0.2 General solutions of simultaneous equations  (Page 4/4)

• If $\mathbf{A}$ is case 1a, (square and nonsingular), then
  $${\mathbf{A}}^{+}={\mathbf{A}}^{-\mathbf{1}}$$
• If $\mathbf{A}$ is case 2a or 2b, (over specified) then
  $${\mathbf{A}}^{+}={\left[{\mathbf{A}}^{\mathbf{T}}\mathbf{A}\right]}^{-\mathbf{1}}{\mathbf{A}}^{\mathbf{T}}$$
• If $\mathbf{A}$ is case 3a, (under specified) then
  $${\mathbf{A}}^{+}={\mathbf{A}}^{\mathbf{T}}{\left[\mathbf{A}{\mathbf{A}}^{\mathbf{T}}\right]}^{-\mathbf{1}}$$

Figure 2. Four Cases with Analytical Solutions

Fortunately, most practical cases are one of these four but even then, it is generally faster and less error prone to use special techniques on the normal equationsrather than directly calculating the inverse matrix. Note the matrices to be inverted above are all $r$ by $r$ ( $r$ is the rank) and nonsingular. In the other six cases from the ten in Figure 1, these would be singular, so alternate methods suchas SVD must be used [link] , [link] , [link] .

In addition to these four cases with “analytical” solutions, we can pose a more general problem by asking for an optimal approximation with a weighted norm [link] to emphasize or de-emphasize certain components or range of equations.

• If $\mathbf{A}$ is case 2a or 2b, (over specified) then the weighted error pseudoinverse is
  $${\mathbf{A}}^{+}={\left[{\mathbf{A}}^{\mathbf{T}*}{\mathbf{W}}^{\mathbf{T}*}\mathbf{W}\mathbf{A}\right]}^{-\mathbf{1}}{\mathbf{A}}^{\mathbf{T}*}{\mathbf{W}}^{\mathbf{T}*}\mathbf{W}$$
• If $\mathbf{A}$ is case 3a, (under specified) then the weighted norm pseudoinverse is
  $${\mathbf{A}}^{+}={\left[{\mathbf{W}}^{\mathbf{T}}\mathbf{W}\right]}^{-\mathbf{1}}{\mathbf{A}}^{\mathbf{T}}{\left[\mathbf{A},{\left[{\mathbf{W}}^{\mathbf{T}}\mathbf{W}\right]}^{-\mathbf{1}},{\mathbf{A}}^{\mathbf{T}}\right]}^{-\mathbf{1}}$$

Figure 3. Three Cases with Analytical Solutions and Weights

These solutions to the weighted approxomation problem are useful in their own right but also serve as the foundation to the Iterative Reweighted Least Squares (IRLS)algorithm developed in the next chapter.

Geometric interpretation and least squares approximation

A particularly useful application of the pseudo-inverse of a matrix is to various least squared error approximations [link] , [link] . A geometric view of the derivation of the normal equations can be helpful. If $\mathbf{b}$ does not lie in the range space of $\mathbf{A}$ , an error vector is defined as the difference between $\mathbf{A}\mathbf{x}$ and $\mathbf{b}$ . A geometric picture of this vector makes it clear that for the length of $\epsilon$ to be minimum, it must be orthogonal to the space spanned by the columns of $\mathbf{A}$ . This means that ${\mathbf{A}}^{*}\epsilon =\mathbf{0}$ . If both sides of [link] are multiplied by ${\mathbf{A}}^{*}$ , it is easy to see that the normal equations of [link] result in the error being orthogonal to the columns of $\mathbf{A}$ and, therefore its being minimal length. If $\mathbf{b}$ does lie in the range space of $\mathbf{A}$ , the solution of the normal equations gives the exact solution of [link] with no error.

For cases 1b, 1c, 2c, 2d, 3a, 3b, and 3c, the homogeneous equation [link] has non-zero solutions. Any vector in the space spanned by these solutions (the null space of $\mathbf{A}$ ) does not contribute to the equation error $\epsilon$ defined in [link] and, therefore, can be added to any particular generalized solution of [link] to give a family of solutions with the same approximation error. If the dimension of the nullspace of $\mathbf{A}$ is $d$ , it is possible to find a unique generalized solution of [link] with $d$ zero elements. The non-unique solution for these seven cases can be written in the form [link] .

where $\mathbf{y}$ is an arbitrary vector. The first term is the minimum norm solution given by the Moore-Penrose pseudo-inverse ${\mathbf{A}}^{+}$ and the second is a contribution in the null space of $\mathbf{A}$ . For the minimum $\left|\right|x\left|\right|$ , the vector $\mathbf{y}=0$ .

Derivations

To derive the necessary conditions for minimizing $q$ in the overspecified case, we differentiate $q={ϵ}^{\mathbf{T}}ϵ$ with respect to $\mathbf{x}$ and set that to zero. Starting with the error

and taking the gradient or derivative gives

which are the normal equations in [link] and the pseudoinverse in [link] and [link] .

If we start with the weighted error problem

using the same steps as before gives the normal equations for the minimum weighted squared error as

and the pseudoinverse as

To derive the necessary conditions for minimizing the Euclidian norm ${\left|\right|x\left|\right|}_2$ when there are few equations and many solutions to [link] , we define a Lagrangian

take the derivatives in respect to both $\mathbf{x}$ and $\mu$ and set them to zero.

and

Solve these two equation simultaneously for $\mathbf{x}$ eliminating $\mu$ gives the pseudoinverse in [link] and [link] result.

Because the weighting matrices $\mathbf{W}$ are diagonal and real, multiplication and inversion is simple. These equations are used in the Iteratively Reweighted LeastSquares (IRLS) algorithm described in the next chapter.

Regularization

To deal with measurement error and data noise, a process called “regularization" is sometimes used [link] , [link] , [link] .

Least squares approximation with constraints

The solution of the overdetermined simultaneous equations is generally a least squared error approximation problem. A particularly interesting anduseful variation on this problem adds inequality and/or equality constraints. This formulation has proven very powerful in solving theconstrained least squares approximation part of FIR filter design [link] . The equality constraints can be taken into account by using Lagrange multipliers and the inequality constraints can use theKuhn-Tucker conditions [link] , [link] , [link] . The iterative reweighted least squares (IRLS) algorithm described in the next chaptercan be modified to give results which are an optimal constrained least p-power solution [link] , [link] , [link] .

Conclusions

There is remarkable structure and subtlety in the apparently simple problem of solving simultaneous equations and considerable insight can begained from these finite dimensional problems. These notes have emphasized the $l_2$ norm but some other such as $l_{\infty }$ and $l_1$ are also interesting. The use of sparsity [link] is particularly interesting as applied in Compressive Sensing [link] , [link] and in the sparse FFT [link] . There are also interesting and important applications ininfinite dimensions. One of particular interest is in signal analysis using wavelet basis functions [link] . The use of weighted error and weighted norm pseudoinverses provide a base for iterative reweighted leastsquares (IRLS) algorithms.