def square_dist(self, X, Xs=None):
X2 = aet.sum(aet.square(X), 1)
if Xs is None:
sqd = -2.0 * aet.dot(X, aet.transpose(X)) + (
aet.reshape(X2, (-1, 1)) + aet.reshape(X2, (1, -1)))
else:
Xs2 = aet.sum(aet.square(Xs), 1)
sqd = -2.0 * aet.dot(X, aet.transpose(Xs)) + (
aet.reshape(X2, (-1, 1)) + aet.reshape(Xs2, (1, -1)))
return aet.clip(sqd, 0.0, np.inf)
def full(self, X, Xs=None):
X, Xs = self._slice(X, Xs)
rx = self.lfunc(at.as_tensor_variable(X), self.args)
if Xs is None:
rz = self.lfunc(at.as_tensor_variable(X), self.args)
r2 = self.square_dist(X, X)
else:
rz = self.lfunc(at.as_tensor_variable(Xs), self.args)
r2 = self.square_dist(X, Xs)
rx2 = at.reshape(at.square(rx), (-1, 1))
rz2 = at.reshape(at.square(rz), (1, -1))
return at.sqrt((2.0 * at.outer(rx, rz)) / (rx2 + rz2)) * at.exp(-1.0 * r2 / (rx2 + rz2))
def square_dist(self, X, Xs):
X = at.mul(X, 1.0 / self.ls)
X2 = at.sum(at.square(X), 1)
if Xs is None:
sqd = -2.0 * at.dot(X, at.transpose(X)) + (at.reshape(X2,
(-1, 1)) +
at.reshape(X2, (1, -1)))
else:
Xs = at.mul(Xs, 1.0 / self.ls)
Xs2 = at.sum(at.square(Xs), 1)
sqd = -2.0 * at.dot(X, at.transpose(Xs)) + (
at.reshape(X2, (-1, 1)) + at.reshape(Xs2, (1, -1)))
return at.clip(sqd, 0.0, np.inf)
def kron_matrix_op(krons, m, op):
r"""Apply op to krons and m in a way that reproduces ``op(kronecker(*krons), m)``
Parameters
-----------
krons : list of square 2D array-like objects
D square matrices :math:`[A_1, A_2, ..., A_D]` to be Kronecker'ed
:math:`A = A_1 \otimes A_2 \otimes ... \otimes A_D`
Product of column dimensions must be :math:`N`
m : NxM array or 1D array (treated as Nx1)
Object that krons act upon
Returns
-------
numpy array
"""
def flat_matrix_op(flat_mat, mat):
Nmat = mat.shape[1]
flat_shape = flat_mat.shape
mat2 = flat_mat.reshape((Nmat, -1))
return op(mat, mat2).T.reshape(flat_shape)
def kron_vector_op(v):
return reduce(flat_matrix_op, krons, v)
if m.ndim == 1:
m = m[:, None] # Treat 1D array as Nx1 matrix
if m.ndim != 2: # Has not been tested otherwise
raise ValueError(f"m must have ndim <= 2, not {m.ndim}")
res = kron_vector_op(m)
res_shape = res.shape
return at.reshape(res, (res_shape[1], res_shape[0])).T
def test_local_reshape_dimshuffle():
reshape_dimshuffle = out2in(local_reshape_dimshuffle)
x = tensor.matrix("x")
y = x.dimshuffle("x", 0, "x", 1)
out = tensor.reshape(y, (1, x.shape[0] * x.shape[1], 1))
g = FunctionGraph([x], [out])
reshape_dimshuffle(g)
topo = g.toposort()
assert any([not isinstance(x, DimShuffle) for x in topo])
def reshape(self, shape, ndim=None):
"""Return a reshaped view/copy of this variable.
Parameters
----------
shape
Something that can be converted to a symbolic vector of integers.
ndim
The length of the shape. Passing None here means for
Aesara to try and guess the length of `shape`.
.. warning:: This has a different signature than numpy's
ndarray.reshape!
In numpy you do not need to wrap the shape arguments
in a tuple, in aesara you do need to.
"""
if ndim is not None:
if not isinstance(ndim, int):
raise ValueError("Expected ndim to be an integer, is " +
str(type(ndim)))
return at.reshape(self, shape, ndim=ndim)
def jacobian_det(self, x):
grad = aet.reshape(gradient(aet.sum(self.backward(x)), [x]), x.shape)
return aet.log(aet.abs_(grad))
def normal(
self,
size,
avg=0.0,
std=1.0,
ndim=None,
dtype=None,
nstreams=None,
truncate=False,
**kwargs,
):
"""
Sample a tensor of values from a normal distribution.
