ginolhac, Guillaume; Renaux, Alexandre; Pascal, Frédéric

doi:10.1109/tsp.2021.3054237

Cited by 7 publications

(12 citation statements)

References 43 publications

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“…A "good" clustering algorithm finds a partition whose associated φ is close to the minimum φ OPT of the WCSS (36). In the Euclidean case, [21] establishes that the Euclidean WCSS of a partition produced by K-means++ is upper bounded with respect to φ OPT (minimum of the Euclidian WCSS).…”

Section: Theoretical Propertiesmentioning

confidence: 99%

“…As stated in [40], this result in the Euclidean case holds for any distance (thus for d M p,k,n ) and does not rely on the mean computation. Hence, the WCSS (36)…”

Section: Theoretical Propertiesmentioning

confidence: 99%

“…where the expectation is taken with respect to the seeding procedure. Moreover, "Assignment step" and "Update step" from Algorithm 2 decrease WCSS (36). Indeed, the "Assignment step" directly decreases the WCSS (36) by assigning points {θ i } M i=1 to the closest centers.…”

Section: Theoretical Propertiesmentioning

confidence: 99%

“…It follows that ∀S j ∈ S, (38) where θ (t) and θ (t+1) are the means taken before and after the "Update step" respectively. Hence, the "Update step" decreases the WCSS (36). This implies that the final clustering returned by K-means++ M p,k,n satisfies However, this clustering is not necessarily a global minimum of WCSS (36).…”

Section: Theoretical Propertiesmentioning

confidence: 99%

“…Hence, the "Update step" decreases the WCSS (36). This implies that the final clustering returned by K-means++ M p,k,n satisfies However, this clustering is not necessarily a global minimum of WCSS (36). Hence, a standard practice is to run the algorithm several times with different initializations and then to keep the clustering with the lowest inertia (36).…”

Section: Theoretical Propertiesmentioning

confidence: 99%

See 4 more Smart Citations

Probabilistic PCA From Heteroscedastic Signals: Geometric Framework and Application to Clustering

Collas¹,

Bouchard²,

Breloy³

et al. 2021

IEEE Trans. Signal Process.

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This paper studies a statistical model for heteroscedastic (i.e., power fluctuating) signals embedded in white Gaussian noise. Using the Riemannian geometry theory, we propose an unified approach to tackle several problems related to this model. The first axis of contribution concerns parameters (signal subspace and power factors) estimation, for which we derive intrinsic Cramér-Rao bounds and propose a flexible Riemannian optimization algorithmic framework in order to compute the maximum likelihood estimator (as well as other cost functions involving the parameters). Interestingly, the obtained bounds are in closed forms and interpretable in terms of problem's dimensions and SNR. The second axis of contribution concerns the problem of clustering data assuming a mixture of heteroscedastic signals model, for which we generalize the Euclidean K-means++ to the considered Riemannian parameter space. We propose an application of the resulting clustering algorithm on the Indian Pines segmentation problem benchmark.

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Section: Theoretical Propertiesmentioning

confidence: 99%

“…As stated in [40], this result in the Euclidean case holds for any distance (thus for d M p,k,n ) and does not rely on the mean computation. Hence, the WCSS (36)…”

Section: Theoretical Propertiesmentioning

confidence: 99%

Section: Theoretical Propertiesmentioning

confidence: 99%