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Mathematical Structures and Modeling 2013. N. 1(27). PP. 38-41
UDC 519.6
FOR DESCRIBING UNCERTAINTY, ELLIPSOIDS ARE BETTER THAN GENERIC POLYHEDRA AND PROBABLY BETTER THAN BOXES: A REMARK
O. Kosheleva, V. Kreinovich
For a single quantity, the set of all possible values is usually an interval. An interval is easy to represent in a computer: e.g., we can store its two endpoints. For several quantities, the set of possible values may have an arbitrary shape. An exact description of this shape requires infinitely many parameters, so in a computer, we have to use a finite-parametric approximation family of sets. One of the widely used methods for selecting such a family is to pick a symmetric convex set and to use its images under all linear transformations. If we pick a unit ball, we end up with ellipsoids; if we pick a unit cube, we end up with boxes and parallelepipeds; we can also pick a polyhedron. In this paper, we show that ellipsoids lead to better approximations of actual sets than generic polyhedra; we also show that, under a reasonable conjecture, ellipsoids are better approximators than boxes.
1. Formulation of the Problem
Need for describing sets of possible values. Measurement and estimates are never 100% accurate. As a result, we usually do not know the exact value of a physical quantity; we usually know the set of possible values of this quantity. For a single quantity, this set is usually an interval. Representing an interval in a computer is easy: e.g., we can represent an interval by its endpoints; see, e.g., [7,10].
For several quantities xx,... , xn, in addition to interval bounds on each of these quantities, we often have additional restrictions on their combinations; as a result, the set of possible values of x = (xi,... ,xn) can have different shapes. The space of all possible sets is infinite-dimensional, meaning that we need infinitely many real-valued parameters to represent a generic set. In a computer, at any given time, we can only store finitely many parameters; so, we cannot represent generic sets exactly, we need to approximate them by sets from a finite-parametric family.
Copyright © 2013 O. Kosheleva, V. Kreinovich
University of Texas at El Paso (USA)
E-mail: [email protected], [email protected]
Mathematical Structures and Modeling. 2013. N 1(27).
39
Convex set-based representation of sets of possible values. In many practical situations, e.g., when x are spatial coordinates, the selection of the quantities is rather arbitrary: we can use a different coordinate system in which, instead of
m
the original quantities x,, we use linear combinations y = Tx, i.e., y = tj • Xj.
j=i
In view of this, a reasonable way to select a finite-parametric set is to pick a bounded symmetric convex set S0 with non-empty interior, and to use images TS0 of this set S0 under arbitrary linear transformations T.
If we start with a Euclidean unit ball S0 = B =f |x : ^ xf < l|, we get
the family of ellipsoids (see, e.g., [1-4,11-14,16]); if we start with a unit cube S0 = C = {x : |x,| < l for all i}, we get the family of all boxes (plus the corresponding parallelepipeds); alternatively, we can also start with a symmetric convex polyhedron P.
Which set S0 should we choose? Once we pick a set S0, we can (precisely) represent sets S of the type TS0. If we start with such a set S, we enclose it into a set TS0 = S, and then, if we want to enclose TS0 in a set A • S corresponding to the original S-based representations, we get the same original set S = TS0 back, with A = l.
For sets S which are different from TS0, the S0-based representation is only approximate. We start with a set S, and we enclose it in a set TS0 D S for an appropriate linear transformation T. If we then try to enclose TS0 in a set of the type A • S, then we inevitably get A > l.
The smaller A, the better the approximation. It is therefore reasonable, as a measure d(S0,S) of accuracy of approximating S by S0, to use the smallest A corresponding to all possible T:
d(S0, S) = inf{A : 3T (S c TS0 C A • S)}.
This quantity is known as a Banach-Mazur distance between the convex sets S and S0; see, e.g., [15,17].
For each "standard" set S0, we get different values A(S0,S) for different sets S. As a measure of quality Q(S0) of choosing S0, it is reasonable to select the worst-case approximation accuracy
Q(S0) =f sup d(S0,S),
S
where the supremum is taken over all possible bounded symmetric convex sets S with non-empty interior.
40 O. Kosheleva, V. Kreinovich. For Describing Uncertainty, Ellipsoids.
2. Main Results
Main conclusion: ellipsoids are better than generic polyhedra. According to the well-known John's Theorem [8,15,17], for the Euclidean unit ball B, we have d(B,S) < jn for all symmetric convex sets S. Thus, we have Q(B) < jn.
On the other hand, according to Gluskin's theorem [6,15,17], there exists a constant c > 0 such that for each dimension n, there exist polyhedra P and P' for which d(P, P') > c■ n and for which, therefore, d(P) > c■ n. Moreover, if we take a convex hull P of 2n points randomly selected from a unit Euclidean sphere, then, with high probability, we get Q(P) > c ■ n. Since for large n, we have c ■ n > jn and therefore, Q(B) c Q(P), this shows that for large dimensions, ellipsoids are indeed better than generic polyhedra.
Additional conclusion: ellipsoids are probably better than boxes. A Euclidean unit ball B (corresponding to ellipsoids) and a unit cube C (corresponding to boxes) can be viewed as particular cases of unit balls Bp =f {x : ||x||p < 1}
def ( n \ 1/p
in the €p-metric ||x||p = I |x^|M : B is a unit ball in the ¿2-metric while
C is a unit ball in the ¿^-metric: B = B2 and C = B^. The exact values of d(Bp, Bq) are known only when both p and q are on the same side of 2; in this case, d(Bp, Bq) = n|1/p-1/q|. In particular, for p =1 and q = 2, we get d(B1, B2) = jn.
These values have the property that when p < q, then d(Bp,Bq) strictly increases when p decreases or when q increases; in other words, the larger the difference between p and q, the larger the value d(Bp,Bq). For values p and q on different sides of 2, this monotonicity does not hold for n = 2, since in this case, B1 (rhombus) and B^ (square) are linearly equivalent and thus, d(B1, B^) = 0. However, for n > 3, we do not have this anomaly and therefore, it is reasonable to conjecture that for n > 3, this monotonicity holds. Under this hypothesis, d(B^,B1) > d(B2,B1) = jn, and thus, Q(B^) > d(B^,B1) > jn. Since Q(B2) = jn, we therefore conclude that Q(B2) < Q(B^) and thus, ellipsoids are better than boxes.
Comment. These results are in line with a general result according to which, under certain conditions, ellipsoids are the best approximators [5,9].
Acknowledgments. This work was supported in part by the National Science Foundation grants HRD-0734825 and HRD-1242122 (Cyber-ShARE Center of Excellence) and DUE-0926721, by Grants 1 T36 GM078000-01 and 1R43TR000173-01 from the National Institutes of Health, and by a grant on F-transforms from the Office of Naval Research.
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