The thesis explores sparse machine learning algorithms for supervised (classification and regression) and unsupervised (subspace methods) learning. For classification, we review the set covering machine (SCM) and propose new algorithms that directly minimise the SCMs sample compression generalisation error bounds during the training phase. Two of the resulting algorithms are proved to produce optimal or near-optimal solutions with respect to the loss bounds they minimise. One of the SCM loss bounds is shown to be incorrect and a corrected derivation of the sample compression bound is given along with a framework for allowing asymmetrical loss in sample compression risk bounds. In regression, we analyse the kernel matching pursuit (KMP) algorithm and derive a loss bound that takes into account the dual sparse basis vectors. We make connections to a sparse kernel principal components analysis (sparse KPCA) algorithm and bound its future loss using a sample compression argument. This investigation suggests a similar argument for kernel canonical correlation analysis (KCCA) and so the application of a similar sparsity algorithm gives rise to the sparse KCCA algorithm. We also propose a loss bound for sparse KCCA using the novel technique developed for KMP. All of the algorithms and bounds proposed in the thesis are elucidated with experiments.