I want to compute the following derivative with respect to $n\times1$ vector $\mathbf x$. $$g = \left\lVert \mathbf x - A \mathbf x \right\rVert_1 $$

My work:

$$g = \left\lVert \mathbf x - A \mathbf x \right\rVert_1 = \sum_{i=1}^{n} \lvert x_i - (A\mathbf x)_i\rvert = \sum_{i=1}^{n} \lvert x_i - A_i \cdot \mathbf x \rvert = \sum_{i=1}^{n} \lvert x_i - \sum_{j=1}^n a_{ij} x_j\rvert$$ So the $k$th element of derivative is:

$$\frac{\partial g}{\partial x_k} = \frac{\partial }{\partial x_k}\sum_{i=1}^n \lvert x_i - \sum_{j=1}^n a_{ij} x_j\rvert $$ $$= \frac{\partial }{\partial x_k}\bigg(\lvert x_1 - \sum_{j=1}^n a_{1j} x_j\rvert +\cdots+ \lvert x_k - \sum_{j=1}^n a_{kj} x_j\rvert + \cdots\lvert x_n - \sum_{j=1}^n a_{nj} x_j\rvert \bigg)$$ $$ =-a_{1k}sign(x_1 - \sum_{j=1}^n a_{1j} x_j)-\cdots+(1-a_{kk})sign(x_k - \sum_{j=1}^n a_{kj} x_j)-\cdots -a_{nk}sign(x_n - \sum_{j=1}^n a_{nj} x_j)$$

And my questions:

  • Is this derivation correct?
  • How I can represent the answer compactly?
  • Can you introduce me a source to master this material?

Apart from a sign error, your result looks correct. The term with $(1-a_{1k})$ should have a positive sign. Also note that $\text{sgn}(x)$ as the derivative of $|x|$ is of course only valid for $x\neq 0$. If you take this into account, you can write the derivative in vector/matrix notation if you define $\text{sgn}(\mathbf{a})$ to be a vector with elements $\text{sgn}(a_i)$:

$$\nabla g=(\mathbf{I}-\mathbf{A}^T)\text{sgn}(\mathbf{x}-\mathbf{Ax})$$

where $\mathbf{I}$ is the $n\times n$ identity matrix.

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    $\begingroup$ nice, but I wonder in what way this is really related to DSP ;) $\endgroup$ – Marcus Müller Feb 8 '16 at 21:41
  • $\begingroup$ Thanks a lot. I correct the sign error. But in the paper I study, there is $A^T$ instead $A$ in the first parenthesis. $\endgroup$ – user153245 Feb 9 '16 at 5:16
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    $\begingroup$ @Marcus Müller, $L_1$ norm is used as a regularization term in reconstructing signal and image. $\endgroup$ – user153245 Feb 9 '16 at 5:18
  • $\begingroup$ @user153245: It should indeed be $A^T$; I corrected it. $\endgroup$ – Matt L. Feb 9 '16 at 7:37
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    $\begingroup$ @PeterK., user153245: That question came out of interest about the background of the original question; I'm very well aware the needs to find a derivate of some norm, metric etc, but usually, when questions like OP's are asked, there's a whole interesting problem to solve behind that :) $\endgroup$ – Marcus Müller Feb 9 '16 at 20:48

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