Chapter 5: Self-organizing maps
Daniela Calvetti, Erkki Somersalo · Society for Industrial and Applied Mathematics eBooks · 2020
A motivation for reducing the dimensionality of the data is to make it possible to visualize high-dimensional data, e.g., by projecting them onto a suitable lower-dimensional subspace. Successful data reduction and visualization by projection methods often require that an orthogonal projection onto a linear subspace capture salient features of the data, such as the presence of clusters. It is easy to find examples where the data are supported on or near a low-dimensional manifold, yet the intrinsic low dimensionality and the organization of the data are not correctly represented by any orthogonal projection onto a linear subspace. This is the case, for instance, when the data are distributed on or near the surface of a sphere in ℝ3; in fact, while the surface is clearly two-dimensional, a projection onto a two-dimensional plane may confound the organization of the data, since any pair of antipodal points along a ray orthogonal to the plane will be represented as being the same point, while when measured along the surface of the sphere, the points are distant.