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Functional magnetic resonance imaging (fMRI) is a notoriously noisy measurement of brain activity because of the large variations between individuals, signals marred by environmental differences during collection, and spatiotemporal averaging required by the measurement resolution. In addition, the data is extremely high dimensional, with the space of the activity typically having much lower intrinsic dimension. In order to understand the connection between stimuli of interest and brain activity, and analyze differences and commonalities between subjects, it becomes important to learn a meaningful embedding of the data that denoises, and reveals its intrinsic structure. Specifically, we assume that while noise varies significantly between individuals, true responses to stimuli will share common, low-dimensional features between subjects which are jointly discoverable. Similar approaches have been exploited previously but they have mainly used linear methods such as PCA and shared response modeling (SRM). In contrast, we propose a neural network called MRMD-AE (manifold-regularized multiple decoder, autoencoder), that learns a common embedding from multiple subjects in an experiment while retaining the ability to decode to individual raw fMRI signals. We show that our learned common space represents an extensible manifold (where new points not seen during training can be mapped), improves the classification accuracy of stimulus features of unseen timepoints, as well as improves cross-subject translation of fMRI signals. We believe this framework can be used for many downstream applications such as guided brain-computer interface (BCI) training in the future.