1、A Geometric Perspective on Machine Learning何晓飞浙江大学计算机学院1Machine Learning: the problemf 何晓飞Information(training data)f: X YX and Y are usually considered as a Euclidean spaces.2Manifold Learning: geometric perspectiveo The data space may not be a Euclidean space, but a nonlinear manifold. Euclidean d
2、istance. f is defined on Euclidean space. ambient dimension geodesic distance. f is defined on nonlinear manifold. manifold dimension.instead3Manifold Learning: the challenges The manifold is unknown! We have only samples!o How do we know M is a sphere or a torus, or else?o How to compute the distan
3、ce on M?o versusThis is unknown:This is what we have:? ? or else? TopologyGeometryFunctional analysis4Manifold Learning: current solutiono Find a Euclidean embedding, and then perform traditional learning algorithms in the Euclidean space.5Simplicity6Simplicity7Simplicity is relative8Manifold-based
4、Dimensionality Reductiono Given high dimensional data sampled from a low dimensional manifold, how to compute a faithful embedding? o How to find the mapping function ?o How to efficiently find the projective function ?9A Good Mapping Function o If xi and xj are close to each other, we hope f(xi) and f(xj) preserve the local structure (distance, similarity )o k-nearest neighbor graph:o Objective function:n Different algorithms have different concerns10
