Math friendly syntax¶
pykeops.numpy
 Mathfriendly aliases:
Summary

Alias for 

Alias for 

Alias for 

Alias for 
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: dim3 vector per line ... 'y = Vj(3)') # 2nd input: dim3 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_iy_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) – Scalarvalued 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
Loglikelihood of a Gaussian Mixture Model,
\[\begin{split}a_i~=~f(x_i)~&=~ \log \sum_{j=1}^{N} \exp(\gamma\cdot\x_iy_j\^2)\cdot b_j \\\\ ~&=~ \log \sum_{j=1}^{N} \exp\\big(\gamma\cdot\x_iy_j\^2 \,+\, \log(b_j) \\big).\end{split}\]>>> log_likelihood = generic_logsumexp( ... '((g * SqNorm2(x  y))) + b', # Formula ... 'a = Vi(1)', # Output: 1 scalar per line ... 'x = Vi(3)', # 1st input: dim3 vector per line ... 'y = Vj(3)', # 2nd input: dim3 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_iy_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) – Scalarvalued 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: dim100 vector per line ... 'y = Vj(100)') # 2nd input: dim100 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) – Scalarvalued 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 Knearest 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: dim100 vector per line ... 'y = Vj(100)') # 2nd input: dim100 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]