Unsupervised Metric Learning in Presence of Missing Data
This addresses a practical issue for machine learning practitioners dealing with real-world datasets that are often incomplete, though it appears incremental as it builds on existing manifold learning methods.
The paper tackles the problem of computing low-dimensional representations from data with missing entries, which existing methods like ISOMAP and Laplacian Eigenmaps cannot handle directly, by introducing MR-MISSING, an algorithm that extends previous techniques to work with missing data and demonstrates effectiveness through experiments on synthetic manifolds and MNIST, including classification tasks.
For many machine learning tasks, the input data lie on a low-dimensional manifold embedded in a high dimensional space and, because of this high-dimensional structure, most algorithms are inefficient. The typical solution is to reduce the dimension of the input data using standard dimension reduction algorithms such as ISOMAP, LAPLACIAN EIGENMAPS or LLES. This approach, however, does not always work in practice as these algorithms require that we have somewhat ideal data. Unfortunately, most data sets either have missing entries or unacceptably noisy values. That is, real data are far from ideal and we cannot use these algorithms directly. In this paper, we focus on the case when we have missing data. Some techniques, such as matrix completion, can be used to fill in missing data but these methods do not capture the non-linear structure of the manifold. Here, we present a new algorithm MR-MISSING that extends these previous algorithms and can be used to compute low dimensional representation on data sets with missing entries. We demonstrate the effectiveness of our algorithm by running three different experiments. We visually verify the effectiveness of our algorithm on synthetic manifolds, we numerically compare our projections against those computed by first filling in data using nlPCA and mDRUR on the MNIST data set, and we also show that we can do classification on MNIST with missing data. We also provide a theoretical guarantee for MR-MISSING under some simplifying assumptions.