Re: [eigen] Does anyone know of a good l1-solver?

[Márton Danóczy <marton78@xxxxxxxxx>]

Hi Anil,

you mentioning L1 magic and primal-dual solvers makes me guess that
you actually want to solve argmin ||Ax-b||^2_2 + a ||x||_1? That is,
quadratic loss with L1-penalty, aka LASSO? In this case, I can
recommend the orthant-wise limited-memory quasi-Newton (OWLQN)
algorithm. Actually, it can solve f(Ax-b) + ||W x||_1, where f is a
smooth convex loss and W is diagonal. There is a good implementation
as part of the libBFGS [2] library. I have taken this implementation
and eigenified it, if you (or somebody else) is interested, I can send
it per Email. Or even push it to Eigen/unsupported? I have used it
from MATLAB in the past (there's also a mex file wrapper) and found it
to work ok, as long as there are not too many variables (then,
convergence can be slow). A better behaved method is DAL [3], however,
to my knowledge, there is only a MATLAB implementation.

In case your loss function is really 1-norm, I think the standard
method is called "column generation" but I have never used it.

[1]: http://nlp.stanford.edu/~pupochik/papers/andrew07scalable.pdf
[2]: http://www.chokkan.org/software/liblbfgs/
[3]: http://www.ibis.t.u-tokyo.ac.jp/ryotat/dal/

HTH,
Marton

On 17 April 2012 14:38, cr.anil@xxxxxxxxx <cr.anil@xxxxxxxxx> wrote:
>> Either that, or maybe find x, s.t. ||Ax - b||_1=min ?
>>
> Yes this is what I want to solve although I want to add a l2 regulariser
> sometime later to this.
>
>> Do you know anything about A? (sparse, square, symmetric, rank-deficient,
>> ill-conditioned, ...)
>>
> No. A is a dense matrix and full rank but I do know that it has 6 columns
> and 3N rows.
>
>> Anyways, afaik, Eigen itself does not provide L1-solvers, so you need to
>> find an external library anyways (or find an algorithm and implement it). It
>> is quite easy to interact with other libraries by using Eigen's Map and
>> .data() functionality.
>
> Hmmm... I thought it would be nice if there was a efficient library which
> did this since I run the optimisation in a loop.
>
> Also is the primal dual technique the most efficient for this problem?
> That's what I found in a matlab library called l1 magic.
>
> Anil