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An image $I$ is computed by performing convolution and summation:
$$ \sum_{k=0}^{K-1} z_k * f_k = I $$
Given only the feature maps $z_k$ and the resulting image $I$, how do I compute the filters $f_k$ ?
This is a linear system, therefore it must be possible to solve it with conjugate gradient. How to formulate it as $\mathbf{Ax} = \mathbf{b}$ ?
$z_k$ and $I$ have both the same quadratic size $N \times N$.
This is a typical deconvolution problem that you can solve either by transforming to the frequency domain where convolution is a simple multiplication:
\begin{equation} F\left \{ {\bf{I}} \right \}= F\left \{ {\bf{z}} \right \} F\left \{ {\bf{f}} \right \}, \end{equation}
where ${\mathcal{F}}$ denotes the Fourier transform (or DFT) so $f$ will be:
\begin{equation} {\bf{f}}={\mathcal{F}}^{-1}\left \{ \frac{{\mathcal{F}}\left \{ {\bf{I}} \right \}}{{\mathcal{F}}\left \{ {\bf{z}} \right \}} \right \} \end{equation}, where ${\mathcal{F}}^{-1}$ is the inverse Fourier transform.
Otherwise, you can also represent in a matrix form, using a Circulant (kind of Toeplitz) for the convolution operation. The frequency domain calculation though does not involve inverting (or pseudo inverting) a matrix.
If there's additive noise in your problem, please see Wiener deconvolution (both time and frequency domain implementation)
