We propose a statistical learning-based traffic speed estimation method that uses sparse vehicle trajectory information. Using a convolutional encoder-decoder based architecture, we show that a well trained neural network can learn spatiooral traffic speed dynamics from time-space diagrams. We demonstrate this for a homogeneous road section using simulated vehicle trajectories and then validate it using real-world data from the Next Generation Simulation (NGSIM) program. Our results show that with probe vehicle penetration levels as low as 5%, the proposed estimation method can provide a sound reconstruction of macroscopic traffic speeds and reproduce realistic shockwave patterns, implying applicability in a variety of traffic conditions. We further discuss the model's reconstruction mechanisms and confirm its ability to differentiate various traffic behaviors such as congested and free-flow traffic states, transition dynamics, and shockwave propagation. We also provide a comparison against a widely used adaptive smoothing technique used for the same purpose and demonstrate the superiority of the proposed approach, even with probe vehicle lower penetration levels.