Extremal products of matrices
3.Computation algorithms and the finiteness property
Vincent Blondel, Raphaël Jungers
UCL
Cant 2009, Liège, Belgium
Outline
• NP-hardness/polynomial
time algorithm
• Computation algorithms
– Geometric methods
– Conic programming methods
– Particular cases
• The finiteness property
– A counterproof
– Binary matrices
• Turing undecidability
• Sturmian words
• Interior point methods
Semidefinite programming
Outline
• Computation algorithms
– Geometric methods
– Conic programming methods
– Particular cases
• The finiteness property
– A counterproof
– Binary matrices
Computation algorithms
How to compute
Theorem. Computing or approximating ρ is NP-hard.
Theorem. The problem ρ≤1 is algorithmically undecidable.
[Tsitsiklis, Blondel, 1997]
[Blondel, Tsitsiklis, 2000]
Conjecture. The problem ρ<1 is algorithmically undecidable.
Computation algorithms
• How to compute
• Idea: find a common good norm on
• Theorem: For all
there exists a norm such that
[Rota Strang, 60]
• So it suffices to find a good norm!
Extremal norms
• Is there a vector norm such that
Extremal norms
• Definition: A norm
is extremal if
• In this case, knowing the extremal norm does the job:
• Imagine an extremal vector norm such that
• Then, the induced matrix norm is extremal
Extremal norms
Does there exist an extremal norm?
Theorem: If
is irreducible, then there exist an extremal norm
Irreducibility is decidable
Extremal norms
How to find an extremal norm?
Any convex set symmetric around the origin defines a norm:
A geometric algorithm…
• Idea:
– Start with a norm (a convex set)
– Apply recursively all the matrices
– Take the convex hull to get a new norm
• Theorem: This algorithm converges to an extremal (or almost
extremal) norm
Moreover each
step allows us to
approximate
better and better
Geometric methods
• Pros:
–
–
–
–
If one knows approximately the extremal norm
Geometric interpretation
Guaranteed convergence Æ at each step, bounds on the accuracy
Can converge in finite time (finiteness property) Æ gives theoretical
proofs
• Cons:
– Implementation tedious
– Extremal norms can have very complex shapes (fractals)
Outline
• Computation algorithms
– Geometric methods
– Conic programming methods
– Particular cases
• The finiteness property
– A counterproof
– Binary matrices
Conic programming methods
• John’s ellipsoid Theorem: Let K be a compact convex set with
nonempty interior symmetric about the origin. Then there is an
ellipsoid E such that
[John 1948]
• So we can approximate the unit ball K of any extremal norm with
an ellipsoid
Conic programming methods
• Theorem: The best ellipsoidal norm
spectral radius up to a factor
Proof:
approximates the joint
K
• Theorem: The best ellipsoidal norm of a set of m (nonnegative)
matrices also approximates the joint spectral radius up to a factor
:
Conic programming methods
• Problem: How to compute the best ellipsoidal norm?
is the set of positive semidefinite matrices (ellipsoids)
• Computable in polynomial time (interior point methods)
Conic programming methods
• How to improve this
-factor approximation scheme?
• We will lift the set of matrices in a bigger dimensional space,
where we will do better approximations
• Idea:
Let us look for a new set of matrices such that
So that
Conic programming methods
•
does the job:
• More generally
• So, if we want an approximation of
approximation factor of
where
up to an arbitrary
, return the solution of
Conic programming methods
• As a conclusion, for any accuracy there is an algorithm that runs in
polynomial time, and that returns an -approximation of
.
However this algorithm is not polynomial in
.
•
There is a kind of duality between n and m:
Recall that
Is there also a way to lift
by leaving m constant?
Conic programming methods
summary
• Pros:
–
–
–
–
–
Leads to theoretical questions (duality, interpretations)
Allows to tune the algorithm/exploit the duality
Generalisations in conic programming
very easy to implement
Also works for the subradius
• Cons:
– Always approximate solutions
Outline
• Computation algorithms
– Geometric methods
– Conic programming methods
– Particular cases
• The finiteness property
– A counterproof
– Binary matrices
Particular cases
Theorem. There is a polynomial time algorithm to decide whether the joint spectral
radius is equal to zero.
Proof:
[Gurvits 1993]
Theorem. If the matrices in the set are normal, then there is a polynomial time
algorithm to compute the joint spectral radius.
[J. 2008]
Proof: The Euclidean norm is extremal.
Theorem. For matrices with nonnegative integer entries, the cases
ρ= 0, ρ=1, ρ > 1 can be distinguished in polynomial time.
[J. Protassov Blondel 2006]
Proof:
Particular cases
Theorem. For matrices with nonnegative integer entries, the cases
ρ= 0, ρ=1, ρ > 1 can be distinguished in polynomial time.
