CS621: Artificial Intelligence Lecture 13: region counting; perceptron training

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CS621 Artificial IntelligenceLecture 13 region
counting perceptron training

• Pushpak Bhattacharyya
• Computer Science and Engineering Department
• IIT Bombay

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• 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

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• After all this expansion,
• since other two terms become zero

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

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Comparing co-efficients we get
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Applied To Perceptron

• No. of dimensions (m1)
• No. of hyperplanes 2m

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Calculating Upper Bound On Total Threshold
Functions (1/4)
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Upper bound contd. (2/4)

• Using summation of GP formula,

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Upper bound contd. (3/4)
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Upper bound contd. (4/4)

• No. of hyperplanes n 2m
• No. of threshold functions R(n,d)

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Functional Mapping of Brain
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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
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Functional Map of the Brain