example with numbers - vision.in.tum.de · • regression learns a function, classiﬁcation...

PD Dr. Rudolph TriebelComputer Vision Group

Machine Learning for Computer Vision

“ lowers the probability that the door is open”

Example with Numbers

Assume we have a second sensor:

Then:

(from above)

!1



General Form

Measurements:

Markov assumption: and are conditionally independent given the state .

Recursion

!2



Example: Sensing and Acting

Now the robot senses the door state and acts (it opens or closes the door).

!3



If the door is open, the action “close door” succeeds in 90% of all cases.

State Transitions

The outcome of an action is modeled as a random variable where in our case means “state after closing the door”.

State transition example:

!4



If the state space is continuous:

If the state space is discrete:

For a given action we want to know the probability . We do this by integrating over all possible previous states .

!5

The Outcome of Actions


Machine Learning for Computer Vision !6

Back to the Example



Definition 2.1: Let be a sequence of sensor measurements and actions until time . Then the belief of the current state is defined as

Sensor Update and Action Update

So far, we learned two different ways to update the system state: • Sensor update: • Action update: • Now we want to combine both:

!7



This incorporates the following Markov assumptions:

We can describe the overall process using a Dynamic Bayes Network:

(measurement)

(state)

!8

Graphical Representation

p(xt | x0:t�1, u1:t, z1:t�1) = p(xt | xt�1, ut)

p(zt | x0:t, u1:t, z1:t) = p(zt | xt)



(Bayes)

(Markov)

(Tot. prob.)

(Markov)

(Markov)

!9

The Overall Bayes Filter


Algorithm Bayes_filter :

1. if is a sensor measurement then 2. 3. for all do 4. 5. 6. for all do 7. else if is an action then 8. for all do 9. return


The Bayes Filter Algorithm

!10



Bayes Filter Variants

The Bayes filter principle is used in • Kalman filters • Particle filters • Hidden Markov models • Dynamic Bayesian networks • Partially Observable Markov Decision Processes

(POMDPs)

!11



Summary• Probabilistic reasoning is necessary to deal with

uncertain information, e.g. sensor measurements • Using Bayes rule, we can do diagnostic reasoning

based on causal knowledge • The outcome of a robot‘s action can be described by a

state transition diagram • Probabilistic state estimation can be done recursively

using the Bayes filter using a sensor and a motion update • A graphical representation for the state estimation

problem is the Dynamic Bayes Network

!12

Computer Vision Group Prof. Daniel Cremers

2. Regression



Categories of Learning (Rep.)

no supervision, but a reward function

Learning

Unsupervised Learning

Supervised Learning

Reinforcement Learning

clustering, density estimation

learning from a training data set, inference on

the test data

!14

Regression Classification

target set is discrete, e.g.

Y = [1, . . . , C]

target set is continuous, e.g.

Y = R



Mathematical Formulation (Rep.)

Suppose we are given a set of objects and a set oof 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

!15



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.).

!16

RM



Simple Example: Linear Regression

• Assume: (identity) • Given: data points • Goal: predict the value t of a new example x • Parametric formulation:

x1 x2 x3 x4 x5

t5 t3 t4 t1 t2

!17

ff(x,w) = w0 + w1x

f(x,w⇤)



Linear RegressionTo determine the function f, we need an error function:

We search for parameters s.th. is minimal:

“Sum of Squared Errors”

!18

E(w) =1

2

NX

i=1

(f(xi,w)� ti)2

rE(w) =NX

i=1

(f(xi,w)� ti)rf(xi,w)·= (0 0)

f(x,w) = w0 + w1x ) rf(xi,w) = (1 xi)





Using vector notation:


!19

E(w) =1

2

NX

i=1

(f(xi,w)� ti)2

rE(w) =NX

i=1

(f(xi,w)� ti)rf(xi,w)·= (0 0)

f(x,w) = w0 + w1x ) rf(xi,w) = (1 xi)

f(xi,w) = wTxixi = (1 xi)T )





Using vector notation:


!20

| {z }=:AT

rE(w) =NX

i=1

wTxixTi �

NX

i=1

tixTi = (0 0) ) wT

NX

i=1

xixTi =

NX

i=1

tixTi

| {z }=:bT

E(w) =1

2

NX

i=1

(f(xi,w)� ti)2

rE(w) =NX

i=1

(f(xi,w)� ti)rf(xi,w)·= (0 0)

f(x,w) = w0 + w1x ) rf(xi,w) = (1 xi)

f(xi,w) = wTxixi = (1 xi)T )



