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Few calculations with huge matrices vs. lots of calculations with small matrices

I am working on a Java project which has thousands of matrix calculations. But the matrices are at most 10x10 matrices.
I wonder if it is better to use a matrix library or use write the simple functions (determinant(), dotproduct() etc.) Because when small matrices are used, it is advised not to use libraries but do the operations by custom functions.
I know that matrix libraries like JAMA provides high performance when it comes to 10000x10000 matrices or so.
Instead making 5-6 calculations with 10000x10000 matrices, I make 100000 calculations with 10x10 matrices. Number of primitive operations are nearly the same.
Are both cases same in terms of performance? Should I treat myself as if I'm working with huge matrices and use a library?

I suspect for a 10x10 matrix you won't see much difference.
In tests I have done for hand coding a 4x4 matrix the biggest overhead was loading the data into the L1 cache and how you did it didn't matter very much. For a 3x3 matrix and smaller it did appear to make a significant difference.

Getting the maximum possible speed (with lots of effort)
For maximum possible speed I would suggest writing a C function that uses vector math intrinsics such as Streaming SIMD Extensions (SSE) or Advanced Vector Extensions (AVX) operations, together with multithreading (e.g. via OpenMP).
Your Java program would pass all 100k matrices to this native function, which would then handle all the calculations. Portability becomes an issue, e.g. AVX instructions are only supported on recent CPUs. Developer effort, especially if you are not familiar with SSE/AVX increases a lot too.
Reasonable speed without too much effort
You should use multiple threads by creating a class that extends java.lang.Thread or implements java.lang.Runnable. Each thread iterates through a subset of the matrices, calling your maths routine(s) for each matrix. This part is key to getting decent speed on multi-core CPUs. The maths could be your own Java function to do the calculations on a single matrix, or you could use a library's functions.
I wonder if it is better to use a matrix library or use write the
simple functions (determinant(), dotproduct() etc.) Because when small
matrices are used, it is advised not to use libraries but do the
operations by custom functions.
...
Are both cases same in terms of performance? Should I treat myself as
if I'm working with huge matrices and use a library?
No, using a library and writing your own function for the maths are not the same performance-wise. You may be able to write a faster function that is specialised to your application, but consider this:
The library functions should have fewer bugs than code you will write.
A good library will use implementations that are efficient (i.e. least amount of operations). Do you have the time to research and implement the most efficient algorithms?
You might find the Apache Commons Math library useful. I would encourage you to benchmark Apache Commons Math and JAMA to choose the fastest.

Sensibility of converting Matlab program in Java to improve performance

His
I have a somewhat hypothetical question. We've just programmed some code implementing genetic algorithm to find a solution to a sudoku game as part of the Computational Intelligence course project. Unfortunately it runs very slowly which limits our ability to perform adequate number of runs to find the optimal parameters. The question is whether reprogramming the whole thing - the code basis is not that big - into java would be a viable solution to boost up the speed of the software. Like we need 10x performance improvement really and i am doubtful that a Java version would be so much snappier. Any thoughts?
Thanks
=== Update 1 ===
Here is the code of the function that is computationally most expensive. It's a GA fitness function, that iterates through the population (different sudoku boards) and computes for each row and column how many elements are duplicates. The parameter n is passed, and is currently set to 9. That is, the function computes how many elements a row has that come up within the range 1 to 9 more then once. The higher the number the less is the fitness of the board, meaning that it is a weak candidate for the next generation.
The profiler reports that the two lines calling intersect in the for loops causing the poor performance, but we don't know how to really optimize the code. It follows below:
function [fitness, finished,d, threshold]=fitness(population_, n)
finished=false;
threshold=false;
V=ones(n,1);
d=zeros(size(population_,2),1);
s=[1:1:n];
for z=1:size(population_,2)
board=population_{z};
t=0;
l=0;
for i=1:n
l=l+n-length(intersect(s,board(:,i)'));
t=t+n-length(intersect(s,board(i,:)));
end
k=sum(abs(board*V-t));
f=t+l+k/50;
if t==2 &&l==2
threshold=true;
end
if f==0
finished=true;
else
fitness(z)=1/f;
d(z)=f;
end
end
end
=== Update 2 ===
Found a solution here: http://www.mathworks.com/matlabcentral/answers/112771-how-to-optimize-the-following-function
Using histc(V, 1:9), it's much faster :)

