1. A method for hierarchical visualization of multi-dimensional data, comprising: receiving, with a computer processor, a multi-dimensional data set of data points dispersed into a plurality of categories, each respective category containing data points that share a common characteristic unique to the respective category; applying, with a computer processor, a first dimension-reduction process to the multi-dimensional data set to obtain a current dimension-reduced visualization of the multi-dimensional data set; displaying the current dimension-reduced visualization to a user, wherein displaying the current dimension-reduced visualization includes: indicating, for each of the data points in the data set, a location based on a first set of dimensions associated with the current visualization; and indicating a categorization of a plurality of the data points in the data set; receiving user input that graphically selects a user-defined region of the current visualization, wherein the selected region comprises a mixed region that contains data points of different categories, including a first data point and a second data point, wherein a location of the first data point in the current visualization and a location of a second data point in the current visualization are separated by a first distance; selecting, with a computer processor, a subset of the multi-dimensional data set, the subset including the data points contained within the selected region of the current dimension-reduced visualization; applying, with a computer processor, an additional dimension-reduction process to the selected subset of the multi-dimensional data set to obtain a subsequent visualization; and displaying the subsequent visualization to the user, wherein displaying the subsequent visualization includes indicating, for each of the data points located in the mixed region, a location based on a second set of dimensions associated with the subsequent visualization, and wherein a location of the first data point in the subsequent visualization and a location of the second data point in the subsequent visualization are separated by a second distance that is different from the first distance.