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More efficient nested sum in numpy

I am trying to calculate a vectorised nested sum
(so effectively doing a separate calculation for each row k)
The fastest way I have come up with is to define a lower triangular matrix of ones to take into account the range on the inner sum
O = np.tril(np.ones((N,N),uint8),-1)
and then evaluate the inner sum with range j=1..N, allowing the use of a dot product
np.einsum('ij,ij->i',V,np.dot(W,O))
This works well, but even with uint8 data type, O requires a lot of memory for N>>10000. Is there a better way using standard numpy/scipy functions? My plan is to run this on a GPU using cupy.

As suggested by #hpaulj, you can use cumsum to completely get rid of O. YOu can write sum W as total sum - cumsum.
O = np.sum(W,axis=1)[:,None]-np.cumsum(W,axis=1)
np.einsum('ij,ij->i',V,O)

Matrix of "for x in vectors: for y in vectors:", without two for loops

Can one get the cartesian outer product of two set of vectors without using two for loops? It is slow because the data is large.
[[f(x,y) for x in vectors] for y in vectors]
I am trying to do an agglomerative clustering project in python and for this, I need to create a distance matrix.
This is the code that I have to define the function for the distance matrix:
def distance_matrix(vectors):
s = np.zeros((len(vectors), len(vectors)))
for i in range(len(vectors)):
for v in range(len(vectors)):
s[i, v] = dissimilarity(vectors[i], vectors[v])
return s
What it should do is take a take a list of NumPy arrays and return a 2D NumPy array d where the entry d[i,j] should contain the dissimilarity between vectors[i] and vectors[j].
In this case, vectors is the list of NumPy arrays and the dissimilarity is calculated by:
def dissimilarity(v1, v2):
return (1-(v1.dot(v2)/(np.linalg.norm(v1)*np.linalg.norm(v2))))
or in other words, the dissimilarity is the cosine dissimilarity between two 1D NumPy arrays.
My goal is to find a way to get the distance matrix without the double for loops but still have the computational time be very small.

In general one cannot do this. In general (but not in this case), the computation time will not change because the work is physically there and exists and nomatter the accounting tricks one might try to rewrite it (a single for loop that acts like two for loops, or a recursive implementation), at the end of the day the work must still be done irrespective of the ordering you do it.
Is there a way to get rid of the for loops? Even if it slows down the run time a bit. – Cat_Smithyyyy03
Your case is special. While one normally has no reason to consider a speedup is possible, you are asking to consider tensor products, or in this case a matrix product. When you think about general matrix multiplication AB=C, a matrix product C is basically the outer product {a dot b} of all vectors a in the rowspace of A and b in the colspace of B.
So, for your case, first normalize all your vectors (so the normalization is not required later), then stack them to form A, then let B=transp(A), then do a matrix multiply.
[[a0 a1 a2] [[a0 [b0 [z0 [aa ab ac .. az
[b0 b1 b2] a1 b1 ... z1 ba bb bc .. bz
( . )( a2] b2] z2]] ) = ( . . )
. . .
. . .
[z0 z1 z2]] za zb zc zz]
Interestingly, now you can then plug this into a fast matrix-multiply algorithm, which is actually faster than O(dim * #vecs^2). Hopefully it's also optimized for self-transpose matrices that would generate symmetric matrices (which might save a factor of 2 work... maybe it has some flag like matrixmult(a,b,outputWillBeSymmetric)).
This is faster than "O(N^2)", unintuitively: This rewrite exposed a substructure in the problem, which can be leveraged to get faster than O(dim * #vecs^2). The leveragable substructure is namely the fact that you are computing the outer product of THE SAME vectors. The fast matrix-multiply algorithm will leverage this.
edit: my original answer was wrong
You have a set of size N and you wish to compute f(a,b) for all a and b in the set.
Unless you know some values are trivial, there is no way to be asymptotically faster than this, because you have to imagine the worst case: Every pair f(a,b) may be unique... so there's no way to do less than roughly N^2 work.
However since your function f is symmetric, you could do half the work, then duplicate it:
N = len(vectors)
for i in range(N):
for v in range(N):
dissim = #...
s[i,v] = dissim
s[v,i] = dissim
(You can avoid calculating your metric in the reflexive case f(a,a) because it's trivial, but that doesn't make things asymptotically faster since the fraction of such work N/N^2 tends to zero as N increases, so it's not that great an optimization... it is reasonable only if you are working with a very small number of vectors.)
Whether you should optimize further depends on whether you need to. Such code should be able to easily handle millions of small vectors. Next steps:
Is there something fishy that is making my code very slow? What could it be? We don't have enough information to comment.
If there is nothing fishy of the sort, you can try rephrasing as matrix operations, in order to stay as much as possible in numpy's optimized C routines instead of bouncing back and forth into python. This is ugly and I would avoid doing it, because your code readability will decrease.
If you are dealing with hundreds of millions of vectors, perhaps consider a more cache-friendly approach where you do for blockI in range(N//10**6): for blockV in range(N//10**6): for i in range(blockI*10**6, (blockI+1)*10**6): for v in range(blockV*...):
If you're dealing with billions of vectors, then look into leveraging gpgpu. This is quite ideal for the gpu and might be a factor of thousands speedup.

