cs621: artificial intelligence lecture 13: region counting; perceptron training
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CS621: Artificial Intelligence Lecture 13: region counting; perceptron training. Pushpak Bhattacharyya Computer Science and Engineering Department IIT Bombay. Number of regions founded by n hyperplanes in d-dim passing through origin is given by the following recurrence relation - PowerPoint PPT PresentationTRANSCRIPT
CS621: Artificial IntelligenceLecture 13: region counting;
perceptron training Pushpak Bhattacharyya
Computer Science and Engineering DepartmentIIT Bombay
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
1,1,, 1 dnddn RRR n
2
2
1,
,1
n
d
R
R
d
n d
ndn yxRyxf
1 1
,),(
22 2
,1
2
1,1
2 2
,1
2 2
1,11
1 1
1,
2 1
,1
1 1
1,
1 1
,
2
),(
),(
),(
),(
n
n
n d
dndn
n
nn
n d
dndn
d
n d
ndn
d
n d
ndn
d
n d
ndn
n d
dndn
d
n d
ndn
yxyxR
yxRyxRyxfx
yxRyxRyxfxy
yxRyxRyxfx
yxRyxf
After all this expansion,
since other two terms become zero
xyyxyxyxyxR
xyRyxRxyRyxR
yxRyxf
n
n
d
dd
n d
ndn
n
nn
d
dd
d
n d
ndn
d
n d
ndn
222
),(
112 2
,
1,1
1
1,
1
,1
2 2
,
1 1
,
1
121
2 2
1,1,1,
2
2222
)(
d
d
d
d
nn
d
d
n d
ndndndn
yx
yxxyxyxy
yxRRR
),(),(),( yxfxyyxfxyxf
This implies
also we have,
Comparing coefficients of each term in RHS we get,
].....)1(...)1()1(1[
]........[2
2)]1(1[
1),(
2),(]1[
22
32
1
1
dd
d
d
d
d
d
yxyxyx
yyyyx
yxyx
yxf
yxyxfxyx
d
n d
ndn yxRyxf
1 1
,),(
1
0
12d
i
n
iC
Comparing co-efficients we get
dnR ,
Applied To Perceptron
• No. of dimensions = (m+1)• No. of hyperplanes = 2m
)1(,2,
mdn mRR
m
iiC
m
0
)12(2
Calculating Upper Bound On Total Threshold Functions (1/4)
m
iimCR
m
m
0
)12(
)1(,22
]2...2221[2
]!
2...
!3
2
!2
2
!12
1[2
]...1[2
2
2
32
32
)12(2
)12(1
)12(
mmmm
mmmm
m
m
CCCmmm
Upper bound contd. (2/4)
]2...2221[2232
)1(,2
mmmm
mmR
Using summation of GP formula,
]12
12[2
)1(
)1(,2
m
m
m
m
mR
Upper bound contd. (3/4)
12
122
)1(
)1(,2 m
m
m
m
mR
)2(
2
2
22
12
122
2
2
2
2
1
m
m
m
mm
m
mm
O
Upper bound contd. (4/4)
• No. of hyperplanes = n = 2m
• No. of threshold functions = R(n,d) = )2(2mO
Functional Mapping of Brain
The human brain
Pre-frontal cortex: Self control
Insula:Intuition &empathy
Anterior CingulateCortex:Anxiety
Hippocampus:Emotional memory
Hypothalamus:Control of HormonesPituitary gland:
Mother instincts
Amygdala:Aggresion
Functional Map of the Brain