Probabilistic Sparse Subspace Clustering Using Delayed Association

Discovering and clustering subspaces in high-dimensional data is a fundamental problem of machine learning with a wide range of applications in data mining, computer vision, and pattern recognition. Earlier methods divided the problem into two separate stages of finding the similarity matrix and finding clusters. Similar to some recent works, we integrate these two steps using a joint optimization approach. We make the following contributions: (i) we estimate the reliability of the cluster assignment for each point before assigning a point to a subspace. We group the data points into two groups of "certain" and "uncertain", with the assignment of latter group delayed until their subspace association certainty improves. (ii) We demonstrate that delayed association is better suited for clustering subspaces that have ambiguities, i.e. when subspaces intersect or data are contaminated with outliers/noise. (iii) We demonstrate experimentally that such delayed probabilistic association leads to a more accurate self-representation and final clusters. The proposed method has higher accuracy both for points that exclusively lie in one subspace, and those that are on the intersection of subspaces. (iv) We show that delayed association leads to huge reduction of computational cost, since it allows for incremental spectral clustering

Hallucinating optimal high-dimensional subspaces

Linear subspace representations of appearance variation are pervasive in computer vision. This paper addresses the problem of robustly matching such subspaces (computing the similarity between them) when they are used to describe the scope of variations within sets of images of different (possibly greatly so) scales. A naive solution of projecting the low-scale subspace into the high-scale image space is described first and subsequently shown to be inadequate, especially at large scale discrepancies. A successful approach is proposed instead. It consists of (i) an interpolated projection of the low-scale subspace into the high-scale space, which is followed by (ii) a rotation of this initial estimate within the bounds of the imposed ``downsampling constraint''. The optimal rotation is found in the closed-form which best aligns the high-scale reconstruction of the low-scale subspace with the reference it is compared to. The method is evaluated on the problem of matching sets of (i) face appearances under varying illumination and (ii) object appearances under varying viewpoint, using two large data sets. In comparison to the naive matching, the proposed algorithm is shown to greatly increase the separation of between-class and within-class similarities, as well as produce far more meaningful modes of common appearance on which the match score is based.Comment: Pattern Recognition, 201

Multiple pattern classification by sparse subspace decomposition

A robust classification method is developed on the basis of sparse subspace decomposition. This method tries to decompose a mixture of subspaces of unlabeled data (queries) into class subspaces as few as possible. Each query is classified into the class whose subspace significantly contributes to the decomposed subspace. Multiple queries from different classes can be simultaneously classified into their respective classes. A practical greedy algorithm of the sparse subspace decomposition is designed for the classification. The present method achieves high recognition rate and robust performance exploiting joint sparsity.Comment: 8 pages, 3 figures, 2nd IEEE International Workshop on Subspace Methods, Workshop Proceedings of ICCV 200

Discrimination on the Grassmann Manifold: Fundamental Limits of Subspace Classifiers

We present fundamental limits on the reliable classification of linear and affine subspaces from noisy, linear features. Drawing an analogy between discrimination among subspaces and communication over vector wireless channels, we propose two Shannon-inspired measures to characterize asymptotic classifier performance. First, we define the classification capacity, which characterizes necessary and sufficient conditions for the misclassification probability to vanish as the signal dimension, the number of features, and the number of subspaces to be discerned all approach infinity. Second, we define the diversity-discrimination tradeoff which, by analogy with the diversity-multiplexing tradeoff of fading vector channels, characterizes relationships between the number of discernible subspaces and the misclassification probability as the noise power approaches zero. We derive upper and lower bounds on these measures which are tight in many regimes. Numerical results, including a face recognition application, validate the results in practice.Comment: 19 pages, 4 figures. Revised submission to IEEE Transactions on Information Theor

Subspace clustering of dimensionality-reduced data

Subspace clustering refers to the problem of clustering unlabeled high-dimensional data points into a union of low-dimensional linear subspaces, assumed unknown. In practice one may have access to dimensionality-reduced observations of the data only, resulting, e.g., from "undersampling" due to complexity and speed constraints on the acquisition device. More pertinently, even if one has access to the high-dimensional data set it is often desirable to first project the data points into a lower-dimensional space and to perform the clustering task there; this reduces storage requirements and computational cost. The purpose of this paper is to quantify the impact of dimensionality-reduction through random projection on the performance of the sparse subspace clustering (SSC) and the thresholding based subspace clustering (TSC) algorithms. We find that for both algorithms dimensionality reduction down to the order of the subspace dimensions is possible without incurring significant performance degradation. The mathematical engine behind our theorems is a result quantifying how the affinities between subspaces change under random dimensionality reducing projections.Comment: ISIT 201

