Vladimir Protasov (Moscow State University, Russia) riant polyhedra for families of linear opera
Jan 11, 2016
Vladimir Protasov (Moscow State University, Russia)
Invariant polyhedra for families of linear operators
d1, , are linear operators in mA A R
11
1/
1,..., {1,..., }
ˆ ( , , ) lim maxk
k
k
m d dk d d m
A A A A
The geometric sense:
ˆ 1 there exists a norm in
such that 1 for all 1, ... ,
d
iA i m
g fR
Taking the unit ball in that norm:
ˆ 1 there exists a symmetric convex body such that int , 1, ... , diM A M M i m g fR
M2A M
1A M
1 11
ˆ inf { 0 | , , are all contractions in some norm}mA A
The Joint spectral radius (JSR)
1960 Rota, Strang (normed algebras)
1988-90 Barabanov, Kozyakin, Gurvits (linear switching systems)
1991 Daubechies, Lagarias, Cohen, Heil, Villemoes,…. (wavelets)
1989-92 Micchelli, Prautzsch, Dyn, Levin, Dahmen, … (approximation theory)
distribution of random series (probability), asymptotics of the partition function (combinatorics, number theory),capacity of codes, counting of non-overlaping words, graph tractability problem, etc.
1/
1,...,ˆ1. For one operator (A) = (A) = lim max | | (Gelfand's formula)
kkjk j d
m A
1If ,... , are commutative, or all symmetric, or all upper (lower) triangular,
or all orthogonal, or all stochastic, then
mA A
1 1ˆ ( ,..., ) max ( ) ,..., ( )m mA A A A
1 1ˆIn general, however ( ,..., ) max ( ) ,..., ( )m mA A A A
Basic properties of JSR
How to compute or estimate ?
11
1/
,..., {1,..., }ˆmax ,
Daubechies, Lagarias, Heil, Collela, Gripenberg.
Exponential complexit
By exhaust
y. Collela
ion ( by definiti
and Heil (1994) to compute JSR for spec
o )
ial
nk
k
k
d dd d m
A A k
2 2-matrices
with relative accuracy = 0.05 looked over all matrix products up to the length k = 19.
Blondel, Tsitsiclis (1997-2000). The problem of JSR computing for rational matrices in NP-hard
The problem, whether JSR is less than 1 (for rational nonnegative matrices) is algorithmicallyundecidable in the dimension d = 47.
There is no polynomial-time algorithm, with respect to both the dimension d and the accuracyfor estimating JSR with the relative deviation
1
1However, there are algorithms polynomial separately in d or in (we shall s .ee)
The convergence to JSR is very slow
11
1/
,..., {1,..., }
d
The convergence to JSR is slow:
ˆmax ,
because our initial norm in may not be suitable for our operators.
The rate of convergence is ,
where C is a consta
kk
k
d dd d m
C
k
A A k
dR
nt, that may be very large.
Extremal norms
Theorem 1 (N. Barabanov, 1988)
1
1
1
a) For any irreducible family of operators ,..., there exists 0
and a norm such that
A , ... , A =
for any
ˆb) For any such a norm one has
max
( ,..., ).
m
m
d
m
A A
x
A A
x x x
RN
F(N)
f
f
1,...,
Let ( ) be the set of all norms in . is an infinite-dimensional pointed convex cone.
: , ( || . || )[ ] max || || ,
For any || . || N the functional ( | . | ) is also
d d
dj
j m
N N N
F N N F u A u u
F
R R
R
a norm.
The geometric sense:
1
1a) For an irreducible pair of operators ,..., there exists 0
and a symmetric convex body (invariant body) s.t.
ˆb) For any invariant body one has
( ,...,
(
)
def
m
m
AM Conv A M
A A
A M M
M
1 ,..., ).mA A
1A M
mA M
M
1 ( ,... , )def
mAM Conv A M A M
Independently. The ‘’dual’’ fact:
M
Theorem 2 (A.Dranishnikov, S.Konyagin, V.Protasov, 1996)
*
* *1
M = B (the polar to B), where B is the unit ball of Barabanov's norm for
the adjoint opeartors ,.
Du
..,
ali
ty:
mA A
How to determine M ?
1
is an arbitrary point , 0,
( ) { , 1,2, 1, , }
( ) , { } , ( ) , lim
is the set of accomulation points of the orbit .
