Dealing with matrix inversion efficiently

I have two matrix X and Y. Both are of dimension [bs, hidden_size].
I want to take inverse of matrix obtained by[X, Y], dim=1).

Pseudo inverse is not stable (docs say the same). I obtain inverse to deal with this. So I used a Linear layer of dimension (hidden_size, batch_size//2) to convert X and Y of dimensions [bs, bs//2]. After concat, I get a matrix of form [bs, bs]. The downside of this approach is that it requires me to use a batch size = hidden_size/2 to perform well, otherwise the network doesn’t learn anything well.

Is there a mathematical elegant way you can suggest as a workaround ?

Hello kl!

Could you explain conceptually what meaning this matrix inverse
is supposed to have? Your construction seems odd.

As you have recognized, a matrix must be square to be invertible
(although not all square matrices are invertible, of course). But
your batch size (bs) is something of a 'technical" parameter that
doesn’t really have anything to do with your data or the structure of
your network. As you point out, you have to adjust your batch size
to match hidden_size / 2. But how does your construction make
sense if it breaks when you change your batch size?


K. Frank

Thanks for taking interest. The matrix inversion is part of a formula that is supposed to project features orthogonally.

$F_{G}$ (comes from concatenation of two vectors) is not a square matrix, that’s why I make this modification.

The issue is, that my network performs much better if I reduce second dimensions of X and Y to half of what it was (768->384), in all other cases, it doesn’t learn anything. There might be better ways of projection, which I’ve not explored. (For instance, if bs=512, 768->512, then I don’t notice any improvement.)

This is an unstable method. The main issue is to perform inversion. That implies I need square matrices. That’s why I’m asking if there’s a better mathematical way to perform this.
Yes, I realize that my implementation is not batch size agnostic once model is trained. As a result, I am unable to train bigger models (because batch_size = hidden_size/2 won’t fit in memory).

So in general sense, what is a better way to perform inversion when the matrix is not square (pinverse isn’t stable) ?