The SOI Algorithm

In [34] Abbott et al. have proposed the so-called Stable Order Ideal (SOI) algorithm which is also concerned with the computation of border basis of ideals of points in the presence of (measurement) errors in the input data. However, the SOI algorithm is only concerned with the computation of a stable order ideal O and does not compute (almost) vanishing polynomials.

In this section we will present the underlying ideas of the SOI algorithm and point out the

dierences to the algorithms which were introduced in this thesis. Additionally, we apply the algorithms to the same input data sets and compare the results.

As in [34] and [49], we introduce the following denitions:

Denition 4.7.16. [Empirical point]

Let p ∈ Rⁿ be a real point and let ε = (ε1, ..., εn), with each εi ∈ R⁺0 , be a vector. We call the entries of ε the component-wise tolerances. An empirical point is the pair (p, ε) and will be denoted by p^(ε). The point p is called the specied value and ε is called the tolerance.

Denition 4.7.17. Let p^(ε) be an empirical point. Its ellipsoid of perturbations is dened as

N (p^(ε)) = {˜p ∈ Rⁿ | k˜p − pk_W ≤ 1 }

where k·k_W = kW ·k is the weighted 2-norm (see [36] for details) dened by the diagonal weight matrix

W = diag (¹/ε1, ...,¹/εn) ∈ Mat_n(R) . Denition 4.7.18. Let X^(ε) = n

p^(ε)₁ , ..., p^(ε)s

o be a nite set of empirical points. Each set of points ˜X = {˜p1, ..., ˜p_s} that satises ˜pi ∈ N

p^(ε)_i 

for all 1 ≤ i ≤ s is called an admissible perturbation of X^(ε).

Denition 4.7.19. A nite set of empirical points X^(ε)=n

p^(ε)₁ , ..., p^(ε)s o is called distinct if N

p^(ε)_i 

∩ N p^(ε)_j 

= ∅ for all 1 ≤ i < j ≤ s.

Denition 4.7.20. [Stable order ideal]

An order ideal O is called stable w.r.t. X^(ε) if the evaluation matrix evalX˜(O) has full rank for each admissible perturbation ˜X of X^(ε).

Denition 4.7.21. [Stable border basis]

Let X^(ε) be a nite set of distinct empirical points, let X be the set of specied values of X^(ε), and let O be a quotient basis for the vanishing ideal I (X). If O is stable w.r.t. X^(ε), then the O-border basis for I (X) is called stable w.r.t. X^(ε).

The following denitions are related to rst order approximation and rst order error analysis.

Fur further details please consider [34].

Let e = (e1, ..., em) be indeterminates and let F = R (e) be the eld of rational functions.

Denition 4.7.22. [Multivariate Taylor Expansion]

Using multi-index notation the formal Taylor expansion of f ∈ F at 0 is given by

f = X

|α|≥0

D^αf (0) e^α α! ,

where α = (α1, ..., α_m) ∈ N^m₀ , |α| = α1 + ... + α_m, and α! = α1!...α_m!. Furthermore, D^α = D₁^α1...D^α_m^m with D^j_i = ∂^j/∂e^j_i and e^α = e^α₁¹...e^α_m^m.

Each f ∈ F can be decomposed into components of homogeneous degree such that f =X

k≥0

fk where fk= X

|α|=k

D^αf (0) e^α α! ,

where D^(0...0)f = f. Each polynomial fk is called the homogeneous component of degree k of f .

This concept can also be extended to matrices that contain entries from F .

Denition 4.7.23. Let M ∈ Matr,c(F ) and let us denote the entries of M by mij. We dene Mk, the homogeneous component of degree k of M , as the matrix whose (i, j) entry is the homogeneous component of degree k of mij.

Let X^(ε)=n

p^(ε)₁ , ..., p^(ε)s o be a nite set of distinct empirical points with specied values X ⊂ Rⁿ. It is possible to express admissible perturbations of X^(ε) with the help of sn (error) variables e = (e₁₁, ..., e_s1, ..., e_1n, ..., e_sn).

For this purpose, we let

X (e) = { ˜˜ p₁(e) , ..., ˜p_s(e)} ,

where ˜pi(e) = (p_i1+ e_i1, ..., p_in+ e_in). The coordinates of each perturbed point ˜pi(e) are elements of the polynomial ring R [e]. Naturally, ˜X is an admissible perturbation of X^(ε) if the condition k(ei1, ..., ein)k_W ≤ 1 on the values of the ekj holds for all 1 ≤ i ≤ s, where W = diag (¹/ε1, ...,¹/εn).

