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https://issues.apache.org/jira/browse/MAHOUT-672?page=com.atlassian.jira.plugin.system.issuetabpanels:comment-tabpanel&focusedCommentId=13134503#comment-13134503
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Dmitriy Lyubimov edited comment on MAHOUT-672 at 10/24/11 9:23 PM:
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bq. I think it's a bit extreme to say we need to have nearly O(1)
These may mean quite different things in practice. the devil is in the details.
by ~O(1) i meant ok, if in practice it grows so slow that it's enought to
process several billion rows (m) of input by having 40 iterations, that's
perhaps still 'nearly' O(1) in my definiton.
I just said that i seem to have gleaned in the javadoc explanation that with
this patch
num of iterations ~ n (num of columns) so if for million columns it means a
million iterations, that's probably not cool. On the other side, conversion
will unlikely require a million. Then what? I kind of still did not get a clear
clarifications on the estimate of num iterations (except that it is not exactly
O(1)).
was (Author: dlyubimov):
bq. I think it's a bit extreme to say we need to have nearly O(1)
These may mean quite different things in practice. the devil is in the details.
by ~O(1) i meant ok, if in practice it grows so slow that it's enought to
process several billion rows (m) of input by having 40 iterations, that's
perhaps still 'nearly' O(1) in my definiton.
I just said that i seem to have gleaned in the javadoc explanation that with
this patch
num of iterations ~ n (num of columns) so if for million columns it means a
million iterations, that's probably not cool. On the other side, conversion
will unlikely require a million. Than what? I kind of still did not get a clear
clarifications on the estimate of num iterations (except that it is not exactly
O(1)).
> Implementation of Conjugate Gradient for solving large linear systems
> ---------------------------------------------------------------------
>
> Key: MAHOUT-672
> URL: https://issues.apache.org/jira/browse/MAHOUT-672
> Project: Mahout
> Issue Type: New Feature
> Components: Math
> Affects Versions: 0.5
> Reporter: Jonathan Traupman
> Priority: Minor
> Fix For: 0.6
>
> Attachments: 0001-MAHOUT-672-LSMR-iterative-linear-solver.patch,
> 0001-MAHOUT-672-LSMR-iterative-linear-solver.patch, MAHOUT-672.patch,
> mahout-672-111023.patch, mahout-672.patch
>
> Original Estimate: 48h
> Remaining Estimate: 48h
>
> This patch contains an implementation of conjugate gradient, an iterative
> algorithm for solving large linear systems. In particular, it is well suited
> for large sparse systems where a traditional QR or Cholesky decomposition is
> infeasible. Conjugate gradient only works for matrices that are square,
> symmetric, and positive definite (basically the same types where Cholesky
> decomposition is applicable). Systems like these commonly occur in statistics
> and machine learning problems (e.g. regression).
> Both a standard (in memory) solver and a distributed hadoop-based solver
> (basically the standard solver run using a DistributedRowMatrix a la
> DistributedLanczosSolver) are included.
> There is already a version of this algorithm in taste package, but it doesn't
> operate on standard mahout matrix/vector objects, nor does it implement a
> distributed version. I believe this implementation will be more generically
> useful to the community than the specialized one in taste.
> This implementation solves the following types of systems:
> Ax = b, where A is square, symmetric, and positive definite
> A'Ax = b where A is arbitrary but A'A is positive definite. Directly solving
> this system is more efficient than computing A'A explicitly then solving.
> (A + lambda * I)x = b and (A'A + lambda * I)x = b, for systems where A or A'A
> is singular and/or not full rank. This occurs commonly if A is large and
> sparse. Solving a system of this form is used, for example, in ridge
> regression.
> In addition to the normal conjugate gradient solver, this implementation also
> handles preconditioning, and has a sample Jacobi preconditioner included as
> an example. More work will be needed to build more advanced preconditioners
> if desired.
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