Title: Maximum Margin Principal Components,
Abstract: Principal Component Analysis (PCA) is a very successful dimensionality
reduction technique, widely used in predictive modeling. A key factor in its
widespread use in this domain is the fact that the projection of a dataset onto
its first $K$ principal components minimizes the sum of squared errors between
the original data and the projected data over all possible rank $K$
projections. Thus, PCA provides optimal low-rank representations of data for
least-squares linear regression under standard modeling assumptions. On the
other hand, when the loss function for a prediction problem is not the
least-squares error, PCA is typically a heuristic choice of dimensionality
reduction -- in particular for classification problems under the zero-one loss.
In this paper we target classification problems by proposing a straightforward
alternative to PCA that aims to minimize the difference in margin distribution
between the original and the projected data. Extensive experiments show that
our simple approach typically outperforms PCA on any particular dataset, in
terms of classification error, though this difference is not always
statistically significant, and despite being a filter method is frequently
competitive with Partial Least Squares (PLS) and Lasso on a wide range of
datasets.