Explain how to modify the CAE architecture to prevent it from degenerating into a simple scaling operation, and what tradeoff this modification would impose on the model’s ability to learn accurate tangent vectors.

To prevent the CAE from degenerating into a simple scaling operation, we can introduce a penalty term that encourages the encoder to drive the contractive penalty Ω(h) away from , but not necessarily towards 0. This modification would impose a tradeoff on the model’s ability to learn accurate tangent vectors, as it would require more complex and expressive architectures.