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Chap5.6-MixDensityNetworks

# Chap5.6-MixDensityNetworks - Machine Learning Srihari...

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Machine Learning Srihari Mixture Density Networks Sargur Srihari 1

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Machine Learning Srihari Mixture Density Networks In some problems distribution can be multimodal Particularly in inverse problems Gaussian assumption can lead to poor results In regression p( t|x ) i s typically assumed to be Gaussian i.e. , p (t|x)= N (t| y (x,w) , β -1 ) 2
Machine Learning Srihari Kinematics of a robot arm Robot arm with two links This is a regression problem with two inputs (desired location of arm) and two outputs (angles for links) 3 Forward problem: Find end effector position given joint angles Has a unique solution Inverse kinematics has two solutions: Elbow-up and elbow-down

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Machine Learning Srihari Forward and Inverse Problems 4 Forward problems correspond to causality in a physical system have a unique solution If forward problem is a many-to-one mapping, inverse has multiple solutions Inverse classification problem symptoms caused by disease
Machine Learning Srihari Data Set for Forward and Inverse Problems 5 Least squares corresponds to Maximum likelihood under a Gaussian assumption Leads to a poor result for highly non-Gaussian inverse problem Seek a general framework for modeling conditional probability distributions Achieved by using a mixture model p( t|x ) Forward problem data set: x is sampled uniformly over (0,1) to give values { x n } Target t n obtained by function x n +0.3sin(2 π x n ) Then add noise over (-0.1,0.1) Red curve is result of fitting a two-layer neural network by minimizing sum-of-squared error Corresponding inverse problem by reversing x and t Very poor fit to data

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Machine Learning Srihari A Mixture Density Model with K components Components can be Gaussian for
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