1.
Principal Sensitivity Analysis
Sotetsu Koyamada (Presenter), Masanori Koyama, Ken Nakae, Shin Ishii
Graduate School of Informatics, Kyoto University
@PAKDD2015
May 20, 2015
Ho Chi Minh City, Viet Nam

2.
Table of contents
2
3
4
Sensitivity analysis and PSA
Results
Conclusion
1 Motivation

3.
Prediction and Recognition tasks at high accuracy
Machine learning is awesome
Horikawa et al., 2014
Taigman et al., 2014
Machines can carry out the tasks beyond human capability
Deep Learning matches human in the accuracy of face
Recognition tasks
Predicting the dream contents from Brain activities You can’t do
this unless you are Psychic!

4.
How can the machines carry out the tasks beyond our capability?
How can we learn the machine’s
“secret” knowledge?
In the process of training, machines must have
learned the knowledge not in our natural scope

5.
Machine is a black box
Neural Networks, Nonlinear kernel SVM, …
Input Classification
result
?

6.
Visualizing the knowledge of Linear Model
The knowledge of the linear classifiers like Logistic Regression are
expressible in terms of weight parameters w = (w1,…, wd)
Classifier
Input x = (x1,…, xd)
Classification labels: {0, 1}
wi : weight parameter b: bias parameter
σ : sigmoid activation function
Meaning of wi = Importance of i-th input dimension
within machine’s knowledge

7.
It is extremely difficult to make sense out of weight parameters in
Neural Networks (nonlinear composition of logistic regressions)
Visualizing the knowledge of non Linear Model
Our proposal
We shall directly analyze the behavior of f in the input space!
Meaning of wij
(k) = ???????
h: nonlinear activation function

8.
Table of contents
2
3
4
Sensitivity analysis and PSA
Results
Conclusion
1 Motivation

9.
Sensitivity analysis
Sensitivity analysis compute the sensitivity of f with respect to i-th input dimension
Def. Sensitivity with respect to i-th input dimension
Note
Def. Sensitivity map
Zurada et al., 1994, 97, Kjems et al., 2002
q: true distribution of x
In the case of linear model (e.g. logistic regression)

10.
c.f. Sensitivity with respect to i-th input dimension
PSM: Principal Sensitivity Map
Define the directional sensitivity in the arbitrary direction
and seek the the direction to which the machine is most sensitive
Def. (1st) Principal Sensitivity Map
Def. Directional sensitivity in the direction v
Recall
ei: standard basis of

11.
PSA: Principal Sensitivity Analysis
Define the kernel metric K as:
Def. (1st) Principal Sensitivity Map (PSM)
1st PSM is the dominant eigen vector of K!
When K is covariance matrix,
1st PSM is same as the 1st PC
Recall PSA vs PCA

12.
PSA: Principal Sensitivity Analysis
Define the kernel metric K as:
Def. (1st) Principal Sensitivity Map (PSM)
1st PSM is the dominant eigen vector of K!
k-th PSM is the k-th dominant eigen vector of K!
Def. (k-th) Principal Sensitivity Map (PSM)
When K is covariance matrix,
1st PSM is same as the 1st PC
Recall
k-th PSM := analogue of k-th PC
PSA vs PCA

13.
Table of contents
2
3
4
Sensitivity analysis and PSA
Numerical Experiments
Conclusion
1 Motivation

14.
Digit classification
! Artificial Data Each pixel have the same meaning
! Classifier
– Neural Network (one hidden layer)
– Error percentage: 0.36%
– We applied the PSA to the log of each output from NN
(b) Noisy samples
(a) Templates
c = 0, …, 9
c = 0, ….9

15.
Strength of PSA (relatively signed map)
(b) 1st PSMs (proposed)
(a) (Conventional) sensitivity maps

16.
visualize the
values of
Strength of PSA (relatively signed map)
(Conventional) sensitivity map cannot distinguish
the set of the edges whose presence characterizes the class 1
and
The set of the edges whose absence characterizes the class 1
On the other hand, 1st PSM (proposed) can!
(b) 1st PSMs (proposed)
(a) (Conventional) sensitivity maps
visualize the
values of

17.
Strength of the PSA (sub PSM)
PSMs of f9 (c = 9)
What is the meaning of sub PSMs
(Same as the previous slide)

18.
Strength of the PSA (sub PSM)
PSMs (c = 9)
By definition, globally
important knowledge
Perhaps locally
important knowledge?

19.
Local Sensitivity
Def. Local sensitivity in the region A
sA(v) := EA
∂fc(x)
∂v
2
Expectation over the region A

20.
Local Sensitivity
Measure of the contribution of k-th PSM in the classification of class c
in the subset A
Def. Local sensitivity in the region A
sk
A := sA(vk)
k-th PSM
sA(v) := EA
∂fc(x)
∂v
2
Def. Local sensitivity in the direction of k-th PSM

21.
Local Sensitivity
Measure of the contribution of k-th PSM in the classification of class c
in the subset A
Def. Local sensitivity in the region A
c = 9, k = 1, A = A(9,4) := set of all the samples of the classes 9 and 4
SA
k is The contribution of 1st PSM in the classification of 9 in the data
containing class 9 and 4
= The contribution of 1st PSM in distinguishing 9 from 4.
sk
A := sA(vk)
k-th PSM
sA(v) := EA
∂fc(x)
∂v
2
Def. Local sensitivity in the direction of k-th PSM
Example

22.
Strength of the PSA (sub PSM)
Let’s look at what the knowledge of f9 is doing in distinguishing the
pairs of classes (class 9 vs the other class)
Local sensitivity of k-th PSM of f9 on the subdata
containing class 9 and class c’ ( = A(9, c’) )
c = 9

23.
Strength of the PSA (sub PSM)
Let’s look at what the knowledge of f9 is doing in distinguishing the
pairs of classes (class 9 vs the other class)
Local sensitivity of k-th PSM of f9 on the subdata
containing class 9 and class c’
c = 9
Example: c = 9, c’ = 4, k = 1
Recall
This indicates the contribution of
1st PSM in distinguishing 9 from 4

24.
Strength of the PSA (sub PSM)
Let’s look at what the knowledge of f9 is doing in distinguishing the
pairs of classes (class 9 vs the other class
Local sensitivity of k-th PSM of f9 on the subdata
containing class 9 and class C’ 3rd PSM contributes MUCH
more than the 1st PSM in the
classification of 9 against 4!!!c = 9

25.
In fact….!
PSM
(c = 9, k = 3)
We can visually confirm that the 3rd PSM of f9 is indeed the knowledge
of the machine that helps (MUCH!) in distinguishing 9 from 4!
94

26.
When PSMs are difficult to interpret
• PSMs of NN trained from MNIST data in classifying 10 digits
• Each pixel have different meaning
• In order to applying PSA, Data should be registered

27.
Table of contents
2
3
4
Sensitivity analysis and PSA
Numerical Experiments
Conclusion
1 Motivation

28.
Conclusion
Can identify the sets of input dimensions that acts
oppositely in characterizing the classes
Sub PSMs provide additional information of the machines
(possibly local)
PSA is different from the original sensitivity analysis in that it identifies
the weighted combination of the input dimensions that are essential in
the machine’s knowledge
2
1
Made possible with the definition of the PSMs that allows
negative elements
Merits of PSA
Sotetsu Koyamada
koyamada-s@sys.i.kyoto-u.ac.jpThank you