Multivariate Categorical Data

Hi data-enthusiasts,

Recently, I stumbled across an irritatingly odd problem at work concerning the visualization of multidimensional nominal data. In my Job as a data-scientist, I had to create a visualization for our powerpoint reports that provide an overview of our current projects and their status. As I started to deal with the data, I realized how many attributes/tags our projects have and how much information we could provide about our projects. These multidimensional project data got me thinking if there is a way to visualize this data more efficiently. Since there are some very effective data preprocessing & analysis tools for dimensionality reduction with ordinal and numeric data (such as PCA or LDA), I was wondering if there exists a dimensionality reduction technique for nominal data? Does anyone of you know of such a thing or some further information/research about this?
However, the goal of dimensionality reduction remains to be to reduce the dimensionality while preserving as much information as possible. Hence it is difficult for me to imagine how this could work with categorical data. But anyway, let me know what you think because our powerpoint reports really could need a breath of fresh air.
Oh and by the way a late happy new year!