计算Vandermonde矩阵的有效方法

Question

I'm calculating Vandermonde matrix for a fairly large 1D array. The natural and clean way to do this is using np.vander() . However, I found that this is approx. 2.5x slower than a list comprehension based approach.

In [43]: x = np.arange(5000)
In [44]: N = 4

In [45]: %timeit np.vander(x, N, increasing=True)
155 µs ± 205 ns per loop (mean ± std. dev. of 7 runs, 10000 loops each)

# one of the listed approaches from the documentation
In [46]: %timeit np.flip(np.column_stack([x**(N-1-i) for i in range(N)]), axis=1)
65.3 µs ± 235 ns per loop (mean ± std. dev. of 7 runs, 10000 loops each)

In [47]: np.all(np.vander(x, N, increasing=True) == np.flip(np.column_stack([x**(N-1-i) for i in range(N)]), axis=1))
Out[47]: True

I'm trying to understand where the bottleneck is and the reason why does the implementation of native np.vander() is ~ 2.5x slower.

Efficiency matters for my implementation. So, even faster alternatives are also welcome!

Answer 1

How about broadcasted exponentiation?

%timeit (x ** np.arange(N)[:, None]).T
43 µs ± 348 ns per loop (mean ± std. dev. of 7 runs, 10000 loops each)

Sanity check -

np.all((x ** np.arange(N)[:, None]).T == np.vander(x, N, increasing=True))
True

The caveat here is that this speedup is possible only if your input array x has a dtype of int . As @Warren Weckesser pointed out in a comment, the broadcasted exponentiation slows down for floating point arrays.

---

As for why np.vander is slow, take a look at the source code -

x = asarray(x)
if x.ndim != 1:
    raise ValueError("x must be a one-dimensional array or sequence.")
if N is None:
    N = len(x)

v = empty((len(x), N), dtype=promote_types(x.dtype, int))
tmp = v[:, ::-1] if not increasing else v

if N > 0:
    tmp[:, 0] = 1
if N > 1:
    tmp[:, 1:] = x[:, None]
    multiply.accumulate(tmp[:, 1:], out=tmp[:, 1:], axis=1)

return v

The function has to cater to a lot more use cases besides yours, so it uses a more generalized method of computation which is reliable, but slower (I'm specifically pointing to multiply.accumulate ).

---

As a matter of interest, I found another way of computing the Vandermonde matrix, ending up with this:

%timeit x[:, None] ** np.arange(N)
150 µs ± 230 ns per loop (mean ± std. dev. of 7 runs, 10000 loops each)

It does the same thing, but is so much slower. The answer lies in the fact that the operations are broadcast, but inefficiently .

On the flip side, for float arrays, this actually ends up performing the best.

Answer 2

Here are some more methods some of which are quite a bit faster (on my computer) than what has been posted so far.

The most important observation I think is that it really depends a lot on how many degrees you want. Exponentiation (which I believe is special cased for small integer exponents) only makes sense for small exponent ranges. The more exponents the better multiplication based approaches fare.

I'd like to highlight a multiply.accumulate based method ( ma ) which is similar to numpy's builtin approach but faster (and not because I skimped on checks - nnc , numpy-no-checks demonstrates this). For all but the smallest exponent ranges it is actually the fastest for me.

For reasons I do not understand, the numpy implementation does three things that are to the best of my knowledge slow and unnecessary: (1) It makes quite a few copies of the base vector. (2) It makes them non-contiguous. (3) It does the accumulation in-place which I believe forces buffering.

Another thing I'd like to mention is that the fastest for small ranges of ints ( out_e_1 essentially a manual version of ma ), is slowed down by a factor of more than two by the simple precaution of promoting to a larger dtype ( safe_e_1 arguably a bit of a misnomer).

The broadcasting based methods are called bc_* where * indicates the broadcast axis (b for base, e for exp) 'cheat' means the result is noncontiguous.

Timings (best of 3):

