Much of our work involves deciphering patterns in high-dimensional distributions. We would often like to understand features of these distributions in low-dimensional subspaces, which depends on the specific projection used to reduce dimensionality. Furthermore, if the distributions have a high density of data points, direct visualization of such low-dimensional projections can be misleading because many points are overlapping, and a disproportionate amount of "visual weight" is dedicated to outliers.
To aid in the quantitative analysis of high-dimensional datasets, we are developing tools to better visualize the underlying features of these distributions. These techniques can be used to resolve density variations in low-dimensional projections, and they can be used to easily compare across different projections.
In some cases, we are interested in understanding categorical relationships in our datasets. For example, given recordings of olfactory cells responding to different odor mixtures, we are interested in understanding how relationships between mixtures are represented in neural firing patterns. Is the mixture A+B+C most similar to A+B, A+C, or B+C? And how do these submixtures relate to the original components A, B, and C? We are developing hierarchical clustering techniques for understanding such relationships across different categorical scales.