Computer Vision Group Prof. Daniel Cremers. 3. Regression

Transcription

1 Prof. Daniel Cremers 3. Regression

2 Categories of Learning (Rep.) Learnin g Unsupervise d Learning Clustering, density estimation Supervised Learning learning from a training data set, inference on the test data Reinforceme nt Learning No supervision, but a reward function Discriminan t Function No prob. formulation, earns a function from objects to Discriminati ve Model Estimates the posterior prob. for each class labels. 2 Generative Model Est. the likelihoods and use Bayes rule for the post.

3 Categories of Learning Learning Unsupervised Learning clustering, density estimation Supervised Learning learning from a training data set, inference on the test data Reinforcement Learning no supervision, but a reward function Discriminant Function no prob. formulation, learns a function from objects to labels. Discriminative Model estimates the posterior for each class Generative Model est. the likelihoods and use Bayes rule for the post. Computer Vision 3

4 Mathematical Formulation (Rep.) Suppose we are given a set of objects and a set o of object categories (classes). In the learning task we search for a mapping such that similar elements in are mapped to similar elements in. Difference between regression and classification: In regression, is continuous, in classification it is discrete Regression learns a function, classification usually learns class labels For now we will treat regression 4

5 Basis Functions In principal, the elements of can be anything (e.g. real numbers, graphs, 3D objects). To be able to treat these objects mathematically we need functions that map from to. We call these the basis functions. We can also interpret the basis functions as functions that extract features from the input data. Features reflect the properties of the objects (width, height, etc.). 5

6 Simple Example: Linear Regression Assume: Given: data points (identity) Goal: predict the value t of a new example x Parametric formulation: t 5 t 3 t 4 t 1 t 2 x 1 x 2 x 3 x 4 x 5 6

7 Linear Regression To evaluate the function y, we need an error function: Sum of Squared Errors We search for parameters s.th. is minimal: y(x i, w) =w 0 + w 1 x i ) ry(x i, w) =(1 x i ) Using vector notation: x i := (1 x i ) T y(x i, w) =w T x i re(w) = NX w T x i x T i NX t i x T i = (0 0) ) w T N X x i x T i = NX t i x T i i=1 i=1 i=1 {z } =:A T i=1 {z } =:b T 7

8 Now we have: Given: data points Polynomial Regression Data Set Size Model Complexity 8

9 Polynomial Regression We define: And obtain: Basis functions E(w) = 1 2 NX (w T (x i ) t i ) 2 re(w) =w T N X i=1 (x i ) (x i ) T! NX t i (x i ) T i=1 i=1 T 1 Outer Product 1 9

10 Polynomial Regression We define: And obtain: E(w) = 1 2 NX (w T (x i ) t i ) 2 re(w) =w T N X i=1 (x i ) (x i ) T! NX t i (x i ) T i=1 i=1 T 1 T

11 Polynomial Regression We define: And obtain: E(w) = 1 2 NX (w T (x i ) t i ) 2 re(w) =w T N X i=1 (x i ) (x i ) T! NX t i (x i ) T i=1 i=1 11

12 Polynomial Regression Thus, we have: where It follows: Pseudoinverse + 12

13 Computing the Pseudoinverse Mathematically, a pseudoinverse every matrix. + exists for However: If solution of is (close to) singular the direct is numerically unstable. Therefore: Singular Value Decomposition (SVD) is used: = UDV T where matrices U and V are orthogonal matrices D is a diagonal matrix + = VD + U T Then: where contains the D + reciprocal of all non-zero elements of D 13

14 A Simple Example j(x) =x j 14

15 Varying the Sample Size 15

16 The Resulting Model Parameters E+05 6E+05 4E+05 2E+05 0E+00-2E+05-4E+05-6E+05-8E+05 16

17 Other Basis Functions Other basis functions are possible: Gaussian basis function: Sigmoidal basis function: where mean val scale where In both cases a set of mean values is required. These define the locations of the basis functions. 17

18 Gaussian Basis Functions 18

19 Sigmoidal Basis Functions 19

20 Observations The higher the model complexity grows, the better is the fit to the data If the model complexity is too high, all data points are explained well, but the resulting model oscillates very much. It can not generalize well. This is called overfitting. By increasing the size of the data set (number of samples), we obtain a better fit of the model More complex models have larger parameters Problem: How can we find a good model complexity for a given data set with a fixed size? 20

21 Regularization We observed that complex models yield large parameters, leading to oscillation. Idea: Minimize the error function and the magnitude of the parameters simultaneously We do this by adding a regularization term : where λ rules the influence of the regularization. 21

22 Regularization As above, we set the derivative to zero: With regularization, we can find a complex model for a small data set. However, the problem now is to find an appropriate regularization coefficient λ. 22

23 Regularized Results 23

24 The Problem from a Different View Assume that y is affected by Gaussian noise : Thus, we have where 24

25 Maximum Likelihood Estimation Aim: we want to find the w that maximizes p. is the likelihood of the measured data given a model. Intuitively: Find parameters w that maximize the probability of measuring the already measured data t. Maximum Likelihood Estimation We can think of this as fitting a model w to the data t. Note: σ is also part of the model and can be estimated. For now, we assume σ is known. 25

