Wide-field probes of weak gravitational lensing have the potential to address fundamental questions about the nature of the universe. Measures such as the correlation function, power spectrum, or statistics of shear peaks can be compared with theoretical predictions to answer substantive question about the nature of dark matter, dark energy, gravity, and primordial perturbations. Comparison of the data to the theoretical model, however, can be subject to systematic effects due to survey geometry, selection functions, and other biases. This can be framed as a machine learning problem: given a sparse set of noisy observations, how can one best recover the underlying signal of interest? We propose to address these challenges using a compressed-sensing approach based on a Karhunen-Loeve (KL) model of the signal. This approach can efficiently recover the shear signal from noisy data with arbitrary masking and survey geometry. The signal-to-noise-ranked KL vectors allow effective noise filtration, leading to a 30% decrease in B-mode contamination for simulated data. Furthermore, because the KL model is based on covariance matrices, it naturally encapsulates the two-point information of the field and provides a framework for efficient Bayesian likelihood analysis of the two-point statistics of a cosmological shear