The instructor asks us to look for patterns in the space, and the students find one: the movies seem to have separated by genre. Romances are on one side, while action movies are on the other. A student asks how the algorithm knew about genre, when the input only contained rating data. "It seems like magic, I know," the instructor replies, "Magically, we've measured two secret things about these movies." Those two secrets are the two dimensions of our space, although we could have instructed the computer to generate more of them. Our horizontal dimension appears to correspond to genre, as though information about the movies genres were hidden in the data, waiting to be revealed. The significance of the vertical dimension is less obvious, although the instructor suggests that it may reflect how "serious" the movies are.

What made matrix factorization "magical," like McDonald's alchemy, was its ability to uncover such cultural secrets from data that appeared to be about something else. We were being instructed not only in the fundamental spatiality of data but in what the sociologist of science Catelijne Coopmans (2014) has called "artful revelation" -- the rhetorical use of visualization to make manifest hidden patterns in data. Revelations like the one performed in class are a common way to claim authority for data analytic practices. Coordinate spaces are often presented as ways to make data intuitively understandable by people -- an easy means to see, rather than to calculate, the similarities among a set of data points. The geographers Martin Dodge and Rob Kitchin (2001, 30) call images like these "spatializations," to distinguish them from proper "maps," which they reserve for visualizations that have geographical referents. But, as we will see, this distinction is not a significant one for most people working in machine learning. Spatializations effectively summarize differences along various axes into singular, readily comparable distances, appealing to intuitions about the relationship between distance and similarity. When visualized, these spaces are almost always described as "maps."