Laplacian Eigenmaps Dimension Reduction:If x1 and x2 are close in a high density region,z1 and z2 are close to each other. s=∑ue-z Any problem?How about zi=z=0? Giving some constraints to z: If the dim of z is M,Span(z1,z2,...N)=RM Spectral clustering:clustering on z Belkin,M.,Niyogi,P.Laplacian eigenmaps and spectral techniques for embedding and clustering.Advances in neural information processing systems.2002 Laplacian Eigenmaps • Dimension Reduction: If 𝑥 1 and 𝑥 2 are close in a high density region, 𝑧 1 and 𝑧 2 are close to each other. 𝑆 = 1 2 ෍ 𝑖,𝑗 𝑤𝑖,𝑗 𝑧 𝑖 − 𝑧 𝑗 2 Spectral clustering: clustering on z Any problem? How about 𝑧 𝑖 = 𝑧 𝑗 = 𝟎? Giving some constraints to z: If the dim of z is M, Span{z1 , z2 , … z N} = RM Belkin, M., Niyogi, P. Laplacian eigenmaps and spectral techniques for embedding and clustering. Advances in neural information processing systems . 2002