Dimensionality reduction

In statistics, dimension reduction is the process of reducing the number of random variables under consideration, and can be divided into feature selection and feature extraction. In physics, dimension reduction is a widely discussed phenomenon, whereby a physical system exists in three dimensions, but its properties behave like those of a lower-dimensional system. It has been experimentally realised at the quantum critical point in an insulating magnet made of the pigment Han Purple.

Feature selection
Feature selection approaches try to find a subset of the original variables (also called features or attributes). Two strategies are filter (e.g. information gain) and wrapper (e.g. genetic algorithm) approaches. See also combinatorial optimization problems.

It is sometimes the case that data analysis such as regression or classification can be done in the reduced space more accurately than in the original space.

Feature extraction
Feature extraction is applying a mapping of the multidimensional space into a space of fewer dimensions. This means that the original feature space is transformed by applying e.g. a linear transformation via a principal components analysis.

Consider a string of beads, first 100 black and then 100 white. If the string is wadded up, a classification boundary between black and white beads will be very complicated in three dimensions. However, there is a mapping from three dimensions to one dimension, namely distance along the string, which makes the classification trivial. Unfortunately, a simplification as dramatic as that is rarely possible in practice.

The main linear technique for dimensionality reduction, principal components analysis (PCA), performs a linear mapping of the data to a lower dimensional space in such a way, that the variance of the data in the low-dimensional representation is maximized. In practice, the correlation matrix of the data is constructed and the eigenvectors on this matrix are computed. The eigenvectors that correspond to the largest eigenvalues (the principal components) can now be used to reconstruct a large fraction of the variance of the original data. Moreover, the first few eigenvectors can often be interpreted in terms of the large-scale physical behaviour of the system. The original space (with dimension of the number of points) has been reduced (with data loss, but hopefully retaining the most important variance) to the space spanned by a few eigenvectors.

Principal component analysis can be employed in a nonlinear way by means of the kernel trick. The resulting techniques is capable of constructing nonlinear mappings that maximize the variance in the data. The resulting technique is entitled Kernel PCA. Other nonlinear techniques include techniques for locally linear embedding (such as locally linear embedding (LLE), Hessian LLE, Laplacian eigenmaps, and LTSA). These techniques construct a low-dimensional data representation using a cost function that retains local properties of the data (actually these techniques can be viewed upon as defining a graph-based kernel for Kernel PCA). In this way, the techniques are capable of unfolding datasets such as the Swiss roll. Techniques that employ neighborhood graphs in order to retain global properties of the data include Isomap and maximum variance unfolding.The neighbourhood preservation can also be achieved through the minimisation of the weighted difference between distances in the input and output spaces (i.e. curvilinear component analysis (CCA) and data-driven high-dimensional scaling (DD-HDS)).

Principal curves and manifolds give another framework for PCA generalization and extend the geometric interpretation of PCA by explicitly constructing a low-dimensional embedded manifold for data approximation.

A completely different approach to nonlinear dimensionality reduction is through the use of autoencoders, a special kind of feed-forward neural networks. Although the idea of autoencoders is quite old, training of the encoders has only recently become possible through the use of Restricted Boltzmann machines.