Similarity matrix framework for data from union of subspaces

Aldroubi, Akram; Sekmen, Ali; Koku, Ahmet Buğra; Çakmak, Ahmet Faruk

doi:10.1016/j.acha.2017.08.006

Cited by 20 publications

(21 citation statements)

References 18 publications

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“…C. Basis Framework of [31] As a final note, the CUR framework proposed here gives a broader vantage point for obtaining similarity matrices than that of [31]. Consider the following, which is the main result therein:…”

Section: B Low-rank Representation Algorithmmentioning

confidence: 99%

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CUR Decompositions, Similarity Matrices, and Subspace Clustering

Aldroubi

Hamm

Koku

et al. 2019

Front. Appl. Math. Stat.

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A general framework for solving the subspace clustering problem using the CUR decomposition is presented. The CUR decomposition provides a natural way to construct similarity matrices for data that come from a union of unknown subspaces U = M i=1 S i . The similarity matrices thus constructed give the exact clustering in the noise-free case. Additionally, this decomposition gives rise to many distinct similarity matrices from a given set of data, which allow enough flexibility to perform accurate clustering of noisy data. We also show that two known methods for subspace clustering can be derived from the CUR decomposition. An algorithm based on the theoretical construction of similarity matrices is presented, and experiments on synthetic and real data are presented to test the method.Additionally, an adaptation of our CUR based similarity matrices is utilized to provide a heuristic algorithm for subspace clustering; this algorithm yields the best overall performance to date for clustering the Hopkins155 motion segmentation dataset.

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Section: B Low-rank Representation Algorithmmentioning

confidence: 99%

“…The next lemma is a special case of[31, Lemma 1].Lemma 2. Suppose that U ∈ K m×n has columns which are generic for the subspace S of K m from which they are drawn.…”

mentioning

confidence: 99%

CUR Decompositions, Similarity Matrices, and Subspace Clustering

Aldroubi

Hamm

Koku

et al. 2019

Front. Appl. Math. Stat.

Self Cite

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“…Using the same argument again, we can conclude that with probability at least 1 − 2 exp(−c/δ), rank (Ĉ) = k. Thus with probability at least (1 − 2 exp(−c/δ)) 2 , rank (C) = rank (R) = k, and so A = CU † R. The moreover statement follows from the fact that repeated columns and rows do not affect the validity of the statement A = CU † R as mentioned subsequent to Theorem 3.1.…”

Section: Samplingmentioning

confidence: 68%

“…Matrix factorization methods have been used to good effect in solving the Subspace Clustering Problem [1,2,7] as have associated low-rank based optimization methods [13,22]. In particular, it is known that under certain subspace configurations, the truncated SVD, any basis factorization with basis vectors coming from the subspaces S i , and CUR decompositions can all be used to give a valid clustering of the data.…”

Section: 3mentioning

confidence: 99%

Stability of sampling for CUR decompositions

Hamm¹,

Huang²

2020

Foundations of Data Science

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This article studies how to form CUR decompositions of low-rank matrices via primarily random sampling, though deterministic methods due to previous works are illustrated as well. The primary problem is to determine when a column submatrix of a rank k matrix also has rank k. For random column sampling schemes, there is typically a tradeoff between the number of columns needed to be chosen and the complexity of determining the sampling probabilities. We discuss several sampling methods and their complexities as well as stability of the method under perturbations of both the probabilities and the underlying matrix. As an application, we give a high probability guarantee of the exact solution of the Subspace Clustering Problem via CUR decompositions when columns are sampled according to their Euclidean lengths.

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“…More recent works [31], [32], [33] use a different subset selection method for subspace clustering. In particular, the method named Scalable and Robust SSC (SR-SSC) [33] selects a few sets of anchor points using a randomized hierarchical clustering method.…”

Section: A Fast and Scalable Subspace Clustering Methodsmentioning

confidence: 99%

A fast and accurate similarity-constrained subspace clustering algorithm for land cover segmentation

Hinojosa¹,

Vera²,

Argüello³

2021

Preprint

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Accurate land cover segmentation of spectral images is challenging and has drawn widespread attention in remote sensing due to its inherent complexity. Although significant efforts have been made for developing a variety of methods, most of them rely on supervised strategies. Subspace clustering methods, such as Sparse Subspace Clustering (SSC), have become a popular tool for unsupervised learning due to their high performance. However, the computational complexity of SSC methods prevents their use on large spectral remotely sensed datasets. Furthermore, since SSC ignores the spatial information in the spectral images, its discrimination capability is limited, hampering the clustering results' spatial homogeneity. To address these two relevant issues, in this paper, we propose a fast algorithm that obtains a sparse representation coefficient matrix by first selecting a small set of pixels that best represent their neighborhood. Then, it performs spatial filtering to enforce the connectivity of neighboring pixels and uses fast spectral clustering to get the final segmentation. Extensive simulations with our method demonstrate its effectiveness in land cover segmentation, obtaining remarkable high clustering performance compared with state-of-the-art SSC-based algorithms and even novel unsupervised-deep-learning-based methods. Besides, the proposed method is up to three orders of magnitude faster than SSC when clustering more than 2x10<sup>4</sup> spectral pixels.

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Similarity matrix framework for data from union of subspaces

Cited by 20 publications

References 18 publications

CUR Decompositions, Similarity Matrices, and Subspace Clustering

CUR Decompositions, Similarity Matrices, and Subspace Clustering

Stability of sampling for CUR decompositions

A fast and accurate similarity-constrained subspace clustering algorithm for land cover segmentation

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