[J. Protassov Blondel 2006]
?
or
?
⎛O
⎞
⎜
⎟
L
L
1
⎟
A=⎜
⎜
⎟
L O
⎜⎜
⎟⎟
O⎠
⎝
⎛O
⎜
1
⎜
A=
⎜
L
⎜⎜
⎝
⎛O
⎜
2
⎜
A=
⎜
L
⎜⎜
⎝
L
O
L
O
⎞
⎟
1⎟
⎟
⎟
1 ⎟⎠
⎞
⎟
⎟
⎟
⎟
O ⎟⎠
<==>
<==>
i
(i, i, j )
( j ', k ', l ')
(i ', j ')
<==>
(i, i )
(i, j , j )
(i '', j '')
Outline
• Computation algorithms
– Geometric methods
– Conic programming methods
– Particular cases
• The finiteness property
– A counterproof
– Binary matrices
The Finiteness property
ρ ( A) = lim || At || 1/ t
t →∞
ρ (Σ) = limmax {|| A || 1/ t : A ∈Σt }
t →∞
ρ ( A ) = ρ ( A)
ρ (Σ ) = ρ (Σ)
ρ ( A) = max {| λi |}
ρ = limsup ρt
t
t
t
t
t →∞
1/ t
The Finiteness property
Question: are these limits reached in finite time?
We cannot hope this for the maximum norm:
⎧⎛ 1 1⎞⎫
Σ = ⎨⎜
⎟⎬ , ρ (Σ) = 1
⎩⎝ 0 1⎠⎭
t 1/ t
but
⎛ 1 1⎞
⎜
⎟
⎝ 0 1⎠
⎛1 t ⎞
= ⎜
⎟
⎝ 0 1⎠
1/ t
≠1
So, could the lower bound reach ρ (Σ) with a finite product?
∃t, A ∈Σt : ρ = ρ ( A)1/ t ?
[Lagarias Wang, 1995]
This would be nice, because it would imply that the stability
Problem ρ <1 is decidable!
The Finiteness property
ρ(A,A) = limk maxAi=A,A |A1 … Ak |1/k
·
A=
3 0
1 3
¸
k
·
A=
k
3
∗
0
3k
¸
A=
·
3 −3
0 −1
¸
k
A=
·
3
0
3.298 < ρ(A,A) < 3.356
Is there an optimal periodic product of the type AAAAAAAAAAAAAAA …
period
In case there is, then ρ(A,A) = ρ
1/5
(AAAAA)
k
∗
(−1)k
¸
Theorem. In a heap of two pieces, a minimal growth rate can always be
obtained with a Sturmian sequence.
[Gaubert, Mairesse, 1999]
Sturmian sequence
The infinite sequence
is a sturmian sequence. If the slope is
rational, the sequence is periodic, otherwise it is not.
Finiteness property
Theorem. Let be a scalar. The maximal growth rate of the matrices
·
1 0
1 1
¸
a
·
1 1
0 1
¸
can always be obtained by a Sturmian product sequence. There are
values of for which no periodic sequence is optimal.
[Blondel, Theys, Vladimirov, 2003]
[Bousch, Mairesse, 2002]
Outline
• Computation algorithms
– Geometric methods
– Conic programming methods
– Particular cases
• The finiteness property
– A counterproof
– Binary matrices
Finiteness property for binary matrices
So it is not the case [Bousch Mairesse 2001, B. et al. 2002, Kozyakin 2005], but…
What about matrices with Binary entries? rational entries?
Nonnegative rational entries?
Conjecture:
Pairs of binary matrices have the finiteness
property
[B. J. Protasov 06]
Finiteness property for binary matrices
• Theorem: There exists an infinite product
1/ t
such that
ρ = lim || Ai Ai ... Ai ||
0
1
Ai0 Ai1 ...
t
So, the conjecture is: « is there such a product that is (ultimately)
periodic? »
ρ = ρ ( A)1/ t ⇔ ρ = lim || AAA... ||1/ kt
k →∞
ω
• For an (infinite) word = i0i1... define its « complexity function » f(t) as
the number of its factors of length t
Theorem : A word is ultimately periodic
iff its complexity function is bounded
iff f(t)=t for one particular t
The finiteness property:why?
• Intuitively it seems reasonable:
Only non-majorated subwords in the optimal product
ex.: If
A1 A0 ≤ A0 A0 → A0 A0 ... A0 A1 A1...
So if there are less than n products of length n
that are not majorated, the finiteness property holds.
• All sets for which we know ρ exactly satisfy it.
