Re: [math] matrix multiplication surprisingly slow?
On 7 April 2016 at 18:41, Gilles wrote:
>
> It's certainly worth ascertaining with more data points (to make a nice
> plot).
>
I can run mult on both for {10, 100, 500, 1000, 1500, 2000, 2500, 5000} if
that would help.
>
> Naturally, better asymptotic
>> performance would be nice,
>>
>
> What do you mean?
>
When I wrote that I was thinking it might make sense to use Strassen's
algorithm (or something else with better asymptotic performance; this isn't
my area of expertise) for large square matrices. After looking a bit at the
literature it seems unlikely to be worth it, given numerical stability
concerns etc.
>
> but I recognize that the AMC devs likely have
>> better things to do.
>>
>> Observation 2: When multiplying new matrices, BlockMatrix offers some
>> improvement when multiplying the same matrices repeatedly
>>
>
> Why would someone do that?
>
Good question! Maybe if matrices are being selected/changed at runtime and
someone can't be bothered to check/cache?
>
> Observation 3: OJAlgo doesn't scale better asymptotically from what I can
>> tell. That's no great surprise based on
>> https://github.com/optimatika/ojAlgo/wiki/Multiply.
>> <https://github.com/optimatika/ojAlgo/wiki/Multiply> Scala Breeze
>> w/native
>> libs scales closer to O(N^2.4).
>>
>
> Details on how you arrive to those numbers would really be a useful
> addition
> to the CM documentation.
>
I can share code snippets. I'd guess that breeze actually scales w/N^3 in a
single thread and that the faster-seeming scaling is related to
parallelization, e.g., adding threads for larger matrices.
>
> Observation 4: Breeze with native libraries does better than one might
>> guess based on some previous benchmarks. People still might want to use
>> math commons, not least because those native libs might be encumbered with
>> undesirable licenses.
>>
>
> Do they use multi-threading?
>
Yes, they do. I suppose that's important to note given that there are some
applications where running entire separate pipelines in parallel will be
more efficient than having parallel matrix ops.
>
> Regards,
> Gilles
>
>
>
>> [^1]:
>>
>>
>> https://git-wip-us.apache.org/repos/asf?p=commons-math.git;a=blob;f=src/main/java/org/apache/commons/math4/linear/Array2DRowRealMatrix.java;h=3778db56ba406beb973e1355234593bc006adb59;hb=HEAD
>>
>>
>> On 7 April 2016 at 16:50, Gilles wrote:
>>
>> On Thu, 7 Apr 2016 16:17:52 +0100, Chris Lucas wrote:
>>>
>>> Thanks for suggesting the BlockRealMatrix. I've run a few benchmarks
>>>> comparing the two, along with some other libraries.
>>>>
>>>>
>>> The email is not really suited for tables (see your message below).
>>> What are the points which you want to make?
>>>
>>> As said before, "reports" on CM features (a.o. benchmarks) are welcome
>>> but
>>> should be formatted for inclusion in the userguide (for now, until we
>>> might
>>> create a separate document for data that is subject to change more or
>>> less
>>> rapidly).
>>>
>>> If you suspect a problem with the code, then I suggest that you file a
>>> JIRA
>>> report, where files can be attached (or tables can be viewed without
>>> being
>>> deformed thanks to the "{noformat}" tag).
>>>
>>> Regards,
>>> Gilles
>>>
>>>
>>> I'm using jmh 1.11.3, JDK 1.8.0_05, VM 25.5-b02, with 2 warmups and 10
>>> test
>>>
>>>> iterations.
>>>>
>>>> The suffixes denote what's being tested:
>>>> MC: AMC 3.6.1 using Array2DRowRealMatrix
>>>> SB: Scala Breeze 0.12 with native libraries
>>>> OJ: OJAlgo 39.0. Some other benchmarks suggest OJAlgo is an especially
>>>> speedy pure-java library.
>>>> BMC: MC using a BlockRealMatrix.
>>>>
>>>> I'm using the same matrix creation/multiplication/inverse code as I
>>>> mentioned in my previous note. When testing BlockReadMatrices, I'm using
>>>> fixed random matrices and annotating my class with
>>>> @State(Scope.Benchmark).
