import numpy as np
import nengo
from nengo.exceptions import ValidationError
from nengo.networks.product import Product
def transform_in(dims, align, invert):
"""Create a transform to map the input into the Fourier domain.
See CircularConvolution docstring for more details.
Parameters
----------
dims : int
Input dimensions.
align : 'A' or 'B'
How to align the real and imaginary components; the alignment
depends on whether we're doing transformA or transformB.
invert : bool
Whether to reverse the order of elements.
"""
if align not in ("A", "B"):
raise ValidationError("'align' must be either 'A' or 'B'", "align")
dims2 = 4 * (dims // 2 + 1)
tr = np.zeros((dims2, dims))
dft = dft_half(dims)
for i in range(dims2):
row = dft[i // 4] if not invert else dft[i // 4].conj()
if align == "A":
tr[i] = row.real if i % 2 == 0 else row.imag
else: # align == 'B'
tr[i] = row.real if i % 4 == 0 or i % 4 == 3 else row.imag
remove_imag_rows(tr)
return tr.reshape((-1, dims))
def transform_out(dims):
dims2 = dims // 2 + 1
tr = np.zeros((dims2, 4, dims))
idft = dft_half(dims).conj()
for i in range(dims2):
row = idft[i] if i == 0 or 2 * i == dims else 2 * idft[i]
tr[i, 0] = row.real
tr[i, 1] = -row.real
tr[i, 2] = -row.imag
tr[i, 3] = -row.imag
tr = tr.reshape(4 * dims2, dims)
remove_imag_rows(tr)
# IDFT has a 1/D scaling factor
tr /= dims
return tr.T
def remove_imag_rows(tr):
"""Throw away imaginary row we don't need (since they're zero)"""
i = np.arange(tr.shape[0])
if tr.shape[1] % 2 == 0:
tr = tr[(i == 0) | (i > 3) & (i < len(i) - 3)]
else:
tr = tr[(i == 0) | (i > 3)]
def dft_half(n):
x = np.arange(n)
w = np.arange(n // 2 + 1)
return np.exp((-2.0j * np.pi / n) * (w[:, None] * x[None, :]))
[docs]class CircularConvolution(nengo.Network):
r"""Compute the circular convolution of two vectors.
The circular convolution :math:`c` of vectors :math:`a` and :math:`b`
is given by
.. math:: c[i] = \sum_j a[j] b[i - j]
where negative indices on :math:`b` wrap around to the end of the vector.
This computation can also be done in the Fourier domain,
.. math:: c = DFT^{-1} ( DFT(a) \odot DFT(b) )
where :math:`DFT` is the Discrete Fourier Transform operator, and
:math:`DFT^{-1}` is its inverse. This network uses this method.
Parameters
----------
n_neurons : int
Number of neurons to use in each product computation.
dimensions : int
The number of dimensions of the input and output vectors.
invert_a : bool, optional
Whether to reverse the order of elements in first input.
invert_b : bool, optional
Whether to reverse the order of elements in the second input.
Flipping exactly one input will make the network perform circular
correlation instead of circular convolution which can be treated as an
approximate inverse to circular convolution.
input_magnitude : float, optional
The expected magnitude of the vectors to be convolved.
This value is used to determine the radius of the ensembles
computing the element-wise product.
**kwargs : dict
Keyword arguments to pass through to the `nengo.Network` constructor.
Attributes
----------
input_a : nengo.Node
The first vector to be convolved.
input_b : nengo.Node
The second vector to be convolved.
product : nengo.networks.Product
Network created to do the element-wise product of the :math:`DFT`
components.
output : nengo.Node
The resulting convolved vector.
Examples
--------
A basic example computing the circular convolution of two 10-dimensional
vectors represented by ensemble arrays::
A = EnsembleArray(50, n_ensembles=10)
B = EnsembleArray(50, n_ensembles=10)
C = EnsembleArray(50, n_ensembles=10)
cconv = nengo_spa.networks.CircularConvolution(50, dimensions=10)
nengo.Connection(A.output, cconv.input_a)
nengo.Connection(B.output, cconv.input_b)
nengo.Connection(cconv.output, C.input)
Notes
-----
The network maps the input vectors :math:`a` and :math:`b` of length
:math:`N` into the Fourier domain and aligns them for complex
multiplication.
Letting :math:`F = DFT(a)` and :math:`G = DFT(b)`, this is given by::
[ F[i].real ] [ G[i].real ] [ w[i] ]
[ F[i].imag ] * [ G[i].imag ] = [ x[i] ]
[ F[i].real ] [ G[i].imag ] [ y[i] ]
[ F[i].imag ] [ G[i].real ] [ z[i] ]
where :math:`i` only ranges over the lower half of the spectrum, since
the upper half of the spectrum is the flipped complex conjugate of
the lower half, and therefore redundant. The input transforms are
used to perform the DFT on the inputs and align them correctly for
complex multiplication.
The complex product :math:`H = F * G` is then
.. math:: H[i] = (w[i] - x[i]) + (y[i] + z[i]) I
where :math:`I = \sqrt{-1}`. We can perform this addition along with the
inverse DFT :math:`c = DFT^{-1}(H)` in a single output transform, finding
only the real part of :math:`c` since the imaginary part
is analytically zero.
"""
def __init__(
self,
n_neurons,
dimensions,
invert_a=False,
invert_b=False,
input_magnitude=1.0,
**kwargs
):
super().__init__(**kwargs)
tr_a = transform_in(dimensions, "A", invert_a)
tr_b = transform_in(dimensions, "B", invert_b)
tr_out = transform_out(dimensions)
with self:
self.input_a = nengo.Node(size_in=dimensions, label="input_a")
self.input_b = nengo.Node(size_in=dimensions, label="input_b")
self.product = Product(
n_neurons,
tr_out.shape[1],
input_magnitude=2 * input_magnitude / np.sqrt(2.0),
)
self.output = nengo.Node(size_in=dimensions, label="output")
nengo.Connection(
self.input_a, self.product.input_a, transform=tr_a, synapse=None
)
nengo.Connection(
self.input_b, self.product.input_b, transform=tr_b, synapse=None
)
nengo.Connection(
self.product.output, self.output, transform=tr_out, synapse=None
)