This is rather impossible to say without viewing your code, knowing if you use parallelization, etc. Indeed, as MrAzzaman says, profiling is the first thing to do. If you find a single bottleneck, especially if it is loop-heavy, it might be sufficient to write that part in C and connect it to Matlab via MEX.
In genetics algorithms, I'd believe that a 10x speed increase could be obtained rather than not. I do not quite agree with MrAzzaman here - in some cases (for loops, working with dynamic objects) is much, much slower than C/C++/Java. That is not to say that Matlab is always slow, for it is not, but there is plenty of algorithms where it would be slow.
I.e., I'd say that if you don't spend so much time looping over things, don't use objects, are not limited by Matlab's data structures, you might be ok with Matlab. That said, if I was to write GAs in Java or Matlab, I'd rather pick the former (and I'm using Matlab a lot more than Java these days, it's not just a matter of habit).
Btw. if you don't want to program it yourself, have a look at JGAP, it's a rather useful Java library for GAs.

OK, the first step is just to write a faster MATLAB function. Save the new languages for later.
I'm going to make the assumption that the board is full of valid guesses: that is, each entry is in [1, 9]. Now, what we're really looking for are duplicate entries in each row/column. To find duplicates, we sort. On a sorted row, if any element is equal to its neighbor, we have a duplicate. In MATLAB, the diff function does sliding pairwise differencing, and a zero in its output means that two neighboring values are equal. Both sort and diff operate on entire matrices, so no need for looping. Here's the code for the columnwise check:
l=sum(sum(diff(sort(board)) == 0));
The rowwise check is exactly the same, just using the transpose. Now let's put that in a test harness to compare results and timing with the previous version:
n = 9;
% Generate a test board. Random integers numbers from 1:n
board = randi(n, n);
s = 1:n;
K=1000; % number of iterations to use for timing
% Repeat current code for comparison
tic
for k=1:K
t=0;
l=0;
for i=1:n
l=l+n-length(intersect(s,board(:,i)'));
t=t+n-length(intersect(s,board(i,:)));
end
end
toc
% New code based on sort/diff for finding repeated values
tic
for k=1:K
l2=sum(sum(diff(sort(board)) == 0));
t2=sum(sum(diff(sort(board.')) == 0));
end
toc
% Check that reported values match
disp([l l2])
disp([t t2])
I encourage you to break down the sort/diff/sum code, and build it up on a sample board right at the command line, and try to understand exactly how it works.
On my system, the new code is about 330x faster.

For traditional GA applications for studying and research purposes it is better to use a native machine compiled source code programming language, like C, C++. Which I used when working with Genetic
Programming in the past and it is really fast.
However if you are planning to put this inside a more modern type of application that can be deployed in a web container or run in a mobile device, different OS, etc. Then Java is your best alternative as it is platform independent.
Another thing that can be important is about concurrency. For example lets us suppose that you want to put your GA in the Internet and you will have a growing number of users that are connected concurrently and all of them want to solve a different sudoku, Java applications are very good for scaling horizontally and works great with big number of concurrent connections.
Other thing that can be good if you migrate to Java is the number of libraries and frameworks that you can use, the Java universe is so big that you can find useful tools for almost any kind of application.
Java is a Virtual Machine compiled language, but it is important to note that currently the JVMs are very good in performance and are able to optimize the programs, for example they will find which methods are being more heavily used and compile them to native code, which means that for some applications you will find a Java program to be almost same fast than a native compiled from C.
Matlab is a platform that is very useful for engineering training and math, vector, matrix based calculations, also for some control stuff with Simulink. I used these products when in my electrical engineering bachelor, however those product's goal is to be mainly a tool for academic purposes I won't definitely go for Matlab if I am wanting to build a production application for the real world. It is not scalable, it is expensive to maintain and fine-tune it, also there are not lot of infrastructure providers that will support this kind of technology.
About the complexity of rewriting your code to Java, the Matlab code and Java code syntax is pretty similar, they also live in the same paradigm: Procedural OOP, even if you are not using OO in your code it can be easy rewritten in Java, the painful stuff will be when working with Matlab shortcuts to Math structures like matrix and passing functions as parameters.
For the matrix stuff, there are lot of java libraries like EJML that will make your life easier. About assigning functions to variables and then pass them as parameters to another functions, Java is not currently able to do that (Java 8 will be with Lambda Expressions) but you can have a equivalent functionality by using Class closures. Maybe these will be the only little painful things that you will find if migrating.

