09_Classification Models
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1.Computer vision: models, learning and inference Chapter 9 Classification Models
2.Structure 2 2 Computer vision: models, learning and inference. ©2011 Simon J.D. Prince Logistic regression Bayesian logistic regression Nonlinear logistic regression Kernelization and Gaussian process classification Incremental fitting, boosting and trees Multiclass classification Random classification trees Nonprobabilistic classification Applications
3.Models for machine vision 3 Computer vision: models, learning and inference. ©2011 Simon J.D. Prince
4.Example application: Gender Classification 4 Computer vision: models, learning and inference. ©2011 Simon J.D. Prince
5.Type 1: Model Pr( w  x )  Discriminative How to model Pr( w  x )? Choose an appropriate form for Pr( w ) Make parameters a function of x Function takes parameters q that define its shape Learning algorithm : learn parameters q from training data x , w Inference algorithm : just evaluate Pr( wx ) 5 Computer vision: models, learning and inference. ©2011 Simon J.D. Prince
6.Logistic Regression Consider two class problem. Choose Bernoulli distribution over world. Make parameter l a function of x Model activation with a linear function creates number between . Maps to with 6 Computer vision: models, learning and inference. ©2011 Simon J.D. Prince
7.Two parameters Learning by standard methods (ML,MAP, Bayesian) Inference: Just evaluate Pr( wx ) 7 Computer vision: models, learning and inference. ©2011 Simon J.D. Prince
8.Neater Notation To make notation easier to handle, we Attach a 1 to the start of every data vector Attach the offset to the start of the gradient vector f New model: 8 Computer vision: models, learning and inference. ©2011 Simon J.D. Prince
9.Logistic regression 9 Computer vision: models, learning and inference. ©2011 Simon J.D. Prince
10.Maximum Likelihood 10 Computer vision: models, learning and inference. ©2011 Simon J.D. Prince Take logarithm Take derivative:
11.Derivatives Unfortunately, there is no closed form solution– we cannot get an expression for f in terms of x and w Have to use a general purpose technique: “iterative nonlinear optimization” 11 Computer vision: models, learning and inference. ©2011 Simon J.D. Prince
12.Optimization Goal: How can we find the minimum? Basic idea: Start with estimate Take a series of small steps to Make sure that each step decreases cost When can’t improve, then must be at minimum Cost function or Objective function 12 Computer vision: models, learning and inference. ©2011 Simon J.D. Prince
13.Local Minima 13 Computer vision: models, learning and inference. ©2011 Simon J.D. Prince
14.Convexity If a function is convex, then it has only a single minimum. Can tell if a function is convex by looking at 2 nd derivatives 14 Computer vision: models, learning and inference. ©2011 Simon J.D. Prince
15.15 Computer vision: models, learning and inference. ©2011 Simon J.D. Prince
16.Gradient Based Optimization Choose a search direction s based on the local properties of the function Perform an intensive search along the chosen direction. This is called line search Then set 16 Computer vision: models, learning and inference. ©2011 Simon J.D. Prince
17.Gradient Descent Consider standing on a hillside Look at gradient where you are standing Find the steepest direction downhill Walk in that direction for some distance (line search) 17 Computer vision: models, learning and inference. ©2011 Simon J.D. Prince
18.Finite differences What if we can’t compute the gradient? Compute finite difference approximation: where e j is the unit vector in the j th direction 18 Computer vision: models, learning and inference. ©2011 Simon J.D. Prince
19.Steepest Descent Problems Close up 19 Computer vision: models, learning and inference. ©2011 Simon J.D. Prince
20.Second Derivatives In higher dimensions, 2 nd derivatives change how much we should move in the different directions: changes best direction to move in. 20 Computer vision: models, learning and inference. ©2011 Simon J.D. Prince
21.Newton’s Method Approximate function with Taylor expansion Take derivative Rearrange 21 Computer vision: models, learning and inference. ©2011 Simon J.D. Prince Adding line search (derivatives taken at time t )
22.Newton’s Method Matrix of second derivatives is called the Hessian. Expensive to compute via finite differences. If positive definite, then convex 22 Computer vision: models, learning and inference. ©2011 Simon J.D. Prince
23.Newton vs. Steepest Descent 23 Computer vision: models, learning and inference. ©2011 Simon J.D. Prince
24.Line Search Gradually narrow down range 24 Computer vision: models, learning and inference. ©2011 Simon J.D. Prince
25.Optimization for Logistic Regression Derivatives of log likelihood: Positive definite! 25 Computer vision: models, learning and inference. ©2011 Simon J.D. Prince
26.26 Computer vision: models, learning and inference. ©2011 Simon J.D. Prince
27.Maximum likelihood fits 27 Computer vision: models, learning and inference. ©2011 Simon J.D. Prince
28.Structure 28 28 Computer vision: models, learning and inference. ©2011 Simon J.D. Prince Logistic regression Bayesian logistic regression Nonlinear logistic regression Kernelization and Gaussian process classification Incremental fitting, boosting and trees Multiclass classification Random classification trees Nonprobabilistic classification Applications
29.29 Computer vision: models, learning and inference. ©2011 Simon J.D. Prince

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