Editing Nonlinear dimensionality reduction (section)

==== Modified Locally-Linear Embedding (MLLE) ====

Modified LLE (MLLE)<ref>{{cite journal |first1=Z. |last1=Zhang |first2=J. |last2=Wang |title=MLLE: Modified Locally Linear Embedding Using Multiple Weights |journal=NIPS'06: Proceedings of the 19th International Conference on Neural Information Processing Systems |year=2006 |pages=1593–1600 |url=https://dl.acm.org/doi/abs/10.5555/2976456.2976656 }}</ref> is another LLE variant which uses multiple weights in each neighborhood to address the local weight matrix conditioning problem which leads to distortions in LLE maps. Loosely speaking the multiple weights are the local [[orthogonal projection]] of the original weights produced by LLE. The creators of this regularised variant are also the authors of Local Tangent Space Alignment (LTSA), which is implicit in the MLLE formulation when realising that the global optimisation of the orthogonal projections of each weight vector, in-essence, aligns the local tangent spaces of every data point. The theoretical and empirical implications from the correct application of this algorithm are far-reaching.<ref name=borntolose>{{cite journal|author-last=Sidhu|author-first=Gagan|title=Locally Linear Embedding and fMRI feature selection in psychiatric classification|journal=IEEE Journal of Translational Engineering in Health and Medicine|year=2019|volume=7|pages=1–11|doi=10.1109/JTEHM.2019.2936348|pmid=31497410|pmc=6726465|arxiv=1908.06319|s2cid=201832756}}</ref>