The problem of clustering noisy and incompletely observed high-dimensional data points into a union of low-dimensional subspaces and a set of outliers is considered. The number of subspaces, their dimensions, and their orientations are assumed unknown. We propose a simple low-complexity subspace clustering algorithm, which applies spectral clustering to an adjacency matrix obtained by thresholding the correlations between data points. In other words, the adjacency matrix is constructed from the nearest neighbors of each data point in spherical distance. A statistical performance analysis shows that the algorithm exhibits robustness to additive noise and succeeds even when the subspaces intersect. Specifically, our results reveal an explicit tradeoff between the affinity of the subspaces and the tolerable noise level. We furthermore prove that the algorithm succeeds even when the data points are incompletely observed with the number of missing entries allowed to be (up to a log-factor) linear in the ambient dimension. We also propose a simple scheme that provably detects outliers, and we present numerical results on real and synthetic data.

Robust Subspace Clustering via Thresholding

Geometry and topology Metric spaces Differential geometry Hyperbolic space Groups and their actions Median spaces and spaces with measured walls Finitely generated and finitely presented groups Coarse geometry Coarse topology Ultralimits of metric spaces Gromov-hyperbolic spaces and groups Lattices in Lie groups Solvable groups Geometric aspects of solvable groups The Tits alternative Gromov's theorem The Banach-Tarski paradox Amenability and paradoxical decomposition Ultralimits, fixed point properties, proper actions Stallings's theorem and accessibility Proof of Stallings's theorem using harmonic functions Quasiconformal mappings Groups quasiisometric to $\mathbb{H}^n$ Quasiisometries of nonuniform lattices in $\mathbb{H}^n$ A survey of quasiisometric rigidity Appendix: Three theorems on linear groups Bibliography Index

Geometric Group Theory

Smarandache initiated neutrosophic sets (NSs) which can be used as a mathematical tool for dealing with indeterminate and inconsistent information. In order to apply NSs conveniently, single valued neutrosophic sets (SVNSs) were proposed by Wang et al. In this paper, we propose single valued neutrosophic relations (SVNRs) and study their properties. The notions of anti-reflexive kernel, symmetric kernel, reflexive closure, and symmetric closure of a SVNR are introduced, respectively. Their accurate calculate formulas and some properties are explored. Some examples are also given. Finally, single valued neutrosophic relation mappings and inverse single valued neutrosophic relation mappings are introduced, and some interesting properties are also obtained.

On single valued neutrosophic relations

https://openresearch-repository.anu.edu.au/bitstream/1885/12859/2/Barnsley%20and%20Massopust%20Bilinear%20Fractal%20Interpolation%202015.pdf

Bilinear fractal interpolation and box dimension

Topodynamics of metastable brains.

General Topology-I

In this paper, new characterizations of pre-open (pre-closed, pre-continuous) maps will be obtained using the map η# induced via a map η between any two topological spaces. Further, these maps are characterized in terms of saturated sets under some sufficient conditions.

https://ojs.wiserpub.com/index.php/CM/article/download/2886/1458

Nitakshi Goyal

Papers

General Topology-I

New Type of Characterizations of Pre-Open (Closed and Continuous) Mappings