SLIDE 1
Solving polynomial systems
A well studied problem
solving polynomial systems exact methods Gröbner bases numerical methods real roots
- ptimization
Finding one root of a polynomial system Smales 17th problem Pierre - - PowerPoint PPT Presentation
Finding one root of a polynomial system Smales 17th problem Pierre - - PowerPoint PPT Presentation
Finding one root of a polynomial system Smales 17th problem Pierre Lairez Inria Saclay FoCM 2017 Foundations of computational mathematics 15 july 2017, Barcelona Solving polynomial systems subdivision finding one root finding all roots
SLIDE 2
SLIDE 3
Finding one root: a purely numerical question
Bézout bound vs. input size n polynomial equations n variables, degree D degree input size #roots D n (D+n
n
) Dn 2 ∼ 1
2n3
2n n ∼
1 πn
1 2 4n
nn D ≫ n ∼
1 (n−1)!Dn
Dn #roots ≫ input size To compute a single root, do we have to pay for #roots? Exact computation Having one root is having them all (generically). Numerical computation One may approximate one root disregarding the others. Polynomial complexity? Maybe, but only with numerical methods.
2
SLIDE 4
Smale 17th problem
“Can a zero of n complex polynomial equations in n unknowns be found approximately, on the average, in polynomial time with a uniform algorithm?” — S. Smale, 1998 approximate root A point from which Newton’s iteration converges quadratically. polynomial time with respect to the input size.
- n the average with respect to some input distribution.
uniform algorithm A Blum–Shub–Smale machine (a.k.a. real random access machine):
- registers store exact real numbers,
- unit cost arithmetic operations,
- branching on positivity testing.
Infinite precision?! Yes, but we still have to deal with stability issues. The model is very relevant for this problem.
3
SLIDE 5
Another brick in the wall
Problem solved!
Shub, Smale (1990s) Quantitative theory of Newton’s iteration Complexity of numerical continuation Beltrán, Pardo (2009) Randomization Bürgisser, Cucker (2011) Deterministic polynomial average time when D ≪ n or D ≫ n Smoothed analysis Lairez (2017) Derandomization
4
SLIDE 6
Numerical continuation
SLIDE 7
Newton's iteration
z F(z) −dzF −1
- utput
F : Cn → Cn a polynomial map, zk+1 = zk −dzkF −1 ·F(zk).
- Convergerges quadratically fast close to a
regular root.
- May diverge on a open set of initial point.
5
SLIDE 8
The geometry of the basins of attraction is complex...
Convergence of Newton’s iteration for the polynomial z3 −2z +2. In red, the points from which Newton’s iteration do not converge.
(Picture by Henning Makholm.)
6
SLIDE 9
... but we can give sufgicient conditions
Smale (1986)
F : Cn → Cn, a polynomial map. γ(F,x) ≜ sup
k>1
- 1
k!dxF −1 ·dk xF
- 1
k−1 .
γ-Theorem If F(ζ) = 0 and ∥z −ζ∥γ(F,ζ) 3−
- 7
2
then
- Newton(k)(z)−ζ
- 21−2k.
7
SLIDE 10
Numerical continuation
z Ft(z) δt t −dzF †
t
- utput
Ft : Cn → Cn a polynomial system depending continuously on t ∈ [0,1]; z0 a root of F0. zk+1 = zk −dzkF †
tk ·Ftk(zk)
tk+1 = tk +δtk
- Solves any generic system
- How to set the step size δt ?
- How to choose the start system F0?
- How to choose a path?
- How many steps do we need to go
from F0 to F1?
8
SLIDE 11
Condition number of a root
a corner stone
H the space of homogeneous polynomial systems of n equations of degree D in n +1 variables, embedded with some Hermitian norm that is invariant under unitary change of variables. F a polynomial system in H z a root of F in Pn µ(F,z) = sup dP(z,z′) ∥F ′ −F∥ with F ′ ∼ F and F ′(z′) = 0 =
- (dzF)†
- =
1 least singular value of dzF ≃ sup 1 ∥F −F ′∥ where z is a singular root of F ′ 2D− 3
2 γproj(F,z).
9
SLIDE 12
Complexity of numerical computation
Choosing the step size
Ft = 0 ζ z Ft+δt = 0 ζ′ z′ Newton’s iteration We have d(ζ,ζ′) ≲ µ(Ft,ζ)∥ ˙ Ft∥δt. We need d(z,ζ′) ≲ 1 D
3 2 µ(Ft+δt,ζ′)
. It sufgices that δt ≲ 1 D
3 2 µ(Ft,ζ)2 .
