I need to multiply two big matrices and sort their columns.
import numpy
a= numpy.random.rand(1000000, 100)
b= numpy.random.rand(300000,100)
c= numpy.dot(b,a.T)
sorted = [argsort(j)[:10] for j in c.T]
This process takes a lot of time and memory. Is there a way to fasten this process? If not how can I calculate RAM needed to do this operation? I currently have an EC2 box with 4GB RAM and no swap.
I was wondering if this operation can be serialized and I dont have to store everything in the memory.
One thing that you can do to speed things up is compile numpy with an optimized BLAS library like e.g. ATLAS, GOTO blas or Intel's proprietary MKL.
To calculate the memory needed, you need to monitor Python's Resident Set Size ("RSS"). The following commands were run on a UNIX system (FreeBSD to be precise, on a 64-bit machine).
> ipython
In [1]: import numpy as np
In [2]: a = np.random.rand(1000, 1000)
In [3]: a.dtype
Out[3]: dtype('float64')
In [4]: del(a)
To get the RSS I ran:
ps -xao comm,rss | grep python
[Edit: See the ps manual page for a complete explanation of the options, but basically these ps options make it show only the command and resident set size of all processes. The equivalent format for Linux's ps would be ps -xao c,r, I believe.]
The results are;
After starting the interpreter: 24880 kiB
After importing numpy: 34364 kiB
After creating a: 42200 kiB
After deleting a: 34368 kiB
Calculating the size;
In [4]: (42200 - 34364) * 1024
Out[4]: 8024064
In [5]: 8024064/(1000*1000)
Out[5]: 8.024064
As you can see, the calculated size matches the 8 bytes for the default datatype float64 quite well. The difference is internal overhead.
The size of your original arrays in MiB will be approximately;
In [11]: 8*1000000*100/1024**2
Out[11]: 762.939453125
In [12]: 8*300000*100/1024**2
Out[12]: 228.8818359375
That's not too bad. However, the dot product will be way too large:
In [19]: 8*1000000*300000/1024**3
Out[19]: 2235.1741790771484
That's 2235 GiB!
What you can do is split up the problem and perfrom the dot operation in pieces;
load b as an ndarray
load every row from a as an ndarray in turn.
multiply the row by every column of b and write the result to a file.
del() the row and load the next row.
This wil not make it faster, but it would make it use less memory!
Edit: In this case I would suggest writing the output file in binary format (e.g. using struct or ndarray.tofile). That would make it much easier to read a column from the file with e.g. a numpy.memmap.
What DrV and Roland Smith said are good answers; they should be listened to. My answer does nothing more than present an option to make your data sparse, a complete game-changer.
Sparsity can be extremely powerful. It would transform your O(100 * 300000 * 1000000) operation into an O(k) operation with k non-zero elements (sparsity only means that the matrix is largely zero). I know sparsity has been mentioned by DrV and disregarded as not applicable but I would guess it is.
All that needs to be done is to find a sparse representation for computing this transform (and interpreting the results is another ball game). Easy (and fast) methods include the Fourier transform or wavelet transform (both rely on similarity between matrix elements) but this problem is generalizable through several different algorithms.
Having experience with problems like this, this smells like a relatively common problem that is typically solved through some clever trick. When in a field like machine learning where these types of problems are classified as "simple," that's often the case.
YOu have a problem in any case. As Roland Smith shows you in his answer, the amount of data and number of calculations is enormous. You may not be very familiar with linear algebra, so a few words of explanation might help in understanding (and then hopefully solving) the problem.
Your arrays are both a collection of vectors with length 100. One of the arrays has 300 000 vectors, the other one 1 000 000 vectors. The dot product between these arrays means that you calculate the dot product of each possible pair of vectors. There are 300 000 000 000 such pairs, so the resulting matrix is either 1.2 TB or 2.4 TB depending on whether you use 32 or 64-bit floats.
On my computer dot multiplying a (300,100) array with a (100,1000) array takes approximately 1 ms. Extrapolating from that, you are looking at a 1000 s calculation time (depending on the number of cores).
The nice thing about taking a dot product is that you can do it piecewise. Keeping the output is then another problem.
If you were running it on your own computer, calculating the resulting matrix could be done in the following way:
create an output array as a np.memmap array onto the disk
calculate the results one row at a time (as explained by Roland Smith)
This would result in a linear file write with a largish (2.4 TB) file.
This does not require too many lines of code. However, make sure everything is transposed in a suitable way; transposing the input arrays is cheap, transposing the output is extremely expensive. Accessing the resulting huge array is cheap if you can access elements close to each other, expensive, if you access elements far away from each other.
Sorting a huge memmapped array has to be done carefully. You should use in-place sort algorithms which operate on contiguous chunks of data. The data is stored in 4 KiB chunks (512 or 1024 floats), and the fewer chunks you need to read, the better.
