# Math friendly syntax¶

pykeops.numpy - Math-friendly aliases:

Summary

 generic_sum(formula, output, *aliases, **kwargs) Alias for numpy.Genred with a “Sum” reduction. generic_logsumexp(formula, output, *aliases, …) Alias for numpy.Genred with a “LogSumExp” reduction. generic_argmin(formula, output, *aliases, …) Alias for numpy.Genred with an “ArgMin” reduction. generic_argkmin(formula, output, *aliases, …) Alias for numpy.Genred with an “ArgKMin” reduction.

Syntax

pykeops.numpy.generic_sum(formula, output, *aliases, **kwargs)[source]

Alias for numpy.Genred with a “Sum” reduction.

Parameters
• formula (string) – Symbolic KeOps expression, as in 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 $$i$$ along axis 0; reduction is performed along axis 1.

• Vj: indexation by $$j$$ 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 numpy.Genred.

Keyword Arguments

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 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)

pykeops.numpy.generic_logsumexp(formula, output, *aliases, **kwargs)[source]

Alias for numpy.Genred with a “LogSumExp” reduction.

Parameters
• formula (string) – Scalar-valued symbolic KeOps expression, as in 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 $$i$$ along axis 0; reduction is performed along axis 1.

• Vj: indexation by $$j$$ along axis 1; reduction is performed along axis 0.

• *aliases (strings) – List of identifiers, as in numpy.Genred.

Keyword Arguments

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 numpy.Genred.

Example

Log-likelihood of a Gaussian Mixture Model,

$\begin{split}a_i~=~f(x_i)~&=~ \log \sum_{j=1}^{N} \exp(-\gamma\cdot\|x_i-y_j\|^2)\cdot b_j \\\\ ~&=~ \log \sum_{j=1}^{N} \exp\\big(-\gamma\cdot\|x_i-y_j\|^2 \,+\, \log(b_j) \\big).\end{split}$
>>> 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 = np.random.randn(1000000, 3)
>>> y = np.random.randn(2000000, 3)
>>> g = np.array([.5])            # Parameter of our GMM
>>> b = np.random.rand(2000000, 1)  # Positive weights...
>>> b = b / b.sum()               # Normalized to get a probability measure
>>> a = log_likelihood(x, y, g, np.log(b))  # a_i = log sum_j exp(-g*|x_i-y_j|^2) * b_j
>>> print(a.shape)
(1000000, 1)

pykeops.numpy.generic_argmin(formula, output, *aliases, **kwargs)[source]

Alias for numpy.Genred with an “ArgMin” reduction.

Parameters
• formula (string) – Scalar-valued symbolic KeOps expression, as in 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 $$i$$ along axis 0; reduction is performed along axis 1.

• Vj: indexation by $$j$$ along axis 1; reduction is performed along axis 0.

• *aliases (strings) – List of identifiers, as in numpy.Genred.

Keyword Arguments

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 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]

pykeops.numpy.generic_argkmin(formula, output, *aliases, **kwargs)[source]

Alias for numpy.Genred with an “ArgKMin” reduction.

Parameters
• formula (string) – Scalar-valued symbolic KeOps expression, as in 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 $$i$$ along axis 0; reduction is performed along axis 1.

• Vj: indexation by $$j$$ 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 numpy.Genred.

Keyword Arguments

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 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]