26 Maximum Likelihood Estimation Given data points: Assumption: points are drawn independently from p: where: Instead of maximizing p we can also maximize its logarithm (monotonicity of the logarithm) 26

27 Maximum Likelihood Estimation i i Constant for all w Is equal to The parameters that maximize the likelihood are equal to the minimum of the sum of squared errors 27

28 Maximum Likelihood Estimation i i w ML := arg max w ln p(t x, w, ) = arg min w E(w) =( T ) 1 T t The ML solution is obtained using the Pseudoinverse 28

29 Maximum A-Posteriori Estimation So far, we searched for parameters w, that maximize the data likelihood. Now, we assume a Gaussian prior: Using this, we can compute the posterior (Bayes): Posterior Likelihood Prior Maximum A-Posteriori Estimation (MAP) 29

30 Maximum A-Posteriori Estimation So far, we searched for parameters w, that maximize the data likelihood. Now, we assume a Gaussian prior: Using this, we can compute the posterior (Bayes): strictly: p(w x, t, 1, 2) = p(t x, w, 1)p(w 2) R p(t x, w, 1)p(w 2)dw but the denominator is independent of w and we want to maximize p. 30

31 Maximum A-Posteriori Estimation This is equal to the regularized error minimization. The MAP Estimate corresponds to a regularized error minimization where λ = (σ 1 / σ 2 ) 2 31

32 Summary Regression is a method to find a mathematical model (function) for a given data set Regression can be done by minimizing the sum of squared (SSE) errors, i.e. the distances to the data Maximum-likelihood estimation uses a probabilis-tic representation to fit a model into noisy data Maximum-likelihood under Gaussian noise is equivalent to SSE regression. Maximum-a-posteriori (MAP) estimation assumes a (Gaussian) prior on the model parameters MAP is solved by regularized regression 32

33 Prof. Daniel Cremers Bayesian Linear Regression

34 Bayesian Linear Regression Using MAP, we can find optimal model parameters, but for practical applications two questions arise: What happens in the case of sequential data, i.e. the data points are observed subsequently? Can we model the probability of measuring a new data point, given all old data points? This is called the predictive distribution: New target New data Old targets Old data 34

35 Sequential Data Given: Prior mean and covariance, noise covariance 1.Set 2.Observe data point 3.Formulate the likelihood as a function of w (= Gaussian with mean and covariance ) 4.Multiply the likelihood with the prior normalize (= Gaussian with and ) and 5.This results in a new prior 6.Go back to 1. if there are still data points available 35

36 A Simple Example Our aim to fit a straight line into a set of data points. Assume we have: Basis functions are equal to identity (x) =x Prior mean is zero, prior covariance is, noise variance is Ground truth is where Data points are sampled from ground truth Thus: We want to recover and from the sequentially incoming data points 36

37 Bayesian Line Fitting No data points observed Prior Data Space From: C.M. Bishop Hough Space 37 Line examples drawn from the prior

38 Bayesian Line Fitting One data point observed Ground Truth Likelihood Prior Data Space From: C.M. Bishop Hough Space 38 Line examples drawn from the prior

39 Bayesian Line Fitting Two data points observed Likelihood Prior Data Space From: C.M. Bishop Hough Space 39 Line examples drawn from the prior

40 Bayesian Line Fitting 20 data points observed Likelihood Prior Data Space From: C.M. Bishop Hough Space 40 Line examples drawn from the prior

41 The Predictive Distribution We obtain the predictive distribution by integrating over all possible model parameters: New data likelihood Old data posterior As before the posterior is prop. to the likelihood times the prior. But now, we don t maximize. The posterior can be computed analytically, as the prior is Gaussian. where Prior cov Prior mean 41

42 The Predictive Distribution (2) Example: Sinusoidal data, 9 Gaussian basis functions, 1 data point The predictive distribution From: C.M. Bishop 42 Some samples from the posterior

43 Predictive Distribution (3) Example: Sinusoidal data, 9 Gaussian basis functions, 2 data points The predictive distribution From: C.M. Bishop 43 Some samples from the posterior

44 Predictive Distribution (4) Example: Sinusoidal data, 9 Gaussian basis functions, 4 data points The predictive distribution From: C.M. Bishop 44 Some samples from the posterior

45 Predictive Distribution (5) Example: Sinusoidal data, 9 Gaussian basis functions, 25 data points The predictive distribution From: C.M. Bishop 45 Some samples from the posterior

46 Summary A model that has been found using regression can be evaluated in different ways (e.g. loss function, crossvalidation, leave-one-out, BIC) This can be used to adjust model parameters such as using a validation data set Bayesian Linear Regression operates on sequential data and provides the predictive distribution When using Gaussian priors (and Gaussian noise), all computations can be done analytically, as all probabilities remain Gaussian 46