Hessian Eigenmaps

The Hessian Eigenmaps (Hessian LLE, HLLE) method adapts the weights in LLE to minimize the Hessian operator. Like LLE, it requires careful setting of the nearest neighbor parameter. The main advantage of Hessian LLE is the only method designed for non-convex data sets [1].

This package defines a HLLE type to represent a Hessian LLE results, and provides a set of methods to access its properties.

HLLE{NN <: AbstractNearestNeighbors, T <: Real} <: AbstractDimensionalityReduction

The HLLE type represents a Hessian eigenmaps model constructed for T type data with a help of the NN nearest neighbor algorithm.

fit(HLLE, data; k=12, maxoutdim=2, nntype=BruteForce)

Fit a Hessian eigenmaps model to data.


  • data: a matrix of observations. Each column of data is an observation.

Keyword arguments

  • k: a number of nearest neighbors for construction of local subspace representation
  • maxoutdim: a dimension of the reduced space.
  • nntype: a nearest neighbor construction class (derived from AbstractNearestNeighbors)


M = fit(HLLE, rand(3,100)) # construct Hessian eigenmaps model
R = transform(M)          # perform dimensionality reduction


  • 1Donoho, D. and Grimes, C. "Hessian eigenmaps: Locally linear embedding techniques for high-dimensional data", Proc. Natl. Acad. Sci. USA. 2003 May 13; 100(10): 5591–5596. DOI:10.1073/pnas.1031596100