Polynomial Regression

Now we have: Given: data points Assume we are given M basis functions

Model Complexity

Data Set Size

!21

f(x,w) = w0 +MX

i=1

wj�j(x) = wT�(x)

f

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We have defined:

Therefore:

Outer Product

!22

E(w) =1

2

NX

i=1

(wT�(xi)� ti)2

rE(w) = wT

NX

i=1

�(xi)�(xi)T

!�

NX

i=1

ti�(xi)T

�T1

�1

“Basis functions”

T

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�T1

�1

�T2

�2




!23

E(w) =1

2

NX

i=1

(wT�(xi)� ti)2

rE(w) = wT

NX

i=1

�(xi)�(xi)T

!�

NX

i=1

ti�(xi)T

We have defined:

Therefore:

T

f(x,w) = wT�(x)




!24

E(w) =1

2

NX

i=1

(wT�(xi)� ti)2

rE(w) = wT

NX

i=1

�(xi)�(xi)T

!�

NX

i=1

ti�(xi)T

We have defined:

Therefore:

T

f(x,w) = wT�(x)




Thus, we have:

where

It follows:“Pseudoinverse”

!25

�+

“Normal Equation”



Computing the Pseudoinverse

Mathematically, a pseudoinverse exists for every matrix . However: If is (close to) singular the direct solution of is numerically unstable. Therefore: Singular Value Decomposition (SVD) is used: where

• matrices U and V are orthogonal matrices

•D is a diagonal matrix Then: where contains the reciprocal of all non-zero elements of D

!26

�+

�

�

�

D+�+ = V D+UT

� = UDV T

�j(x) = xj



A Simple Example

!27

�j(x) = xj



A Simple Example

!28

“Overfitting”



Varying the Sample Size

!29



The Resulting Model Parameters

0.000

0.050

0.100

0.150

0.200

-0.60-0.45-0.30-0.150.000.150.300.450.60

-30

-23

-15

-8

0

8

15

23

-8E+05-6E+05-4E+05-2E+050E+002E+054E+056E+058E+05

!30



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?

!31



RegularizationWe 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.

!32

E(w) =1

2

NX

i=1

(wT�(xi)� ti)2 +

�

2kwk2



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.

!33

rE(w) =NX

i=1

(wT�(xi)� ti)�(xi)T + �wT .

= 0T



Regularized Results

!34



The Problem from a Different View Point

Assume that y is affected by Gaussian noise : where

Thus, we have

!35

t = f(x,w) + ✏

p(t | x,w,�) = N (t; f(x,w),�2)



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.

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.

“Maximum Likelihood Estimation”

!36




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)

!37

p(t | x,w,�) =NY

i=1

p(ti | xi,w,�)

=NY

i=1

N (ti;wT�(xi),�

2)




Constant for all w Is equal to

The parameters that maximize the likelihood are equal to the minimum of the sum of squared errors

!38

i

i2�2

ln p(ti | xi,w,�)

wML := argmaxw

ln p(t | x,w,�) = argminw

E(w) = (�T�)�1�T t




!39

i

i

The ML solution is obtained using the Pseudoinverse

ln p(ti | xi,w,�)



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):

“Maximum A-Posteriori Estimation (MAP)”

!40

Likelihood Prior Posterior

p(w | x, t) / p(t | w,x)p(w)




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:

but the denominator is independent of w and we want to maximize p.

!41

p(w | x, t,�1,�2) =p(t | x,w,�1)p(w | �2)Rp(t | x,w,�1)p(w | �2)dw




This is equal to the regularized error minimization. The MAP Estimate corresponds to a regularized

error minimization where λ = (σ1 / σ2 )2

!42

2�21

2�21



Summary: MAP Estimation

To summarize, we have the following optimization problem:

The same in vector notation:

!43

J(w) =1

2

NX

n=1

(wT�(xn)� tn)2 +

�

2wTw �(xn) 2 RM

J(w) =1

2wT�T�w �w�T t+

1

2tT t+

�

2wTw t 2 RN

� 2 RN⇥M

“Feature

Matrix”



Summary: MAP Estimation

To summarize, we have the following optimization problem:

The same in vector notation:

And the solution is

!44

J(w) =1

2

NX

n=1

(wT�(xn)� tn)2 +

�

2wTw �(xn) 2 RM

J(w) =1

2wT�T�w �w�T t+

1

2tT t+

�

2wTw t 2 RN

w⇤ = (�IM + �T�)�1�T t

Identity matrix of size M by M

E(w) =1

2

NX

i=1

(wT�(xi)� ti)2 +

�

2kwk2

E(w) =1

2

NX

i=1

(wT�(xi)� ti)2 +

�

2kwk2



MLE And MAP

• The benefit of MAP over MLE is that prediction is less sensitive to overfitting, i.e. even if there is only little data the model predicts well.

