Beyond Mode Hunting . Joint work with SAMSI Nonlinear WG members: - PowerPoint PPT Presentation

. Beyond Mode Hunting . Joint work with SAMSI Nonlinear WG members: Bobrowski, Kim, Marron, Noh, and Sommerfeld Giseon Heo CANSSI-SAMSI Workshop-Fields Institute May 23, 2014 . . . . . . . . . . . . . . . . . . . . .. . .. . .. . .. . .. . .. . .. . .. . .. . . .. .. . .. . .. . .. . .. . .. . .. . .. . .. . .. . 1 / 35

Outline . . Brief literature review 1 Scale-space theory in computer vision SiZer in statistics Persistent homology in computational topology . . From 1D to 2D and higher 2 . . Application to real data: persistence landscape and hypothesis test 3 . . . . . . . . . . . . . . . . . . . . .. . .. . .. . .. . .. . .. . .. . .. . . .. . .. . .. .. . .. . .. . .. . .. . .. . .. . .. . .. . 2 / 35

Picture at different scale of resolution . . . . . . . . . . . . . . . . . . . . .. . .. . .. . .. . .. . .. . .. . .. . .. . . .. .. . .. . .. . .. . .. . .. . .. . .. . .. . .. . 3 / 35

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Scale-space for signals [Witkin (1983), Koenderink (1984), Lindeberg (1994)] Given a signal f : R d → R , the scale-space representation u : R d × R + → R defined such that the representation at zero scale is equal to the original signal u ( x , 0) = f ( x ) , and the representation at coarser scales are given by convolution of the signal with Gaussian kernels of increasing bandwidth R d f ( y ) e −|| y − x || 2 / 2 t ∫ u ( x , t ) = g ( x , t ) ∗ f ( x ) = dy , | 2 π t | d / 2 Gaussian mean x ∈ R d , variance matrix t I d × d . . . . . . . . . . . . . . . . . . . . . .. . .. . .. . .. . .. . .. . .. . .. . . .. .. . .. . .. . .. . .. . .. . .. . .. . .. . .. . .. . 7 / 35

Connection with heat equation The scale-space representation can equivalently be defined as the solution to heat equation with initial condition u ( x , 0) = f ( x ) . d ∂ 2 ∂ t u ( x , t ) = 1 ∂ 2∆ u ( x , t ) := 1 ∑ u ( x , t ) ∂ x 2 2 i i =1 . . . . . . . . . . . . . . . . . . . . .. . .. . .. . .. . .. . .. . .. . .. . . .. .. . .. . .. . .. . .. . .. . .. . .. . .. . .. . .. . 8 / 35

Two properties Non-enhancement of local extrema At a certain scale t 0 ∈ R + , a point x 0 ∈ R d is a local maximum for the mapping x �→ u ( x , t 0 ) , then ∆ u ( x 0 , t 0 ) < 0, which means ∂ ∂ t u ( x 0 , t 0 ) < 0 . A hot spot will not become warmer and a cold spot will not become cooler (true for all dimensions). Non-creation of new features (Causality) Fine-scale features disappear monotonically with increasing scale. (true for only d = 1) . . . . . . . . . . . . . . . . . . . . .. . .. . .. . .. . .. . .. . .. . .. . .. . .. . . .. .. . .. . .. . .. . .. . .. . .. . .. . .. . 9 / 35

Nonparametric curve estimation in statistics The kernel density estimator based on data x 1 , . . . , x n is n f ( x , h ) = 1 K ( x − x i ˆ ∑ ) , x ∈ R , h ∈ R + nh h i =1 √ 2 π ) exp( − x 2 / 2) , ? proved Note: With Gaussian kernel, K ( x ) = (1 / causality. . . . . . . . . . . . . . . . . . . . . .. . .. . .. . .. . .. . .. . .. . .. . . .. .. . . .. .. . .. . .. . .. . .. . .. . .. . .. . .. . 10 / 35

Scale space surface [Chaudhuri and Marron (2000)] { ˆ f ( x , t ) : x ∈ R , t ∈ R + } . . . . . . . . . . . . . . . . . . . . .. . .. . .. . .. . .. . .. . .. . .. . . .. . .. .. . .. . .. . .. . .. . .. . .. . .. . .. . .. . 11 / 35

SiZer map (SIgnificant ZERo crossings of derivatives) [Chaudhuri and Marron (1999)] . . . . . . . . . . . . . . . . . . . . .. . .. . .. . .. . .. . .. . .. . .. . .. . .. . .. . .. . .. . .. . .. . .. . .. . .. . .. . .. . 12 / 35