Found a solution here: http://www.mathworks.com/matlabcentral/answers/112771-how-to-optimize-the-following-function
Using histc(V, 1:9), it's much faster :)

JAMA Matrix performance

First of all, sorry for my bad English, but I need your help.
I have developed a simulation program with java swing where I have used lots of matrix calculations. My program is just finished but I need to speed up my performance. So I have used the java visual vm profiler to identify performance problems. I recognized that the initialization of Jama Matrices needs a lot of time. After running my program I had over 3 MB allocated Objects by JAMA. That's a lot of, isn't it? I think that's why the performance is bad.
Is there any better library than jama for matrices? I am using 3x3 matrices and I need multiplication and inverse operations or is there anything else i can do?

Usually matrix math libraries are not optimized for speed on small matrices.
You can see for yourself by taking a few stackshots, which are liable to show a large fraction of time in overhead functions like memory allocation and option checking.
What you can do instead (which I've done) is write special-purpose routines to do the multiplication and inverse, since you know the matrices are 3x3.
Multiplication is trivial and you can unroll the whole thing.
Inverse of a 3x3 matrix can also be done in less code than coffee :)
Wikipedia gives you the formula.
Whatever you do, try to minimize memory allocation.

LSH Libraries in Java

I'm looking for a lightweight Java library that supports Nearest Neighbor Searches by Locality Sensitive Hashing for nearly equally distributed data in a high dimensional (in my case 32) dataset with some hundreds of thousands data points.
It's totally good enough to get all entries in a bucket for a query. Which ones i really need could then be processed in a different way under consideration of some filter parameters my problem include.
I already found likelike but hope that there is something a bit smaller and without need of any other tools (like Apache Hadoop in the case of likelike).

Maybe this one:
"TarsosLSH is a Java library implementing Locality-sensitive Hashing (LSH), a practical nearest neighbour search algorithm for multidimensional vectors that operates in sublinear time. It supports several Locality Sensitive Hashing (LSH) families: the Euclidean hash family (L2), city block hash family (L1) and cosine hash family. The library tries to hit the sweet spot between being capable enough to get real tasks done, and compact enough to serve as a demonstration on how LSH works."
Code can be found here

Apache Spark has an LSH implementation: https://spark.apache.org/docs/2.1.0/ml-features.html#locality-sensitive-hashing (API).
After having played with both the tdebatty and TarsosLSH implementations, I'll likely use Spark, as it supports sparse vectors as input. The tdebatty requires a non-sparse array of booleans or int's, and the TarsosLSH Vector implementation is a non-sparse array of doubles. This severely limits the number of dimensions one can reasonably support.
This page provides links to more projects, as well as related papers and information: https://janzhou.org/lsh/.

There is this one:
http://code.google.com/p/lsh-clustering/
I haven't had time to test it but at least it compiles.

Here another one:
https://github.com/allenlsy/knn
It uses LSH for KNN. I'm currently investigating it's usability =)

The ELKI data mining framework comes with an LSH index. It can be used with most algorithms included (anything that uses range or nn searches) and sometimes works very well.
In other cases, LSH doesn't seem to be a good approach. It can be quite tricky to get the LSH parameters right: if you choose some parameters too high, runtime grows a lot (all the way to a linear scan). If you choose them too low, the index becomes too approximative and loses to many neighbors.
It's probably the biggest challenge with LSH: finding good parameters, that yield the desired speedup and getting a good enough accuracy out of the index...

