Manifold Assumption

machine-learning unsupervised-learning dimensionality-reduction

Definition

Manifold Assumption

The manifold assumption states that high-dimensional data points $x\in {\mathbb{R}}^d$ reside on or near a lower-dimensional manifold $\mathcal{M}$ of dimension $k\ll d$. Formally, the data is assumed to be sampled from a distribution whose support is restricted to this manifold, allowing the learner to discover a mapping $f:\mathcal{M}\to {\mathbb{R}}^k$ that captures the intrinsic geometric structure of the data.

Dimensionality Reduction

This assumption provides the theoretical justification for dimensionality reduction techniques. By assuming the data has a lower intrinsic dimensionality, algorithms such as PCA or manifold learning methods (e.g., t-SNE, Isomap) can identify the latent coordinates that best represent the underlying structure, effectively mitigating the curse of dimensionality.