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Given an affine set $$ \mathbf{S}=\left\{ \mathbf{x|Ax}=\mathbf{y} \right\} \subseteq \mathbb{R}^N $$ where the matrix $\mathbf{A}\in\mathbb{R}^{M\times N} (M\ll N)$ satisfying the orthogonal form: $\mathbf{AA}^\top=\mathbf{I}_M,$ and vector $\mathbf{y}\in\mathbb{R}^{M}$ are both known.

Question: For any point $\mathbf{z}\in\mathbb{R}^N$, find $\mathbf{r}\in\mathbf{S}$ with a shortest Euclidean distance $l$ from $\mathbf{z}$.


My Efforts:

My main concern is the non-trivial case where the linear equation $\mathbf{Ax}=\mathbf{y}$ has infinite solutions.

Firstly, I turn the above question into the following optimization problem: $$ \underset{\mathbf{r}}{\min}f\left( \mathbf{r};\mathbf{A},\mathbf{y},\mathbf{z}, \lambda \right), $$

where $$ f\left( \mathbf{r};\mathbf{A},\mathbf{y},\mathbf{z}, \lambda \right) =\frac{1}{2}\lVert \mathbf{r}-\mathbf{z} \rVert _{2}^{2}+\frac{\lambda}{2}\lVert \mathbf{Ar}-\mathbf{y} \rVert _{2}^{2}, \lambda\in \mathbb{R}^+. $$

Let $$ \frac{\partial f}{\partial \mathbf{r}}=\left( \mathbf{r}-\mathbf{z} \right) +\lambda \mathbf{A}^{\top}\left( \mathbf{Ar}-\mathbf{y} \right) =\mathbf{0}_N, $$

and I got:

$$ \mathbf{r}=\left( \mathbf{I}_N+\lambda \mathbf{A}^{\top}\mathbf{A} \right) ^{-1}\left( \lambda \mathbf{A}^{\top}\mathbf{y}+\mathbf{z} \right), $$

then I actually do not know what to do next, and if my idea was correct.


Update:

I notice that:

(1) when $\lambda \rightarrow 0$, we have $\mathbf{r} \rightarrow \mathbf{z}$, and

(2) when $\lambda \rightarrow +\infty$, we have $\mathbf{r} \rightarrow \left( \mathbf{A}^{\top}\mathbf{A} \right) ^{-1} \mathbf{A}^{\top}\mathbf{y}$.

I can understand (1), but am confused by (2).

After a long struggle, I give up solving this problem and guess the solution is: $$ \mathbf{r}=\mathbf{A}^\top\mathbf{y}+(\mathbf{I}_N-\mathbf{A}^{\top}\mathbf{A})\mathbf{z}. $$

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    $\begingroup$ A member of over a year, with as many questions as you, should already know this, but I'll say it again: his is a site that encourages and helps with learning, it is best if you show your own ideas and efforts in solving the question. Please, edit your question to add your thoughts and ideas about it. Don't worry if it's wrong - that's what we're here for. Here's a quick guide. $\endgroup$
    – 5xum
    Nov 4 '21 at 8:48
  • $\begingroup$ Thanks for your reminder, I've add my efforts. @5xum $\endgroup$
    – BinChen
    Nov 4 '21 at 9:11
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It seems that I solved this problem by myself.


My Solution:

For any $\mathbf{x}\in \mathbb{R}^N$, there exists a unique range-nullspace decomposition (RND): $$ \mathbf{x}=\mathbf{A^\top Ax}+(\mathbf{I}_N-\mathbf{A^\top A})\mathbf{x}, $$

so for any $\mathbf{s}\in \mathbf{S}$, we have $\mathbf{s}=\mathbf{A^\top y}+(\mathbf{I}_N-\mathbf{A^\top A})\mathbf{s}$, and $\mathbf{z}=\mathbf{A^\top Az}+(\mathbf{I}_N-\mathbf{A^\top A})\mathbf{z}$.

Let

$$ f(\mathbf{s};\mathbf{z})=\frac{1}{2}\lVert \mathbf{s}-\mathbf{z}\rVert^2_2\\ =\frac{1}{2}\lVert \mathbf{A}^\top(\mathbf{y}-\mathbf{Az})+(\mathbf{I}_N-\mathbf{A^\top A})(\mathbf{s}-\mathbf{z})\rVert^2_2, $$

and

$$ \frac{\partial f}{\partial \mathbf{s}}=(\mathbf{I}_N-\mathbf{A^\top A})(\mathbf{s}-\mathbf{z}) = \mathbf{0}_N. $$

The above equation is the necessary condition for an optimal point, so assume $\mathbf{r}$ is a solution, we have

$$ \mathbf{r}=\mathbf{A^\top y}+(\mathbf{I}_N-\mathbf{A^\top A})\mathbf{r}\\ =\mathbf{A^\top y}+(\mathbf{I}_N-\mathbf{A^\top A})(\mathbf{r}-\mathbf{z}+\mathbf{z})\\ =\mathbf{A^\top y}+(\mathbf{I}_N-\mathbf{A^\top A})\mathbf{z}, $$

and the shortest Euclidean distance

$$ l=\lVert \mathbf{r}-\mathbf{z}\rVert_2=\lVert \mathbf{y}-\mathbf{Az}\rVert_2, $$

i.e., the final unique solution is: $\mathbf{r}=\mathbf{A^\top y}+(\mathbf{I}_N-\mathbf{A^\top A})\mathbf{z}$, and $l=\lVert \mathbf{y}-\mathbf{Az}\rVert_2$.

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