Data transformation maps points or vectors from one geometric space to another. In machine learning, a feature vector represents an input as a point in feature space; transforming those points can make classes easier to separate or remove dimensions with little variation. section 2.10 section 2.1.1
A matrix provides a concrete transformation: an m × n matrix maps n-dimensional vectors to m-dimensional vectors by multiplication. Matrix transformations preserve collinearity, while the input and output spaces may differ. section 2.10
Common geometric interpretations include:
- Rotation maps points within the same two-dimensional plane. section 2.10
- Projection maps three-dimensional points onto a two-dimensional plane. section 2.10
- A neural network maps an input manifold through successive layers toward a representation where classes can be separated by a hyperplane. section 12.1
Unrolling a classification space
Suppose two classes lie on a cylindrical surface and cannot be separated by a plane in that space. Unrolling the cylinder into a plane maps the points to a representation where a linear separator can distinguish the classes.
The geometric view treats a model itself as a transformation from input points to output points in a high-dimensional space. Manifolds extend the idea of curves and surfaces to arbitrary dimensions and are locally Euclidean, allowing small neighborhoods to be treated like familiar flat spaces. section 2.1.1 section 12.1
