  We narrow the gap between previous literature on quantum linear algebra and
useful data analysis on a quantum computer, providing quantum procedures that
speed-up the solution of eigenproblems for data representation in machine
learning. The power and practical use of these subroutines is shown through new
quantum algorithms, sublinear in the input matrix's size, for principal
component analysis, correspondence analysis, and latent semantic analysis. We
provide a theoretical analysis of the run-time and prove tight bounds on the
randomized algorithms' error. We run experiments on multiple datasets,
simulating PCA's dimensionality reduction for image classification with the
novel routines. The results show that the run-time parameters that do not
depend on the input's size are reasonable and that the error on the computed
model is small, allowing for competitive classification performances.
