I have p-dimensional data and I need to determine if that data has significant modes or if it’s clustered in any way. Here p=50, (dense embedding), we have n samples and p <<< n.

What are some ways to do that? I can think of creating sample covarince matrix of 50*50. Not sure how to analyze that full matrix (eigenvalues, etc) for conclusions.

I can also try to do kmeans and see if it converges or do GMM but I needs to try on many number of clusters. The data could have many clusters in theory, possibly residing on lower dim manifolds.

Any thoughts? Thanks.