def generic_argkmin(formula, output, *aliases, **kwargs):
r"""Alias for :class:`numpy.Genred <pykeops.numpy.Genred>` with an "ArgKMin" reduction.
Args:
formula (string): Scalar-valued symbolic KeOps expression, as in :class:`numpy.Genred <pykeops.numpy.Genred>`.
output (string): An identifier of the form ``"AL = TYPE(K)"``
that specifies the category and dimension of the output variable. Here:
- ``AL`` is a dummy alphanumerical name.
- ``TYPE`` is a *category*. One of:
- ``Vi``: indexation by :math:`i` along axis 0; reduction is performed along axis 1.
- ``Vj``: indexation by :math:`j` along axis 1; reduction is performed along axis 0.
- ``K`` is an integer, the number of values to extract.
*aliases (strings): List of identifiers, as in :class:`numpy.Genred <pykeops.numpy.Genred>`.
Keyword Args:
dtype (string, default = ``"float64"``): Specifies the numerical **dtype** of the input and output arrays.
The supported values are:
- **dtype** = ``"float32"``,
- **dtype** = ``"float64"``.
Returns:
A generic reduction that can be called on arbitrary
NumPy arrays, as documented in :class:`numpy.Genred <pykeops.numpy.Genred>`.
Example:
Bruteforce K-nearest neighbors search in dimension 100:
>>> knn = generic_argkmin(
... 'SqDist(x, y)', # Formula
... 'a = Vi(3)', # Output: 3 scalars per line
... 'x = Vi(100)', # 1st input: dim-100 vector per line
... 'y = Vj(100)') # 2nd input: dim-100 vector per line
>>> x = np.random.randn(5, 100)
>>> y = np.random.randn(20000, 100)
>>> a = knn(x, y)
>>> print(a)
[[ 9054., 11653., 11614.],
[13466., 11903., 14180.],
[14164., 8809., 3799.],
[ 2092., 3323., 18479.],
[14433., 11315., 11841.]]
>>> print( np.linalg.norm(x - y[ a[:,0].astype(int) ], axis=1) ) # Distance to the nearest neighbor
[10.7933, 10.3235, 10.1218, 11.4919, 10.5100]
>>> print( np.linalg.norm(x - y[ a[:,1].astype(int) ], axis=1) ) # Distance to the second neighbor
[11.3702, 10.6550, 10.7646, 11.5676, 11.1356]
>>> print( np.linalg.norm(x - y[ a[:,2].astype(int) ], axis=1) ) # Distance to the third neighbor
[11.3820, 10.6725, 10.8510, 11.6071, 11.1968]
"""
_, cat, k, _ = get_type(output)
return Genred(formula,
list(aliases),
reduction_op='ArgKMin',
axis=cat2axis(cat),
opt_arg=k,
**kwargs)
def generic_logsumexp(formula, output, *aliases, **kwargs):
r"""Alias for :class:`torch.Genred <pykeops.torch.Genred>` with a "LogSumExp" reduction.
Args:
formula (string): Scalar-valued symbolic KeOps expression, as in :class:`torch.Genred <pykeops.torch.Genred>`.
output (string): An identifier of the form ``"AL = TYPE(1)"``
that specifies the category and dimension of the output variable. Here:
- ``AL`` is a dummy alphanumerical name.
- ``TYPE`` is a *category*. One of:
- ``Vi``: indexation by :math:`i` along axis 0; reduction is performed along axis 1.
- ``Vj``: indexation by :math:`j` along axis 1; reduction is performed along axis 0.
*aliases (strings): List of identifiers, as in :class:`torch.Genred <pykeops.torch.Genred>`.
Keyword Args:
dtype (string, default = ``"float32"``): Specifies the numerical **dtype** of the input and output arrays.
The supported values are:
- **dtype** = ``"float16"`` or ``"half"``.
- **dtype** = ``"float32"`` or ``"float"``.
- **dtype** = ``"float64"`` or ``"double"``.
Returns:
A generic reduction that can be called on arbitrary
Torch tensors, as documented in :class:`torch.Genred <pykeops.torch.Genred>`.
Example:
Log-likelihood of a Gaussian Mixture Model,
.. math::
a_i~=~f(x_i)~&=~ \log \sum_{j=1}^{N} \exp(-\gamma\cdot\|x_i-y_j\|^2)\cdot b_j \\\\
~&=~ \log \sum_{j=1}^{N} \exp\big(-\gamma\cdot\|x_i-y_j\|^2 \,+\, \log(b_j) \big).
>>> log_likelihood = generic_logsumexp(
... '(-(g * SqNorm2(x - y))) + b', # Formula
... 'a = Vi(1)', # Output: 1 scalar per line
... 'x = Vi(3)', # 1st input: dim-3 vector per line
... 'y = Vj(3)', # 2nd input: dim-3 vector per line
... 'g = Pm(1)', # 3rd input: vector of size 1
... 'b = Vj(1)') # 4th input: 1 scalar per line
>>> x = torch.randn(1000000, 3, requires_grad=True).cuda()
>>> y = torch.randn(2000000, 3).cuda()
>>> g = torch.Tensor([.5]).cuda() # Parameter of our GMM
>>> b = torch.rand(2000000, 1).cuda() # Positive weights...
