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

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

After all this expansion, since other two terms become zero

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

Comparing co-efficients we get

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

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

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

Upper bound contd. (3/4)

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

Functional Mapping of Brain

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

Functional Map of the Brain