# 0.3 Approximation with other norms and error measures  (Page 3/3)

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${||x||}_{p}={\left(\sum _{n},{|x\left(n\right)|}^{p}\right)}^{1/p}$

and finding $\mathbf{x}$ to minimizing this p-norm while satisfying $\mathbf{Ax}=\mathbf{b}$ .

It has been shown this is equivalent to solving a least weighted norm problem for specific weights.

${||x||}_{p}={\left(\sum _{n},w,{\left(n\right)}^{2},{|x\left(n\right)|}^{2}\right)}^{1/2}$

The development follows the same arguments as in the previous section but using the formula [link] , [link] derived in [link]

$\mathbf{x}={\left[{\mathbf{W}}^{\mathbf{T}}\mathbf{W}\right]}^{-\mathbf{1}}{\mathbf{A}}^{\mathbf{T}}{\left[\mathbf{A},{\left[{\mathbf{W}}^{\mathbf{T}}\mathbf{W}\right]}^{-\mathbf{1}},{\mathbf{A}}^{\mathbf{T}}\right]}^{-\mathbf{1}}\mathbf{b}$

with the weights, $w\left(n\right)$ , being the diagonal of the matrix, $\mathbf{W}$ , in the iterative algorithm to give the minimum weighted solution norm in the same way as [link] gives the minimum weighted equation error.

A Matlab program that implements these ideas applied to our pseudoinverse problem with more unknowns than equations (case 3a) is:

% m-file IRLS2.m to find the optimal solution to Ax=b %  minimizing the L_p norm ||x||_p, using IRLS. %  Newton iterative update of solution, x, for  M < N. %  For 2<p<infty, use homotopy parameter K = 1.01 to 2 %  For 0<p<2, use K = approx 0.7 to 0.9 %  csb 10/20/2012 function x = IRLS2(A,b,p,K,KK) if nargin < 5, KK= 10;  end; if nargin < 4, K = .8;  end; if nargin < 3, p = 1.1; end; pk = 2;                                 % Initial homotopy value x  = pinv(A)*b;                         % Initial L_2 solution E = []; for k = 1:KK    if p >= 2, pk = min([p, K*pk]);      % Homotopy update of p       else pk = max([p, K*pk]); end    W  = diag(abs(x).^((2-pk)/2)+0.00001);  % norm weights for IRLS    AW = A*W;                            % applying new weights    x1 = W*AW'*((AW*AW')\b);             % Weighted L_2 solution    q  = 1/(pk-1);                       % Newton's parameter    if p >= 2, x = q*x1 + (1-q)*x; nn=p; % Newton's partial update for p>2       else x = x1; nn=1; end            % no Newton's partial update for p<2    ee = norm(x,nn);  E = [E ee];        % norm at each iteration end; plot(E)

This approach is useful in sparse signal processing and for frame representation.

## The chebyshev, minimax, or ${L}_{\infty }$ Appriximation

The Chebyshev optimization problem minimizes the maximum error:

${ϵ}_{m}=\underset{n}{max}|ϵ\left(n\right)|$

This is particularly important in filter design. The Remez exchange algorithm applied to filter design as the Parks-McClellan algorithm is very efficient [link] . An interesting result is the limit of an ${||\mathbf{x}||}_{p}$ optimization as $p\to \infty$ is the Chebyshev optimal solution. So, the Chebyshev optimal, the minimax optimal, and the ${L}_{\infty }$ optimal are all the same [link] , [link] .

A particularly powerful theorem which characterizes a solution to $\mathbf{Ax}=\mathbf{b}$ is given by Cheney [link] in Chapter 2 of his book:

• A Characterization Theorem: For an $M$ by $N$ real matrix, $\mathbf{A}$ with $M>N$ , every minimax solution $\mathbf{x}$ is a minimax solution of an appropriate $N+1$ subsystem of the $M$ equations. This optimal minimax solution will have at least $N+1$ equal magnitude errors and they will be larger than any of the errors of the other equations.

This is a powerful statement saying an optimal minimax solution will have out of $M$ , at least $N+1$ maximum magnitude errors and they are the minimum size possible. What this theorem doesn't state is which of the $M$ equations are the $N+1$ appropriate ones. Cheney develops an algorithm based on this theorem which finds these equations and exactly calculates this optimal solution in a finite numberof steps. He shows how this can be combined with the minimum ${||\mathbf{e}||}_{p}$ using a large $p$ , to make an efficient solver for a minimax or Chebyshev solution.

This theorem is similar to the Alternation Theorem [link] but more general and, therefore, somewhat more difficult to implement.

## The ${L}_{1}$ Approximation and sparsity

The sparsity optimization is to minimize the number of non-zero terms in a vector. A “pseudonorm", ${||\mathbf{x}||}_{0}$ , is sometimes used to denote a measure of sparsity. This is not convex, so is not really a norm but the convex (in the limit) norm ${||\mathbf{x}||}_{1}$ is close enough to the ${||\mathbf{x}||}_{0}$ to give the same sparsity of solution [link] . Finding a sparse solution is not easy but interative reweighted least squares (IRLS) [link] , [link] , weighted norms [link] , [link] , and a somewhat recent result is called Basis Pursuit [link] , [link] are possibilities.

This approximation is often used with an underdetermined set of equations (Case 3a) to obtain a sparse solution $\mathbf{x}$ .

Using the IRLS algorithm to minimize the ${l}_{p}$ equation error often gives a sparse error if one exists. Using the algorithm in the illustrated Matlab program with $p=1.1$ on the problem in Cheney [link] gives a zero error in equation 4 while using no larger $p$ gives any zeros.

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