Numpy ‘smart’ symmetric matrix

Is there a smart and space-efficient symmetric matrix in numpy which automatically (and transparently) fills the position at [j][i] when [i][j] is written to?

import numpy
a = numpy.symmetric((3, 3))
a[0][1] = 1
a[1][0] == a[0][1]
# True
print(a)
# [[0 1 0], [1 0 0], [0 0 0]]

assert numpy.all(a == a.T) # for any symmetric matrix

An automatic Hermitian would also be nice, although I won’t need that at the time of writing.

Solution 1:

If you can afford to symmetrize the matrix just before doing calculations, the following should be reasonably fast:

def symmetrize(a):
    """
    Return a symmetrized version of NumPy array a.

    Values 0 are replaced by the array value at the symmetric
    position (with respect to the diagonal), i.e. if a_ij = 0,
    then the returned array a' is such that a'_ij = a_ji.

    Diagonal values are left untouched.

    a -- square NumPy array, such that a_ij = 0 or a_ji = 0, 
    for i != j.
    """
    return a + a.T - numpy.diag(a.diagonal())

This works under reasonable assumptions (such as not doing both a[0, 1] = 42 and the contradictory a[1, 0] = 123 before running symmetrize).

If you really need a transparent symmetrization, you might consider subclassing numpy.ndarray and simply redefining __setitem__:

class SymNDArray(numpy.ndarray):
    """
    NumPy array subclass for symmetric matrices.

    A SymNDArray arr is such that doing arr[i,j] = value
    automatically does arr[j,i] = value, so that array
    updates remain symmetrical.
    """

    def __setitem__(self, (i, j), value):
        super(SymNDArray, self).__setitem__((i, j), value)                    
        super(SymNDArray, self).__setitem__((j, i), value)                    

def symarray(input_array):
    """
    Return a symmetrized version of the array-like input_array.

    The returned array has class SymNDArray. Further assignments to the array
    are thus automatically symmetrized.
    """
    return symmetrize(numpy.asarray(input_array)).view(SymNDArray)

# Example:
a = symarray(numpy.zeros((3, 3)))
a[0, 1] = 42
print a  # a[1, 0] == 42 too!

(or the equivalent with matrices instead of arrays, depending on your needs). This approach even handles more complicated assignments, like a[:, 1] = -1, which correctly sets a[1, :] elements.

Note that Python 3 removed the possibility of writing def …(…, (i, j),…), so the code has to be slightly adapted before running with Python 3: def __setitem__(self, indexes, value): (i, j) = indexes…

Solution 2:

The more general issue of optimal treatment of symmetric matrices in numpy bugged me too.

After looking into it, I think the answer is probably that numpy is somewhat constrained by the memory layout supportd by the underlying BLAS routines for symmetric matrices.

While some BLAS routines do exploit symmetry to speed up computations on symmetric matrices, they still use the same memory structure as a full matrix, that is, n^2 space rather than n(n+1)/2. Just they get told that the matrix is symmetric and to use only the values in either the upper or the lower triangle.

Some of the scipy.linalg routines do accept flags (like sym_pos=True on linalg.solve) which get passed on to BLAS routines, although more support for this in numpy would be nice, in particular wrappers for routines like DSYRK (symmetric rank k update), which would allow a Gram matrix to be computed a fair bit quicker than dot(M.T, M).

(Might seem nitpicky to worry about optimising for a 2x constant factor on time and/or space, but it can make a difference to that threshold of how big a problem you can manage on a single machine...)

Solution 3:

There are a number of well-known ways of storing symmetric matrices so they don't need to occupy n^2 storage elements. Moreover, it is feasible to rewrite common operations to access these revised means of storage. The definitive work is Golub and Van Loan, Matrix Computations, 3rd edition 1996, Johns Hopkins University Press, sections 1.27-1.2.9. For example, quoting them from form (1.2.2), in a symmetric matrix only need to store A = [a_{i,j} ] fori >= j. Then, assuming the vector holding the matrix is denoted V, and that A is n-by-n, put a_{i,j} in

V[(j-1)n - j(j-1)/2 + i]

This assumes 1-indexing.

Golub and Van Loan offer an Algorithm 1.2.3 which shows how to access such a stored V to calculate y = V x + y.

Golub and Van Loan also provide a way of storing a matrix in diagonal dominant form. This does not save storage, but supports ready access for certain other kinds of operations.