Python matrix multiplication 3d Array

I tried to solve a PDE numerically and in the course of this I faced the problem of a triple-nested for loop resembling the 3 spatial dimension. This construct is nested in another time loop, so you can imagine that the computing takes forever for sufficient large node numbers. The code block looks like this
for jy in range(0,cy-1):
for jx in range(0,cx-1):
for jz in range(0,cz-1):
T[n+1,jx,jy,jz] = T[n,jx,jy,jz] + s*(T[n,jx-1,jy,jz] - 2*T[n,jx,jy,jz] + T[n,jx+1,jy,jz]) + s*(T[n,jx,jy-1,jz] - 2*T[n,jx,jy,jz] + T[n,jx,jy+1,jz]) + s*(T[n,jx,jy,jz-1] - 2*T[n,jx,jy,jz] + T[n,jx,jy,jz+1])
It might look intimidating at first, but is quite easy. I have a 3 dimensional matrix representing a solid bulk material, where each point represents the current temperature. The iteratively calculated next temperature at each point is calculated taking into account each point next to that point - so 6 in total. In the case of a 1-dimensional solid the solution is just a simple matrix multiplication. Is there any chance to represent the 3-loop-system above in a simple matrix solution like in the 1D case?
Best regards!

With numpy you can easily do these kinds of matrix operations,
e.g for a 3x3 matrix
import numpy as np
T = np.random.random((3,3,3))
T = T*T - 2*T ... etc.

First off, you need to be a bit more careful with your terminology. A "matrix" is a 2-Dimensional array of numbers. So you are really talking about an array. Numpy, or better yet Scipy, has an data type called an ndarray. You need to be very careful manipulating them, because although they are sometimes used to represent matrices, there are operations that can be performed on 2-D arrays that are not mathematically legal for matrices.
I strongly recommend you use # and not * to perform multiplication of 1- or 2-D matrices, and be sure to add code to check that the operations you are doing are legal mathematically. As a trivial example, Python lets you add a 1 x n or an n x 1 vector to an n x n matrix, even though that is not mathematically correct. The reason it allows it is, as intimated above, because there is no true matrix type in Python.
It very well may be that you can reformulate your problem to use a 3-D array, and by experimentation find the particular operation you are trying to perform. Just keep in mind that the rules of linear algebra are only casually applied in Python.

Efficient implementation of the transition matrix for page rank

I'm trying to implement PageRank. I'm reading the description here: http://nlp.stanford.edu/IR-book/html/htmledition/markov-chains-1.html
Everything is very clear to me, however I'm concerned about the construction of the matrix $P$. I find that constructing $P$ the naive way would be very expensive. For example: to implement step 1, one would need to check every row of $A$ and then check every element of that row to see if all elements are zero. For step 2 one would need to compute the number of ones for each row. I can imagine my code to have nasty slow loops. I was wondering if there are smart linear algebra techniques that could efficiently construct $P$. I will be using python numpy for my coding.
EDIT: one way I'm thinking now to solve this is by doing a summation element wise over the columns of $A$. By that I would have a column vector. Now I will go through each element of this vector to check which elements are zeros. Thus I can now know which rows has no 1s and I can multiply those rows with $1/N$.

