Use of Grothendieck s Inequality in Interval Computations: Quadratic Terms are Estimated Accurately (Modulo a Constant Factor)

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1 Use of Grothendieck s Inequality in Interval Computations: Quadratic Terms are Estimated Accurately (Modulo a Constant Factor) Olga Kosheleva and Vladik Kreinovich University of Texas at El Paso El Paso, TX 79968, USA olgak@utep.edu, vladik@utep.edu Page 1 of 14

2 1. Interval Computations (IC): Brief Reminder One of the main problem of interval computations: Given: a function f(x 1,..., x n ) and intervals Compute: the range x i = [x i, x i ] = [ x i i, x i + i ], y = f(x 1,..., x n ) = {f(x 1,..., x n ) : x i x i }. Computing the exact range is known to be NP-hard, even for quadratic f(x 1,..., x n ). So, instead, we compute an enclosure Y excess width wid(y ) wid(y) > 0. y, with One of the most widely used methods of efficiently computing Y is the Mean Value (MV) method: n f Y = f( x 1,..., x n ) + (x 1... x n ) [ i, i ]. x i i=1 Page 2 of 14

3 2. Interval Computations: Reminder (cont-d) Mean Value (MV) method (reminder): n f Y = f( x 1,..., x n ) + (x 1... x n ) [ i, i ]. x i i=1 can be esti- def The ranges of the derivatives f,i = f x i mated, e.g., by using straightforward IC: parse the expression f,i, i.e., represent it as a sequence of elementary arithmetic operations, and replace each operation with numbers by the corresponding operation of interval arithmetic. The Mean Value method has excess width O( 2 ), where def = max i. Page 3 of 14

4 3. Can We Get Better Enclosures? The Mean Value method has excess width O( 2 ) Can we come up with more accurate enclosures? We cannot get too drastic an improvement: even for quadratic functions f(x 1..., x n ), computing the interval range is NP-hard and therefore (unless P=NP), a feasible algorithm with excess width O( 2+ε ) is impossible. What we can do is try to decrease the overestimation of the quadratic term. It turns out that such a possibility follows from an inequality proven by A. Grothendieck in Page 4 of 14

5 4. The MV method is based on the 1st order Mean Value Theorem (MVT): f( x+ x) = f( x)+ f,i ( x+η) x i for some η i [ i, i ]. Instead, we propose to use 3rd order MVT: f( x+ x) = f( x)+ f,i ( x) x i f,ij ( x) x i x j f,ijk ( x + η) x i x j x k. Specifically, we propose to add estimates for ranges of linear, quadratic, and cubic terms. The range of the cubic term is estimated via straightforward interval comp.; the estimate is O( 3 ). The range of the linear term f( x) + f,i ( x) x i can be explicitly described as [ỹ, ỹ + ], where ỹ def = f( x) and = f,i ( x) i. Page 5 of 14

6 5. (cont-d) Reminder: we use the 3rd order MVT: f( x+ x) = f( x)+ f,i ( x) x i f,ij ( x) x i x j f,ijk ( x + η) x i x j x k. Specifically, we propose to add estimates for ranges of linear, quadratic, and cubic terms. The range of the linear term can be computed exactly. The range of the cubic term is O( 3 ) O( 2 ). What remains is to estimate the ( range [ Q, Q] of the quadr. term n def a ij x i x j a ij = 1 ) 2 f,ij( x) on [ 1, 1 ]... [ n, n ]. Page 6 of 14

7 6. Relation to Grothendieck Inequality Problem: estimating the range [ Q, Q] of n a ij x i x j on [ 1, 1 ]... [ n, n ]. def Re-scaling: for z i = x i / i, we have z i [ 1, 1], x i = i z i, and the quadratic form becomes: n def b ij z i z j, with b ij = a ij i j. Thus: Q = max { n b ij z i z j : z i [ 1, 1] Grothendieck s inequality enables us to estimate the maximum Q of a related bilinear function n b(z, t) def = b ij z i t j, z i, t j { 1, 1}. }. Page 7 of 14

