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非线性降维I:局部线性嵌入

非线性降维I:局部线性嵌入

陈傲天
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1809
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Failure of PCA and all other linear methods for nonlinear structures in data; spirals, for example. Approximate success of linear methods on small parts of nonlinear structures. Manifolds: smoothly curved surfaces embedded in higher-dimensional Euclidean spaces. Every manifold looks like a linear subspace on a sufficiently small scale, so we should be able to patch together many small local linear approximations into a global manifold. Local linear embedding: approximate each vector in the data as a weighted linear combination of its k nearest neighbors, then find the low-dimensional vectors best reconstructed by these weights. Turning the optimization problems into linear algebra.
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