Consider a matrix, $\mathbf{V}$, where each row corresponds to one of $m$ electoral district and each column corresponds to one of $n$ candidates or political parties. Each element, $V_{ij}$, is the proportion of votes cast in district $i$ for party $j$.Now imagine taking leave of your senses and deciding to carry out a low-rank matrix factorisation of $\mathbf{V}$. It turns out that regular non-negative matrix factorisation minimising generalised KL-Divergence will preserve the row and column sums in the approximation1, so where the rank of your factorisation, $k$, is less than $n$, it's straightforward to find $k$ latent variables for each electoral district that describe contributions to the share of each party's vote in that district.My questions:1) Are there any obvious reasons I shouldn't be doing this?2) Is there a sensible interpretation of these $k$ variables?3) Is there a more canonical method for factorising matrices like this? Or a more canonical method for finding latent variables for data like this?
Matrix factorisation of vote share data
matrix factorisation