Dear numpy community
I would like to propose a mew feature in np.linalg to compute weighted
gram matrix (sandwich product), see
https://github.com/numpy/numpy/issues/29559.
Proposed new feature or change:
The solvers for many important statistical models like Generalized
Linear Models often need to compute |X.T @ W @ X| where |X| is a
2-dimensional array of features (features in columns, observations in
rows) and |W| is very often a diagonal array of (current) weights. This
computation is usually the main computational bottleneck.
It would be great if numpy could provide an efficient implementation of
it, e.g. |np.linalg.sandwicht_product(X, weight=w)|.
Why numpy? It seems even more unrealistic to me, to get it into BLAS
implementations. Numpy has support for SIMD via Highways.
Computational alternatives
Drawbacks of |(X.T * diag(W)) @ X|:
* Additional memory allocation to compute |X.T @ diag(W)| , same size
as (usually large) |X|.
* The result is symmetric, but this fact is not used. So at least a
factor of 2 is possible.
Note that without weights, |X.T @ X| uses BLAS syrk. But the weights
are crucial.
Drawback of |Z = np.sqrt(diag(W))[:, None] * X| and then |Z @ Z|:
* Additional memory allocation for |Z|, same size as (usually large) |X|.
* Taking square roots for possibly large |diag(W)|
Additional information
https://github.com/Quantco/tabmat has an implementation of it with XSIMD.
Best
Christian Lorentzen
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