• This is achieved by using prior information, i.e. model assumptions that are not based on any observations (= data)

• But: both methods only give the most likely model, there is no notion of uncertainty yet

Idea 1: Find a distribution over model parameters (“parameter posterior”)

!45



MLE And MAP

• The benefit of MAP over MLE is that prediction is less sensitive to overfitting, i.e. even if there is only little data the model predicts well.

• This is achieved by using prior information, i.e. model assumptions that are not based on any observations (= data)

• But: both methods only give the most likely model, there is no notion of uncertainty yet

Idea 1: Find a distribution over model parameters Idea 2: Use that distribution to estimate prediction

uncertainty (“predictive distribution”)

!46



When Bayes Meets Gauß

Theorem: If we are given this: I. II. Then it follows (properties of Gaussians):

III. IV.

where

!47

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 µ),⌃)

⌃ = (⌃�11 +AT⌃�1

2 A)�1

”Linear Gaussian Model”

linear dependency

on x

See Bishop’s book for the proof!



When Bayes Meets Gauß

Thus: When using the Bayesian approach, we can do even more than MLE and MAP by using these formulae.

This means: If the prior and the likelihood are Gaussian then the posterior and the normalizer are also Gaussian and we can compute them in closed form.

This gives us a natural way to compute uncertainty!

!48



The Posterior Distribution

Remember Bayes Rule:

With our theorem, we can compute the posterior in closed form (and not just its maximum)! The posterior is also a Gaussian and its mean is the MAP solution.

!49

Likelihood Prior Posterior

p(w | x, t) / p(t | w,x)p(w)



The Posterior Distribution

We have and

From this and IV. we get the posterior covariance:

and the mean: So the entire posterior distribution is

!50

p(t | w,x) = N (t;�w,�21IM )

p(w) = N (w;0,�22IM )

⌃ = (��22 IM + ��2

1 �T�)�1

= �21(�21

�22

IM + �T�)�1

p(w | t,x) = N (w;µ,⌃)

µ = ��21 ⌃�T t

N

Note: So far we only used the training data!

(x, t)



The Predictive Distribution

We obtain the predictive distribution by integrating over all possible model parameters (“inference”):

This distribution can be computed in closed form, because both terms on the RHS are Gaussian. From above we have where and

Parameter posterior Test data likelihood

!51

p(w | t,x) = N (w;µ,⌃)

µ = ��21 ⌃�T t

= �21(�21

�22

IM + �T�)�1⌃

** **

Test data




We obtain the predictive distribution by integrating over all possible model parameters (“inference”):

This distribution can be computed in closed form, because both terms on the RHS are Gaussian. From above we have where and

Parameter posterior Test data likelihood

!52

p(w | t,x) = N (w;µ,⌃)

µ = ��21 ⌃�T t

= �21(�21

�22

IM + �T�)�1⌃w⇤ = (�IM + �T�)�1�T t ) µ

MAP solution

Test data

** **




Using formula III. from above (linear Gaussian),

where

!53

=

ZN (t;�(x)Tw,�)N (w;µ,⌃)dw

= N (t;�(x)Tµ,�2N (x))

�2N (x) = �2 + �(x)T⌃�(x)

** **

**

**

*



The Predictive Distribution (2)

• Example: Sinusoidal data, 9 Gaussian basis functions, 1 data point

From: C.M. Bishop

Some samples from the posterior

The predictive distribution

!54



Predictive Distribution (3)

• Example: Sinusoidal data, 9 Gaussian basis functions, 2 data points

From: C.M. Bishop



!55





From: C.M. Bishop



!56





From: C.M. Bishop



!57



Summary

• Regression can be expressed as a least-squares problem

• To avoid overfitting, we need to introduce a regularisation term with an additional parameter λ • Regression without regularisation is equivalent to


• Regression with regularisation is Maximum A-Posteriori

• When using Gaussian priors (and Gaussian noise), all computations can be done analytically

• This gives a closed form of the parameter posterior and the predictive distribution

!58

example with numbers - vision.in.tum.de · • regression learns a function, classiﬁcation...

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