For Describing Uncertainty, Which Set S 0 Should . . . Ellipsoids - PowerPoint PPT Presentation

Need for Describing . . . Need for Describing . . . Convex Set-Based . . . Which Set S 0 Should . . . For Describing Uncertainty, Which Set S 0 Should . . . Ellipsoids Are Better than Main Conclusion: . . . Ellipsoids Are . . . Generic Polyhedra and Ellipsoids Are . . . Acknowledgments Probably Better than Boxes: Home Page A Remark Title Page ◭◭ ◮◮ Olga Kosheleva and Vladik Kreinovich ◭ ◮ University of Texas at El Paso Page 1 of 10 El Paso, TX 79968, USA Go Back olgak@utep.edu, vladik@utep.edu Full Screen Close Quit

Need for Describing . . . Need for Describing . . . 1. Need for Describing Sets of Possible Values Convex Set-Based . . . • Measurement and estimates are never 100% accurate. Which Set S 0 Should . . . Which Set S 0 Should . . . • As a result, we usually do not know the exact value of Main Conclusion: . . . a physical quantity. Ellipsoids Are . . . • We usually know the set of possible values of this quan- Ellipsoids Are . . . tity. Acknowledgments Home Page • For a single quantity, this set is usually an interval. Title Page • Representing an interval in a computer is easy: e.g., we can represent an interval by its endpoints. ◭◭ ◮◮ • For several quantities x 1 , . . . , x n : ◭ ◮ Page 2 of 10 – in addition to interval bounds on each of these quantities, Go Back – we often have additional restrictions on their com- Full Screen binations. Close Quit

Need for Describing . . . Need for Describing . . . 2. Need for Describing Sets of Possible Values Convex Set-Based . . . (cont-d) Which Set S 0 Should . . . • In addition to interval bounds on quantities x 1 , . . . , x n , Which Set S 0 Should . . . we often have restrictions on their combinations. Main Conclusion: . . . Ellipsoids Are . . . • As a result, the set of possible values of x = ( x 1 , . . . , x n ) Ellipsoids Are . . . can have different shapes. Acknowledgments • The space of all possible sets is infinite-dimensional. Home Page • This means that we need infinitely many real-valued Title Page parameters to represent a generic set. ◭◭ ◮◮ • In a computer, at any given time, we can only store ◭ ◮ finitely many parameters. Page 3 of 10 • So, we cannot represent generic sets exactly. Go Back • We need to approximate them by sets from a finite- Full Screen parametric family. Close Quit

Need for Describing . . . Need for Describing . . . 3. Convex Set-Based Representation of Sets Convex Set-Based . . . • When x i are spatial coordinates, we can use a different Which Set S 0 Should . . . m Which Set S 0 Should . . . coordinate system y i = � t ij · x j . j =1 Main Conclusion: . . . • In view of this, a reasonable way to select a finite- Ellipsoids Are . . . parametric set is: Ellipsoids Are . . . Acknowledgments – to pick a bounded symmetric convex set S 0 with Home Page non-empty interior, and – to use images TS 0 of this set S 0 under arbitrary Title Page linear transformations T . ◭◭ ◮◮ def • If we start with a Euclidean unit ball S 0 = B = ◭ ◮ � n � x 2 x : � i ≤ 1 , we get the family of ellipsoids. Page 4 of 10 i =1 Go Back • If we start with a unit cube C , we get the family of all boxes (plus the corresponding parallelepipeds. Full Screen • We can also pick a symmetric convex polyhedron P . Close Quit

Need for Describing . . . Need for Describing . . . 4. Which Set S 0 Should We Choose? Convex Set-Based . . . • Once we pick a set S 0 , we can (precisely) represent sets Which Set S 0 Should . . . S of the type TS 0 . Which Set S 0 Should . . . Main Conclusion: . . . • If we start with such a set S , we enclose it into a set Ellipsoids Are . . . TS 0 = S . Ellipsoids Are . . . • Then, we want to enclose TS 0 in a set λ · S correspond- Acknowledgments ing to the original S -based representations. Home Page • We get the same original set S = TS 0 back, with λ = 1. Title Page • For sets S which are different from TS 0 , the S 0 -based ◭◭ ◮◮ representation is only approximate. ◭ ◮ • We start with a set S . Page 5 of 10 • We enclose it in a set TS 0 ⊇ S for an appropriate T . Go Back • We then try to enclose TS 0 in a set of the type λ · S . Full Screen • Then we inevitably get λ > 1. Close Quit

Need for Describing . . . Need for Describing . . . 5. Which Set S 0 Should We Choose? ✬✩ Convex Set-Based . . . Which Set S 0 Should . . . ✫✪ Which Set S 0 Should . . . Main Conclusion: . . . Ellipsoids Are . . . • We have S ⊆ TS 0 ⊆ λ · S . Ellipsoids Are . . . • The smaller λ , the better the approximation. Acknowledgments Home Page • As a measure d ( S 0 , S ) of accuracy of approximating S by S 0 , we use the smallest λ : Title Page d ( S 0 , S ) = inf { λ : ∃ T ( S ⊆ TS 0 ⊆ λ · S ) } . ◭◭ ◮◮ ◭ ◮ • This quantity is known as a Banach-Mazur distance between the convex sets S and S 0 . Page 6 of 10 • As a measure of quality Q ( S 0 ) of choosing S 0 , we select Go Back the worst-case approximation accuracy Full Screen def Q ( S 0 ) = sup d ( S 0 , S ) . Close S Quit

