Document on Nonlinear Approximation
Added on 2020-05-08
8 Pages926 Words61 Views
Nonlinear ApproximationQuestion 1a)A necessary condition ||x-^xk*||22≤c2k−2β+1 given any y ∈∨RNweconsiderthesetʄ (y) ∆(y)=ArgminZ∈f(y)K−2β+1We shall prove that N∈∞||x-^xk*||22≤c2k−2β+1Indeed k−2β+1=x−∆(Φx) is in N||x-^xk*||22≤c22β−1∗(x−∆(Φx))≤(c22β−1)x−(c22β−1)∆(Φx)= C2xThe last inequality uses the that fact that ∆(Φx) minimizes k−2β+1(z) over f(y) To prove, let ∆beanydecoderforwhichholdsLet Ƞ be any element in N = N(Φ)∧letȠ0 in xLetting Ƞ0 = Ƞ1 + Ƞ2 be any splitting of Ƞ0 into the vector of size support kȠ0 = Ƞ1 + Ƞ2 + Ƞ3Ƞ3 = Ƞ + Ƞ0 since - Ƞ0 = Ƞ1∈∑khence -ΦȠ1= Φ(Ƞ2 + Ƞ3)||x-^xk*||22 = || Ƞ2 + Ƞ3 - ∆¿(Ƞ2 + Ƞ3))||x < C2(Ƞ2 + Ƞ3)≤c2∨¿Ƞ3||x = c2k−2β+1(Ƞ)Where lp space the best k -term , approximation is obtained by leaving K largest component of x ,unchanged and setting all other to 0||x-^xk*||22≤c2k−2β+1(Ƞ)x
Q1b) ||x|Ar = maxKrσk(x)x||x||wlg2 = sup∈q¿{ij|xi|>∈¿Q= (r +1r¿-1B0||x||wlg≤||A||Ar≤B1r-1/p||x||wlg x∈RNTherefore, x∈Ar is equivalent to x ∈wlgσk(x)lq≤∨|x|∨lqk-1/q where k= 1,2....N∈≤∨|x|∨¿wlgK-1/q≤∨|x|∨lgk-1/qσk(x)plg = ∑i∉¿xi|p≤∈p-q∑i∉¿xi|q≤K-(p-q)/q||x||lgp-q||x||lgqFrom this we consider K⋃(lgN)σk(k)lg≤k-rdn(k)x = infsupY{∨|x|∨¿¿x:x∈k⋂Y}the equation is equivalent to En(K)xQ1c) Let ɸ be any matrix which satisfies RIP of order 2k + ́k with δ2k+k≤δ<1 and ́k = k(NK¿2-2/pThen for any 1 ≤p<z,ɸsatifiesthenullspaceproperty∈lp of order 2k with contact c = 21/p-1/21+δ1−δ||ȠT0||l2≤(1+δ)(1−δ)−1∑j=2s¿∨¿¿ȠT1||l2 if j≥1It follows that ||ȠTj+1||l2≤( ́k)Ƞ2−1p||ȠTj||lp||ȠT||lp≤(2k)1p−12||ȠT||l2≤(1+δ)(1−δ)−1(2k)1p−12 ́k12−1p∑j=1sȠTj||lp
≤21p−12(1+δ)(1−δ)−1∨|ȠTc|∨lpQuestion 2Let assume the %dictionary size is 5x10 hereDict=[ 1 6 11 16 21 26 31 36 41 46 2 7 12 17 22 27 32 37 42 47 3 8 13 18 23 28 33 38 43 48 4 9 14 19 24 29 34 39 44 49 5 10 15 20 25 30 35 40 45 50 ];Dict = Dict./repmat(sum(Dict,1),5,1);b1 = [6;7;8;9;10];%size of the dictionaryn1 = size(Dict);A1 = zeros(n1);R1 = b1;H1 = 100;%if iterations are lesser than zero pop the errorif(H1 <= 0) error('No. of iterations has to be greater than zero!')end;
End of preview
Want to access all the pages? Upload your documents or become a member.