1. CS621: Artificial Intelligence Lecture 13: region counting; perceptron training
Pushpak Bhattacharyya
Computer Science and Engineering Department
IIT Bombay

2. Reduce to n points thru origin The generating function is
Number of regions founded by n hyperplanes in d-dim passing through origin is given by the following recurrence relation
we use generating function as an operating function
Boundary condition:
1 hyperplane in d-dim
n hyperplanes in 1-dim,
Reduce to n points thru origin
The generating function is

4. After all this expansion,
since other two terms become zero

5. This implies also we have, Comparing coefficients of each term in RHS we get,

6. Comparing co-efficients we get

7. Applied To Perceptron No. of dimensions = (m+1)
No. of hyperplanes = 2m

8. Calculating Upper Bound On Total Threshold Functions (1/4)

9. Upper bound contd. (2/4)
Using summation of GP formula,

10. Upper bound contd. (3/4)

11. Upper bound contd. (4/4) No. of hyperplanes = n = 2m
No. of threshold functions = R(n,d) =

12. Functional Mapping of Brain

13. The human brain Insula: Intuition & empathy Pre-frontal cortex:
Self control
Amygdala:
Aggresion
Hippocampus:
Emotional memory
Anterior
Cingulate
Cortex:
Anxiety
Hypothalamus:
Control of Hormones
Pituitary gland:
Mother instincts

14. Functional Map of the Brain