LLE is an unsupervised learning algorithm that computes low-dimensional, neighborhood-preserving embeddings of high-dimensional inputs.
Steps
-
select neighbors
\[\boldsymbol{X}_m = \sum_{i\in S_m} W_{mi} \boldsymbol{X}_i\] -
reconstruct with weights
\[\mathcal{F} = \sum^N_{m=1} \| \boldsymbol{X}_m-\sum^N_{i=1} W_{mi} \boldsymbol{X}_i \|^2\] -
map to embedded coordinates
\[\mathcal{G}(\boldsymbol{\xi}_1,...,\boldsymbol{\xi}_N) = \sum^N_{m=1} \|\boldsymbol{\xi}_i-\sum_{i=1}^N W_{mi}\boldsymbol{\xi}_i\|^2\]