Solving linear equations

[Martin Enzinger <ma...@gmail.com>]

Hi,

I'm wondering how to use Mllib for solving equation systems following this
pattern

2*x1 + x2 + 3*x3 + .... + xn = 0
x1 + 0*x2 + 3*x3 + .... + xn = 0
..........
..........
0*x1 + x2 + 0*x3 + .... + xn = 0

I definitely still have some reading to do to really understand the direct
solving techniques, but at the current state of "knowledge" SVD could help
me with this right?

Can you point me to an example or a tutorial?

best regards

Re: Solving linear equations

[Shivaram Venkataraman <sh...@eecs.berkeley.edu>]

We have been working on large scale QR decompositions which would fit
this problem well -- TSQR
[https://github.com/amplab/ml-matrix/blob/master/src/main/scala/edu/berkeley/cs/amplab/mlmatrix/TSQR.scala]
for instance can be used to solve the least squares system if you have
more equations than variables (over-determined)

We plan to merge some of these into Spark as a part of
https://issues.apache.org/jira/browse/SPARK-3434 as well

Thanks
Shivaram

On Thu, Oct 23, 2014 at 1:54 AM, Sean Owen <so...@cloudera.com> wrote:
> The 0 vector is a trivial solution. Is the data big, such that it
> can't be computed on one machine? if so I assume this system is
> over-determined. You can use a decomposition to find a least-squares
> solution, but the SVD is overkill and in any event distributed
> decompositions don't exist in the project. You can solve it a linear
> regression as Mr Das says.
>
> If it's small enough to fit locally you should just use a matrix
> library to solve Ax = b with the QR decomposition or something, with
> Breeze or Commons Math or octave or R. Lots of options if it's
> smallish.
>
> On Thu, Oct 23, 2014 at 12:15 AM, Martin Enzinger
> <ma...@gmail.com> wrote:
>> Hi,
>>
>> I'm wondering how to use Mllib for solving equation systems following this
>> pattern
>>
>> 2*x1 + x2 + 3*x3 + .... + xn = 0
>> x1 + 0*x2 + 3*x3 + .... + xn = 0
>> ..........
>> ..........
>> 0*x1 + x2 + 0*x3 + .... + xn = 0
>>
>> I definitely still have some reading to do to really understand the direct
>> solving techniques, but at the current state of "knowledge" SVD could help
>> me with this right?
>>
>> Can you point me to an example or a tutorial?
>>
>> best regards
>
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Re: Solving linear equations

[Sean Owen <so...@cloudera.com>]

The 0 vector is a trivial solution. Is the data big, such that it
can't be computed on one machine? if so I assume this system is
over-determined. You can use a decomposition to find a least-squares
solution, but the SVD is overkill and in any event distributed
decompositions don't exist in the project. You can solve it a linear
regression as Mr Das says.

If it's small enough to fit locally you should just use a matrix
library to solve Ax = b with the QR decomposition or something, with
Breeze or Commons Math or octave or R. Lots of options if it's
smallish.

On Thu, Oct 23, 2014 at 12:15 AM, Martin Enzinger
<ma...@gmail.com> wrote:
> Hi,
>
> I'm wondering how to use Mllib for solving equation systems following this
> pattern
>
> 2*x1 + x2 + 3*x3 + .... + xn = 0
> x1 + 0*x2 + 3*x3 + .... + xn = 0
> ..........
> ..........
> 0*x1 + x2 + 0*x3 + .... + xn = 0
>
> I definitely still have some reading to do to really understand the direct
> solving techniques, but at the current state of "knowledge" SVD could help
> me with this right?
>
> Can you point me to an example or a tutorial?
>
> best regards

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Re: Solving linear equations

[Debasish Das <de...@gmail.com>]

Hi Martin,

This problem is Ax = B where A is your matrix [2 1 3 ... 1; 1 0 3 ...;....]
and x is what you want to find..B is 0 in this case...For mllib normally
this is label....basically create a labeledPoint where label is 0 always...

Use mllib's linear regression and solve the following problem:

min ||Ax - B||_{2}^{2} + lambda||x||_{2}^{2}

Put a small regularization to condition the problem (~1e-4)...and play with
some options for learning rate in linear regression...

The parameter vector that you get out of mllib linear regression is the
answer to your linear equation solver...

Thanks.
Deb



On Wed, Oct 22, 2014 at 4:15 PM, Martin Enzinger <ma...@gmail.com>
wrote:

> Hi,
>
> I'm wondering how to use Mllib for solving equation systems following this
> pattern
>
> 2*x1 + x2 + 3*x3 + .... + xn = 0
> x1 + 0*x2 + 3*x3 + .... + xn = 0
> ..........
> ..........
> 0*x1 + x2 + 0*x3 + .... + xn = 0
>
> I definitely still have some reading to do to really understand the direct
> solving techniques, but at the current state of "knowledge" SVD could help
> me with this right?
>
> Can you point me to an example or a tutorial?
>
> best regards
>