Particle accelerators are complex machines, and often differ from their idealized designs, necessitating fine-tuning on the real machine to achieve optimal performance. However, with hundreds or thousands of magnets available for doing this optimization, this process can be quite challenging. In these cases, it becomes useful to reduce the dimensionality of the problem in order to speed up the tuning procedure. Many high-dimensional systems have low-dimensional approximations which maintain almost all of the information contained in the full model. We explore the applicability of dimension reduction to various problems in accelerator physics. We find that using just 8 knobs created from 81 possible corrector magnets enables us to significantly reduce the vertical emittance of the electron beam at the Cornell Electron Storage Ring (CESR) in both simulation and experiment, using both one-dimensional knob scans and a multi-objective genetic algorithm. We also find that such a reduction allows us to use 8 groups of sextupole magnets to increase the injection efficiency of a positron beam into CESR. We also explore the extent to which we may use random combinations of magnets to achieve this dimension reduction, and find that, while we retain comparable ability to improve the parameter of interest, we need greater overall changes in the relevant magnet strengths.