High Dimensional Statistics
Mathematically, manifold learning assumes that high-dimensional data points lie on or near a lower-dimensional manifold
One of the key goals of manifold learning techniques is to preserve local geometric properties, such as distances or angles, during this mapping. Given two data points
Techniques such as Principal Component Analysis (PCA), Isomap, Locally Linear Embedding (LLE), and t-SNE seek to discover these lower-dimensional structures by projecting the data onto meaningful manifolds. These methods are crucial in tasks like clustering, classification, and visualization, where meaningful patterns can only be observed after reducing the data to its intrinsic dimensions.
The reduction to intrinsic dimensions also helps overcome the curse of dimensionality, which refers to the challenges posed by sparse data distributions in high-dimensional spaces. By learning the low-dimensional manifold
In summary, high-dimensional statistical learning leverages the geometry of data manifolds to perform dimensionality reduction, revealing latent structures and facilitating tasks like clustering, pattern recognition, and visualization:
where