>>> b = b / b.sum() # Normalized to get a probability measure
>>> a = log_likelihood(x, y, g, b.log()) # a_i = log sum_j exp(-g*|x_i-y_j|^2) * b_j
>>> print(a.shape)
torch.Size([1000000, 1])
"""
_, cat, _, _ = get_type(output)
axis = cat2axis(cat)
return Genred(formula,
aliases,
reduction_op='LogSumExp',
axis=axis,
**kwargs)
def generic_argmin(formula, output, *aliases, **kwargs):
r"""Alias for :class:`numpy.Genred <pykeops.numpy.Genred>` with an "ArgMin" reduction.
Args:
formula (string): Scalar-valued symbolic KeOps expression, as in :class:`numpy.Genred <pykeops.numpy.Genred>`.
output (string): An identifier of the form ``"AL = TYPE(1)"``
that specifies the category and dimension of the output variable. Here:
- ``AL`` is a dummy alphanumerical name.
- ``TYPE`` is a *category*. One of:
- ``Vi``: indexation by :math:`i` along axis 0; reduction is performed along axis 1.
- ``Vj``: indexation by :math:`j` along axis 1; reduction is performed along axis 0.
*aliases (strings): List of identifiers, as in :class:`numpy.Genred <pykeops.numpy.Genred>`.
Keyword Args:
dtype (string, default = ``"float64"``): Specifies the numerical **dtype** of the input and output arrays.
The supported values are:
- **dtype** = ``"float16"``,
- **dtype** = ``"float32"``,
- **dtype** = ``"float64"``.
Returns:
A generic reduction that can be called on arbitrary
NumPy arrays, as documented in :class:`numpy.Genred <pykeops.numpy.Genred>`.
Example:
Bruteforce nearest neighbor search in dimension 100:
>>> nearest_neighbor = generic_argmin(
... 'SqDist(x, y)', # Formula
... 'a = Vi(1)', # Output: 1 scalar per line
... 'x = Vi(100)', # 1st input: dim-100 vector per line
... 'y = Vj(100)') # 2nd input: dim-100 vector per line
>>> x = np.random.randn(5, 100)
>>> y = np.random.randn(20000, 100)
>>> a = nearest_neighbor(x, y)
>>> print(a)
[[ 8761.],
[ 2836.],
[ 906.],
[16130.],
[ 3158.]]
>>> dists = np.linalg.norm(x - y[ a.view(-1).long() ], axis=1) # Distance to the nearest neighbor
>>> print(dists)
[10.5926, 10.9132, 9.9694, 10.1396, 10.1955]
"""
_, cat, _, _ = get_type(output)
return Genred(formula,
list(aliases),
reduction_op='ArgMin',
axis=cat2axis(cat),
**kwargs)
def generic_sum(formula, output, *aliases, **kwargs):
r"""Alias for :class:`numpy.Genred <pykeops.numpy.Genred>` with a "Sum" reduction.
Args:
formula (string): Symbolic KeOps expression, as in :class:`numpy.Genred <pykeops.numpy.Genred>`.
output (string): An identifier of the form ``"AL = TYPE(DIM)"``
that specifies the category and dimension of the output variable. Here:
- ``AL`` is a dummy alphanumerical name.
- ``TYPE`` is a *category*. One of:
- ``Vi``: indexation by :math:`i` along axis 0; reduction is performed along axis 1.
- ``Vj``: indexation by :math:`j` along axis 1; reduction is performed along axis 0.
- ``DIM`` is an integer, the dimension of the output variable; it should be compatible with **formula**.
*aliases (strings): List of identifiers, as in :class:`numpy.Genred <pykeops.numpy.Genred>`.
Keyword Args:
dtype (string, default = ``"float64"``): Specifies the numerical **dtype** of the input and output arrays.
The supported values are:
- **dtype** = ``"float16"``,
- **dtype** = ``"float32"``,
- **dtype** = ``"float64"``.
Returns:
A generic reduction that can be called on arbitrary
NumPy arrays, as documented in :class:`numpy.Genred <pykeops.numpy.Genred>`.
Example:
>>> my_conv = generic_sum( # Custom Kernel Density Estimator
... 'Exp(-SqNorm2(x - y))', # Formula
... 'a = Vi(1)', # Output: 1 scalar per line
... 'x = Vi(3)', # 1st input: dim-3 vector per line
... 'y = Vj(3)') # 2nd input: dim-3 vector per line
>>> # Apply it to 2d arrays x and y with 3 columns and a (huge) number of lines
>>> x = np.random.randn(1000000, 3)
>>> y = np.random.randn(2000000, 3)
>>> a = my_conv(x, y) # a_i = sum_j exp(-|x_i-y_j|^2)
>>> print(a.shape)
(1000000, 1)
"""
_, cat, _, _ = get_type(output)
return Genred(formula,
list(aliases),
reduction_op='Sum',
axis=cat2axis(cat),
**kwargs)
