Similarity matrix framework for data from union of subspaces

Date

2018-09-01

Author

Aldroubi, Akram
Sekmen, Ali
Koku, Ahmet Buğra
Cakmak, Ahmet Faruk

Metadata

Show full item record

This work is licensed under a Creative Commons Attribution-NonCommercial-NoDerivatives 4.0 International License.

Item Usage Stats

244
views

0
downloads

This paper presents a framework for finding similarity matrices for the segmentation of data W = [w(1)...w(N)] subset of R-D drawn from a union U = boolean OR(M)(i=1) S-i, of independent subspaces {S-i}(i=1)(M), of dimensions {d(i)}(i=1)(M). It is shown that any factorization of W = BP, where columns of B form a basis for data W and they also come from U, can be used to produce a similarity matrix Xi w. In other words, Xi w(i, j) not equal 0, when the columns w(i) and w(j) of W come from the same subspace, and Xi w(i, j) = 0, when the columns w(i) and w(j), of W come from different subspaces. Furthermore, Xi w = Q(dmax), where d(max) = max {d(i)}(i=1)(M), and Q is an element of R-NxN with Q(i, j) = vertical bar P-T(i, j)vertical bar. It is shown that a similarity matrix obtained from the reduced row echelon form of W is a special case of the theory. It is also proven that the Shape Interaction Matrix defined as VVT, where W = U Sigma V-T is the skinny singular value decomposition of W, is not necessarily a similarity matrix. But, taking powers of its absolute value always generates a similarity matrix. An interesting finding of this research is that a similarity matrix can be obtained using a skeleton decomposition of W. First, a square sub-matrix A is an element of R-rxr of W with the same rank r as W is found. Then, the matrix R corresponding to the rows of W that contain A is constructed. Finally, a power of the matrix (PP)-P-T where P = A(-1) R provides a similarity matrix Xi w. Since most of the data matrices are low-rank in many subspace segmentation problems, this is computationally efficient compared to other constructions of similarity matrices.

Subject Keywords

Skeleton decomposition, Shape interaction matrix, Similarity matrix, Data clustering, Union of subspaces, Subspace segmentation

URI

https://hdl.handle.net/11511/42591

Journal

APPLIED AND COMPUTATIONAL HARMONIC ANALYSIS

DOI

https://doi.org/10.1016/j.acha.2017.08.006

Collections

Department of Mechanical Engineering, Article

Suggestions

OpenMETU
Core

Skeleton Decomposition Analysis for Subspace Clustering Sekmen, Ali; Aldroubi, Akram; Koku, Ahmet Buğra (2016-12-08) This paper provides a comprehensive analysis of skeleton decomposition used for segmentation of data W = [w(1) center dot center dot center dot w(N)] subset of R-D drawn from a union u = U-i=1(M) S-i of linearly independent subspaces {Si}(M)(i=1) of dimensionsof {di}(M)(i=1). Our previous work developed a generalized theoretical framework for computing similarity matrices by matrix factorization. Skeleton decomposition is a special case of this general theory. First, a square sub-matrix A is an element of R...
Comparison of regression techniques via Monte Carlo simulation Mutan, Oya Can; Ayhan, Hüseyin Öztaş; Department of Statistics (2004) The ordinary least squares (OLS) is one of the most widely used methods for modelling the functional relationship between variables. However, this estimation procedure counts on some assumptions and the violation of these assumptions may lead to nonrobust estimates. In this study, the simple linear regression model is investigated for conditions in which the distribution of the error terms is Generalised Logistic. Some robust and nonparametric methods such as modified maximum likelihood (MML), least absolut...
Unsupervised Deep Learning for Subspace Clustering SEKMEN, ali; Koku, Ahmet Buğra; PARLAKTUNA, Mustafa; ABDULMALEK, Ayad; VANAMALA, Nagendrababu (2017-12-14) This paper presents a novel technique for the segmentation of data W = [w(1) . . . w(N)] subset of R-D drawn from a union U = boolean OR(M)(i=1) S-i of subspaces {S-i}(i=1)(M). First, an existing subspace segmentation algorithm is used to perform an initial data clustering {C-i}(i=1)(M), where C-i = {w(i1) . . . w(ik)} subset of W is the set of data from the ith cluster. Then, a local subspace LSi is matched for each C-i and the distance d(ij) between LSi and each point w(ij) is an element of C-i is compute...
Physical subspace identification for helicopters Avcıoğlu, Sevil; Kutay, Ali Türker; Department of Aerospace Engineering (2019) Subspace identification is a powerful tool due to its well-understood techniques based on linear algebra (orthogonal projections and intersections of subspaces) and numerical methods like QR and singular value decomposition. However, the state space model matrices which are obtained from conventional subspace identification algorithms are not necessarily associated with the physical states. This can be an important deficiency when physical parameter estimation is essential. This holds for the area of helico...
Covariance Matrix Estimation of Texture Correlated Compound-Gaussian Vectors for Adaptive Radar Detection Candan, Çağatay; Pascal, Frederic (2022-01-01) Covariance matrix estimation of compound-Gaussian vectors with texture-correlation (spatial correlation for the adaptive radar detectors) is examined. The texture parameters are treated as hidden random parameters whose statistical description is given by a Markov chain. States of the chain represent the value of texture coefficient and the transition probabilities establish the correlation in the texture sequence. An Expectation-Maximization (EM) method based covariance matrix estimation solution is given ...

Citation Formats

A. Aldroubi, A. Sekmen, A. B. Koku, and A. F. Cakmak, “Similarity matrix framework for data from union of subspaces,” APPLIED AND COMPUTATIONAL HARMONIC ANALYSIS, pp. 425–435, 2018, Accessed: 00, 2020. [Online]. Available: https://hdl.handle.net/11511/42591.