A new locally linear embedding scheme in light of Hessian eigenmap

TL;DR

This paper introduces Tangential LLE (TLLE), a simplified and more robust variant of Hessian LLE, based on a new interpretation that replaces the Hessian concept with arbitrary weights, improving manifold learning.

Contribution

The paper provides a new interpretation of HLLE, simplifies its implementation, and proposes TLLE, a more robust LLE-type method for manifold learning.

Findings

TLLE is simpler and more robust than HLLE.

Numerical examples show TLLE effectively captures manifold structure.

Modifications improve embedding quality when target space dimension exceeds data manifold.

Abstract

We provide a new interpretation of Hessian locally linear embedding (HLLE), revealing that it is essentially a variant way to implement the same idea of locally linear embedding (LLE). Based on the new interpretation, a substantial simplification can be made, in which the idea of "Hessian" is replaced by rather arbitrary weights. Moreover, we show by numerical examples that HLLE may produce projection-like results when the dimension of the target space is larger than that of the data manifold, and hence one further modification concerning the manifold dimension is suggested. Combining all the observations, we finally achieve a new LLE-type method, which is called tangential LLE (TLLE). It is simpler and more robust than HLLE.