问题
I'm trying to optimize a piece of code that solves a large sparse nonlinear system using an interior point method. During the update step, this involves computing the Hessian matrix H, the gradient g, then solving for d in H * d = -g to get the new search direction.
The Hessian matrix has a symmetric tridiagonal structure of the form:
A.T * diag(b) * A + C
I've run line_profiler on the particular function in question:
Line # Hits Time Per Hit % Time Line Contents
==================================================
386 def _direction(n, res, M, Hsig, scale_var, grad_lnprior, z, fac):
387
388 # gradient
389 44 1241715 28220.8 3.7 g = 2 * scale_var * res - grad_lnprior + z * np.dot(M.T, 1. / n)
390
391 # hessian
392 44 3103117 70525.4 9.3 N = sparse.diags(1. / n ** 2, 0, format=FMT, dtype=DTYPE)
393 44 18814307 427597.9 56.2 H = - Hsig - z * np.dot(M.T, np.dot(N, M)) # slow!
394
395 # update direction
396 44 10329556 234762.6 30.8 d, fac = my_solver(H, -g, fac)
397
398 44 111 2.5 0.0 return d, fac
Looking at the output it's clear that constructing H is by far the most costly step - it takes considerably longer than actually solving for the new direction.
Hsig and M are both CSC sparse matrices, n is a dense vector and z is a scalar. The solver I'm using requires H to be either a CSC or CSR sparse matrix.
Here's a function that produces some toy data with the same formats, dimensions and sparseness as my real matrices:
import numpy as np
from scipy import sparse
def make_toy_data(nt=200000, nc=10):
d0 = np.random.randn(nc * (nt - 1))
d1 = np.random.randn(nc * (nt - 1))
M = sparse.diags((d0, d1), (0, nc), shape=(nc * (nt - 1), nc * nt),
format='csc', dtype=np.float64)
d0 = np.random.randn(nc * nt)
Hsig = sparse.diags(d0, 0, shape=(nc * nt, nc * nt), format='csc',
dtype=np.float64)
n = np.random.randn(nc * (nt - 1))
z = np.random.randn()
return Hsig, M, n, z
And here's my original approach for constructing H:
def original(Hsig, M, n, z):
N = sparse.diags(1. / n ** 2, 0, format='csc')
H = - Hsig - z * np.dot(M.T, np.dot(N, M)) # slow!
return H
Timing:
%timeit original(Hsig, M, n, z)
# 1 loops, best of 3: 483 ms per loop
Is there a faster way to construct this matrix?
回答1:
I get close to a 4x speed-up in computing the product M.T * D * M out of the three diagonal arrays. If d0 and d1 are the main and upper diagonal of M, and d is the main diagonal of D, then the following code creates M.T * D * M directly:
def make_tridi_bis(d0, d1, d, nc=10):
d00 = d0*d0*d
d11 = d1*d1*d
d01 = d0*d1*d
len_ = d0.size
data = np.empty((3*len_ + nc,))
indices = np.empty((3*len_ + nc,), dtype=np.int)
# Fill main diagonal
data[:2*nc:2] = d00[:nc]
indices[:2*nc:2] = np.arange(nc)
data[2*nc+1:-2*nc:3] = d00[nc:] + d11[:-nc]
indices[2*nc+1:-2*nc:3] = np.arange(nc, len_)
data[-2*nc+1::2] = d11[-nc:]
indices[-2*nc+1::2] = np.arange(len_, len_ + nc)
# Fill top diagonal
data[1:2*nc:2] = d01[:nc]
indices[1:2*nc:2] = np.arange(nc, 2*nc)
data[2*nc+2:-2*nc:3] = d01[nc:]
indices[2*nc+2:-2*nc:3] = np.arange(2*nc, len_+nc)
# Fill bottom diagonal
data[2*nc:-2*nc:3] = d01[:-nc]
indices[2*nc:-2*nc:3] = np.arange(len_ - nc)
data[-2*nc::2] = d01[-nc:]
indices[-2*nc::2] = np.arange(len_ - nc ,len_)
indptr = np.empty((len_ + nc + 1,), dtype=np.int)
indptr[0] = 0
indptr[1:nc+1] = 2
indptr[nc+1:len_+1] = 3
indptr[-nc:] = 2
np.cumsum(indptr, out=indptr)
return sparse.csr_matrix((data, indices, indptr), shape=(len_+nc, len_+nc))
If your matrix M were in CSR format, you can extract d0 and d1 as d0 = M.data[::2] and d1 = M.data[1::2], I modified you toy data making routine to return those arrays as well, and here's what I get:
In [90]: np.allclose((M.T * sparse.diags(d, 0) * M).A, make_tridi_bis(d0, d1, d).A)
Out[90]: True
In [92]: %timeit make_tridi_bis(d0, d1, d)
10 loops, best of 3: 124 ms per loop
In [93]: %timeit M.T * sparse.diags(d, 0) * M
1 loops, best of 3: 501 ms per loop
The whole purpose of the above code is to take advantage of the structure of the non-zero entries. If you draw a diagram of the matrices you are multiplying together, it is relatively easy to convince yourself that the main (d_0) and top and bottom (d_1) diagonals of the resulting tridiagonal matrix are simply:
d_0 = np.zeros((len_ + nc,))
d_0[:len_] = d00
d_0[-len_:] += d11
d_1 = d01
The rest of the code in that function is simply building the tridiagonal matrix directly, as calling sparse.diags with the above data is several times slower.
回答2:
I tried running your test case and had problems with the np.dot(N, M). I didn't dig into it, but I think my numpy/sparse combo (both pretty new) had problems using np.dot on sparse arrays.
But H = -Hsig - z*M.T.dot(N.dot(M)) runs just fine. This uses the sparse dot.
I haven't run a profile, but here are Ipython timings for several parts. It takes longer to generate the data than to do that double dot.
In [37]: timeit Hsig,M,n,z=make_toy_data()
1 loops, best of 3: 2 s per loop
In [38]: timeit N = sparse.diags(1. / n ** 2, 0, format='csc')
1 loops, best of 3: 377 ms per loop
In [39]: timeit H = -Hsig - z*M.T.dot(N.dot(M))
1 loops, best of 3: 1.55 s per loop
H is a
<2000000x2000000 sparse matrix of type '<type 'numpy.float64'>'
with 5999980 stored elements in Compressed Sparse Column format>
来源：https://stackoverflow.com/questions/23143285/fastest-way-to-create-a-sparse-matrix-of-the-form-a-t-diagb-a-c