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I need help in the following question. I'm not sure how to even begin to answer this. What is a possible proof for the following question?

If $A$ is an $m \times n$ matrix and $b$ is an $m$-vector, prove that the solution $x$ of minimum Euclidean norm to the least-squares problem $A\cong B$ is given by $$x=\sum_{\sigma_i \neq 0} \frac{u_i^Tb}{\sigma_i}v_i$$ where the $\sigma_i$, $u_i$, and $v_i$ are the singular values and corresponding singular vectors of A.

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  • $\begingroup$ It would be nice to have the objective function. I have no idea what $\rm A \cong B$ means. $\endgroup$ – Rodrigo de Azevedo Mar 30 '17 at 10:04
  • $\begingroup$ The least squares solution you post is derived here: math.stackexchange.com/questions/772039/… $\endgroup$ – dantopa Mar 30 '17 at 14:55
  • $\begingroup$ Possible duplicate of How does the SVD solve the least squares problem? $\endgroup$ – dantopa Mar 30 '17 at 15:45
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    $\begingroup$ I reported this as unclear what you're asking, but at second look it simply appears to be a malicious edit of a valid question. That is to say, a pseudo-deletion. $\endgroup$ – GPhys Mar 31 '17 at 2:02
  • $\begingroup$ Steve, care to explain why you undo edits by others? It's not like you own this post! Did you not read the fine print? $\endgroup$ – Jyrki Lahtonen Apr 6 '17 at 7:33
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Preliminaries

Start with a tighter specification on the target matrix: $$ \mathbf{A} \in \mathbb{C}^{m\times n}_{\rho} $$ where the rank $\rho < \min (m,n)$. The matrix is rank defective and both null spaces are nontrivial.

We are given the data vector $b\notin\color{red}{\mathcal{N}\left( \mathbf{A}^{*} \right)}$ to insure a least squares solution exists. Defining the vector of residual errors $$ r(x) = \mathbf{A} x - b, $$ the desire is to find the solution vector $x$ which minimizes the total error $r^{2}$. The solution set is given by the minimizers $$ x_{LS} = \left\{ x\in\mathbb{C}^{n} \colon \lVert \mathbf{A} x - b \rVert_{2}^{2} \text{ is minimized} \right\} $$ For each element in this set, $r^{2}\left(x_{LS}\right)$ achieves minimum value. We are searching for the element in this set which has minimum length.

In Laub's book (p. 66), he shows the solution is in general an affine space: Laub excerpt

represented as the dashed red line in the figure below.

a dagger b

The singular value decomposition is $$ \begin{align} \mathbf{A} &= \mathbf{U} \, \Sigma \, \mathbf{V}^{*} \\ % &= % U \left[ \begin{array}{cc} \color{blue}{\mathbf{U}_{\mathcal{R}}} & \color{red}{\mathbf{U}_{\mathcal{N}}} \end{array} \right] % Sigma \left[ \begin{array}{cc} \mathbf{S}_{\rho\times \rho} & \mathbf{0} \\ \mathbf{0} & \mathbf{0} \end{array} \right] % V \left[ \begin{array}{c} \color{blue}{\mathbf{V}_{\mathcal{R}}}^{*} \\ \color{red}{\mathbf{V}_{\mathcal{N}}}^{*} \end{array} \right] \\ % \end{align} $$ The Moore-Penrose pseudoinverse is constructed from the SVD: $$ \begin{align} \mathbf{A}^{\dagger} &= \mathbf{V} \, \Sigma^{\dagger} \, \mathbf{U}^{*} \\ % &= % U \left[ \begin{array}{cc} \color{blue}{\mathbf{V}_{\mathcal{R}}} & \color{red} {\mathbf{V}_{\mathcal{N}}} \end{array} \right] % Sigma \left[ \begin{array}{cc} \mathbf{S}^{-1} & \mathbf{0} \\ \mathbf{0} & \mathbf{0} \end{array} \right] % V \left[ \begin{array}{c} \color{blue}{\mathbf{U}_{\mathcal{R}}}^{*} \\ \color{red} {\mathbf{U}_{\mathcal{N}}}^{*} \end{array} \right] \\ % \end{align} $$ The object of the proof is to show that the pseudoinverse solution has minimum length.

