Short answer: Yes. Long answer: Yes but not for all kind of matrices. What I am proposing here is given a matrix A create a matrix S (using CWT) with the following property with probability of at least 9/10: ||SAx||_2 = (1+-epsilon)||Ax||_2 for all x in R. These matrices (S) are called subspace embeddings. SA can be computed in O(nnz(A)) and the property So basically it is an improvement for overconstrained least-squares regression, low-rank approximation, approximation all leverage scores and l_p-regession. On top of that, implementing Blendenpik would remove the uncertainty (I would use the result of CWT as a preconditioner). Blendenpik beats LAPACK's direct dense least-squares solver by a large margin on essentially any dense *tall *matrix. On 23 March 2017 at 12:24, Matt Haberland <haberland@ucla.edu> wrote:

Do I understand that this is intended to be a faster but otherwise drop-in replacement for scipy.linalg.lstsq or scipy.sparse.linalg.lsqr? I would definitely be interested in an alternative sparse least squares solver. I could use it.

On Mar 21, 2017 11:09 PM, "Jordi Montes" <jomsdev@gmail.com> wrote:

...

*# I started this discussion on GitHub <https://github.com/scipy/scipy/issues/7122>. This email is a digested version of the conversation to check what people think about this on the mailing list.*

TL;DR: *I think having an implementation for some randomize matrix algorithms would be great. I would like to discuss if the community finds it useful too.*

Hello,

I am Jordi. I have been working on a xdata <http://www.darpa.mil/program/xdata> open source project for the last year called libSkylark <https://xdata-skylark.github.io/libskylark/>. The library is suitable for general statistical data analysis and optimization applications, but it is heavily focused on distributed systems.

*Randomized matrix algorithms* have been a hot topic in research the last years. Recent developments have shown their utility in large-scale machine learning and statistical data analysis applications.

Although many ML applications would take advantage of the implementation of these algorithms the scope of them goes far beyond. I believe it would be a good idea integrate some of them in Scipy and I would like to know the opinion of others to check if it would be worth it.

To my eyes, Randomized Numerical Linear Algebra is big/useful enough to has its submodule. However, this is something that the community has to decide.

Meanwhile, to demonstrate that these methods can be useful I would implement them under, scipy.linalg where the scipy linear algebra functions live. The idea is to bring value since the first moment. I will implement Blendenpik <http://epubs.siam.org/doi/10.1137/090767911>, a least-squares solver based on these techniques that outperforms LAPACK by significant factors and scales significantly better than any QR-based solver.

The proposal is:

-

Implement CWT <http://delivery.acm.org/10.1145/2490000/2488620/p81-clarkson.pdf?ip=198.4.83.52&id=2488620&acc=ACTIVE%20SERVICE&key=90259622823F2C32%2E66F8B2C43167A3DD%2E4D4702B0C3E38B35%2E4D4702B0C3E38B35&CFID=736713469&CFTOKEN=50269727&__acm__=1489010918_f1abc3d5c4650844f4503dc1a27484ad> . CWT lets us create a matrix *S* such as *SA* is shorter than *A but conservating some properties*. -

Compute *QR = SA* using scipy. -

From here, there are two potential features: -

Implement Blendenpik <http://epubs.siam.org/doi/10.1137/090767911> for solving Linear Square problems faster without losing any accuracy. -

Compute the squared row norms to obtain leverage scores.

Would be better to start with Blendenpik, because it solves a more common problem by far.

I am thinking about doing it for sparse matrices first because CWT can take advantage of it. However, I wonder if the QR method on script takes advantage of sparsity too.

I have already set up the development environment so I could start as soon as we decide what to do. As I said, I am open to discussion, so anyone who could be interested is welcome.

Thank you,

Jordi.

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