>>>> For the curious, my rationale for building matrices on the spot is that
>>>> I'm
>>>> already caching results pretty heavily and rarely perform the same
>>>> operation on the same matrix twice -- I'm most curious about performance
>>>> in
>>>> the absence of caching benef
Re: [math] matrix multiplication surprisingly slow?
Sorry, you're right that I should have summarized the key implications (at
least as I see them).
The short version:
(1) If you're multiplying small matrices with AMC, BlockRealMatrix is
your friend. For large matrices, stick with Array2DRowRealMatrix (or
something else?).
(2) Scaling isn't great -- O(N^3) where N is the number of rows/cols in
a square matrix) -- but neither is OJAlgo's. The inter-library differences
can mostly be attributed to constant factors.
The longer version:
Observation 1: For Array2DRowRealMatrix, matrix multiplication appears to
scale with ~ O(N^3) (i.e., the naive algorithm). In retrospect, that was a
bit silly to spend time benchmarking because I've just looked at the
source[^1] and it *is* the naive algorithm. Naturally, better asymptotic
performance would be nice, but I recognize that the AMC devs likely have
better things to do.
Observation 2: When multiplying new matrices, BlockMatrix offers some
improvement when multiplying the same matrices repeatedly and for smaller
matrices. However on one person's setup (mine) it doesn't offer better
performance on large random matrices. Follow-up tests with N \in
{1000,1500,2000,2500} indicate it scales worse than the naive algorithm
(didn't look at growth, but factor of ~5 worse on a 2500x2500 matrix).
Observation 3: OJAlgo doesn't scale better asymptotically from what I can
tell. That's no great surprise based on
https://github.com/optimatika/ojAlgo/wiki/Multiply.
<https://github.com/optimatika/ojAlgo/wiki/Multiply> Scala Breeze w/native
libs scales closer to O(N^2.4).
Observation 4: Breeze with native libraries does better than one might
guess based on some previous benchmarks. People still might want to use
math commons, not least because those native libs might be encumbered with
undesirable licenses.
[^1]:
https://git-wip-us.apache.org/repos/asf?p=commons-math.git;a=blob;f=src/main/java/org/apache/commons/math4/linear/Array2DRowRealMatrix.java;h=3778db56ba406beb973e1355234593bc006adb59;hb=HEAD
On 7 April 2016 at 16:50, Gilles wrote:
> On Thu, 7 Apr 2016 16:17:52 +0100, Chris Lucas wrote:
>
>> Thanks for suggesting the BlockRealMatrix. I've run a few benchmarks
>> comparing the two, along with some other libraries.
>>
>
> The email is not really suited for tables (see your message below).
> What are the points which you want to make?
>
> As said before, "reports" on CM features (a.o. benchmarks) are welcome but
> should be formatted for inclusion in the userguide (for now, until we might
> create a separate document for data that is subject to change more or less
> rapidly).
>
> If you suspect a problem with the code, then I suggest that you file a JIRA
> report, where files can be attached (or tables can be viewed without being
> deformed thanks to the "{noformat}" tag).
>
> Regards,
> Gilles
>
>
> I'm using jmh 1.11.3, JDK 1.8.0_05, VM 25.5-b02, with 2 warmups and 10 test
>> iterations.
>>
>> The suffixes denote what's being tested:
>> MC: AMC 3.6.1 using Array2DRowRealMatrix
>> SB: Scala Breeze 0.12 with native libraries
>> OJ: OJAlgo 39.0. Some other benchmarks suggest OJAlgo is an especially
>> speedy pure-java library.
>> BMC: MC using a BlockRealMatrix.
>>
>> I'm using the same matrix creation/multiplication/inverse code as I
>> mentioned in my previous note. When testing BlockReadMatrices, I'm using
>> fixed random matrices and annotating my class with
>> @State(Scope.Benchmark).
>> For the curious, my rationale for building matrices on the spot is that
>> I'm
>> already caching results pretty heavily and rarely perform the same
>> operation on the same matrix twice -- I'm most curious about performance
>> in
>> the absence of caching benefits.