Parameters
----------
size : int_vector_like
Array dimensions for the output tensor.
avg : float_like, optional
The mean value for the truncated normal to sample from (defaults to 0.0).
std : float_like, optional
The standard deviation for the truncated normal to sample from (defaults to 1.0).
truncate : bool, optional
Truncates the normal distribution at 2 standard deviations if True (defaults to False).
When this flag is set, the standard deviation of the result will be less than the one specified.
ndim : int, optional
The number of dimensions for the output tensor (defaults to None).
This argument is necessary if the size argument is ambiguous on the number of dimensions.
dtype : str, optional
The data-type for the output tensor. If not specified,
the dtype is inferred from avg and std, but it is at least as precise as floatX.
kwargs
Other keyword arguments for random number generation (see uniform).
Returns
-------
samples : TensorVariable
A Aesara tensor of samples randomly drawn from a normal distribution.
"""
size = _check_size(size)
avg = undefined_grad(as_tensor_variable(avg))
std = undefined_grad(as_tensor_variable(std))
if dtype is None:
dtype = scal.upcast(config.floatX, avg.dtype, std.dtype)
avg = tensor.cast(avg, dtype=dtype)
std = tensor.cast(std, dtype=dtype)
# generate even number of uniform samples
# Do manual constant folding to lower optiimizer work.
if isinstance(size, aesara.Constant):
n_odd_samples = size.prod(dtype="int64")
else:
n_odd_samples = tensor.prod(size, dtype="int64")
n_even_samples = n_odd_samples + n_odd_samples % 2
uniform = self.uniform(
(n_even_samples, ),
low=0.0,
high=1.0,
ndim=1,
dtype=dtype,
nstreams=nstreams,
**kwargs,
)
# box-muller transform
u1 = uniform[:n_even_samples // 2]
u2 = uniform[n_even_samples // 2:]
r = tensor.sqrt(-2.0 * tensor.log(u1))
theta = np.array(2.0 * np.pi, dtype=dtype) * u2
cos_theta, sin_theta = tensor.cos(theta), tensor.sin(theta)
z0 = r * cos_theta
z1 = r * sin_theta
if truncate:
# use valid samples
to_fix0 = (z0 < -2.0) | (z0 > 2.0)
to_fix1 = (z1 < -2.0) | (z1 > 2.0)
z0_valid = z0[tensor.nonzero(~to_fix0)]
z1_valid = z1[tensor.nonzero(~to_fix1)]
# re-sample invalid samples
to_fix0 = tensor.nonzero(to_fix0)[0]
to_fix1 = tensor.nonzero(to_fix1)[0]
n_fix_samples = to_fix0.size + to_fix1.size
lower = tensor.constant(1.0 / np.e**2, dtype=dtype)
u_fix = self.uniform(
(n_fix_samples, ),
low=lower,
high=1.0,
ndim=1,
dtype=dtype,
nstreams=nstreams,
**kwargs,
)
r_fix = tensor.sqrt(-2.0 * tensor.log(u_fix))
z0_fixed = r_fix[:to_fix0.size] * cos_theta[to_fix0]
z1_fixed = r_fix[to_fix0.size:] * sin_theta[to_fix1]
# pack everything together to a useful result
norm_samples = tensor.join(0, z0_valid, z0_fixed, z1_valid,
z1_fixed)
else:
norm_samples = tensor.join(0, z0, z1)
if isinstance(n_odd_samples, aesara.Variable):
samples = norm_samples[:n_odd_samples]
elif n_odd_samples % 2 == 1:
samples = norm_samples[:-1]
else:
samples = norm_samples
samples = tensor.reshape(samples, newshape=size, ndim=ndim)
samples *= std
samples += avg
return samples
def max_pool(images, imgshp, maxpoolshp):
"""Implements a max pooling layer
Takes as input a 2D tensor of shape batch_size x img_size and
performs max pooling. Max pooling downsamples by taking the max
value in a given area, here defined by maxpoolshp. Outputs a 2D
tensor of shape batch_size x output_size.
:param images: 2D tensor containing images on which to apply convolution.