Theorem (Shub 2009) One can compute an approximate root of F1 given an approximate root of F0 with #steps 136D
3 2
∫1 µ(Ft,ζt)2∥ ˙ Ft∥dt.
10
SLIDE 13
How to choose the path?
linear interpolation Ft = tF1 +(1− t)F0 a better path? We can imagine the notion of adaptative path, but it is difgicult to make it works. F0 F1 singular system better path
11
SLIDE 14
How to choose the start system?
Deterministic start systems
difgiculty It is not enough to control the conditioning of the start system, we need a grasp
- n what happen along the continuation path.
A { xi = 0, 1 i n (homogeneized in degree D) with its root (0,...,0). Works well when D ≫ n (Armentano, Beltrán, Bürgisser, Cucker, Shub 2016). B { xD
i = 1,
1 i n with its Dn roots. Works well when D ≪ n (Bürgisser, Cucker 2011). C F(x1,...,xn)−F(0,...,0) = 0 with its root (0,...,0).
12
SLIDE 15
Randomization of the start system
SLIDE 16
A randomized start system
Conditioning of a random system
- F ∈ H , random polynomial system, uniformly distributed in S(H ).
- ζ a random root of F = 0, uniformly chosen among the Dn roots.
Theorem (Beltrán, Pardo 2011; Bürgisser, Cucker 2011) E(µ(F,ζ)2) n ·dimH
the input size
- How to sample (F,ζ)? Chicken-and-egg problem?
13
SLIDE 17
Complexity of numerical continuation with random endpoints
F0, F1 random polynomial systems of norm 1, uniformly distributed. ζ0 a random root of F0, uniformly distributed. Ft linear interpolation (normalized to have norm 1). ζt continuation of ζ0. lemma ∀t, Ft is uniformly distributed and ζt is uniformly distributed among its roots. #steps 136D
3 2 dS(F0,F1)
∫1 µ(Ft,ζt)2dt (Shub 2009) E[#steps] 136πD
3 2 E
[∫1 µ(Ft,ζt)2dt ] 136πD
3 2
∫1 E [ µ(Ft,ζt)2] dt (Tonelli’s theorem) = O ( nD
3 2 (input size)
) (Beltrán, Pardo 2011; Bürgisser, Cucker 2011)
14
SLIDE 18
How to sample uniformly a random system and a root?
Beltrán, Pardo (2009)
first try Sample ζ ∈ Pn uniformly, sample F uniformly in {F s.t. F(ζ) = 0 and ∥F∥ = 1}.
F is not uniformly distributed.
BP method Sample a linear system L uniformly, compute its unique root ζ ∈ Pn, sample F uniformly in { F s.t. F(ζ) = 0, dζF = L and ∥F∥ = 1 } .
F and ζ are uniformly distributed.
Solves Smale’s problem with randomization.
Total average complexity O ( nD
3 2 (input size)2)
.
15
SLIDE 19
Smoothed analysis
Bürgisser, Cucker (2011)
average analysis gives little information on the complexity of solving one given system. worst-case analysis is irrelevant here (unbounded close to a system with a singular root). smoothed analysis bridges the gap and gives information on a single system F pertubed by a Gaussian noise ε of variance σ2. This models an input data that is
- nly approximate.
sup
system F
E [ cost of computing one root of F +ε ] = O(σ−1nD
3 2 N 2).
average-case w.r.t. the noise worst-case
16
SLIDE 20
Derandomization
SLIDE 21
Truncation and noise extraction
duplication of random variables
x, a random uniformly distributed variable in [0,1]. x = 0.6044025624180895161178081249104686505290197465315910133226678885000016210273 0.6044025624180895161178081249104686 truncation 0.505290197465315910133226678885000016210273 noise extraction
- The truncation is a random variable that is close to x.
- The noise is an independent from x and uniformly distributed in [0,1].
17
SLIDE 22
Truncation and noise extraction on an odd-dimensional sphere
S(H ) ≃ S2n−1 mesh on S2n−1 (truncation) [0,1]2n−1 S2n−1 (noise) S
- S is a measure preserving map due to Sibuya (1962).
- The noise is nearly uniformly distributed and nearly independent from the truncation.
18
SLIDE 23
Derandomization
Beltràn and Pardo’s randomization approximate root numerical continuation start system target system BP randomization randomness Lairez’s derandomization approximate root? numerical continuation start system
- approx. target sys.
BP randomization noise target system truncation noise ext. No? increase truncation order
Solves Smale’s problem with a deterministic algorithm.
Randomness is in Smale’s question from its very formulation asking for an average analysis.