Let ( ) ( ( ), ( )); then ( )
{ }
k
j j j
d
k d d j
dj j k m kj
x x
O x A A x d j k
x y k x O x x x
M x Conv x x x
d
dN
R
R
is nonempty,
dim ( ) and M x d
1 2ˆLet m =2. After possible noramalization we can assume ( , ) 1A A
1 2( ) ( ) ( )A x A x x
1 2fractal or self-similarity property ( )A K A K K
1 2( , )Conv A M A M M
Thus, ( ) ( )x x M x
x
1A x 2A x
1( ) ,O x 2 ( ) ,O x 3( ) ,O x , ( )O x
( )x
approximately with a given relative error 0.
1/ .The algorithm is polynomial w.r.t
The key idea: to compute JSR and the extremal norm (the body) M simultaneously as a polytope.
The geometric algorithm for computing JSR.
Find
The invariant polytope concept
1ˆ We assume ( , ... , ) 1.
We need a polytope such that , 1, ...., .
m
dj
A A
P A P P j m
R
0
1 1 2
1 1
is an arbitrary polytope, centrally-symmetric w.r.t. the origin,
the sequence { } is produced iteratively: ( , ).
The polytope approximates with the relative devi
d
k k k k k
k k
P
P Q AP Conv A P A P
P Q
R
(1 ) / 21 1 1
ation
(1 ) contains at most vertices. dk k k dQ P Q N C
1If we put , then the polytope would may have 2 verices,
and the complexity would be exponential. Actually we do not need to keep
all the vertices of , we always can make a selection so
nk k n
n
P A P P
P
that . n nP Q
kP
1 kA P
m kA P
k kQ AP
(1 ) kQ
1kP
2kP
After 10С Iterations we obtain the desirable approximation 1/ ˆ( ) n
ndiam P
The total number of operations ( 1) / 21 2( , , ) dC d A A
For d=2 the number of operations3/ 2C
For 0.0001 one has to perform 610 arithmetic operations.
In practice it works faster
Reason: in general the convergence1
1
1/
, ,ˆmax
nm
n
d dd d
A A
is very slow.
This is unavoidable, unless we do not know the extremal norm
.
The algorithm iteratively approximates both and the extremal norm.
In many cases this leads to the precise value of JSR
The programm implementations for d =2 with pictures were done by I.Sheipak in 2000 and E.Shatokhin in 2005.
1
0
ˆAssume we conjecture that = ( ).
To prove this conjecture we try to construct an invariant politope , where
= {
The ide
, }
a.
k
A
M
M Conv v v
11
1 1
1
, = { , 1, , },
where = ( ) , is the corresponding eigenvector .
ˆIf for some we have , then is an invarint polytope, and = .
k j k
k k k
M Conv A M j m
A v A v v
k M M M
0
0
A paremeter (0 ,1/2) and a polygon are given.
Iterative ``cutting-angle'' algorithm with parameter .
Converges to a continuous curve ( , ).
De Rham (1949-53). Implemented in curve design,
P
P
numerical methods of extrapolations, approximation theory.
Generalized to ``subdivision schemes'', studied since 1985
by N.Dyn, A.Levin, S.Dubuc, C.Micchelli, W.Dahmen, H.Prautzsch, K.DeBoor, etc.
1 2
1 2
De Rh
0
am ma
; , (0, 0.5
tri
)
ce
0 2
s
1
A A
X
Y
( )M
v
Example 1. De Rham curves.
0What is the smoothness (Holder exponent) of Proble (m ) ?: , P
is reduced to computing JSR of two special 2 x The answer 2 -matric es.
1
21 2
For 0.25 we have
For 0.25 we have
Computation of JSR:
ˆ = ( ) max {1 2 , }.
1ˆ = ( ) 4
For all the polygon
7 .
M has 6 vertices depending2
on .
A
A A
Extremal polytope: , 1, , .iA P P i m
0P
0( , )P
0( , )P
( ) is the tEuler binary otal number opartition f binary f eunct xpanion sionsdb k
1 2 10 1 2 1 , where 2 2 2 {0,1, , 1 }m
m jk d d d d d d
2Clearly, ( ) 1. For 3 one needs to estimate the growth of ( ) as . db k d b k k
2 L.( E) u1 l er , 1( 8)2 7b k
3 (S te( ) rn, 1858( 1 ))b k s k
4 Klosinsky, Alexanderson, Hill( ) / man ( ), 19842 1b k k
What is the asymptotic growth of ( ) as ?db k k
L.Euler (1728), A.Tanturri (1918), K.Mahler (1940), N.de Bruijn (1948) L.Carlitz (1965), D.Knuth (1966), R.Churchhouse (1969), B.Reznick (1990)
The asymptotics of the binary Euler partitioExample 2. n func tion.
