Computer Vision Group Prof. Daniel Cremers · l = f =1, n =0.1 l =0.3, f =1.08, n =0.0005 n =0.89 f...
Transcript of Computer Vision Group Prof. Daniel Cremers · l = f =1, n =0.1 l =0.3, f =1.08, n =0.0005 n =0.89 f...
Computer Vision Group Prof. Daniel Cremers
7. Gaussian Processes (contd.)
PD Dr. Rudolph TriebelComputer Vision Group
Machine Learning for Computer Vision
Prediction with a Gaussian Process
In the case of only one test point we have
Now we compute the conditional distribution
where
This defines the predictive distribution.!2
x⇤
K(X,x⇤) =
0
B@k(x1,x⇤)
...k(xN ,x⇤)
1
CA = k⇤
µ⇤ = kT⇤ K
�1t
⌃⇤ = k(x⇤,x⇤)� kT⇤ K
�1k⇤
p(y⇤ | x⇤, X,y) = N (y⇤ | µ⇤,⌃⇤)
PD Dr. Rudolph TriebelComputer Vision Group
Machine Learning for Computer Vision
Implementation
•Cholesky decomposition is numerically stable •Can be used to compute inverse efficiently
!3
Algorithm 1: GP regression
Data: training data (X,y), test data x⇤Input: Hyper parameters �2
f , l, �2n
Kij k(xi,xj)
L cholesky(K + �2yI)
↵ LT \(L\y)E[f⇤] kT
⇤ ↵v L\k⇤var[f⇤] k(x⇤,x⇤)� vTvlog p(y | X) � 1
2yT↵�
Pi logLii � N
2 log(2⇡)
Precomputed during Training
Test Phase
n
l = �f = 1, �n = 0.1
l = 0.3,
�f = 1.08,
�n = 0.0005
�n = 0.89
�f = 1.16
l = 3
PD Dr. Rudolph TriebelComputer Vision Group
Machine Learning for Computer Vision
Varying the Hyperparameters
•20 data samples •GP prediction with
different kernelhyper parameters
!4
−8 −6 −4 −2 0 2 4 6 8−3
−2
−1
0
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−8 −6 −4 −2 0 2 4 6 8−3
−2
−1
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−8 −6 −4 −2 0 2 4 6 8−3
−2
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PD Dr. Rudolph TriebelComputer Vision Group
Machine Learning for Computer Vision
Varying the Hyperparameters
The squared exponential covariance function can be generalized to
where M can be: • : this is equal to the above case
• : every feature dimension has its own length scale parameter
• : here Λ has less than D columns
!5
k(xp,xq) = �2f exp(�
1
2(xp � xq)
TM(xp � xq)) + �2n�pq
M = l�2I
M = diag(l1, . . . , lD)�2
M = ⇤⇤T + diag(l1, . . . , lD)�2
M = I M = diag(1, 3)�2
M =
✓1 �1�1 1
◆+ diag(6, 6)�2
PD Dr. Rudolph TriebelComputer Vision Group
Machine Learning for Computer Vision
Varying the Hyperparameters
!6
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input x1input x2
outp
ut y
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input x1input x2
outp
ut y
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input x1input x2
outp
ut y
PD Dr. Rudolph TriebelComputer Vision Group
Machine Learning for Computer Vision
Estimating the Hyperparameters
To find optimal hyper parameters we need the marginal likelihood:
This expression implicitly depends on the hyper parameters, but y and X are given from the training data. It can be computed in closed form, as all terms are Gaussians. We take the logarithm, compute the derivative and set it to 0. This is the training step.
!7
p(y | X) =
Zp(y | f , X)p(f | X)df
PD Dr. Rudolph TriebelComputer Vision Group
Machine Learning for Computer Vision
Estimating the Hyperparameters
To find optimal hyper parameters we need the marginal likelihood:
!8
p(y | X) =1p
(2⇡)n|K|exp
✓�1
2yTK�1y
◆
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PD Dr. Rudolph TriebelComputer Vision Group
Machine Learning for Computer Vision
Estimating the Hyperparameters
To find optimal hyper parameters we need the marginal likelihood:
!9
p(y | X) =1p
(2⇡)n|K|exp
✓�1
2yTK�1y
◆
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log p(y | X) = �1
2log((2⇡)n|K|)� 1
2yTK�1y
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PD Dr. Rudolph TriebelComputer Vision Group
Machine Learning for Computer Vision
Estimating the Hyperparameters
To find optimal hyper parameters we need the marginal likelihood:
!10
p(y | X) =1p
(2⇡)n|K|exp
✓�1
2yTK�1y
◆
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log p(y | X) = �1
2log((2⇡)n|K|)� 1
2yTK�1y
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@ log p(y | X)
@✓i=
1
2yTK�1 @K
@✓iy � 1
2tr
✓K�1 @K
@✓i
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<latexit sha1_base64="KbLzMChXMKXna/o0wuiG7N97qgg=">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</latexit><latexit sha1_base64="KbLzMChXMKXna/o0wuiG7N97qgg=">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</latexit><latexit sha1_base64="KbLzMChXMKXna/o0wuiG7N97qgg=">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</latexit><latexit sha1_base64="KbLzMChXMKXna/o0wuiG7N97qgg=">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</latexit>
PD Dr. Rudolph TriebelComputer Vision Group
Machine Learning for Computer Vision
Estimating the Hyperparameters
The log marginal likelihood is not necessarily concave, i.e. it can have local maxima.
