Having constructed various autoencoder architectures, our focus now shifts to the core of what these models learn: the latent space. This compressed representation, often denoted as $z$, captures the essential variations within the input data.

This chapter provides methods to analyze and interact with this learned space. We will cover:

• Techniques for visualizing the high-dimensional latent space, such as t-SNE and UMAP, to gain intuition about its structure.
• The concept of disentanglement, where individual dimensions of the latent space correspond to distinct factors of variation in the data, and methods like $\beta$-VAE to encourage it.
• Practical ways to manipulate latent vectors through interpolation and arithmetic operations to generate meaningful outputs or understand semantic relationships.
• Metrics and approaches for evaluating the quality of the learned representations.

Understanding the latent space is key to interpreting autoencoder behavior and effectively applying these models for generation, modification, and analysis tasks. We will explore how to probe these internal representations using common deep learning frameworks.