Robust locally linear embedding using penalty functions

We introduce a modified version of locally linear embedding (LLE) which is more robust to noise. This is accomplished by adding a regularization term to the reconstruction weight cost function. We propose two alternative regularization terms, the ℓ₂-norm and the elastic-net function; a weighted average of the ℓ₂- and ℓ₁-norm. Adding the ℓ₂-norm to the cost function produces more uniform weights. With noise in the data, a more uniform weight structure provides a better representation of the linear patch surrounding each data point. In the case of the elastic-net function, the addition of the ℓ₁-norm produces sparse weights; eliminating possible outliers from the reconstruction. We use several examples to show that these methods are able to outperform LLE and are comparable to other dimensionality reduction algorithms.