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The optic nerve transmits visual information to the brain as trains of discrete events, a low-power, low-bandwidth communication channel also exploited by silicon retina cameras. Extracting highfidelity visual input from retinal event trains is thus a key challenge for both computational neuroscience and neuromorphic engineering. Here, we investigate whether sparse coding can enable the reconstruction of high-fidelity images and video from retinal event trains. Our approach is analogous to compressive sensing, in which only a random subset of pixels are transmitted and the missing information is estimated via inference. We employed a variant of the Locally Competitive Algorithm to infer sparse representations from retinal event trains, using a dictionary of convolutional features optimized via stochastic gradient descent and trained in an unsupervised manner using a local Hebbian learning rule with momentum. We used an anatomically realistic retinal model with stochastic graded release from cones and bipolar cells to encode thumbnail images as spike trains arising from ON and OFF retinal ganglion cells. The spikes from each model ganglion cell were summed over a 32 msec time window, yielding a noisy rate-coded image. Analogous to how the primary visual cortex is postulated to infer features from noisy spike trains arising from the optic nerve, we inferred a higher-fidelity sparse reconstruction from the noisy rate-coded image using a convolutional dictionary trained on the original CIFAR10 database. To investigate whether a similar approachworks on non-stochastic data, we demonstrate that the same procedure can be used to reconstruct high-frequency video from the asynchronous events arising from a silicon retina camera moving through a laboratory environment.