Chap6.3-GaussianProcesses

# Chap6.3-GaussianProcesses - Machine Learning Srihari...

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Unformatted text preview: Machine Learning Srihari Gaussian Processes Sargur Srihari 1 Machine Learning Srihari Topics in Gaussian Processes 1. Examples of use of GP 2. Duality: From Basis Functions to Kernel Functions 3. GP De¡nition and Intuition 4. Linear regression revisited 5. Gaussian processes for regression 6. Learning the hyperparameters Automatic Relevance Determination 7. Gaussian processes for classi¡cation Laplace approximation 8. Connection to neural networks 2 Machine Learning Srihari National Academy of Sciences: Keck Center 3 Machine Learning Srihari Regression Problem: Carbon Dioxide in Atmosphere 1960 1980 2000 2010 2020 320 340 360 380 400 CO 2 Concentration ppm ? Year Dec 2011 We will see a Gaussian Process solution to this regression problem Machine Learning Srihari Fingerprint Core Point using GP Regression 0 0 0 34 32 33 32 38 42 35 29 32 24 24 22 21 17 10 5 169 143 131 135 131 131 127 139 146 0 0 0 0 0 0 0 32 33 33 30 34 39 45 27 33 31 22 22 29 15 11 7 176 150 141 134 130 136 119 129 131 118 0 0 0 0 0 29 30 31 35 36 35 39 40 33 33 36 24 21 34 25 17 10 1 173 151 140 132 136 122 129 113 110 0 0 0 0 0 34 32 33 31 30 39 41 43 37 32 31 19 23 26 16 9 8 178 161 146 136 125 129 116 134 127 121 0 0 0 0 0 41 38 36 30 33 44 43 43 39 35 31 26 23 26 19 5 1 176 164 156 136 121 129 114 121 110 102 0 0 0 · · · · · · Core point (x, y, θ ): (253, 221, 85) Gradients at each pixel NIPS 2010 Su and Srihari Machine Learning Srihari Paper at NIPS 2010 Evaluation of Rarity of Fingerprints in Forensics Chang Su and Sargur N. Srihari University at Buffalo, The State University of New York Process Flow for a Latent Print (Four Steps) 3. A Generative Model Distribution of minutiae : mixture model with bivariate Gaussian for location and von Mises for Orientation Latent print with predicted core point within print 4. Specific n PRC Calculation with Minutiae Confidence (a) (b) (a) minutiae sequencing and (b) dependency For minutia set X joint distribution is Graphical model Examples of Rarity Evaluation Latent case “b115” Predicted Core Latent case “g73” Predicted Core Minutiae confidence manual y assigned by visual inspection. Tolerance set at 10 pixels and ! /8. Specific nPRCs for the latent prints, with n = 100, 000 . Specific nPRC given by Values of specific nPRC are largely dependent on given latent print where g ! M AX is orientation map with the maximum predictive probability where p e (X, ˆm) is the probability that ˆm minutiae match between latent print and a random one among n knowns Original latent print level 2 details . Mayfield inked print Charted features on latent print 1. Core Point Prediction (Gaussian Process) Given orientation map of latent print, core point is predicted by 2. Coordinate Transformation ABSTRACT Since the earliest days of forensics, importance of considering the rarity of features used in the comparison of prints has been known. Yet methods to compute rarity of features has been elusive due to the large number of variables and the complexity of the distributions. When a latent print, typical y found in a crime number of variables and the complexity of the distributions....
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Chap6.3-GaussianProcesses - Machine Learning Srihari...

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