>>
>> Test 1a: Creating and multiplying two random/uniform (i.e., all entries
>> drawn from math.random()) 100x100 matrices:
>> [info] MatrixBenchmarks.buildMultTestMC100 thrpt 10 836.579
>> ± 7.120 ops/s
>> [info] MatrixBenchmarks.buildMultTestSB100 thrpt 101649.089
>> ± 170.649 ops/s
>> [info] MatrixBenchmarks.buildMultTestOJ100 thrpt 10 1485.014 ± 44.158
>> ops/s
>> [info] MatrixBenchmarks.multBMC100thrpt 10 1051.055 ± 2.290 ops/s
>>
>> Test 1b: Creating and multiplying two random/uniform 500x500 matrices:
>> [info] MatrixBenchmarks.buildMultTestMC500 thrpt 10 8.997
>> ± 0.064 ops/s
>> [info] MatrixBenchmarks.buildMultTestSB500 thrpt 10 80.558
>> ± 0.627 ops/s
>> [info] MatrixBenchmarks.buildMultTestOJ500 thrpt 10 34.626
>> ± 2.505 ops/s
>
Re: [math] matrix multiplication surprisingly slow?
Thanks for suggesting the BlockRealMatrix. I've run a few benchmarks comparing the two, along with some other libraries. I'm using jmh 1.11.3, JDK 1.8.0_05, VM 25.5-b02, with 2 warmups and 10 test iterations. The suffixes denote what's being tested: MC: AMC 3.6.1 using Array2DRowRealMatrix SB: Scala Breeze 0.12 with native libraries OJ: OJAlgo 39.0. Some other benchmarks suggest OJAlgo is an especially speedy pure-java library. BMC: MC using a BlockRealMatrix. I'm using the same matrix creation/multiplication/inverse code as I mentioned in my previous note. When testing BlockReadMatrices, I'm using fixed random matrices and annotating my class with @State(Scope.Benchmark). For the curious, my rationale for building matrices on the spot is that I'm already caching results pretty heavily and rarely perform the same operation on the same matrix twice -- I'm most curious about performance in the absence of caching benefits. Test 1a: Creating and multiplying two random/uniform (i.e., all entries drawn from math.random()) 100x100 matrices: [info] MatrixBenchmarks.buildMultTestMC100 thrpt 10 836.579 ± 7.120 ops/s [info] MatrixBenchmarks.buildMultTestSB100 thrpt 101649.089 ± 170.649 ops/s [info] MatrixBenchmarks.buildMultTestOJ100 thrpt 10 1485.014 ± 44.158 ops/s [info] MatrixBenchmarks.multBMC100thrpt 10 1051.055 ± 2.290 ops/s Test 1b: Creating and multiplying two random/uniform 500x500 matrices: [info] MatrixBenchmarks.buildMultTestMC500 thrpt 10 8.997 ± 0.064 ops/s [info] MatrixBenchmarks.buildMultTestSB500 thrpt 10 80.558 ± 0.627 ops/s [info] MatrixBenchmarks.buildMultTestOJ500 thrpt 10 34.626 ± 2.505 ops/s [info] MatrixBenchmarks.multBMC500thrpt 10 9.132 ± 0.059 ops/s [info] MatrixBenchmarks.multCacheBMC500 thrpt 10 13.630 ± 0.107 ops/s [no matrix creation] --- Test 2a: Creating a single 500x500 matrix (to get a sense of whether the mult itself is driving differences: [info] MatrixBenchmarks.buildMC500 thrpt 10 155.026 ± 2.456 ops/s [info] MatrixBenchmarks.buildSB500 thrpt 10 197.257 ± 4.619 ops/s [info] MatrixBenchmarks.buildOJ500 thrpt 10 176.036 ± 2.121 ops/s Test 2b: Creating a single 1000x1000 matrix (to show it scales w/N): [info] MatrixBenchmarks.buildMC1000 thrpt 1036.112 ± 2.775 ops/s [info] MatrixBenchmarks.buildSB1000 thrpt 1045.557 ± 2.893 ops/s [info] MatrixBenchmarks.buildOJ1000 thrpt 1047.438 ± 1.370 ops/s [info] MatrixBenchmarks.buildBMC1000 thrpt 1037.779 ± 0.871 ops/s --- Test 3a: Inverting a random 100x100 matrix: [info] MatrixBenchmarks.invMC100 thrpt 10 1033.011 ± 9.928 ops/s [info] MatrixBenchmarks.invSB100 thrpt 10 1664.833 ± 15.170 ops/s [info] MatrixBenchmarks.invOJ100 