1 2
1 2
The asympotic growth is expressed by the JSR of two matrices , .
, are ( 1
Answer:
1, 2 2 1( )
) ( 1) matrices of zeros and ones:
iff
otherwise. 0,i j k
T T
T T
k j i dT
d d
1
1 1 1 0
0 1 1 0
0 1 1 1
0 0 1 1
T
2
1 1 0 0
1 1 1 0
0 1 1 0
0 1 1 1
T
For 5d Example.
1 2 1It appears that either or .
For every the polytope has 4 ( 2) dimensional
ˆ ˆ
face
( )
.
)
s
(
d M d d
T T T
1
Adjasency matrices in the problem of capacity of codes.
The dimension is 2 .
For {(0, , )} we have two 4 4-matrices of zeros and ones. For t
Example 3.
ˆ (1hem
the extremal polyt
5) / 2,
ope
md
D
M
4
1
5 ( ), ( 2, 5 1 ,2 , 5 1 ) , has 32 vertices.
For {(0, , )} we have the extremal polytope ( ),
(2, 5 1, 5 1, 2), has 40 vertices.
For {( , , , )} we have 8 8-matrices,
ˆ (1 5) / 2,
ˆ (
M x v M
D M M x
v M
AD A
1
2
1the extremal polytope ( ), it has 528 vertices (E.S
) 1.8668
hatokhi
...
n, 200
,
5).M M x
1
Does the extremal polytope norm always exist ?
ˆWe assume ( , ... , ) 1. We need a polytope such that , 1, ...., .dm jA A P A P P j m R
Necessary conditions:
1(1) There exists a finite product ... such that ( ) = 1 (the finiteness property).
kd dA A
1(2) The family is product bounded, i.e., || ... || C for all .
kd dA A k
These conditions are still not sufficient. Example: A is a rotation of the plane by an irrational angle.
Guglielmi, Wirth and Zennaro (2005) applied the concept of complex polytope norm.
11,...,
Let , ... , , then , , | | 1 , 1,....,
is a complex polytope.
dk j j j j
j k
a a P z a z z j k
R C
The CPE conjecture. Are conditions (1) and (2) sufficient for the existence of the invariant complex polytope: , 1, ...., ?jA P P j m
This is true for one operator. Guglielmi, Wirth and Zennaro (2005) proved the conjecture for some special cases.
The answer is negative. Counterexamples are already for d=3 (Jungers, Protasov, 2009)
1ˆWe assume ( , ... , ) 1. We need a polytope such that , 1, ...., .dm jA A P A P P j m R
The cyclic tree algorithm (N.Guglielmi, V.Protasov, 2010):
1
1
1/
We look over all products ... of length k N.
Take the product ... , for which ( ) is maximal.
Normalze the opertors so tha
Step 1.
t ( ) = 1.
k
k
d d
kd d
A A
A A
1ˆWe try to prove that ( ,..., ) = . 1mA A
It appears that in practice the invariant polytope ‘’almost always’’ exists.
For more than 99 % of randomly generated matrices
1 We construct an invariant polytope for St ,.ep .., 2. .mP A A
1
2
1
j 1
We take the leading eigenvector of .
Set v , 2, , .k
k j k
d d
d d
v A A
A A v j k
1v 2v
kv
s j rv A v
p i qv A v
Every time we check if the new vertex is in the convex hull of the previous ones (this is a linear programming problem). The algorithm terminates, when there are no new vertices.
The invariant polytope P is the convex hull of all vertices produced by the algorithm
The ‘’dead’’ branches
3v…..
Thank you !
1 be the closed semigroup generated by the operators ,..., mLet A A A
The algorithm converges within finite time, i.e., the invariant polytope
exists, iff all eigenvalues of operators from , except for the leading eigenvaues of
the product and of
Theorem.
its c li
yc
A
c permutations (that equal to 1) are strictly less than 1.
This holds for the vast majority of practical cases (more than 99% of randomly generated matrices).
The dimension d is up to 30-40.