Autoencoder forward pass

Background

An autoencoder learns to compress and reconstruct its own input. An encoder maps the input $x$ down to a low-dimensional latent code $z$ (the bottleneck), and a decoder maps $z$ back up to a reconstruction $\hat x$. Because the latent layer is smaller than the input, the network is forced to keep only the most informative structure — making autoencoders useful for dimensionality reduction, denoising, and representation learning.

Problem statement

Implement autoencoder_forward(x, W_enc, b_enc, W_dec, b_dec) for one forward pass with a ReLU encoder and a linear decoder:

$$z = \operatorname{ReLU}(W_{\text{enc}}\, x + b_{\text{enc}})$$ $$\hat x = W_{\text{dec}}\, z + b_{\text{dec}}$$

Return the tuple (reconstruction, latent) = $(\hat x, z)$.

Input

• x — input vector, shape (n,).
• W_enc, b_enc — encoder weights (d, n) and bias (d,), where d < n is the latent size.
• W_dec, b_dec — decoder weights (n, d) and bias (n,).

Output

A tuple (reconstruction, latent):

• reconstruction — np.ndarray of shape (n,), the decoded $\hat x$.
• latent — np.ndarray of shape (d,), the bottleneck code $z$.

Examples

Example 1

Input:  x = [1.0, 2.0]
        W_enc = [[0.5, 0.5]], b_enc = [0.0]      # latent size 1
        W_dec = [[1.0], [1.0]], b_dec = [0.0, 0.0]
Output: reconstruction = [1.5, 1.5], latent = [1.5]

Explanation: the encoder maps $[1,2]$ to $z=\operatorname{ReLU}(0.5\cdot1+0.5\cdot2)=1.5$; the decoder broadcasts that single number back to both outputs, giving the (lossy) reconstruction $[1.5,1.5]$.

Constraints

• Apply ReLU only in the encoder; the decoder is linear (no activation).
• Latent dimension is W_enc.shape[0]; reconstruction dimension is W_dec.shape[0] (= n).
• Return (reconstruction, latent) in that order.

Notes

• The bottleneck being smaller than the input is what makes the autoencoder learn — an identity-sized latent could just copy the input.
• For data normalized to $[0,1]$ (e.g. pixels) a sigmoid decoder is common; here the decoder is linear so the reconstruction can take any real value.