Your concern is correct. Since the number of web pages (vertices in the representing graph) is huge, it is impossible to actually generate such A and work on it.
The matrix calculation of page rank can be much more efficiently calculated using sparse matrix implementations, since the matrix is very sparse. Most webpages are not actually connected to each other, so most entries in the matrix are 0.
The sparse matrix is built as follows:
Build matrix A as described A_ij = 1 if (i,j) is an edge, otherwise A_ij = 0
Step 1 is usually not made, and instead we remove 'sinks' iteratively. This is done to prevent the matrix being dense, some alternatives are also linking 'sinks' back to the nodes that linked to them, or link a sink to itself.
Divide each 1 in A as described in (2)
Let's denote the resulting matrix as M, and this is the resulting matrix we will work on, in order to get a column vector p (which is initialized with 1/n for each entry).
x = [1/n, 1/n, ... , 1/n]^T //a column vector
p = [1/n, 1/n, ... , 1/n]^T //a column vector with the initial ranks
M = genSparseMatrix() //as described above
do until p converge:
p = (1-\alpha)* M*p + (\alpha) * x
return p
In the end, this yields p, the column vector that holds the page rank value for each node.

Scipy LinearOperator With Multiple Inputs

I need to invert a large, dense matrix which I hoped to use Scipy's gmres to do. Fortunately, the dense matrix A follows a pattern and I do not need to store the matrix in memory. The LinearOperator class allows us to construct an object which acts as the matrix for GMRES and can compute directly the matrix vector product A*v. That is, we write a function mv(v) which takes as input a vector v and returns mv(v) = A*v. Then, we can use the LinearOperator class to create A_LinOp = LinearOperator(shape = shape, matvec = mv). We can put the linear operator into the Scipy gmres command to evaluate the matrix vector products without ever having to fully load A into memory.
The documentation for the LinearOperator is found here: LinearOperator Documentation.
Here is my problem: to write the routine to compute the matrix vector product mv(v) = A*v, I need another input vector C. The entries in A are of the form A[i,j] = f(C[i] - C[j]). So, what I really want is for mv to be of two inputs, one fixed vector input C, and one variable input v which we want to compute A*v.
MATLAB has a similar setup, where would write x = gmres(#(v) mv(v,C),b) where b is the right hand side of the problem Ax = b, , and mv is the function that takes as variable input v which we want to compute A*v and C is the fixed, known vector which we need for the assembly of A.
My problem is that I can't figure out how to allow the LinearOperator class to accept two inputs, one variable and one "fixed" like I can in MATLAB.
Is there a way to do the analogous operation in SciPy? Alternatively, if anyone knows of a better way of inverting a large, dense matrix (50000, 50000) where the entries follow a pattern, I would greatly appreciate any suggestions.
Thanks!
EDIT: I should have stated this information actually. The matrix is actually (in block form) [A C; C^T 0], where A is N x N (N large) and C is N x 3, and the 0 is 3 x 3 and C^T is the transpose of C. This array C is the same array as the one mentioned above. The entries of A follow a pattern A[i,j] = f(C[i] - C[j]).
I wrote mv(v,C) to go row by row construct A*v[i] for i=0,N, by computing sum f(C[i]-C[j)*v[j] (actually, I do numpy.dot(FC,v) where FC[j] = f(C[i]-C[j]) which works well). Then, at the end doing the computations for the C^T rows. I was hoping to eventually replace the large for loop with a multiprocessing call to parallelize the for loop, but that's a future thing to consider. I will also look into using Cython to speed up the computations.

This is very late, but if you're still interested...
Your A matrix must be very low rank since it's a nonlinearly transformed version of a rank-2 matrix. Plus it's symmetric. That means it's trivial to inverse: get the truncated eigenvalue decompostion with, say, 5 eigenvalues: A = U*S*U', then invert that: A^-1 = U*S^-1*U'. S is diagonal so this is inexpensive. You can get the truncated eigenvalue decomposition with eigh.
That takes care of A. Then for the rest: use the block matrix inversion formula. Looks nasty, but I will bet you 100,000,000 prussian francs that it's 50x faster than the direct method you were using.

I faced the same situation (some years later than you) of trying to use more than one argument to LinearOperator, but for another problem. The solution I found was the use of global variables, to avoid passing the variables as arguments to the function.