1/ 2
For instance:
⎧⎛1 0 ⎞ ⎛ 1 1⎞ ⎫
⎛ ⎛ 1 0 ⎞ ⎛ 1 1⎞ ⎞
1+ 5
Σ = ⎨⎜
⎟,⎜
⎟⎬ : ρ = ρ ⎜ ⎜
⎟ .⎜
⎟⎟
1
1
0
1
1
1
0
1
⎝
⎠
⎝
⎠
⎝
⎠
⎝
⎠⎠
⎩
⎭
⎝
=
2
• No explicit counterexample is known, even with irrational entries.
Finiteness property: Why?
• Graph theoretical interpretation:
Let us represent matrices by bipartite digraphs:
⎛ 0 0 0⎞
⎜
⎟
A0 = ⎜ 0 0 0 ⎟
⎜ 0 1 1⎟
⎝
⎠
⎛ 0 0 0⎞
⎜
⎟
A1 = ⎜ 1 1 1 ⎟
⎜ 0 0 1⎟
⎝
⎠
The norm 1T A1 A1 A0 A1 A1 A0 1 is the number of paths in the corresponding
graph concatenation:
=> If we want to maximize the asymptotic number of paths, is there
always a periodic strategy?
Finiteness property: Why?
• Binary matrices are known to be much easier: recall that the questions
" ρ (Σ) < 1?"
" ρ (Σ) = 1?"
" ρ (Σ) > 1?"
become decidable
• The finiteness property is known to hold in a bunch of cases:
– for symmetric matrices
– If the Lie algebra generated is solvable [liberzon et al. 99]
– For binary matrices, if
ρ (Σ ) ≤ 1
Finiteness property: Some results
Theorem: The finiteness property holds for any set of nonnegative rational
matrices if and only if it holds for any pair of binary matrices
Theorem: The finiteness property holds for any set of rational matrices if
and only if it holds for any pair of matrices with entries in the set {0,1,-1}
[J. B. 2008]
So, if the conjecture is true for {0,-1,1} entries ({0,1} entries)
the stability problem for (nonnegative) rational matrices is
decidable!
Finiteness property: Some results
Theorem: The finiteness property holds for sets of 2 by 2
binary matrices
[J. B. 2008]
Proof: by use of symmetries and simple rules, one reduces to 5 cases,
and then by inspection.
Example:
⎛ 1 1⎞
A1 = ⎜
⎟
0
1
⎝
⎠
⎛ 0 1⎞
A0 = ⎜
⎟
1
0
⎝
⎠
The finiteness propery holds. The optimal product is A1 A1 A1 A0
ρ = ρ ( A1 A1 A1 A0 )
1/ 4
So, what about 3 by 3 matrices?
3 + 13 1/ 4
=(
)
2
>50000
pairs of
matrices!
Finiteness property: Some results
• Pros:
– experiments and calculations
– graphical interpretation
– 2 by 2 matrices
• Cons:
–
is undecidable even for nonnegative rational matrices, so one could
doubt…
– It can be shown that even if the conjecture holds, the length of the period
cannot be a computable function of the dimension
Finiteness property: Some results
Theorem: The finiteness property holds for sets of 2 by 2
Matrices with entries in {0,1,-1}.
[Cicone et al. 2008]
Proof: For all of these sets, there exists a polytopic extremal norm.
Proposition:
The « finiteness
property » does not
hold for the joint
spectral subradius.
Example:
Outline
• Computation algorithms
– Geometric methods
– Conic programming methods
– Particular cases
• The finiteness property
– A counterproof
– Binary matrices
• Conclusion
Conclusion
• Approximation algorithms:
– If the matrices are irreducible, there is an extremal norm
– If not, one can block-triangularize
– Most algorithms then approximate the extremal norm
– Often very impressive results
• The geometric algorithm can be useful for theoretical results
(cfr. Finiteness property)
•
•
Importance of the conjecture: the stability question, that is,
" ρ (Σ) < 1?", would be decidable
Ok for 2X2 matrices
Conclusion
• Joint spectral quantities are intuitive notions
• Many diverse applications:
– Control
– Network design
– Combinatorics
– Information theory
– Curve design
– Numerical analysis
– Number theory
– …
• Very difficult to handle theoretically, but in practice
– Efficient algorithms
– Lead to interesting theoretical questions
Many open questions
Conclusion
• Finiteness property
– Explicit counterexample?
– Decidability?
– Binary matrices?
• Stability
• Convergence/ understanding of algorithms
• Trackable graph design
• Repetition-free words
• Characterize languages that can be expressed with joint spectral quantities
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Enquiries:
Thank you!
Vincent Blondel, Université catholique de Louvain, Belgium
[email protected]
Questions?
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References: http://www.inma.ucl.ac.be/~jungers/
Thank you!
Questions?