rep=100 n_b=5000 n_e=4 b_tp=<class 'numpy.int32'> e_tp=<class 'numpy.int32'>
vander                0.16699657 ms
bc_b                  0.09595204 ms
bc_e                  0.07959786 ms
ma                    0.10755240 ms
nnc                   0.16459018 ms
out_e_1               0.02037535 ms
out_e_2               0.02656622 ms
safe_e_1              0.04652272 ms
safe_e_2              0.04081079 ms
cheat bc_e_cheat            0.04668466 ms
rep=100 n_b=5000 n_e=8 b_tp=<class 'numpy.int32'> e_tp=<class 'numpy.int32'>
vander                0.25086462 ms
bc_b             apparently failed
bc_e             apparently failed
ma                    0.15843041 ms
nnc                   0.24713077 ms
out_e_1          apparently failed
out_e_2          apparently failed
safe_e_1              0.15970622 ms
safe_e_2              0.19672418 ms
bc_e_cheat       apparently failed
rep=100 n_b=5000 n_e=4 b_tp=<class 'float'> e_tp=<class 'numpy.int32'>
vander                0.16225773 ms
bc_b                  0.53315020 ms
bc_e                  0.56200830 ms
ma                    0.07626799 ms
nnc                   0.16059748 ms
out_e_1               0.03653416 ms
out_e_2               0.04043702 ms
safe_e_1              0.04060494 ms
safe_e_2              0.04104209 ms
cheat bc_e_cheat            0.52966076 ms
rep=100 n_b=5000 n_e=8 b_tp=<class 'float'> e_tp=<class 'numpy.int32'>
vander                0.24542852 ms
bc_b                  2.03353578 ms
bc_e                  2.04281270 ms
ma                    0.11075758 ms
nnc                   0.24212880 ms
out_e_1               0.14809043 ms
out_e_2               0.19261359 ms
safe_e_1              0.15206112 ms
safe_e_2              0.19308420 ms
cheat bc_e_cheat            1.99176601 ms

Code:

import numpy as np
import types
from timeit import repeat

prom={np.dtype(np.int32): np.dtype(np.int64), np.dtype(float): np.dtype(float)}

def RI(k, N, dt, top=100):
    return np.random.randint(0, top if top else N, (k, N)).astype(dt)

def RA(k, N, dt, top=None):
    return np.add.outer(np.zeros((k,), int), np.arange(N)%(top if top else N)).astype(dt)

def RU(k, N, dt, top=100):
    return (np.random.random((k, N))*(top if top else N)).astype(dt)

def data(k, N_b, N_e, dt_b, dt_e, b_fun=RI, e_fun=RA):
    b = list(b_fun(k, N_b, dt_b))
    e = list(e_fun(k, N_e, dt_e))
    return b, e

def f_vander(b, e):
    return np.vander(b, len(e), increasing=True)

def f_bc_b(b, e):
    return b[:, None]**e

def f_bc_e(b, e):
    return np.ascontiguousarray((b**e[:, None]).T)

def f_ma(b, e):
    out = np.empty((len(b), len(e)), prom[b.dtype])
    out[:, 0] = 1
    np.multiply.accumulate(np.broadcast_to(b, (len(e)-1, len(b))), axis=0, out=out[:, 1:].T)
    return out

def f_nnc(b, e):
    out = np.empty((len(b), len(e)), prom[b.dtype])
    out[:, 0] = 1
    out[:, 1:] = b[:, None]
    np.multiply.accumulate(out[:, 1:], out=out[:, 1:], axis=1)
    return out

def f_out_e_1(b, e):
    out = np.empty((len(b), len(e)), b.dtype)
    out[:, 0] = 1
    out[:, 1] = b
    out[:, 2] = c = b*b
    for i in range(3, len(e)):
        c*=b
        out[:, i] = c
    return out

def f_out_e_2(b, e):
    out = np.empty((len(b), len(e)), b.dtype)
    out[:, 0] = 1
    out[:, 1] = b
    out[:, 2] = b*b
    for i in range(3, len(e)):
        out[:, i] = out[:, i-1] * b
    return out

def f_safe_e_1(b, e):
    out = np.empty((len(b), len(e)), prom[b.dtype])
    out[:, 0] = 1
    out[:, 1] = b
    out[:, 2] = c = (b*b).astype(prom[b.dtype])
    for i in range(3, len(e)):
        c*=b
        out[:, i] = c
    return out

def f_safe_e_2(b, e):
    out = np.empty((len(b), len(e)), prom[b.dtype])
    out[:, 0] = 1
    out[:, 1] = b
    out[:, 2] = b*b
    for i in range(3, len(e)):
        out[:, i] = out[:, i-1] * b
    return out

def f_bc_e_cheat(b, e):
    return (b**e[:, None]).T

for params in [(100, 5000, 4, np.int32, np.int32),
               (100, 5000, 8, np.int32, np.int32),
               (100, 5000, 4, float, np.int32),
               (100, 5000, 8, float, np.int32)]:
    k = params[0]
    dat = data(*params)
    ref = f_vander(dat[0][0], dat[1][0])
    print('rep={} n_b={} n_e={} b_tp={} e_tp={}'.format(*params))
    for name, func in list(globals().items()):
        if not name.startswith('f_') or not isinstance(func, types.FunctionType):
            continue
        try:
            assert np.allclose(ref, func(dat[0][0], dat[1][0]))
            if not func(dat[0][0], dat[1][0]).flags.c_contiguous:
                print('cheat', end=' ')
            print("{:16s}{:16.8f} ms".format(name[2:], np.min(repeat(
                'f(b.pop(), e.pop())', setup='b, e = data(*p)', globals={'f':func, 'data':data, 'p':params}, number=k)) * 1000 / k))
        except:
            print("{:16s} apparently failed".format(name[2:]))