After these denitions we are now able to present the SOI algorithm.

Algorithm 28: Stable Order Ideal (SOI) Algorithm Input: A set of distinct empirical points X^(ε)=n

p^(ε)₁ , ..., p^(ε)_s o with specied values X ⊂ Rⁿ and tolerance ε = (ε1, ..., εn), γ ≥ 0, (error) variables e = (e11, ..., esn), and a degree compatible term ordering σ on Tⁿ

Output: An order ideal O

1 O := [1], M0 := (1, ..., 1)^tr∈ Mat_s,1(R), M1:= (0, ..., 0)^tr∈ Mat_s,1(R [e]);

2 L := [t₁, ..., t_`] = all terms of degree 1 ordered decreasingly w.r.t. σ, C := [ ];

3 while L 6= [ ] do

4 for i := 1 to ` do

5 v₀ := homogeneous components of degree 0 of ti( ˜X(e)) ;

6 v1 := homogeneous components of degree 1 of ti( ˜X(e)) ;

7 α₀:= M₀^trM₀
−1

M₀^trv₀;

8 α1:= M₀^trM0

−1

M₀^trv1+ M₁^trv0− M₀^trM1α0− M₁^trM0α0

;

9 %0:= v0− M₀α0;

10 %₁:= v₁− M₀α₁− M₁α₀;

11 Ct∈ Mat_s,sn:= coecient matrix of %1;

12 k := the maximal integer such that σk, the minimal singular value of C1:k,1:sn, is greater than kεk;

13 % := %˜ _1:k;

14 C˜_t:= C_1:k,1:sn;

// ˜C_t⁺ is the pseudoinverse of ˜Ct 15 δ := ˜˜ C_t⁺%˜;

16 if k˜δk > (1 + γ) kεk then

17 M0 := (v0, M0);

18 M₁ := (v₁, M₁);

19 O :=concat([ti] , O);

20 Add to L all elements of {x1ti, ..., xnti} which are not divisible by an element in L or C ;

21 else

22 C := concat([ti] , C);

23 Remove all multiples of ti from L;

24 end

25 end

26 end

27 return O;

Theorem 4.7.24. This is an algorithm which stops after nitely many steps and returns an order ideal O ⊂ Tⁿ. If γ satises supδ∈Dεk%₂₊(δ)k ≤ γ√

s kεk², then O is an order ideal stable w.r.t. to the set of empirical points X^(ε). If |O| = s, then I (X) has a corresponding stable border basis w.r.t. X^(ε).

Proof. Compare [34, Theorem 15].

Remark 4.7.25. As pointed out in [34, page 891], it is necessary to chose a starting value for γ even though sup_δ∈Dεk%₂₊(δ)k may be unknown. As a heuristic Abbott et al. suggest to use a value of γ  1 in case % is approximated well by its homogeneous components of degree 0 and 1.

The approach of the SOI algorithm is quite dierent compared to the AVI/ABM type algorithms.

The whole concept of stable border bases assumes that all the points which are in the input data set X are meaningful and that it is possible to associate a priori a maximal amount of noise with each coordinate of the points. In reality one can expect the measurement error to have a Gaussian distribution in each coordinate. However, this means that it is not easily possible to assign a maximal tolerance. The SOI algorithm therefore heavily relies on a preprocessing phase of the data points which tries to eliminate outliers and tries to cluster points which are close to each other. Some strategies for data clustering and preprocessing which are supposed to work well together with the SOI algorithm are discussed by Abbott et al. in [33]. It should be noted that preprocessing can be quite costly and depending on the algorithm which was used for clustering it can destroy some relations between the original input points. Additionally, the cost of the SOI algorithm itself is signicantly higher than e.g. the cost of the ABM algorithm which is another reason why preprocessing the data is necessary before the SOI algorithm can be applied.

From a theoretical point of view is is nice that the stability of the border basis can be controlled in a much more direct way compared to the algorithms presented in this thesis. Nevertheless it may be a lot more dicult than for the AVI or ABM algorithm to determine a suitable ε for which a stable border basis actually exists.

Remark 4.7.26. The behaviour we just explained also nds its resemblance in the fact that SOI will in general return an order ideal O which contains about s elements. This is not true for the ABM algorithm where it is expected for practical values of ε that|O|  s.

Remark 4.7.27. In case the SOI algorithm returns a set O such that |O| = s the associated stable border basis for I (X) is an exact border basis in the usual sense.