Trick

Cast the total error in terms of the SVD and perform a unitary transformation $$ r^{2}(x) = \lVert \mathbf{A} x - b\rVert_{2}^{2} = \lVert \mathbf{U} \, \Sigma \, \mathbf{V}^{*} x - b \rVert_{2}^{2} = \lVert \Sigma \, \mathbf{V}^{*} x - \mathbf{U}^{*} b \rVert_{2}^{2} $$ to separate the $\color{blue}{range}$ and $\color{red}{null}$ spaces. $$ \begin{align} r^{2}(x) = \lVert \Sigma \, \mathbf{V}^{*} x - \mathbf{U}^{*} b \rVert_{2}^{2} &= \Bigg\lVert \left[ \begin{array}{cc} \mathbf{S} & \mathbf{0} \\ \mathbf{0} & \mathbf{0} \\ \end{array} \right] \left[ \begin{array}{c} \color{blue}{\mathbf{V}_{\mathcal{R}}}^{*} \\ \color{red} {\mathbf{V}_{\mathcal{N}}}^{*} \end{array} \right] x - \left[ \begin{array}{c} \color{blue}{\mathbf{U}_{\mathcal{R}}}^{*} \\ \color{red} {\mathbf{U}_{\mathcal{N}}}^{*} \end{array} \right] b \Bigg\rVert_{2}^{2} \\ % &= \big\lVert \mathbf{S} \color{blue}{\mathbf{V}_{\mathcal{R}}}^{*} x - \color{blue}{\mathbf{U}_{\mathcal{R}}}^{*} b - \color{red} {\mathbf{U}_{\mathcal{N}}}^{*} b \big\rVert_{2}^{2} \\ % &= \big\lVert \mathbf{S} \color{blue}{\mathbf{V}_{\mathcal{R}}}^{*} x - \color{blue}{\mathbf{U}_{\mathcal{R}}}^{*} b \big\rVert_{2}^{2} + \big\lVert \color{red} {\mathbf{U}_{\mathcal{N}}}^{*} b \big\rVert_{2}^{2} \\ % \end{align} $$ How to minimize the total error now? We control the range space component, and we force that contribution to $0$: $$ r^{2}(x) = \underbrace{\big\lVert \mathbf{S} \color{blue}{\mathbf{V}_{\mathcal{R}}}^{*} x - \color{blue}{\mathbf{U}_{\mathcal{R}}}^{*} b \big\rVert_{2}^{2}}_{\text{controlled}} + \underbrace{\big\lVert \color{red} {\mathbf{U}_{\mathcal{N}}}^{*} b \big\rVert_{2}^{2}}_{\text{uncontrolled}} \\ $$ That is, force $$ \mathbf{S} \color{blue}{\mathbf{V}_{\mathcal{R}}}^{*} x - \color{blue}{\mathbf{U}_{\mathcal{R}}}^{*} b = 0 $$ by setting $$ \color{blue}{x_{LS}} = \color{blue}{\mathbf{V}_{\mathcal{R}}}\mathbf{S}^{-1}\color{blue}{\mathbf{U}_{\mathcal{R}}}^{*} b = \color{blue}{\mathbf{A}^{\dagger} b}. $$ The least value for the sums of the squares of residuals is $$ r^{2}\left( x_{LS} \right) = \big\lVert \color{red} {\mathbf{U}_{\mathcal{N}}}^{*} b \big\rVert_{2}^{2} $$ The derivation deliberately avoided the requirement that $\mathbf{A}$ be of full column rank. When the null space $\color{red}{\mathcal{N}\left( \mathbf{A}^{*} \right)}$ is nontrivial, the least squares solution includes a projection onto that null space. The least squares solution set is $$ x_{LS} = \color{blue}{\mathbf{A}^{\dagger}b} + \color{red} {\left( \mathbf{I}_{n} - \mathbf{A}^{\dagger} \mathbf{A} \right)y}, \quad y\in\mathbb{C}^{n} $$ Refer back to the blue vector and red dashed line in the previous figure.

What are the lengths of the least squares solution vectors?

$$ \lVert x_{LS} \rVert_{2}^{2} = \lVert \color{blue}{\mathbf{A}^{\dagger}b} \rVert_{2}^{2} + \lVert \color{red} {\left( \mathbf{I}_{n} - \mathbf{A}^{\dagger} \mathbf{A} \right)y}\rVert_{2}^{2} $$

What the least squares solution of minimum length?

It is the vector which is entirely in $\color{blue}{\mathcal{N}\left( \mathbf{A}^{*} \right)}$. It is the pseudoinverse solution.

$$ \lVert \color{blue}{x_{LS}} \rVert_{2}^{2} = \lVert \color{blue}{\mathbf{A}^{\dagger}b} \rVert_{2}^{2} $$

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  • $\begingroup$ Why is it possible to separate controlled and uncontrolled parts in the 2-norm? $\endgroup$ – errorist Oct 21 '18 at 14:47
  • $\begingroup$ @errorist: Excuse the sloppy vernacular. The theorem of Pythagorus is used to separate range spaces (blue) from null spaces (red). The range space portion contains the $x$ term which we vary in order to find the best result. We can adjust $x$ to eliminate the range space contribution to the error. Colloquially, we say "we can control the range space contribution" because of the freedom to choose $x$. There are no such variable parameters in the null space term. The residual error term, $r^{2}(x)$ is a function of $x$ which highlights the role of $x$ in reducing the total error. $\endgroup$ – dantopa Oct 22 '18 at 16:14

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