The local maxima can correspond to sub-optimal solutions.
!11
100 101
10−1
100
characteristic lengthscale
nois
e st
anda
rd d
evia
tion
−5 0 5−2
−1
0
1
2
input, x
outp
ut, y
−5 0 5−2
−1
0
1
2
input, x
outp
ut, y
PD Dr. Rudolph TriebelComputer Vision Group
Machine Learning for Computer Vision
Automatic Relevance Determination
•We have seen how the covariance function can be generalized using a matrix M
•If M is diagonal this results in the kernel function
•We can interpret the as weights for each feature dimension
•Thus, if the length scale of an input dimension is large, the input is less relevant
•During training this is done automatically
!12
k(x,x0) = �f exp
1
2
DX
i=1
⌘i(xi � x0i)
2
!
⌘i
li = 1/⌘i
PD Dr. Rudolph TriebelComputer Vision Group
Machine Learning for Computer Vision
Automatic Relevance Determination
During the optimization process to learn the hyper-parameters, the reciprocal length scale for one parameter decreases, i.e.:
This hyper parameter is not very relevant!!13
3-dimensional data, parameters as they evolve during training
⌘1 ⌘2 ⌘3
⌘1
⌘2
⌘3
Computer Vision Group Prof. Daniel Cremers
Gaussian Processes -Classification
PD Dr. Rudolph TriebelComputer Vision Group
Machine Learning for Computer Vision
Remember the Visualisation
!15
Linear Regression
Logistic Regression
Bayesian Logistic Regression
Bayesian Linear Regression
probabilistic reasoning
from regression toclassification
PD Dr. Rudolph TriebelComputer Vision Group
Machine Learning for Computer Vision
Kernelization as a New Dimension
!16
Linear Regression
Logistic Regression
Bayesian Logistic Regression
Bayesian Linear Regression
Kernelization
Kernel Regression
PD Dr. Rudolph TriebelComputer Vision Group
Machine Learning for Computer Vision
Kernelization as a New Dimension
!17
Linear Regression
Logistic Regression
Bayesian Logistic Regression
Bayesian Linear Regression
Kernelization
Kernel Regression
GP Regression
PD Dr. Rudolph TriebelComputer Vision Group
Machine Learning for Computer Vision
Kernelization as a New Dimension
!18
Linear Regression
Logistic Regression
Bayesian Logistic Regression
Bayesian Linear Regression
Kernel Regression
GP Regression GP Classification
Kernel Classification
PD Dr. Rudolph TriebelComputer Vision Group
Machine Learning for Computer Vision
Gaussian Processes For Classification
In regression we have , in binary classification we have To use a GP for classification, we can apply a sigmoid function to the posterior obtained from the GP and compute the class probability as:
If the sigmoid function is symmetric:then we have . A typical type of sigmoid function is the logistic sigmoid:
!19
y 2 Ry 2 {�1; 1}
p(y = +1 | x) = �(f(x))
�(�z) = 1� �(z)
p(y | x) = �(yf(x))
�(z) =1
1 + exp(�z)
PD Dr. Rudolph TriebelComputer Vision Group
Machine Learning for Computer Vision
Application of the Sigmoid Function
Function sampled from a Gaussian Process
!20
Sigmoid function applied to the GP function
Another symmetric sigmoid function is the cumulative Gaussian:
�(z) =
Z z
�1N (x | 0, 1)dx
PD Dr. Rudolph TriebelComputer Vision Group
Machine Learning for Computer Vision
Visualization of Sigmoid Functions
The cumulative Gaussian is slightly steeper than the logistic sigmoid
!21
PD Dr. Rudolph TriebelComputer Vision Group
Machine Learning for Computer Vision
The Latent Variables
In regression, we directly estimated f asand values of f where observed in the training data. Now only labels +1 or -1 are observed and
f is treated as a set of latent variables.
A major advantage of the Gaussian process classifier over other methods is that it marginalizes over all latent functions rather than maximizing some model parameters.
