Things

1
Prof. Neil Gershenfeld Director http//cba.mit.e
du/neilg NSF CCR-0122419
2
Programmable Self-Assembly
code (Butera)
amino acid nanotubes (Zhang)
adlayer on monolayer (Mosley)
building components (Sass)
tiles (Griffith)
3
Graphical Algorithms

• Kalman filter recursive
• Hidden Markov model Baum-Welch
• Trellis decoder Viterbi
• Density estimation Expectation-Maximization
• FFT divide-and-conquer
• ...
• problem statement? efficient algorithm ?
  distributed implementation

4
Graphical Networks
5
Factor Graphs
(Andi Loeliger, )
p(x,y,c) p(yx,c) p(xc) p(c)
p(c)
c
p(xc)
p(yx,c)
y
x
6
Exponential Estimation
marginalization
maximization
7
Sub-Exponential Estimation?
marginalization
maximization
factorization
8
Semirings
(Ali and McEliece)
sum-product
max-sum
(inference)
(coding)
9
Recursion
Fm
fm
xi
Fn
fn
Gj
xj
10
Sum Product
Belief propagation, , (Loeliger, Yedidia, )
xj
fm
xi
fn
11
Resistors
v1
R1
v1
v2
R2
v1-v2 IR1
v3
v1
v1
v2
I
p(v1,v2 ,I)
R
I
v2-v2 IR2
v2
v2
v3
12
Estimation
p(yn1)
p(yn)
yn1
yn
p(yn1xn1)
p(ynxn)
ML


xn1
xn
p(xn1xn)
p(xn2xn1)
recursive
MAP
13
Feedback,Entrainment
(Ben Vigoda)
xn
yn
p(xn1xn)
p(ynxn)
14
PLL
p(ynxn)
yn
yn1
yn1
15
Thermodynamics
(Jonathan Yedidia)
16
Bethe Approximation
(Yedidia)
fn
xi
17
VariableStretching
y
y
f
g
f
g
x
z
x,z
x,z
h
h
y
y
f
g
h
fgh
x,z
x,z
18
SpanningTrees
y
y
f
g
f
g
x
z
x
z
h
h
x,y
f
g
x
x,z
h
19
Fourier Transform Sum-Product
20
Fourier TransformFactor Graph
y2
y1
y0
x0
f
x1
x2
21
GBPFFT
y2
y1
y0
A
B
C
x0
f
E
D
x1
F
x2
E
B
f
F
D
A
x2 ,x1 ,x0
y0 ,x1 ,x0
y0 ,y1 ,x0
y0 ,y1 ,y2
C
f
F
ABC
DE
x2 ,x1 ,x0
y0 ,x1 ,x0
y0 ,y1 ,x0
y0 ,y1 ,y2
log N
22
Proofs

• P
• easy to evaluate
• easy to obtain
• NP
• easy to evaluate
• not necessarily easy to obtain

• coNP
• not possible to obtain

23
coNP Bounds
(Parrilo, Doyle)

• Prove polynomial non-negativity
• NP-hard
• Seek sum of squares decomposition (SOS)
• Hilbert problem
• Apply Positivstellensatz
• Polynomial expansion of candidates
• Generate relaxations
• Enlarge problem space
• Pose as semidefinite program (SDP)
• Convex linear programming with semidefinite
  matrices
• Test with interior point method
• Polynomial time