Statistical analysis of metric graph reconstruction

A metric graph is a 1-dimensional stratified metric space consisting of vertices and edges or loops glued together. Metric graphs can be naturally used to represent and model data that take the form of noisy filamentary structures, such as street maps, neurons, networks of rivers and galaxies. We consider the statistical problem of reconstructing the topology of a metric graph embedded in R^D from a random sample. We derive lower and upper bounds on the minimax risk for the noiseless case and tubular noise case. The upper bound is based on the reconstruction algorithm given in Aanjaneya et al. (2012).

[1]  Sayan Mukherjee,et al.  Local homology transfer and stratification learning , 2012, SODA.

[2]  Bin Yu.  Assouad, Fano, and Le Cam , 1997 .

[3]  Mikhail Belkin,et al.  Data Skeletonization via Reeb Graphs , 2011, NIPS.

[4]  Carola Wenk,et al.  Probabilistic street-intersection reconstruction from GPS trajectories: approaches and challenges , 2012, QUeST '12.

[5]  Larry D. Hostetler,et al.  The estimation of the gradient of a density function, with applications in pattern recognition , 1975, IEEE Trans. Inf. Theory.

[6]  Larry A. Wasserman,et al.  Nonparametric Ridge Estimation , 2012, ArXiv.

[7]  Frédéric Chazal,et al.  Topology guaranteeing manifold reconstruction using distance function to noisy data , 2006, SCG '06.

[8]  Sayan Mukherjee,et al.  Towards Stratification Learning through Homology Inference , 2010, 1008.3572.

[9]  Jian Sun,et al.  Gromov-Hausdorff Approximation of Metric Spaces with Linear Structure , 2013, ArXiv.

[10]  Alexandre B. Tsybakov,et al.  Introduction to Nonparametric Estimation , 2008, Springer series in statistics.

[11]  Frédéric Chazal,et al.  A Sampling Theory for Compact Sets in Euclidean Space , 2006, SCG '06.

[12]  David Cohen-Steiner,et al.  Versions of Intersection and Local Homology , 2008 .

[13]  Stephen Smale,et al.  Finding the Homology of Submanifolds with High Confidence from Random Samples , 2008, Discret. Comput. Geom..

[14]  Larry A. Wasserman,et al.  Minimax Manifold Estimation , 2010, J. Mach. Learn. Res..

[15]  P. Kuchment.  Quantum graphs , 2004 .

[16]  Guangliang Chen,et al.  Spectral clustering based on local linear approximations , 2010, 1001.1323.

[17]  Vitaliy Kurlin,et al.  Reconstructing persistent graph structures from noisy images , 2013 .

[18]  Dorin Comaniciu,et al.  Mean Shift: A Robust Approach Toward Feature Space Analysis , 2002, IEEE Trans. Pattern Anal. Mach. Intell..

[19]  G. Ascoli,et al.  NeuroMorpho.Org: A Central Resource for Neuronal Morphologies , 2007, The Journal of Neuroscience.

[20]  T. Freund,et al.  Total Number and Ratio of Excitatory and Inhibitory Synapses Converging onto Single Interneurons of Different Types in the CA1 Area of the Rat Hippocampus , 1999, The Journal of Neuroscience.

[21]  P. Kuchment.  Quantum graphs: I. Some basic structures , 2004 .

[22]  Leonidas J. Guibas,et al.  Road network reconstruction for organizing paths , 2010, SODA '10.

[23]  Leonidas J. Guibas,et al.  Metric graph reconstruction from noisy data , 2011, SoCG '11.

[24]  I. Hassan.  Embedded , 2005, The Cyber Security Handbook.