Classification via Incoherent Subspaces

This article presents a new classification framework that can extract individual features per class. The scheme is based on a model of incoherent subspaces, each one associated to one class, and a model on how the elements in a class are represented in this subspace. After the theoretical analysis an alternate projection algorithm to find such a collection is developed. The classification performance and speed of the proposed method is tested on the AR and YaleB databases and compared to that of Fisher's LDA and a recent approach based on on $\ell_1$ minimisation. Finally connections of the presented scheme to already existing work are discussed and possible ways of extensions are pointed out.Comment: 22 pages, 2 figures, 4 table

Face Recognition in Color Using Complex and Hypercomplex Representation

The final publication is available at Springer via http://dx.doi.org/10.1007/978-3-540-72847-4_29Color has plenty of discriminative information that can be used to improve the performance of face recognition algorithms, although it is difficult to use it because of its high variability. In this paper we investigate the use of the quaternion representation of a color image for face recognition. We also propose a new representation for color images based on complex numbers. These two color representation methods are compared with the traditional grayscale and RGB representations using an eigenfaces based algorithm for identity verification. The experimental results show that the proposed method gives a very significant improvement when compared to using only the illuminance information.Work supported by the Spanish Project DPI2004-08279-C02-02 and the Generalitat Valenciana - Consellería d’Empresa, Universitat i Ciència under an FPI scholarship.Villegas, M.; Paredes Palacios, R. (2007). Face Recognition in Color Using Complex and Hypercomplex Representation. En Pattern Recognition and Image Analysis: Third Iberian Conference, IbPRIA 2007, Girona, Spain, June 6-8, 2007, Proceedings, Part I. Springer Verlag (Germany). 217-224. https://doi.org/10.1007/978-3-540-72847-4_29S217224Yip, A., Sinha, P.: Contribution of color to face recognition. Perception 31(5), 995–1003 (2002)Torres, L., Reutter, J.Y., Lorente, L.: The importance of the color information in face recognition. In: ICIP, vol. 3, pp. 627–631 (1999)Jones III, C., Abbott, A.L.: Color face recognition by hypercomplex gabor analysis. In: FG2006, University of Southampton, UK, pp. 126–131 (2006)Hamilton, W.R.: On a new species of imaginary quantities connected with a theory of quaternions. In: Proc. Royal Irish Academy, vol. 2, pp. 424–434 (1844)Zhang, F.: Quaternions and matrices of quaternions. Linear Algebra And Its Applications 251(1-3), 21–57 (1997)Pei, S., Cheng, C.: A novel block truncation coding of color images by using quaternion-moment preserving principle. In: ISCAS, Atlanta, USA, vol. 2, pp. 684–687 (1996)Sangwine, S., Ell, T.: Hypercomplex fourier transforms of color images. In: ICIP, Thessaloniki, Greece, vol. 1, pp. 137–140 (2001)Bihan, N.L., Sangwine, S.J.: Quaternion principal component analysis of color images. In: ICIP, Barcelona, Spain, vol. 1, pp. 809–812 (2003)Chang, J.-H., Pei, S.-C., Ding, J.J.: 2d quaternion fourier spectral analysis and its applications. In: ISCAS, Vancouver, Canada, vol. 3, pp. 241–244 (2004)Li, S.Z., Jain, A.K.: 6. In: Handbook of Face Recognition. Springer (2005)Gross, R., Brajovic, V.: An image preprocessing algorithm for illumination invariant face recognition. In: Kittler, J., Nixon, M.S. (eds.) AVBPA 2003. LNCS, vol. 2688, p. 1055. Springer, Heidelberg (2003)Lee, K., Ho, J., Kriegman, D.: Nine points of light: Acquiring subspaces for face recognition under variable lighting. In: CVPR, vol. 1, pp. 519–526 (2001)Zhang, L., Samaras, D.: Face recognition under variable lighting using harmonic image exemplars. In: CVPR, vol. 1, pp. 19–25 (2003)Villegas, M., Paredes, R.: Comparison of illumination normalization methods for face recognition. In: COST 275, University of Hertfordshire, UK, pp. 27–30 (2005)Turk, M., Pentland, A.: Face recognition using eigenfaces. In: CVPR, Hawaii, pp. 586–591 (1991)Bihan, N.L., Mars, J.: Subspace method for vector-sensor wave separation based on quaternion algebra. In: EUSIPCO, Toulouse, France (2002)XM2VTS (CDS00{1,6}), http://www.ee.surrey.ac.uk/Reseach/VSSP/xm2vtsdbLuettin, J., Maître, G.: Evaluation protocol for the extended M2VTS database (XM2VTSDB). IDIAP-COM 05, IDIAP (1998