Need for Describing . . . Need for Describing . . . 6. Main Conclusion: Ellipsoids Are Better Than Convex Set-Based . . . Generic Polyhedra Which Set S 0 Should . . . • John’s Theorem: for the Euclidean unit ball B , Which Set S 0 Should . . . d ( B, S ) ≤ √ n for all symmetric convex sets S . Main Conclusion: . . . • Thus, we have Q ( B ) ≤ √ n (actually Q ( B ) = √ n ). Ellipsoids Are . . . Ellipsoids Are . . . • Gluskin’s Theorem for polyhedra P , P ′ : Acknowledgments ∃ c > 0 ∀ n ∃ P ∃ P ′ ( d ( P, P ′ ) ≥ c · n ) . Home Page Title Page • For this polyhedron P , we have d ( P ) ≥ c · n . ◭◭ ◮◮ • Moreover, if we take a convex hull P of 2 n points ran- domly selected from a unit Euclidean sphere, then: ◭ ◮ Q ( P ) ≥ c · n with high probability . Page 7 of 10 • For large n , c · n ≫ √ n and thus, Q ( B ) ≪ Q ( P ) . Go Back Full Screen • This shows that for large dimensions, ellipsoids are in- deed better than generic polyhedra. Close Quit

Need for Describing . . . Need for Describing . . . 7. Ellipsoids Are Probably Better Than Boxes Convex Set-Based . . . • A unit ball B and a unit cube C are unit balls Which Set S 0 Should . . . � n � 1 /p def def Which Set S 0 Should . . . | x i | p = { x : � x � p ≤ 1 } in the ℓ p -metric � x � p = � : B p Main Conclusion: . . . i =1 • B is a unit ball in the ℓ 2 -metric: B = B 2 ; Ellipsoids Are . . . • C is a unit ball in the ℓ ∞ -metric: C = B ∞ . Ellipsoids Are . . . Acknowledgments • The exact values of d ( B p , B q ) are known only when Home Page both p and q are on the same side of 2. Title Page • In this case, d ( B p , B q ) = n | 1 /p − 1 /q | . • Example: for p = 1 and q = 2, we get d ( B 1 , B 2 ) = √ n . ◭◭ ◮◮ ◭ ◮ • These values have the property that when p < q , then Page 8 of 10 d ( B p , B q ) ↑ when p ↓ or when q ↑ . Go Back • For n = 2, B 1 (rhombus) and B ∞ (square) are linearly equivalent. Full Screen • Thus, d ( B 1 , B ∞ ) = 0 < d ( B 2 , B ∞ ) (no monotonicity). Close Quit

Need for Describing . . . Need for Describing . . . 8. Ellipsoids Are Probably Better Than Boxes Convex Set-Based . . . (cont-d) Which Set S 0 Should . . . • For n = 2, we have an anomaly: B 1 = TB ∞ . Which Set S 0 Should . . . Main Conclusion: . . . • As a result, d ( B p , B q ) is not monotonic in p and q . Ellipsoids Are . . . • For n > 3, we do not have this anomaly. Ellipsoids Are . . . • Therefore, it is reasonable to conjecture that for n > 3, Acknowledgments d ( B p , B q ) is monotonic in p and q . Home Page • Under this hypothesis, d ( B ∞ , B 1 ) > d ( B 2 , B 1 ) = √ n . Title Page • Thus, Q ( B ∞ ) ≥ d ( B ∞ , B 1 ) > √ n . ◭◭ ◮◮ • Since Q ( B 2 ) = √ n , we therefore conclude that ◭ ◮ Q ( B 2 ) < Q ( B ∞ ) . Page 9 of 10 Go Back • Thus, ellipsoids are better than boxes. Full Screen • This is in line with a general result that, under certain conditions, ellipsoids are the best approximators. Close Quit

9. Acknowledgments Need for Describing . . . Need for Describing . . . Convex Set-Based . . . This work was supported in part: Which Set S 0 Should . . . Which Set S 0 Should . . . • by the National Science Foundation grants HRD-0734825 Main Conclusion: . . . and HRD-1242122 (Cyber-ShARE Center of Excellence) Ellipsoids Are . . . Ellipsoids Are . . . and DUE-0926721, Acknowledgments • by Grants 1 T36 GM078000-01 and 1R43TR000173-01 Home Page from the National Institutes of Health, and Title Page • by a grant on F-transforms from the Office of Naval ◭◭ ◮◮ Research. ◭ ◮ Page 10 of 10 Go Back Full Screen Close Quit