Convolutional neural networks (CNNs) are increasingly widely used in psychology and neuroscience to predict how human minds and brains respond to visual images. Typically, CNNs represent these images using thousands of features that are learned through extensive training on image datasets. This raises a question: How many of these features are really needed to model human behavior? Here, we attempt to estimate the number of dimensions in CNN representations that are required to capture human psychological representations in two ways: (1) (...) directly, using human similarity judgments and (2) indirectly, in the context of categorization. In both cases, we find that low-dimensional projections of CNN representations are sufficient to predict human behavior. We show that these low-dimensional representations can be easily interpreted, providing further insight into how people represent visual information. A series of control studies indicate that these findings are not due to the size of the dataset we used and may be due to a high level of redundancy in the features appearing in CNN representations. (shrink)