Modern machine learning architectures often embed their inputs into a lower-dimensional latent space before generating a final output. A vast set of empirical results---and some emerging theory---predicts that these lower-dimensional codes often are highly structured, capturing lower-dimensional variation in the data. Based on this observation, in this talk I will describe efforts in my group to develop lightweight algorithms that navigate, restructure, and reshape learned latent spaces. Along the way, I will consider a variety of practical problems in machine learning, including low-rank adaptation of large models, regularization to promote local latent structure, and efficient training/evaluation of generative models.[-]