Autoencoders

In the previous lesson, we used PCA for linear data compression, but real-world data is often too complex and tangled to be simplified by straight lines alone. Autoencoders represent the next step in dimensionality reduction, using the power of neural networks to learn highly effective nonlinear representations of data. This unique network architecture learns to compress data into a compact code and then accurately reconstruct it, allowing us to capture intricate patterns for tasks like noise removal and anomaly detection.

Nonlinear dimensionality reduction

Datasets might not always conform to a linear subspace. In such cases, employing linear techniques like PCA for dimensionality reduction proves ineffective. To address this, nonlinear dimensionality reduction techniques come into play. In this approach, data points undergo encoding/transforming via a nonlinear function. Let’s consider a scenario with $n$ data points existing in a $d$-dimensional space, organized as columns in a matrix denoted as $X_{d \times n}$. The aim is to derive corresponding $k$-dimensional nonlinear encodings (latent spaceCompressed internal representation of the input data learned by the encoder.), denoted as $Z_{k \times n}$, achieved through the following process:

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