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**Unformatted text preview: **Machine Learning Srihari Mixture Density Networks Sargur Srihari 1 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 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 classi¡cation 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 ¡tting a two-layer neural network by minimizing sum-of-squared error Corresponding inverse problem by reversing x and t Very poor ¡t to data Machine Learning Srihari A Mixture Density • Model with K components • Components can be Gaussian for...

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