We present fast methods for filtering voltage measurements and performing optimal inference of the location and strength of synaptic connections in large dendritic trees. Given noisy, subsampled voltage observations we develop fast l1-penalized regression methods for Kalman state-space models of the neuron voltage dynamics. The value of the l1-penalty parameter is chosen using cross-validation or, for low signal-to-noise ratio, a Mallows’ Cp-like criterion. Using low-rank approximations, we reduce the inference runtime from cubic to linear in the number of dendritic compartments. We also present an alternative, fully Bayesian approach to the inference problem using a spike-and-slab prior. We illustrate our results with simulations on toy and real neuronal geometries. We consider observation schemes that either scan the dendritic geometry uniformly or measure linear combinations of voltages across several locations with random coefficients. For the latter, we show how to choose the coefficients to offset the correlation between successive measurements imposed by the neuron dynamics. This results in a “compressed sensing” observation scheme, with an important reduction in the number of measurements required to infer the synaptic weights.