Now that you are not running the code in our own machine but on a cloud platform, things change a lot. Usually the cloud SSD storage is very fast with random accesses, but IO is expensive (also in terms of money). Probably the least expensive option is to calculate suitable chunks of data and send them to S3 storage for further use. The "suitable chunk" part depends on how you intend to use the data. If you need to process individual columns, then you send one or a few columns at a time to the cloud object storage.
However, a lot depends on your sorting needs. Your code looks as if you are finally only looking at a few first items of each column. If this is the case, then you should only calculate the first few items and not the full output matrix. That way you can do everything in memory.
Maybe if you tell a bit more about your sorting needs, there can be a viable way to do what you want.
Oh, one important thing: Are your matrices dense or sparse? (Sparse means they mostly contain 0's.) If your expect your output matrix to be mostly zero, that may change the game completely.
Related
I am writing a code in Python in which I am doing a lot of matrix multiplications on huge matrices. For the performance as well as the speed I am using numpy.einsum. Anyways, that einum is optimized very well, my code works very slowly. This is a reason that I'm looking for a solution to speed it up. I thought that PyCuda might have been helpful, but after doing some research on the Internet, I figured out that I was mistaken.
Do you have any tips how to improve performance of math operations Python?
For example I have a bunch of this kind of following multiplications:
NewMatrix_krls = np.einsum('kK,krls->Krls', C, Matrx_krls)
where C is a matrix 1000 by 1000 and a Matrx_krls is 1000 by 1000 by 1000 by 1000.
einsum constructs an iterator (ndinter) that loops (in C) over the problem space. With the dimensions and indexes in your expression that space is 1000**5 (5 variables, each of size 1000). That's 10**15 operations.
If the problem involved multiplying 3 or 4 arrays, you might speed things up by splitting it into a couple of einsum calls (or a mix of einsum and dot). But there's not enough information in your description to suggest that kind of partitioning.
Actually you might be able to cast this in a form that uses np.dot. Matrx_krls can be reshaped to a 2d array, combining the rls dimensions into one 1000**3 in size.
E.g. (not tested)
np.dot(C.T, Matrx_krls.reshape(1000,-1))
For a product of 2 2d arrays, np.dot is faster than einsum.
(Note I changed your tags to include numpy - that will bring it to the attention of more knowledgeable posters).
'Matrx_krls is 1000 by 1000 by 1000 by 1000'
So thats a 4-terabyte array, assuming 4-byte floats... I think you are better off investing time in Hadoop than in GPGPU. Even if you stream that data to a GPU, you will see little benefit because of limited PCIE bandwidth.
That, or perhaps explaining why you think you need this kind of operation might help on a higher level.
I have a program which creates an array:
List1 = zeros((x, y), dtype=complex_)
Currently I am using x = 500 and y = 1000000.
I will initialize the first column of the list by some formula. Then the subsequent columns will calculate their own values based on the preceding column.
After the list is completely filled, I will then display this multidimensional array using imshow().
The size of each value (item) in the list is 24 bytes.
A sample value from the code is: 4.63829355451e-32
When I run the code with y = 10000000, it takes up too much RAM and the system stops the run. How do I solve this problem? Is there a way to save my RAM while still being able to process the list using imshow() easily? Also, how large a list can imshow() display?
There's no way to solve this problem (in any general way).
Computers (as commonly understood) have a limited amount of RAM, and they require elements to be in RAM in order to operate on them.
An complex128 array size of 10000000x500 would require around 74GiB to store. You'll need to somehow reduce the amount of data you're processing if you hope to use a regular computer to do it (as opposed to a supercomputer).
A common technique is partitioning your data and processing each partition individually (possibly on multiple computers). Depending on the problem you're trying to solve, there may be special data structures that you can use to reduce the amount of memory needed to represent the data - a good example is a sparse matrix.
It's very unusual to need this much memory - make sure to carefully consider if it's actually needed before you dwell into the extremely complex workarounds.
I am performing dot product of a matrix with 50000 rows and 100 columns with it's transpose. The values of the matrix is in float.
A(50000, 100)
B(100, 50000)
Basically I get the matrix after performing SVD on a larger sparse matrix.
The matrix is of numpy.ndarray type.
I use dot method of numpy for multiplying the two matrices. And I get segmentation fault.
numpy.dot(A, B)
The dot product works well on matrix with 30000 rows but fails for 50000 rows.
Is there any limit on numpy's dot product?
Any problem above while using dot product?
Is there any other good python linear algebra tool which is efficient on large matrices.