!22
f(x) ⇠ GP(m(x), k(x,x0))
PD Dr. Rudolph TriebelComputer Vision Group
Machine Learning for Computer Vision
Class Prediction with a GP
The aim is to compute the predictive distribution
!23
p(y⇤ = +1 | X,y,x⇤) =
Zp(y⇤ | f⇤)p(f⇤ | X,y,x⇤)df⇤
�(f⇤)
PD Dr. Rudolph TriebelComputer Vision Group
Machine Learning for Computer Vision
Class Prediction with a GP
The aim is to compute the predictive distribution
we marginalize over the latent variables from the training data:
!24
p(y⇤ = +1 | X,y,x⇤) =
Zp(y⇤ | f⇤)p(f⇤ | X,y,x⇤)df⇤
p(f⇤ | X,y,x⇤) =
Zp(f⇤ | X,x⇤, f)p(f | X,y)df
predictive distribution of the latent variable (from regression)
PD Dr. Rudolph TriebelComputer Vision Group
Machine Learning for Computer Vision
Class Prediction with a GP
The aim is to compute the predictive distribution
we marginalize over the latent variables from the training data:
we need the posterior over the latent variables:
!25
p(y⇤ = +1 | X,y,x⇤) =
Zp(y⇤ | f⇤)p(f⇤ | X,y,x⇤)df⇤
p(f⇤ | X,y,x⇤) =
Zp(f⇤ | X,x⇤, f)p(f | X,y)df
p(f | X,y) =p(y | f)p(f | X)
p(y | X)
likelihood (sigmoid)
prior
normalizer
PD Dr. Rudolph TriebelComputer Vision Group
Machine Learning for Computer Vision
A Simple Example
•Red: Two-class training data •Green: mean function of •Light blue: sigmoid of the mean function
!26
p(f | X,y)
PD Dr. Rudolph TriebelComputer Vision Group
Machine Learning for Computer Vision
But There Is A Problem...
•The likelihood term is not a Gaussian! •This means, we can not compute the
posterior in closed form. •There are several different solutions in the
literature, e.g.: •Laplace approximation •Expectation Propagation •Variational methods
!27
p(f | X,y) =p(y | f)p(f | X)
p(y | X)
PD Dr. Rudolph TriebelComputer Vision Group
Machine Learning for Computer Vision
Laplace Approximation
Consider a general distribution
where
!28
p(z) =1
Zf(z)
<latexit sha1_base64="CxQntPZAFSCFZVcJEk+flDxQNb4=">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</latexit><latexit sha1_base64="CxQntPZAFSCFZVcJEk+flDxQNb4=">AAACkXicZVHbahsxEJW3l6TuJU4DfemLqDGkUMKuKTR9CJj0pVACKdRJaGyMVjtri+iySLPFzmZ/pq/tD/VvOusuJXEGJA5nNGc0Z9JCq4Bx/KcTPXj46PHW9pPu02fPX+z0dl+eBVd6CWPptPMXqQiglYUxKtRwUXgQJtVwnl59avLnP8AH5ew3XBUwNWJuVa6kQKJmvVfF/vVbfsQnuReySurqe50TM+v144N4Hfw+SFrQZ22cznY7cpI5WRqwKLUI4TKJC5xWwqOSGurupAxQCHkl5nBJ0AoDYVqtB6j5gJiM587TscjX7O2KSpgQViall0bgImzmGvJ/bnA7ictGMWz0x/xwWilblAhW/mufl5qj441HPFMeJOoVASG9ogm4XAgyCMnJO0qN7dk7X2q6pTPN9MGWJgUPGQ2v547KF2YINecDTraQQYG3P+atClc2oNCk0Z0EQOlKS42qE7E8EejVkrYY6moY111aS7K5hPvgbHiQEP76vj86bhe0zV6zN2yfJewDG7HP7JSNmWQ37Cf7xX5He9HHaBS1b6NOW7PH7kT05S9fsMpZ</latexit><latexit sha1_base64="CxQntPZAFSCFZVcJEk+flDxQNb4=">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</latexit><latexit sha1_base64="CxQntPZAFSCFZVcJEk+flDxQNb4=">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</latexit>
Z =
Zf(z)dz
<latexit sha1_base64="KJ+LtQckQPa24U0jctbxbT8D8lA=">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</latexit><latexit sha1_base64="KJ+LtQckQPa24U0jctbxbT8D8lA=">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</latexit><latexit sha1_base64="KJ+LtQckQPa24U0jctbxbT8D8lA=">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</latexit><latexit sha1_base64="KJ+LtQckQPa24U0jctbxbT8D8lA=">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</latexit>
PD Dr. Rudolph TriebelComputer Vision Group
Machine Learning for Computer Vision
Laplace Approximation
Consider a general distribution
where
Aim: approximate this with a normal distribution
!29
p(z) =1
Zf(z)
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Z =
Zf(z)dz
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q(f) = N (f | f̂ , A�1)<latexit sha1_base64="LhwFSevBBtaPgSh2BN28G4QY9cc=">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</latexit><latexit sha1_base64="LhwFSevBBtaPgSh2BN28G4QY9cc=">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</latexit><latexit sha1_base64="LhwFSevBBtaPgSh2BN28G4QY9cc=">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</latexit><latexit sha1_base64="LhwFSevBBtaPgSh2BN28G4QY9cc=">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</latexit>
