# Is spare encoding a special case of spare autoencoding with ignoring non linear activation

we know that Sparse encoding is to minimize the objective function: $$\sum\limits_{n=1}^N\Big(||x_n - Az_n||^2 + \lambda\rho(z_n)\Big).$$ Here

1. $$A = [a_1,\cdots,a_M]$$ is overcomplete Dictionary;

2. $$z = (z^1,\cdots,z^M):\ x = Az$$ is Encoding;

3. $$\rho(z_n)$$ is sparsity function.

And we need to train both $$(A,z_n)$$

Sparse autoencoding is to minimize the objective function:

$$\sum\limits_{n=1}^N\Big(||x_n - \phi\cdot\psi(x_n)||^2 + \lambda\rho(h_n)\Big).$$ here $$\psi(x) =h = \sigma(Ax); \phi(h) = x = \sigma'(A'x')$$ are the non-linear activation functions. We need to train $$A,A'.$$

I am not sure if we ignore the non-linear activation functions $$\sigma/\sigma',$$ is Sparse encoding a special case of Sparse autoencoding?

Since if we represent PCA as encoding-decoding error: $$\sum\limits_{n=1}^N\left(||x_n-AA^Tx_n||^2\right)$$ it is exactly the linear case of autoencoding.