Editing Nonlinear dimensionality reduction (section)

=== Isomap ===

[[Isomap]]<ref>{{cite journal |first1=J B. |last1=Tenenbaum |first2=V. |last2=de Silva |first3=J.C. |last3=Langford |url=http://www.robots.ox.ac.uk/~az/lectures/ml/tenenbaum-isomap-Science2000.pdf |title=A Global Geometric Framework for Nonlinear Dimensionality Reduction |journal=Science |volume=290 |date=2000 |issue=5500 |pages=2319–23|doi=10.1126/science.290.5500.2319 |pmid=11125149 |bibcode=2000Sci...290.2319T |s2cid=221338160 }}</ref> is a combination of the [[Floyd–Warshall algorithm]] with classic [[Multidimensional Scaling]] (MDS). Classic MDS takes a matrix of pair-wise distances between all points and computes a position for each point. Isomap assumes that the pair-wise distances are only known between neighboring points, and uses the Floyd–Warshall algorithm to compute the pair-wise distances between all other points. This effectively estimates the full matrix of pair-wise [[geodesic distance]]s between all of the points. Isomap then uses classic MDS to compute the reduced-dimensional positions of all the points. Landmark-Isomap is a variant of this algorithm that uses landmarks to increase speed, at the cost of some accuracy.

In manifold learning, the input data is assumed to be sampled from a low dimensional [[manifold]] that is embedded inside of a higher-dimensional vector space. The main intuition behind MVU is to exploit the local linearity of manifolds and create a mapping that preserves local neighbourhoods at every point of the underlying manifold.