19
SLIDE 24
Beyond Smale's problem
SLIDE 25
Complexity of numerical algorithms
theory gap applications and observations structured system Can we have interesting complexity bounds, supported by probabilistic analysis, for structured systems, especially sparse systems and low evaluation complexity systems? singular roots Can we design algorithms that find singular roots within nice complexity bounds? better complexity In the setting of Smale’s question, can we reach a quasi-optimal (input size)1+o(1) average complexity?
20
SLIDE 26
Complexity exponent in Smale's problem
total cost = O ( (input size)
- cost of Newton’s iteration
·#steps ) . Beltrán, Pardo (2009) E(#steps) = (input size)1+o(1) Armentano, Beltrán, Bürgisser, Cucker, Shub (2016) E(#steps) = (input size)
1 2 +o(1)
work in progress E(#steps) = poly(n,D) = (input size)o(1)
21
SLIDE 27
Short paths in the condition metric
Beltrán, Shub (2009)
question Given polynomial systems F0 and F1 and a root ζ of F0, how large is inf
path F0 → F1
∫1 µ(Ft,ζt) √ ∥ ˙ Ft∥2 +∥˙ ζt∥2dt? (This upper bounds the minimal number of continuation steps required to go from F0 to F1.) answer Not much! #steps = O ( nD
3 2 +n 1 2 log
( µ(F0,ζ0)µ(F1,ζ1) ))
⇝ E(#steps) = O
( nD3 log(input size) ) with F0 and F1 random but... The construction is not algorithmically useful (one need to know a root of the target system to construct the path).
22
SLIDE 28
Bigger steps with unitary paths
- bservation In H , relatively small pertubation of a typical system F changes everything.
Makes it difgicult to make bigger steps. idea Perform the continuation is a lower dimensional parameter space: We allow only rigid motions of the equations rather than arbitrary deformations. compute one solution
- f each equation
move the hypersurfaces to make the solution match continuously return to the original position
23
SLIDE 29
Unitary paths
More formally... parameter space U(n +1)×···×U(n +1), that is n copy of the unitary group. This has dimension ∼ n3, compare with n · (D+n
n
) . paths Geodesics in the parameter space. randomization Same principle as Beltràn and Pardo’s randomization. complexity E(#steps) = poly(n,D).
24
SLIDE 30
Gràcies! Merci ! ¡Gracias! Thank you! Danke!
Present slides are online at pierre.lairez.fr with bibliographic references.
25
SLIDE 31
References I
Armentano, D., C. Beltrán, P. Bürgisser, F. Cucker, M. Shub (2016). “Condition Length and Complexity for the Solution of Polynomial Systems”. In: Found. Comput. Math. Beltrán, C., L. M. Pardo (2009). “Smale’s 17th Problem: Average Polynomial Time to Compute Afgine and Projective Solutions”. In: J. Amer. Math. Soc. 22.2, pp. 363–385. – (2011). “Fast Linear Homotopy to Find Approximate Zeros of Polynomial Systems”. In:
- Found. Comput. Math. 11.1, pp. 95–129.
Beltrán, C., M. Shub (2009). “Complexity of Bezout’s Theorem. VII. Distance Estimates in the Condition Metric”. In: Found. Comput. Math. 9.2, pp. 179–195. Bürgisser, P., F. Cucker (2011). “On a Problem Posed by Steve Smale”. In: Ann. of Math. (2) 174.3,
- pp. 1785–1836.
Lairez, P. (2017). “A Deterministic Algorithm to Compute Approximate Roots of Polynomial Systems in Polynomial Average Time”. In: Found. Comput. Math.
26
SLIDE 32
References II
Shub, M. (1993). “Some Remarks on Bezout’s Theorem and Complexity Theory”. In: From Topology to Computation: Proceedings of the Smalefest. Springer, New York, pp. 443–455. – (2009). “Complexity of Bezout’s Theorem. VI. Geodesics in the Condition (Number) Metric”. In: Found. Comput. Math. 9.2, pp. 171–178. Shub, M., S. Smale (1993a). “Complexity of Bézout’s Theorem. I. Geometric Aspects”. In: J.
- Amer. Math. Soc. 6.2, pp. 459–501.
– (1993b). “Complexity of Bezout’s Theorem. II. Volumes and Probabilities”. In: Computational Algebraic Geometry (Nice, 1992). Vol. 109. Progr. Math. Birkhäuser Boston, Boston, MA,
- pp. 267–285.
– (1993c). “Complexity of Bezout’s Theorem. III. Condition Number and Packing”. In: J. Complexity 9.1, pp. 4–14. – (1994). “Complexity of Bezout’s Theorem. V. Polynomial Time”. In: Theoret. Comput. Sci. 133.1. Selected papers of the Workshop on Continuous Algorithms and Complexity (Barcelona, 1993), pp. 141–164.
27
SLIDE 33