Solving nonlinear equations numerically

I need to solve nonlinear minimization (least residual squares of N unknowns) problems in my Java program. The usual way to solve these is the Levenberg-Marquardt algorithm. I have a couple of questions
Does anybody have experience on the different LM implementations available? There exist slightly different flavors of LM, and I've heard that the exact implementation of the algorithm has a major effect on the its numerical stability. My functions are pretty well-behaved so this will probably not be a problem, but of course I'd like to choose one of the better alternatives. Here are some alternatives I've found:
FPL Statistics Group's Nonlinear Optimization Java Package. This includes a Java translation of the classic Fortran MINPACK routines.
JLAPACK, another Fortran translation.
Optimization Algorithm Toolkit.
Javanumerics.
Some Python implementation. Pure Python would be fine, since it can be compiled to Java with jythonc.
Are there any commonly used heuristics to do the initial guess that LM requires?
In my application I need to set some constraints on the solution, but luckily they are simple: I just require that the solutions (in order to be physical solutions) are nonnegative. Slightly negative solutions are result of measurement inaccuracies in the data, and should obviously be zero. I was thinking to use "regular" LM but iterate so that if some of the unknowns becomes negative, I set it to zero and resolve the rest from that. Real mathematicians will probably laugh at me, but do you think that this could work?
Thanks for any opinions!
Update: This is not rocket science, the number of parameters to solve (N) is at most 5 and the data sets are barely big enough to make solving possible, so I believe Java is quite efficient enough to solve this. And I believe that this problem has been solved numerous times by clever applied mathematicians, so I'm just looking for some ready solution rather than cooking my own. E.g. Scipy.optimize.minpack.leastsq would probably be fine if it was pure Python..

The closer your initial guess is to the solution, the faster you'll converge.
You said it was a non-linear problem. You can do a least squares solution that's linearized. Maybe you can use that solution as a first guess. A few non-linear iterations will tell you something about how good or bad an assumption that is.
Another idea would be trying another optimization algorithm. Genetic and ant colony algorithms can be a good choice if you can run them on many CPUs. They also don't require continuous derivatives, so they're nice if you have discrete, discontinuous data.

You should not use an unconstrained solver if your problem has constraints. For
instance if know that some of your variables must be nonnegative you should tell
this to your solver.
If you are happy to use Scipy, I would recommend scipy.optimize.fmin_l_bfgs_b
You can place simple bounds on your variables with L-BFGS-B.
Note that L-BFGS-B takes a general nonlinear objective function, not just
a nonlinear least-squares problem.

I agree with codehippo; I think that the best way to solve problems with constraints is to use algorithms which are specifically designed to deal with them. The L-BFGS-B algorithm should probably be a good solution in this case.
However, if using python's scipy.optimize.fmin_l_bfgs_b module is not a viable option in your case (because you are using Java), you can try using a library I have written: a Java wrapper for the original Fortran code of the L-BFGS-B algorithm. You can download it from http://www.mini.pw.edu.pl/~mkobos/programs/lbfgsb_wrapper and see if it matches your needs.

The FPL package is quite reliable but has a few quirks (array access starts at 1) due to its very literal interpretation of the old fortran code. The LM method itself is quite reliable if your function is well behaved. A simple way to force non-negative constraints is to use the square of parameters instead of the parameters directly. This can introduce spurious solutions but for simple models, these solutions are easy to screen out.
There is code available for a "constrained" LM method. Look here http://www.physics.wisc.edu/~craigm/idl/fitting.html for mpfit. There is a python (relies on Numeric unfortunately) and a C version. The LM method is around 1500 lines of code, so you might be inclined to port the C to Java. In fact, the "constrained" LM method is not much different than the method you envisioned. In mpfit, the code adjusts the step size relative to bounds on the variables. I've had good results with mpfit as well.
I don't have that much experience with BFGS, but the code is much more complex and I've never been clear on the licensing of the code.
Good luck.

I haven't actually used any of those Java libraries so take this with a grain of salt: based on the backends I would probably look at JLAPACK first. I believe LAPACK is the backend of Numpy, which is essentially the standard for doing linear algebra/mathematical manipulations in Python. At least, you definitely should use a well-optimized C or Fortran library rather than pure Java, because for large data sets these kinds of tasks can become extremely time-consuming.
For creating the initial guess, it really depends on what kind of function you're trying to fit (and what kind of data you have). Basically, just look for some relatively quick (probably O(N) or better) computation that will give an approximate value for the parameter you want. (I recently did this with a Gaussian distribution in Numpy and I estimated the mean as just average(values, weights = counts) - that is, a weighted average of the counts in the histogram, which was the true mean of the data set. It wasn't the exact center of the peak I was looking for, but it got close enough, and the algorithm went the rest of the way.)
As for keeping the constraints positive, your method seems reasonable. Since you're writing a program to do the work, maybe just make a boolean flag that lets you easily enable or disable the "force-non-negative" behavior, and run it both ways for comparison. Only if you get a large discrepancy (or if one version of the algorithm takes unreasonably long), it might be something to worry about. (And REAL mathematicians would do least-squares minimization analytically, from scratch ;-P so I think you're the one who can laugh at them.... kidding. Maybe.)