Determine if high dimensional data is multimodal 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.

Multimodal is with regard to models, not data. If I understand correctly, you are trying to find the optimal number of clusters given a specific model structure. Typically in this set up selecting "number of clusters" is part of a standard model selection process.
Take GMM for example,

*

*If you use Bayesian GMM, you can try a set of possible "number of clusters", and select the one that minimize BIC on your complete data set;

*If you use GMM (without Bayesian), you need to split the data set into training and validation, train each variation on the training set and select the one that maximize the likelihood on validation set.

