As an effective non-linear dimension reduction method

manifold learning has attracted widespread attention and made great progress. But when sample points are not dense

these algorithms often become worse or even failed just because the points in some neighborhoods do not meet the requirement of local homeomorphism. An effective solution to this question is to increase some new interpolation points. Unfortunately

the points selected by existing interpolation methods nowadays are all linear with the original sample points. From the theory of linear algebra

the subspace spanned by the interpolation points and the original neighbors is the same as the subspace spanned by the original ones; therefore

the interpolation points will not improve the linear approximation error either. Moreover

the interpolation points have no consideration to the native structure and characteristics of the manifold

which deviates from the purpose of data dimensionality reduction. To this end

a new manifold learning algorithm based on a nonlinear interpolation method called Biharmonic is proposed. Experimental results demonstrate the improvement of the neighborhood structure. The effectiveness and stability of this algorithm are further confirmed by applying it to the classical manifold learning algorithms.