why is it useful to walk randomly? lászló lovász mathematical institute eötvös loránd...
Post on 01-Jan-2016
215 Views
Preview:
TRANSCRIPT
1
Why is it useful to walk randomly?
László Lovász
Mathematical InstituteEötvös Loránd University
October 2012
2
Random walk on a graph
October 2012
Graph G=(V,E)
3
Random walk on a graph
October 2012
t(v): probability of being
at node v after t steps
1
( )
1( ) ( )
deg( )t t
j N i
i jj
1deg( ) deg( )( ) ( )
2 2t ti i
i i i im m
Stationary distribution
4
Hitting time
H(s,t) = hitting time from s to t
= expected # of steps, starting at s,
before hitting t
k(s,t) = commute time between s and t
= H(s,t) + H(t,s)
October 2012
( )
1( )
ha
( ).deg(
rmonic in
)
:
j N i
f i f ji
f i V
( )
1( )
ha
( )deg(
s pole i
)
n :
j N i
f i V
f i f ji
Every nonconstant function has at least 2 poles.
Harmonic functions
Every function defined on SV (S) has a unique extension harmonic on V \ S.
G=(V,E) graph,
f: V
October 2012 5
S
2
3
1 f(v)= E(f(Zv))
Zv: (random) point where
random walk from v hits Sv
0v
1
f(v)= P(random walk from
v hits t before s)
s t
Harmonic functions and random walks
October 2012 6
0v
1f(v)=electrical potentials t
Harmonic functions and electrical networks
October 2012 7
f(v) = position of nodes
0 1
Harmonic functions and rubber bands
October 2012 8
2( , ) 2 ( , )
force acting on
mu v mR u v
u
Commute time and resistance
October 2012 9
effective resistence between u and v
( )
11 ( , )
d )( )
eg(,
i N s
H i ts
H s t
( )
deg( ) ( ( , ) ( , )) 0i N s
s H i t H s t
Distance from s to t = H(s,t).
t
weight=degree
strength=1
Hitting time and rubber bands
October 2012 10
11
1{
7
1{
5
}1
}12
7
} 1
7
9
3
5
Hitting time and rubber bands
October 2012 11
12
Random maze
October 2012
13
Random maze
October 2012
14October 2012
We obtain every mazewith the same probability!
Random maze
15
Random spanning tree
October 2012
16
- card shuffling
- statistics
- simulation
- counting
- numerical integration
- optimization
- …
Sampling: a general algorithmic task
October 2012
17
{ : ( ) ( , )}L x y xA y
polynomial time algorithm
certificate
October 2012
L: a „language” (a family of graphs, numbers,...)
Sampling: a general algorithmic task
18
{ : ( ) ( , )}L x y xA y
Find: - a certificate
Given: x
- an optimal certificate
- the number of certificates
- a random certificate
(uniform, or given distribution)
October 2012
L: a „language” (a family of graphs, numbers,...)
Sampling: a general algorithmic task
19
One general method for sampling: Random walks
(+rejection sampling, lifting,…)
Construct regular graph with node set V
Want: sample uniformly from V
Simulate (run) random walk for T steps
Output the final node ????????????
mixing timeOctober 2012
Sampling by random walk
Given: convex body K n
Want: volume of K
Not possible in polynomial time, even if an errorof nn/10 is allowed.
Elekes, Bárány, Füredi
Volume computation
October 2012 20
Dyer-Frieze-Kannan 1989
But if we allow randomization:
There is a polynomial time randomized algorithmthat computes the volume of a convex body
with high probability with arbitrarily small relative error
Volume computation
October 2012 21
B
K
Why not just....
***
*
*
*
*
*
**
* *
**
*
*
*
*
S
| |vol( ) vol( )
| |
S KK B
S
Need exponential size S
to get nonzero!
Volume computation by plain Monte-Carlo
October 2012 22
i iK K B
0B1B
2B
mB
1 10
1 2 0
vol( ) vol( ) vol( )vol( ) vol( )
vol( ) vol( ) vol( )m m
m m
K K KK K
K K K
mK K
0 0K B
1/1 2 n
i iB B
Volume computation by multiphase Monte-Carlo
October 2012 23
1vol( )1 2
vol( )i
i
K
K
Can use Monte-Carlo!
But...Now we have to generate random points from Ki+1.
Need sampling to computethe volume
Volume computation by multiphase Monte-Carlo
October 2012 24
Do sufficiently longrandom walk on centersof cubes in K
Construct sufficiently dense lattice
Pick random point p from little cube
If p is outside K, abort;else return p
Dyer-Frieze-Kannan 1989
Sampling by random walk on lattice
October 2012 25
Sampling by ball walk
October 2012 26
Sampling by hit-and-run walk
October 2012 27
steplength can be large!
Sampling by reflecting walk
October 2012 28
- Stepsize
- Where to start the walk?
- How long to walk?
- How close will be the returned point to random?
Issues with all these walks
October 2012 29
bottleneck
1S 2S1 1 2( ) ( , ' )PS x S x S
1 21
1 2
( , ' )( )
( ) ( ' )
P
P P
x S x SS
x S x S
isoperimetric quantity
inf { ( ) : }S S K
: uniform random point inx K
' : one step fromx x
x
'x
Conductance
October 2012 30
Dyer-Frieze-Kannan 1989 ** 27 * 32 23( ) (( ) )O nO n O n
Polynomial time!
Cost of volume computation
(number of oracle calls)Amortized cost
of sample point
Cost ofsample point
Time bounds
October 2012 31
Dyer-Frieze-Kannan 1989
Lovász-Simonovits 1990
Applegate-Kannan 1990
Lovász 1991
Dyer-Frieze 1991
Lovász-Simonovits 1992,93
Kannan-Lovász-Simonovits 1997
** 27 * 32 23( ) (( ) )O nO n O n
Lovász 1999
** 16 * 41 13( ) (( ) )O nO n O n** 10 * 87(( ) ( ))O n O nO n** 10 * 87(( ) ( ))O n O nO n
* 8 ** 6 7( ) ( )( )O nO n O n* 7 ** 5 6( ) ( )( )O nO n O n* 5 ** 3 4( ) ( )( )O nO n O n
Kannan-Lovász 1999
Lovász-Vempala 2002 * 3( )O n
Lovász-Vempala 2003 * 4( )O n
* 3( )O n
* 3( )O n
Time bounds
October 2012 32
- The Slicing Conjecture
- Reflecting walk
Possibilities for further improvement
October 2012 33
Reflecting random walk in K
v
u
steplength h large
How fast does this mix?
Stationary distribution: uniform
Chain is time-reversible
(e.g. exponentially distributedwith expectation = diam(K))
October 2012 34
Smallest bisecting surface
F H
Smallest bisecting hyperplane
1 1vol ( ) vol ( )n nH F
??
The Slicing Conjecture
October 2012 35
top related