"Latent Space Characterization of Autoencoder Variants"

Autoencoders explained: Why some create smooth transitions while others don't

This paper characterizes the latent space structure of different autoencoders and explains why VAEs form smooth manifolds while CAEs and DAEs have non-smooth structures.

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https://arxiv.org/abs/2412.04755

🔍 Original Problem:

→ Understanding why autoencoders exhibit different latent space behaviors is crucial for deep learning but remains poorly understood

→ Need to mathematically explain why VAEs form smooth latent spaces while CAEs/DAEs don't

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🛠️ Solution in this Paper:

→ Models latent tensors as points on product manifolds of symmetric positive semi-definite matrices

→ Analyzes rank configurations of these matrices under varying noise levels

→ Maps matrix manifold points to Hilbert space using distance-preserving transforms

→ Examines subspace dimensionality changes with input perturbations

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💡 Key Insights:

→ CAE/DAE latent spaces form stratified manifolds - smooth within strata but discontinuous between them

→ VAE latent space is a smooth product manifold of two symmetric positive definite matrices

→ Noise impacts CAE/DAE subspace dimensions more severely than VAE

→ Principal angles between clean/noisy subspaces increase with noise in CAE/DAE but remain zero for VAE

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📊 Results:

→ CAE/DAE show rank variability (S1:5-7, S2:6-7, S3:29-48) while VAE maintains fixed ranks (S1:7, S2:7, S3:48)

→ VAE maintains constant PSNR (~25dB) across noise levels while CAE/DAE drop significantly

→ t-SNE shows VAE points stay tightly clustered while CAE/DAE points diverge with noise