  We propose the nuclear norm penalty as an alternative to the ridge penalty
for regularized multinomial regression. This convex relaxation of reduced-rank
multinomial regression has the advantage of leveraging underlying structure
among the response categories to make better predictions. We apply our method,
nuclear penalized multinomial regression (NPMR), to Major League Baseball
play-by-play data to predict outcome probabilities based on batter-pitcher
matchups. The interpretation of the results meshes well with subject-area
expertise and also suggests a novel understanding of what differentiates
players.