fnew = f � (rr )�1r <latexit sha1_base64="wPiO0NKQB1a58e+E1I1BEGzq9CA=">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</latexit><latexit sha1_base64="wPiO0NKQB1a58e+E1I1BEGzq9CA=">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</latexit><latexit sha1_base64="wPiO0NKQB1a58e+E1I1BEGzq9CA=">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</latexit><latexit sha1_base64="wPiO0NKQB1a58e+E1I1BEGzq9CA=">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</latexit>
A = K�1 +W<latexit sha1_base64="Sl1t+eeH6M9Gx5k/U8ZUBOM8Nas=">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</latexit><latexit sha1_base64="Sl1t+eeH6M9Gx5k/U8ZUBOM8Nas=">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</latexit><latexit sha1_base64="Sl1t+eeH6M9Gx5k/U8ZUBOM8Nas=">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</latexit><latexit sha1_base64="Sl1t+eeH6M9Gx5k/U8ZUBOM8Nas=">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</latexit>
PD Dr. Rudolph TriebelComputer Vision Group
Machine Learning for Computer Vision
Laplace Approximation
where and To compute an iterative approach using Newton’s method has to be used.
The Hessian matrix A can be computed as
where is a diagonal matrix which depends on the sigmoid function.
!30
p(f | X,y) ⇡ q(f | X,y) = N (f | f̂ , A�1)
f̂ = argmaxf
p(f | X,y)
A = �rr log p(f | X,y)|f=f̂
second-order Taylor expansion
f̂
A = K�1 +W
W = �rr log p(y | f)
PD Dr. Rudolph TriebelComputer Vision Group
Machine Learning for Computer Vision
Laplace Approximation
•Yellow: a non-Gaussian posterior •Red: a Gaussian approximation, the mean is
the mode of the posterior, the variance is the negative second derivative at the mode
!31
Now that we have we can compute:
From the regression case we have:
where
This reminds us of a property of Gaussians that we saw earlier!
p(f | X,y)
PD Dr. Rudolph TriebelComputer Vision Group
Machine Learning for Computer Vision
Predictions
!32
p(f⇤ | X,y,x⇤) =
Zp(f⇤ | X,x⇤, f)p(f | X,y)df
⌃⇤ = k(x⇤,x⇤)� kT⇤ K
�1k⇤
p(f⇤ | X,x⇤, f) = N (f⇤ | µ⇤,⌃⇤)
µ⇤ = kT⇤ K
�1f
Linear in f
PD Dr. Rudolph TriebelComputer Vision Group
Machine Learning for Computer Vision
Gaussian Properties (Rep.)
If we are given this: I. II. Then it follows (properties of Gaussians):
III. IV.
where
!33
p(x) = N (x | µ,⌃1)
p(y | x) = N (y | Ax+ b,⌃2)
p(y) = N (y | Aµ+ b,⌃2 +A⌃1AT )
p(x | y) = N (x | ⌃(AT⌃�12 (y � b) + ⌃�1
1 y),⌃)
⌃ = (⌃�11 +AT⌃�1
s A)�1
V[f⇤ | X,y,x⇤] = k(x⇤,x⇤)� kT⇤ (K +W�1)�1k⇤
PD Dr. Rudolph TriebelComputer Vision Group
Machine Learning for Computer Vision
Applying this to Laplace
It remains to compute
Depending on the kind of sigmoid function we • can compute this in closed form (cumulative
Gaussian sigmoid)
• have to use sampling methods or analytical approximations (logistic sigmoid)
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E[f⇤ | X,y,x⇤] = k(x⇤)TK�1f̂
p(y⇤ = +1 | X,y,x⇤) =
Zp(y⇤ | f⇤)p(f⇤ | X,y,x⇤)df⇤
PD Dr. Rudolph TriebelComputer Vision Group
Machine Learning for Computer Vision
A Simple Example
•Two-class problem (training data in red and blue) •Green line: optimal decision boundary •Black line: GP classifier decision boundary •Right: posterior probability
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PD Dr. Rudolph TriebelComputer Vision Group
Machine Learning for Computer Vision
Summary • Kernel methods solve problems by implicitly mapping
the data into a (high-dimensional) feature space
• The feature function itself is not used, instead the algorithm is expressed in terms of the kernel
• Gaussian Processes are Normal distributions over functions
• To specify a GP we need a covariance function (kernel) and a mean function
• More on Gaussian Processes:http://videolectures.net/epsrcws08_rasmussen_lgp/
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