thrpt 10 1174.044 ± 52.083 ops/s [info] MatrixBenchmarks.invBMC100 thrpt 10 1043.858 ± 22.144 ops/s Test 3b: Inverting a random 500x500 matrix: [info] MatrixBenchmarks.invMC500thrpt 10 9.089 ± 0.284 ops/s [info] MatrixBenchmarks.invSB500thrpt 10 43.857 ± 1.083 ops/s [info] MatrixBenchmarks.invOJ500thrpt 10 23.444 ± 1.484 ops/s [info] MatrixBenchmarks.invBMC500 thrpt 10 9.156 ± 0.052 ops/s [info] MatrixBenchmarks.invCacheBMC500 thrpt 10 9.627 ± 0.084 ops/s [no matrix creation] Test 3c: [info] MatrixBenchmarks.invMC1000 thrpt 10 0.704 ± 0.040 ops/s [info] MatrixBenchmarks.invSB1000 thrpt 10 8.611 ± 0.557 ops/s [info] MatrixBenchmarks.invOJ1000 thrpt 10 3.539 ± 0.229 ops/s [info] MatrixBenchmarks.invBMC1000thrpt 10 0.691 ± 0.095 ops/s Also, isn't matrix multiplication at least O(N^2.37), rather than O(N^2)? On 6 April 2016 at 12:55, Chris Lucas wrote: > Thanks for the quick reply! > > I've pasted a small self-contained example at the bottom. It creates the > matrices in advance, but nothing meaningful changes if they're created on a > per-operation basis. > > Results for 50 multiplications of [size]x[size] matrices: >Size: 10, total time: 0.012 seconds, time per: 0.000 seconds >Size: 100, total time: 0.062 seconds, time per: 0.001 seconds >Size: 300, total time: 3.050 seconds, time per: 0.061 seconds >Size: 500, total time: 15.186 seconds, time per: 0.304 seconds >Size: 600, total time: 17.532 seconds, time per: 0.351 seconds > > For comparison: > > Results for 50 additions of [size]x[size] matrices (which should be > faster, be the extent of the difference is nonetheless striking to me): >Size: 10, total time: 0.011 seconds, time per: 0.000 seconds >Size: 100, total time: 0.012 seconds, time per: 0.000 seconds >Size: 300, total time: 0.020 seconds, time per: 0.000 second
Re: [math] matrix multiplication surprisingly slow?
Thanks for the quick reply!
I've pasted a small self-contained example at the bottom. It creates the
matrices in advance, but nothing meaningful changes if they're created on a
per-operation basis.
Results for 50 multiplications of [size]x[size] matrices:
Size: 10, total time: 0.012 seconds, time per: 0.000 seconds
Size: 100, total time: 0.062 seconds, time per: 0.001 seconds
Size: 300, total time: 3.050 seconds, time per: 0.061 seconds
Size: 500, total time: 15.186 seconds, time per: 0.304 seconds
Size: 600, total time: 17.532 seconds, time per: 0.351 seconds
For comparison:
Results for 50 additions of [size]x[size] matrices (which should be faster,
be the extent of the difference is nonetheless striking to me):
Size: 10, total time: 0.011 seconds, time per: 0.000 seconds
Size: 100, total time: 0.012 seconds, time per: 0.000 seconds
Size: 300, total time: 0.020 seconds, time per: 0.000 seconds
Size: 500, total time: 0.032 seconds, time per: 0.001 seconds
Size: 600, total time: 0.050 seconds, time per: 0.001 seconds
Results for 50 inversions of a [size]x[size] matrix, which I'd expect to be
slower than multiplication for larger matrices:
Size: 10, total time: 0.014 seconds, time per: 0.000 seconds
Size: 100, total time: 0.074 seconds, time per: 0.001 seconds
Size: 300, total time: 1.005 seconds, time per: 0.020 seconds
Size: 500, total time: 5.024 seconds, time per: 0.100 seconds
Size: 600, total time: 9.297 seconds, time per: 0.186 seconds
I hope this is useful, and if I'm doing something wrong that's leading to
this performance gap, I'd love to know.
