4,507 research outputs found
Intrinsic dimensionality detection criterion based on Locally Linear Embedding
We revisit in this work the Locally Linear Embedding (LLE) algorithm which is a widely employed technique in dimensionality reduction. With a particular interest on the correspondences of nearest neighbors in the original and em- bedded spaces, we observe that, when prescribing low-dimensional embedding spaces, LLE remains merely a weight preserving, rather than a neighborhood preserving algorithm. We propose thus a ”neighborhood preserving ratio” crite- rion to estimate a minimal intrinsic dimensionality required for neighbourhood preservation. We validate its efficiency on a set of synthetic data, including S-curve, swiss roll, as well as a dataset of grayscale images
Nonlinear Supervised Dimensionality Reduction via Smooth Regular Embeddings
The recovery of the intrinsic geometric structures of data collections is an
important problem in data analysis. Supervised extensions of several manifold
learning approaches have been proposed in the recent years. Meanwhile, existing
methods primarily focus on the embedding of the training data, and the
generalization of the embedding to initially unseen test data is rather
ignored. In this work, we build on recent theoretical results on the
generalization performance of supervised manifold learning algorithms.
Motivated by these performance bounds, we propose a supervised manifold
learning method that computes a nonlinear embedding while constructing a smooth
and regular interpolation function that extends the embedding to the whole data
space in order to achieve satisfactory generalization. The embedding and the
interpolator are jointly learnt such that the Lipschitz regularity of the
interpolator is imposed while ensuring the separation between different
classes. Experimental results on several image data sets show that the proposed
method outperforms traditional classifiers and the supervised dimensionality
reduction algorithms in comparison in terms of classification accuracy in most
settings
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