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speeding up matrix multiplication (newbie question)
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Date: -- (:)
From: Erick Matsen <matsen@b...>
Subject: Re: [Caml-list] speeding up matrix multiplication (newbie question)
Wow, once again I am amazed by the vitality of this list. Thank you
for your suggestions.

Here is the context: we are interested in calculating the likelihood
of taxonomic placement of short "metagenomics" sequence fragments from
unknown organisms in the ocean. We start by assuming a model of
sequence evolution, which is a reversible Markov chain. The taxonomy
is represented as a tree, and the sequence information is a collection
of likelihoods of sequence identities. As we move up the tree, these
sequences "evolve" by getting multiplied by the exponentiated
instantaneous Markov matrix.

The matrices are of the size of the sequence model: 4x4 when looking
at nucleotides, and 20x20 when looking at proteins.

The bottleneck is (I mis-spoke before) that we are multiplying many
length-4 or length-20 vectors by a collection of matrices which
represent the time evolution of those sequences as follows.

Outer loop:
  modify the amount of time each markov process runs
  exponentiate the rate matrices to get transition matrices

  Recur over the tree, starting at the leaves:
    at a node, multiply all of the daughter likelihood vectors together
    return the multiplication of that product by the trasition matrix
(bottleneck!)

The trees are on the order of 50 leaves, and there are about 500
likelihood vectors done at once.

All of this gets run on a big cluster of Xeons. It's not worth
parallelizing because we are running many instances of this process
already, which fills up the cluster nodes.

So far I have been doing the simplest thing possible, which is just to
multiply the matrices out like \sum_j a_ij v_j. Um, this is a bit
embarassing.

let mul_vec m v =
    if Array.length v <> n_cols m then
      failwith "mul_vec: matrix size and vector size don't match!";
    let result = Array.create (n_rows m) N.zero in
    for i=0 to (n_rows m)-1 do
      for j=0 to (n_cols m)-1 do
	result.(i) <- N.add result.(i) (N.mul (get m i j) v.(j))
      done;
    done;
    result

I have implemented it in a functorial way for flexibility. N is the
number class. How much improvement might I hope for if I make a
dedicated float vector multiplication function? I'm sorry, I know
nothing about "boxing." Where can I read about that?


Thank you again,

Erick



On Fri, Feb 20, 2009 at 10:46 AM, Xavier Leroy <Xavier.Leroy@inria.fr> wrote:
>> I'm working on speeding up some code, and I wanted to check with
>> someone before implementation.
>>
>> As you can see below, the code primarily spends its time multiplying
>> relatively small matrices. Precision is of course important but not
>> an incredibly crucial issue, as the most important thing is relative
>> comparison between things which *should* be pretty different.
>
> You need to post your matrix multiplication code so that the regulars
> on this list can tear it to pieces :-)
>
> From the profile you gave, it looks like you parameterized your matrix
> multiplication code over the + and * operations over matrix elements.
> This is good for genericity but not so good for performance, as it
> will result in more boxing (heap allocation) of floating-point values.
> The first thing you should try is write a version of matrix
> multiplication that is specialized for type "float".
>
> Then, there are several ways to write the textbook matrix
> multiplication algorithm, some of which perform less boxing than
> others.  Again, post your code and we'll let you know.
>
>> Currently I'm just using native (double-precision) ocaml floats and
>> the native ocaml arrays for a first pass on the problem.  Now I'm
>> thinking about moving to using float32 bigarrays, and I'm hoping
>> that the code will double in speed. I'd like to know: is that
>> realistic? Any other suggestions?
>
> It won't double in speed: arithmetic operations will take exactly the
> same time in single or double precision.  What single-precision
> bigarrays buy you is halving the memory footprint of your matrices.
> That could result in better cache behavior and therefore slightly
> better speed, but it depends very much on the sizes and number of your
> matrices.
>
> - Xavier Leroy
>