Assumed to be of shape batch_size x img_size
:param imgshp: tuple containing image dimensions
:param maxpoolshp: tuple containing shape of area to max pool over
:return: out1, symbolic result (2D tensor)
:return: out2, logical shape of the output
"""
poolsize = np.int64(np.prod(maxpoolshp))
# imgshp contains either 2 entries (height,width) or 3 (nfeatures,h,w)
# in the first case, default nfeatures to 1
if np.size(imgshp) == 2:
imgshp = (1,) + imgshp
# construct indices and index pointers for sparse matrix, which,
# when multiplied with input images will generate a stack of image
# patches
indices, indptr, spmat_shape, sptype, outshp = convolution_indices.conv_eval(
imgshp, maxpoolshp, maxpoolshp, mode="valid"
)
# print 'XXXXXXXXXXXXXXXX MAX POOLING LAYER XXXXXXXXXXXXXXXXXXXX'
# print 'imgshp = ', imgshp
# print 'maxpoolshp = ', maxpoolshp
# print 'outshp = ', outshp
# build sparse matrix, then generate stack of image patches
csc = aesara.sparse.CSM(sptype)(np.ones(indices.size), indices, indptr, spmat_shape)
patches = sparse.structured_dot(csc, images.T).T
pshape = tensor.stack(
[
images.shape[0] * tensor.as_tensor(np.prod(outshp)),
tensor.as_tensor(imgshp[0]),
tensor.as_tensor(poolsize),
]
)
patch_stack = tensor.reshape(patches, pshape, ndim=3)
out1 = tensor.max(patch_stack, axis=2)
pshape = tensor.stack(
[
images.shape[0],
tensor.as_tensor(np.prod(outshp)),
tensor.as_tensor(imgshp[0]),
]
)
out2 = tensor.reshape(out1, pshape, ndim=3)
out3 = tensor.DimShuffle(out2.broadcastable, (0, 2, 1))(out2)
return tensor.flatten(out3, 2), outshp
def convolve(
kerns,
kshp,
nkern,
images,
imgshp,
step=(1, 1),
bias=None,
mode="valid",
flatten=True,
):
"""Convolution implementation by sparse matrix multiplication.
:note: For best speed, put the matrix which you expect to be
smaller as the 'kernel' argument
"images" is assumed to be a matrix of shape batch_size x img_size,
where the second dimension represents each image in raster order
If flatten is "False", the output feature map will have shape:
.. code-block:: python
batch_size x number of kernels x output_size
If flatten is "True", the output feature map will have shape:
.. code-block:: python
batch_size x number of kernels * output_size
.. note::
IMPORTANT: note that this means that each feature map (image
generate by each kernel) is contiguous in memory. The memory
layout will therefore be: [ <feature_map_0> <feature_map_1>
... <feature_map_n>], where <feature_map> represents a
"feature map" in raster order
kerns is a 2D tensor of shape nkern x N.prod(kshp)
:param kerns: 2D tensor containing kernels which are applied at every pixel
:param kshp: tuple containing actual dimensions of kernel (not symbolic)
:param nkern: number of kernels/filters to apply.
nkern=1 will apply one common filter to all input pixels
:param images: tensor containing images on which to apply convolution
:param imgshp: tuple containing image dimensions
:param step: determines number of pixels between adjacent receptive fields
(tuple containing dx,dy values)
:param mode: 'full', 'valid' see CSM.evaluate function for details
:param sumdims: dimensions over which to sum for the tensordot operation.
By default ((2,),(1,)) assumes kerns is a nkern x kernsize
matrix and images is a batchsize x imgsize matrix
containing flattened images in raster order
:param flatten: flatten the last 2 dimensions of the output. By default,
instead of generating a batchsize x outsize x nkern tensor,
will flatten to batchsize x outsize*nkern
:return: out1, symbolic result
:return: out2, logical shape of the output img (nkern,heigt,width)
:TODO: test for 1D and think of how to do n-d convolutions
"""
# start by computing output dimensions, size, etc
kern_size = np.int64(np.prod(kshp))
# inshp contains either 2 entries (height,width) or 3 (nfeatures,h,w)
# in the first case, default nfeatures to 1
if np.size(imgshp) == 2:
imgshp = (1,) + imgshp
# construct indices and index pointers for sparse matrix, which,
# when multiplied with input images will generate a stack of image
# patches
indices, indptr, spmat_shape, sptype, outshp = convolution_indices.conv_eval(
imgshp, kshp, step, mode
)
# build sparse matrix, then generate stack of image patches
csc = aesara.sparse.CSM(sptype)(np.ones(indices.size), indices, indptr, spmat_shape)
patches = (sparse.structured_dot(csc, images.T)).T
# compute output of linear classifier
pshape = tensor.stack(
[
images.shape[0] * tensor.as_tensor(np.prod(outshp)),
tensor.as_tensor(imgshp[0] * kern_size),
]
)
patch_stack = tensor.reshape(patches, pshape, ndim=2)
# kern is of shape: nkern x ksize*number_of_input_features
# output is thus of shape: bsize*outshp x nkern
output = tensor.dot(patch_stack, kerns.T)
# add bias across each feature map (more efficient to do it now)
if bias is not None:
output += bias
# now to have feature maps in raster order ...
# go from bsize*outshp x nkern to bsize x nkern*outshp
newshp = tensor.stack(
[images.shape[0], tensor.as_tensor(np.prod(outshp)), tensor.as_tensor(nkern)]
)
tensout = tensor.reshape(output, newshp, ndim=3)
output = tensor.DimShuffle((False,) * tensout.ndim, (0, 2, 1))(tensout)
if flatten:
output = tensor.flatten(output, 2)
return output, np.hstack((nkern, outshp))
def jacobian_det(self, rv_var, rv_value):
grad = at.reshape(
gradient(at.sum(self.backward(rv_var, rv_value)), [rv_value]), rv_value.shape
)
return at.log(at.abs_(grad))
def conv2d(
input,
filters,
image_shape=None,
filter_shape=None,
border_mode="valid",
subsample=(1, 1),
**kargs,
):
"""
signal.conv.conv2d performs a basic 2D convolution of the input with the
given filters. The input parameter can be a single 2D image or a 3D tensor,
containing a set of images. Similarly, filters can be a single 2D filter or
a 3D tensor, corresponding to a set of 2D filters.
Shape parameters are optional and will result in faster execution.
Parameters
----------
input : Symbolic aesara tensor for images to be filtered.
Dimensions: ([num_images], image height, image width)
filters : Symbolic aesara tensor for convolution filter(s).
Dimensions: ([num_filters], filter height, filter width)
border_mode: {'valid', 'full'}
See scipy.signal.convolve2d.
subsample
Factor by which to subsample output.
image_shape : tuple of length 2 or 3
([num_images,] image height, image width).
filter_shape : tuple of length 2 or 3
([num_filters,] filter height, filter width).
kwargs
See aesara.tensor.nnet.conv.conv2d.
Returns
-------
symbolic 2D,3D or 4D tensor
Tensor of filtered images, with shape
([number images,] [number filters,] image height, image width).
"""
assert input.ndim in (2, 3)
assert filters.ndim in (2, 3)
# use shape information if it is given to us ###
if filter_shape and image_shape:
if input.ndim == 3:
bsize = image_shape[0]
else:
bsize = 1
imshp = (1, ) + tuple(image_shape[-2:])
if filters.ndim == 3:
nkern = filter_shape[0]
else:
nkern = 1
kshp = filter_shape[-2:]
else:
nkern, kshp = None, None
bsize, imshp = None, None
# reshape tensors to 4D, for compatibility with ConvOp ###
if input.ndim == 3:
sym_bsize = input.shape[0]
else:
sym_bsize = 1
if filters.ndim == 3:
sym_nkern = filters.shape[0]
else:
sym_nkern = 1
new_input_shape = tensor.join(0, tensor.stack([sym_bsize, 1]),
input.shape[-2:])
input4D = tensor.reshape(input, new_input_shape, ndim=4)
new_filter_shape = tensor.join(0, tensor.stack([sym_nkern, 1]),
filters.shape[-2:])
filters4D = tensor.reshape(filters, new_filter_shape, ndim=4)
# perform actual convolution ###
op = conv.ConvOp(
output_mode=border_mode,
dx=subsample[0],
dy=subsample[1],
imshp=imshp,
kshp=kshp,
nkern=nkern,
bsize=bsize,
**kargs,
)
output = op(input4D, filters4D)
# flatten to 3D tensor if convolving with single filter or single image
if input.ndim == 2 and filters.ndim == 2:
if aesara.config.warn.signal_conv2d_interface:
warnings.warn(
"aesara.tensor.signal.conv2d() now outputs a 2d tensor when both"
" inputs are 2d. To disable this warning, set the Aesara flag"
" warn.signal_conv2d_interface to False",
stacklevel=3,
)
output = tensor.flatten(output.T, ndim=2).T
elif input.ndim == 2 or filters.ndim == 2:
output = tensor.flatten(output.T, ndim=3).T
return output
