Applying dimensionality reduction on OneHotEncoded array I have a really large data set with mixed variables. I have converted categorical variables to numerical using OneHotEncoding and it has resulted in more than a couple of thousand different features, combined that is. 

Is it possible to apply dimensionality reduction algorithms on OneHotEncoded data which looks like [[1. 0. 1. 0.]...[0. 0. 0. 0.]] or should it be done by merging with the original data set?
Following your example, you have different points in a 4-dimensional space. So, yes! you can use any dimensionality reduction technique, from PCA to UMAP.

In general, if your data is in a numeric format (and one-hot actually is), all the elements have the same dimensionality, and you don't have undefined values (NAN, inf), you can always use dimensionality reduction.