Sharp generalization error bounds for randomly-projected classifiers

We derive sharp bounds on the generalization error of a generic linear classifier trained by empirical risk minimization on randomly projected data. We make no restrictive assumptions (such as sparsity or separability) on the data: Instead we use the fact that, in a classification setting, the question of interest is really ‘what is the effect of random projection on the predicted class labels?’ and we therefore derive the exact probability of ‘label flipping’ under Gaussian random projection in order to quantify this effect precisely in our bounds

Dimension-adaptive bounds on compressive FLD Classification

Efficient dimensionality reduction by random projections (RP) gains popularity, hence the learning guarantees achievable in RP spaces are of great interest. In finite dimensional setting, it has been shown for the compressive Fisher Linear Discriminant (FLD) classifier that forgood generalisation the required target dimension grows only as the log of the number of classes and is not adversely affected by the number of projected data points. However these bounds depend on the dimensionality d of the original data space. In this paper we give further guarantees that remove d from the bounds under certain conditions of regularity on the data density structure. In particular, if the data density does not fill the ambient space then the error of compressive FLD is independent of the ambient dimension and depends only on a notion of ‘intrinsic dimension'

Learning in high dimensions with projected linear discriminants

The enormous power of modern computers has made possible the statistical modelling of data with dimensionality that would have made this task inconceivable only decades ago. However, experience in such modelling has made researchers aware of many issues associated with working in high-dimensional domains, collectively known as `the curse of dimensionality', which can confound practitioners' desires to build good models of the world from these data. When the dimensionality is very large, low-dimensional methods and geometric intuition both break down in these high-dimensional spaces. To mitigate the dimensionality curse we can use low-dimensional representations of the original data that capture most of the information it contained. However, little is currently known about the effect of such dimensionality reduction on classifier performance. In this thesis we develop theory quantifying the effect of random projection - a recent, very promising, non-adaptive dimensionality reduction technique - on the classification performance of Fisher's Linear Discriminant (FLD), a successful and widely-used linear classifier. We tackle the issues associated with small sample size and high-dimensionality by using randomly projected FLD ensembles, and we develop theory explaining why our new approach performs well. Finally, we quantify the generalization error of Kernel FLD, a related non-linear projected classifier