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Unformatted text preview: EE263s Summer 200910 Laurent Lessard EE263s homework 5 1. Curvesmoothing. We are given a function F : [0 , 1] → R (whose graph gives a curve in R 2 ). Our goal is to find another function G : [0 , 1] → R , which is a smoothed version of F . We’ll judge the smoothed version G of F in two ways: • Meansquare deviation from F , defined as D = integraldisplay 1 ( F ( t ) − G ( t )) 2 dt. • Meansquare curvature, defined as C = integraldisplay 1 G ′′ ( t ) 2 dt. We want both D and C to be small, so we have a problem with two objectives. In general there will be a tradeoff between the two objectives. At one extreme, we can choose G = F , which makes D = 0; at the other extreme, we can choose G to be an affine function ( i.e. , to have G ′′ ( t ) = 0 for all t ∈ [0 , 1]), in which case C = 0. The problem is to identify the optimal tradeoff curve between C and D , and explain how to find smoothed functions G on the optimal tradeoff curve. To reduce the problem to a finitedimensional one, we will represent the functions F and G (approximately) by vectors f, g ∈ R n , where f i = F ( i/n ) , g i = G ( i/n ) . You can assume that n is chosen large enough to represent the functions well. Using this representation we will use the following objectives, which approximate the ones defined for the functions above: • Meansquare deviation, defined as d = 1 n n summationdisplay i =1 ( f i − g i ) 2 . • Meansquare curvature, defined as c = 1 n − 2 n − 1 summationdisplay i =2 parenleftbigg g i +1 − 2 g i + g i − 1 1 /n 2 parenrightbigg 2 . In our definition of c , note that g i +1 − 2 g i + g i − 1 1 /n 2 gives a simple approximation of G ′′ ( i/n ). You will only work with this approximate version of the problem, i.e. , the vectors f and g and the objectives c and d . (a) Explain how to find g that minimizes d + μc , where μ ≥ 0 is a parameter that gives the relative weighting of sumsquare curvature compared to sumsquare deviation. Does your method always work? If there are some assumptions you need to make (say, on rank of some matrix, independence of some vectors, etc.), state them clearly. Explain how to obtain the two extreme cases: μ = 0, which corresponds to minimizing d without regard for c , and also the solution obtained as μ → ∞ ( i.e. , as we put more and more weight on minimizing curvature). (b) Get the file curve smoothing.m from the course web site. This file defines a specific vector f that you will use. Find and plot the optimal tradeoff curve between d and c . Be sure to identify any critical points (such as, for example, any intersection of the curve with an axis). Plot the optimal g for the two extreme cases μ = 0 and μ → ∞ , and for three values of μ in between (chosen to show the tradeoff nicely). On your plots of g , be sure to include also a plot of f , say with dotted line type, for reference. Submit your Matlab code....
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 Summer '08
 BOYD,S
 Rank, Invertible matrix, optimal tradeoﬀ curve

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