import org.apache.commons.math3.linear.LUDecomposition;
import org.apache.commons.math3.linear.Array2DRowRealMatrix;
import org.apache.commons.math3.linear.RealMatrix;
public class AMCMatrices {
public static void main(String[] args) {
miniTest(0);
}
public static void miniTest(int tType) {
int samples = 50;
int sizes[] = { 10, 100, 300, 500, 600 };
for (int sI = 0; sI < sizes.length; sI++) {
int mSize = sizes[sI];
org.apache.commons.math3.linear.RealMatrix m0 = buildM(mSize, mSize);
RealMatrix m1 = buildM(mSize, mSize);
long start = System.nanoTime();
for (int n = 0; n < samples; n++) {
switch (tType) {
case 0:
m0.multiply(m1);
break;
case 1:
m0.add(m1);
break;
case 2:
new LUDecomposition(m0).getSolver().getInverse();
break;
}
}
long end = System.nanoTime();
double dt = ((double) (end - start)) / 1E9;
System.out.println(String.format(
"Size: %d, total time: %3.3f seconds, time per: %3.3f seconds",
mSize, dt, dt / samples));
}
}
public static Array2DRowRealMatrix buildM(int r, int c) {
double[][] matVals = new double[r][c];
for (int i = 0; i < r; i++) {
for (int j = 0; j < c; j++) {
matVals[i][j] = Math.random();
}
}
return new Array2DRowRealMatrix(matVals);
}
}
On 5 April 2016 at 19:36, Gilles wrote:
> Hi.
>
> On Tue, 5 Apr 2016 15:43:04 +0100, Chris Lucas wrote:
>
>> I recently ran a benchmark comparing the performance math commons 3.6.1's
>> linear algebra library to the that of scala Breeze (
>> https://github.com/scalanlp/breeze).
>>
>> I looked at det, inverse, Cholesky factorization, addition, and
>> multiplication, including matrices with 10, 100, 500, and 1000 elements,
>> with symmetric, non-symmetric, and non-square cases where applicable.
>>
>
> It would be interesting to add this to the CM documentation:
> https://issues.apache.org/jira/browse/MATH-1194
>
> In general, I was pleasantly surprised: math commons performed about as
>> well as Breeze, despite the latter relying on native libraries. There was
>> one exception, however:
>>
>> m0.multiply(m1)
>>
>> where m0 and m1 are both Array2DRowRealMatrix instances. It scaled very
>> poorly in math commons, being much slower than nominally more expensive
>> operations like inv and the Breeze implementation. Does anyone have a
>> thought as to what's going on?
>>
>
> Could your provide more information such as a plot of performance vs size?
> A self-contained and minimal code to run would be nice too.
> See the CM micro-benchmarking tool here:
>
> https://github.com/apache/commons-math/blob/master/src/test/java/org/apache/commons/math4/PerfTestUtils.java
> And an example of how to use it:
>
> https://github.com/apache/commons-math/blob/master/src/userguide/java/org/apache/commons/math4/userguide/FastMathTestPerformance.java
>
> In case it's useful, one representative test
>> involves multiplying two instances of
>>
>> new Ar
[math] matrix multiplication surprisingly slow?
I recently ran a benchmark comparing the performance math commons 3.6.1's linear algebra library to the that of scala Breeze ( https://github.com/scalanlp/breeze). I looked at det, inverse, Cholesky factorization, addition, and multiplication, including matrices with 10, 100, 500, and 1000 elements, with symmetric, non-symmetric, and non-square cases where applicable. In general, I was pleasantly surprised: math commons performed about as well as Breeze, despite the latter relying on native libraries. There was one exception, however: m0.multiply(m1) where m0 and m1 are both Array2DRowRealMatrix instances. It scaled very poorly in math commons, being much slower than nominally more expensive operations like inv and the Breeze implementation. Does anyone have a thought as to what's going on? In case it's useful, one representative test involves multiplying two instances of new Array2DRowRealMatrix(matVals) where matVals is 1000x1000 entries of math.random and the second instance is created as part of the loop. This part of the benchmark is not specific to the expensive multiplication step, and takes very little time relative to the multiplication itself. I'm using System.nanotime for the timing, and take the average time over several consecutive iterations, on a 3.5 GHz Intel Core i7, Oracle JRE (build 1.8.0_05-b13). Thanks, Chris
