learning a tangent hyperplane on your data manifold where around every input x there is a distortion radius created around it based on the noise that you're trying to add so it's very easy to understand in the case of additive Gaussian noise because you can think of Gaussian is a spherical distortion around your every input and you can think of the decoder as trying to put the distorted version back onto the tangent hyperplane so you can think of the whole denoising auto-encoder pipeline is trying to learn this tangent