As you have been told, there is a memory problem. You want to do this:
numpy.dot(A, A.T)
which requires a lot of memory for the result (not the operands). However the operation is easy to perform in pieces. You may use a loop-based approach to produce one output row at a time:
def trans_multi(A):
rows = A.shape[0]
result = numpy.empty((rows, rows), dtype=A.dtype)
for r in range(rows):
result[r,:] = numpy.dot(A, A[r,:].T)
return result
This, as such, is just a slower and equally memory consuming version (numpy.dot is well-optimized). However, what you most probably want to do is to write the result into a file, as you do not have the memory to hold the result:
def trans_multi(A, filename):
with open(filename, "wb") as f:
rows = A.shape[0]
for r in range(rows):
f.write(numpy.dot(A, A[r,:].T).tostring())
Yes, it is not exactly lightning-fast. However, it is most probably the fastest you can hope for. Sequential writes are usually well-optimized. I tried:
a=numpy.random.random((50000,100)).astype('float32')
trans_multi(a,"/tmp/large.dat")
This took approximately 60 seconds, but it really depends on your HDD performance.
Why not memmap?
I like mmap, and numpy.memmap is a great thing. However, numpy.memmap is optimized for having large tables and calculating small results from them. There is, for example memmap.dot which is optimized for taking dot products of memmapped arrays. The scenario is that the operands are memmapped, but the result is in RAM. Exactly the other way round, that is.
Memmapping is very useful when you have random access. Here the access is not random access but sequential write. Also, yf you try to use numpy.memmap to create a (50000,50000) array of float32's, it will take some time (for some reason I do not get, maybe it initializes the data even though there is no need).
However, after the file has been created, it is a very good idea to use numpy.memmap to analyze the huge table, as it gives the best random read performance and a very convenient interface.
Circumstances
I have a procedure which will construct a matrix using the given list of values!
and the list starts growing bigger like 100 thousand or million values in a list, which in turn, will result in million x million size matrix.
in the procedure, i am doing some add/sub/div/multiply operations on the matrix, either based on the row, the column or just the element.
Issues
since the matrix is so big that i don`t think doing the whole manipulation in the memory would work.
Questions
therefore, my question would be:
how should i manipulate this huge matrix and the huge value list?
like, where to store it, how to read it etc, so that i could carry out my operations on the matrix and the computer won`t stuck or anything.
I suggest using NumPy. It's quite fast on arithmetic operations.
First and foremost, such matrix would have 10G elements. Considering that for any useful operation you would then need 30G elements, each taking 4-8 bytes, you cannot assume to do this at all on a 32-bit computer using any sort of in-memory technique. To solve this, I would use a) genuine 64-bit machine, b) memory-mapped binary files for storage, and c) ditch python.
Update
And as I calculated below, if you have 2 input matrices and 1 output matrix, 100000 x 100000 32 bit float/integer elements, that is 120 GB (not quite GiB, though) of data. Assume, on a home computer you could achieve constant 100 MB/s I/O bandwidth, every single element of a matrix needs to be accessed for any operation including addition and subtraction, the absolute lower limit for operations would be 120 GB / (100 MB/s) = 1200 seconds, or 20 minutes, for a single matrix operation. Written in C, using the operating system as efficiently as possible, memmapped IO and so forth. For million by million elements, each operation takes 100 times as many time, that is 1.5 days. And as the hard disk is saturated during that time, the computer might just be completely unusable.
Have you considered using a dictionary? If the matrix is very sparse it might be feasible to store it as
matrix = {
(101, 10213) : "value1",
(1099, 78933) : "value2"
}
Your data structure is not possible with arrays, it is too large. If the matrix is for instance a binary matrix you could look at representations for its storage like hashing larger blocks of zeros together to the same bucket.
I need to create about 2 million vectors w/ 1000 slots in each (each slot merely contains an integer).
What would be the best data structure for working with this amount of data? It could be that I'm over-estimating the amount of processing/memory involved.
I need to iterate over a collection of files (about 34.5GB in total) and update the vectors each time one of the the 2-million items (each corresponding to a vector) is encountered on a line.
I could easily write code for this, but I know it wouldn't be optimal enough to handle the volume of the data, which is why I'm asking you experts. :)
Best,
Georgina
You might be memory bound on your machine. Without cleaning up running programs:
a = numpy.zeros((1000000,1000),dtype=int)
wouldn't fit into memory. But in general if you could break the problem up such that you don't need the entire array in memory at once, or you can use a sparse representation, I would go with numpy (scipy for the sparse representation).
Also, you could think about storing the data in hdf5 with h5py or pytables or netcdf4 with netcdf4-python on disk and then access the portions you need.
Use a sparse matrix assuming most entries are 0.
If you need to work in RAM try the scipy.sparse matrix variants. It includes algorithms to efficiently manipulate sparse matrices.