Derivative of an $\ell_1$ norm of transform of a vector.

Question

I have to take derivative of the l-1 norm. L1 is the function R in the following expression: $$ R(\psi Fx) $$

where x is a vector, F is the inverse Fourier transform, and $\psi$ is a wavelet transform. If I define a variable C such that

$$C = \psi F$$

then my $\ell_1$ norm is defined as:

$$\|Cx\|^{1}_{1}$$ I know that taking a derivative of an l1 norm is not possible. The l1 norm is defined as: $$\sum\nolimits|{x_{i}|}^{1}_{1}$$ To take a derivative of the l1 term, I addd a small positive number, call it $\epsilon$. Therefore,

$$\sum\nolimits|{x_{1}|}^{1}_{1} = \sum\nolimits\sqrt{x^*_{i}x_{i} + \epsilon}$$

What is the derivative of the $\ell_1$ norm of $Cx$ and what would be the elements of the matrix $C$?

Answer 1

Solving in coordinates, use the formula $\frac{\partial}{\partial x_k} \|\mathbf{x}\|_p = \frac{x_k |x_k|^{p-2}}{\|\mathbf{x}\|_{p}^{p-1}}$ for $p=1$ and with obvious existence conditions.

See also the answer to Taking derivative of $L_0$-norm, $L_1$-norm, $L_2$-norm.