I am referring to this brilliant blog on Auto encoders

Came across this statement that says vanilla encoders run the risk of overfitting.
“Since the autoencoder learns the identity function, we are facing the risk of “overfitting” when there are more network parameters than the number of data points.”

The parameters (𝜃,𝜙) are learned together to output a reconstructed data sample same as the original input, 𝑥≈𝑓𝜃(𝑔𝜙(𝑥)), or in other words, to learn an identity function.

So we apply encoder(x) = z and then apply a decoder function f (z) which results in x’. Now you use either a MSE loss or a cross entropy loss depending on the activation function used and minimize the loss using SGD until x’ = x.

Identity function is a function which just returns the input f(x) = x.

Can someone please explain what do they mean by ‘same as learning an identity function’ ?
The only difference I see is that in a regular deep neural net, the loss function measures the distance between the predicted label and the training label ( the label can be a text caption, a bool value, a pixel value, whatever ) and in the case of an AE the loss function measures the distance between the predicted pixel value and the training pixel value.

Why does learning an identity function when there are more network parameters than input data points lead to overfitting.

When they refer to input data points are they talking about a single training image or the learned weights ?

Hello,
Hi Pradeep,
Great question! Let’s break it down.
Learning an Identity Function
When we say an autoencoder learns an identity function, we mean that the network is trained to output the same data it receives as input. Mathematically, this is represented as:
x≈fθ(gϕ(x))
Here, ( g_\phi ) is the encoder function that maps the input ( x ) to a latent space representation ( z ), and ( f_\theta ) is the decoder function that reconstructs ( x ) from ( z ).
Overfitting in Autoencoders
Overfitting occurs when a model learns not just the underlying patterns in the data but also the noise. In the context of autoencoders, if the model has more parameters (weights) than the number of data points, it can easily memorize the training data, including any noise, rather than learning a generalizable representation.
Why More Parameters Lead to Overfitting
When there are more network parameters than data points, the model has enough capacity to learn the identity function perfectly. This means it can simply memorize each input and reproduce it exactly, rather than learning meaningful features. This is particularly problematic because:

Lack of Generalization: The model performs well on training data but poorly on unseen data.
Noise Memorization: The model captures noise in the training data, which is not useful for generalization.

Input Data Points vs. Learned Weights
When referring to input data points, we are talking about the number of unique training samples, not the learned weights. The risk of overfitting is higher when the number of parameters (weights) in the network exceeds the number of unique training samples.
Mitigating Overfitting
To mitigate overfitting in autoencoders, several techniques can be employed:
Regularization: Adding a penalty to the loss function to constrain the model complexity.
Dropout: Randomly dropping units during training to prevent co-adaptation.
Denoising Autoencoders: Adding noise to the input data and training the model to reconstruct the original data
Best Regards
esther598.