Riemannian Manifolds clustering via Geometric median.
ABSTRACT In this paper, we propose a new kernel function that makes use of Riemannian geodesic distance s among data points, and present a Geometric median shift algorithm over Riemannian Manifolds. Relying on the geometric median shift, together with geodesic distances, our approach is able to effectively cluster data points distributed on Riemannian manifolds. In addition to improving the clustering results, Using both Riemannian Manifolds and Euclidean spaces, We compare the geometric median shift and mean shift algorithms on synthetic and real data sets for the tasks of clustering.
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Conference Proceeding: Mode-detection via median-shift.[show abstract] [hide abstract]
ABSTRACT: Median-shift is a mode seeking algorithm that relies on computing the median of local neighborhoods, instead of the mean. We further combine median-shift with Locality Sensitive Hashing (LSH) and show that the combined algorithm is suitable for clustering large scale, high dimensional data sets. In particular, we propose a new mode detection step that greatly accelerates performance. In the past, LSH was used in conjunction with mean shift only to accelerate nearest neighbor queries. Here we show that we can analyze the density of the LSH bins to quickly detect potential mode candidates and use only them to initialize the median-shift procedure. We use the median, instead of the mean (or its discrete counterpart - the medoid) because the median is more robust and because the median of a set is a point in the set. A median is well defined for scalars but there is no single agreed upon extension of the median to high dimensional data. We adopt a particular extension, known as the Tukey median, and show that it can be computed efficiently using random projections of the high dimensional data onto 1D lines, just like LSH, leading to a tightly integrated and efficient algorithm.IEEE 12th International Conference on Computer Vision, ICCV 2009, Kyoto, Japan, September 27 - October 4, 2009; 01/2009
Conference Proceeding: Nonlinear Mean Shift for Clustering over Analytic Manifolds.[show abstract] [hide abstract]
ABSTRACT: The mean shift algorithm is widely applied for nonpara- metric clustering in Euclidean spaces. Recently, mean shift was generalized for clustering on matrix Lie groups. We further extend the algorithm to a more general class of nonlinear spaces, the set of analytic manifolds. As exam- ples, two specific classes of frequently occurring parameter spaces, Grassmann manifolds and Lie groups, are consid- ered. When the algorithm proposed here is restricted to ma- trix Lie groups the previously proposed method is obtained. The algorithm is applied to a variety of robust motion seg- mentation problems and multibody factorization. The mo- tion segmentation method is robust to outliers, does not require any prior specification of the number of indepen- dent motions and simultaneously estimates all the motions present.2006 IEEE Computer Society Conference on Computer Vision and Pattern Recognition (CVPR 2006), 17-22 June 2006, New York, NY, USA; 01/2006
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ABSTRACT: Scientists working with large volumes of high-dimensional data, such as global climate patterns, stellar spectra, or human gene distributions, regularly confront the problem of dimensionality reduction: finding meaningful low-dimensional structures hidden in their high-dimensional observations. The human brain confronts the same problem in everyday perception, extracting from its high-dimensional sensory inputs-30,000 auditory nerve fibers or 10(6) optic nerve fibers-a manageably small number of perceptually relevant features. Here we describe an approach to solving dimensionality reduction problems that uses easily measured local metric information to learn the underlying global geometry of a data set. Unlike classical techniques such as principal component analysis (PCA) and multidimensional scaling (MDS), our approach is capable of discovering the nonlinear degrees of freedom that underlie complex natural observations, such as human handwriting or images of a face under different viewing conditions. In contrast to previous algorithms for nonlinear dimensionality reduction, ours efficiently computes a globally optimal solution, and, for an important class of data manifolds, is guaranteed to converge asymptotically to the true structure.Science 01/2001; 290(5500):2319-23. · 31.03 Impact Factor