8 7. Grothendieck Inequality (cont-d) Auxiliary problem: estimating { n } Q = max b ij z i t j : z i, t j { 1, 1}. This problem is known to be NP-hard. General fact: discrete optimization problems are more complex than continuous ones. Observation: the discrete set { 1, 1} is a unit sphere in 1-D Euclidean space. Interesting: for larger dimensions, a unit sphere is connected (hence not discrete). Grothendieck s idea: consider z i and t j from the unit sphere in a Hilbert space (= -dim. Euclidean space). Page 8 of 14

9 8. Grothendieck s Result and Related Algorithm We want to compute: { n } Q = max b ij z i t j : z i, t j { 1, 1}. We estimate instead: { n } Q def = max b ij z i, t j : z i, t j S. Grothendieck s inequality: for some universal constant 1 K G [1, 1.782], we have Q Q Q. K G Comment: the part Q Q is trivial, since we can have all z i and t j equal to ±e for some unit vector e. Computational result: an ellipsoid method similar to linear programming one can feasibly compute Q. Page 9 of 14

10 9. How to Use This Algorithm to Estimate the Range [ Q, Q] of the Quadratic Part We want to estimate: Q = max{b(z) : z i [ 1, 1]}, where B(z) def = b(z, z) and b(z, t) = n b ij z i t j. We know: Q = max{b(z, t) : z i { 1, 1}, t j { 1, 1}}. Fact: a bilinear f-n b(z, t) attains its max at endpoints. Hence: Q = max{b(z, t) : z i [ 1, 1], t j [ 1, 1]}. Since b(z, t) = B((z + t)/2) B((z t)/2), we have Q 2Q. Clearly, Q Q, hence Q /2 Q Q. From K 1 G Q Q Q, we can now conclude that Q 2K G Q Q. Hence: by computing Q, we can feasibly estimate Q accurately modulo a small constant factor 2K G 3.6. Page 10 of 14

11 10. According to the 3rd order Mean Value Theorem, for x i [ i, i ], we have: f( x + x) = T 1 + T 2 + T 3, where: T 1 def = f( x) + f,i ( x) x i ; T 2 def = a ij x i x j, where a ij = 1 2 f,ij( x); and T 3 def = 1 6 f,ijk ( x + η) x i x j x k. As an enclosure for the range of f, we take the sum of enclosures for T 1, T 2, and T 3. For T 1, we compute the exact range in linear time O(n). For T 3, we use straightforward interval computations and get an enclosure of width O( 3 ) O( 2 ). Page 11 of 14

12 11. (cont-d) To estimate the range [ Q, Q] of the quadratic term T 2 = a ij x i x j, we do the following: compute an auxiliary matrix b ij = a ij i j, and use the ellipsoid method to compute { n } Q def = max b ij z i, t j : z i, t j S. Then, Q 2K G Q Q, with 2 2K G 3.6. Why this is better that the Mean Value method: we still get excess width O( 2 ), but this time, we overestimate the quadratic terms by no more than a known constant factor. Page 12 of 14

13 12. This work was supported in part: by the National Science Foundation grants HRD , HRD , and DUE , and by Grant 1 T36 GM from the National Institutes of Health. Page 13 of 14

14 13. Grothendieck s inequality and related algorithms: N. Alon, A. Naor, Approximating the cut-norm via Grothendieck s inequality, SIAM J. Comp., 35 (2006), No. 4, pp A. Grothendieck, Résumé de la théorie métrique des produits tensoriels topologiques, Boll. Soc. Mat. São- Paulo, 8 (1953), pp G. Pisier, Grothendieck s theorem, past and present, Bulletin of the American Mathematical Society, 49 (2012), No. 2, pp Using 3rd order Mean Value Theorem: O. Kosheleva, How to explain usefulness of different results when teaching calculus: example of the Mean Value Theorem, J. Uncertain Systems, 7 (2013), to appear. Interval Computations... Page 14 of 14