Genred¶
This section contains the full API documentation for the numpy Generic reductions.
Summary
Creates a new generic operation. 

Instantiate a new generic operation. 

Apply the routine on arbitrary NumPy arrays. 
Syntax

class
pykeops.numpy.
Genred
[source]¶ Creates a new generic operation.
This is KeOps’ main function, whose usage is documented in the userguide, the gallery of examples and the highlevel tutorials. Taking as input a handful of strings and integers that specify a custom MapReduce operation, it returns a C++ wrapper that can be called just like any other NumPy function.
Note
On top of the Sum and LogSumExp reductions, KeOps supports variants of the ArgKMin reduction that can be used to implement knearest neighbor search. These routines return indices encoded as floating point numbers, and produce no gradient. Fortunately though, you can simply turn them into
LongTensors
and use them to index your arrays, as showcased in the documentation ofgeneric_argmin()
,generic_argkmin()
and in the Kmeans tutorial.Example
>>> my_conv = Genred('Exp(SqNorm2(x  y))', # formula ... ['x = Vi(3)', # 1st input: dim3 vector per line ... 'y = Vj(3)'], # 2nd input: dim3 vector per column ... reduction_op='Sum', # we also support LogSumExp, Min, etc. ... axis=1) # reduce along the lines of the kernel matrix >>> # 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]

__init__
(formula, aliases, reduction_op='Sum', axis=0, dtype='float64', opt_arg=None, formula2=None, cuda_type=None)[source]¶ Instantiate a new generic operation.
Note
Genred
relies on C++ or CUDA kernels that are compiled onthefly, and stored in a cache directory as shared libraries (“.so” files) for later use. Parameters
formula (string) – The scalar or vectorvalued expression that should be computed and reduced. The correct syntax is described in the documentation, using appropriate mathematical operations.
aliases (list of strings) –
A list of identifiers of the form
"AL = TYPE(DIM)"
that specify the categories and dimensions of the input variables. Here:AL
is an alphanumerical alias, used in the formula.TYPE
is a category. One of:Vi
: indexation by \(i\) along axis 0.Vj
: indexation by \(j\) along axis 1.Pm
: no indexation, the input tensor is a vector and not a 2d array.
DIM
is an integer, the dimension of the current variable.
As described below,
__call__()
will expect as input Tensors whose shape are compatible with aliases.
 Keyword Arguments
reduction_op (string, default =
"Sum"
) – Specifies the reduction operation that is applied to reduce the values offormula(x_i, y_j, ...)
along axis 0 or axis 1. The supported values are one of Reductionsaxis (int, default = 0) –
Specifies the dimension of the “kernel matrix” that is reduced by our routine. The supported values are:
axis = 0: reduction with respect to \(i\), outputs a
Vj
or “\(j\)” variable.axis = 1: reduction with respect to \(j\), outputs a
Vi
or “\(i\)” variable.
dtype (string, default =
"float32"
) –Specifies the numerical
dtype
of the input and output arrays. The supported values are:dtype =
"float32"
or"float"
.dtype =
"float64"
or"double"
.
opt_arg (int, default = None) – If reduction_op is in
["KMin", "ArgKMin", "KMinArgKMin"]
, this argument allows you to specify the numberK
of neighbors to consider.

__call__
(*args, backend='auto', device_id=1, ranges=None)[source]¶ Apply the routine on arbitrary NumPy arrays.
Warning
Even for variables of size 1 (e.g. \(a_i\in\mathbb{R}\) for \(i\in[0,M)\)), KeOps expects inputs to be formatted as 2d Tensors of size
(M,dim)
. In practice,a.view(1,1)
should be used to turn a vector of weights into a list of scalar values. Parameters
*args (2d arrays (variables
Vi(..)
,Vj(..)
) and 1d arrays (parametersPm(..)
)) –The input numerical arrays, which should all have the same
dtype
, be contiguous and be stored on the same device. KeOps expects one array per alias, with the following compatibility rules:All
Vi(Dim_k)
variables are encoded as 2darrays withDim_k
columns and the same number of lines \(M\).All
Vj(Dim_k)
variables are encoded as 2darrays withDim_k
columns and the same number of lines \(N\).All
Pm(Dim_k)
variables are encoded as 1darrays (vectors) of sizeDim_k
.
 Keyword Arguments
backend (string) –
Specifies the mapreduce scheme. The supported values are:
"auto"
(default): let KeOps decide which backend is best suited to your data, based on the tensors’ shapes."GPU_1D"
will be chosen in most cases."CPU"
: use a simple C++for
loop on a single CPU core."GPU_1D"
: use a simple multithreading scheme on the GPU  basically, one thread per value of the output index."GPU_2D"
: use a more sophisticated 2D parallelization scheme on the GPU."GPU"
: let KeOps decide which one of the"GPU_1D"
or the"GPU_2D"
scheme will run faster on the given input.
device_id (int, default=1) – Specifies the GPU that should be used to perform the computation; a negative value lets your system choose the default GPU. This parameter is only useful if your system has access to several GPUs.
ranges (6uple of integer arrays, None by default) –
Ranges of integers that specify a blocksparse reduction scheme with Mc clusters along axis 0 and Nc clusters along axis 1. If None (default), we simply loop over all indices \(i\in[0,M)\) and \(j\in[0,N)\).
The first three ranges will be used if axis = 1 (reduction along the axis of “\(j\) variables”), and to compute gradients with respect to
Vi(..)
variables:ranges_i
, (Mc,2) integer array  slice indices \([\operatorname{start}^I_k,\operatorname{end}^I_k)\) in \([0,M]\) that specify our Mc blocks along the axis 0 of “\(i\) variables”.slices_i
, (Mc,) integer array  consecutive slice indices \([\operatorname{end}^S_1, ..., \operatorname{end}^S_{M_c}]\) that specify Mc ranges \([\operatorname{start}^S_k,\operatorname{end}^S_k)\) inredranges_j
, with \(\operatorname{start}^S_k = \operatorname{end}^S_{k1}\). The first 0 is implicit, meaning that \(\operatorname{start}^S_0 = 0\), and we typically expect thatslices_i[1] == len(redrange_j)
.redranges_j
, (Mcc,2) integer array  slice indices \([\operatorname{start}^J_l,\operatorname{end}^J_l)\) in \([0,N]\) that specify reduction ranges along the axis 1 of “\(j\) variables”.
If axis = 1, these integer arrays allow us to say that
for k in range(Mc)
, the output values for indicesi in range( ranges_i[k,0], ranges_i[k,1] )
should be computed using a MapReduce scheme over indicesj in Union( range( redranges_j[l, 0], redranges_j[l, 1] ))
forl in range( slices_i[k1], slices_i[k] )
.Likewise, the last three ranges will be used if axis = 0 (reduction along the axis of “\(i\) variables”), and to compute gradients with respect to
Vj(..)
variables:ranges_j
, (Nc,2) integer array  slice indices \([\operatorname{start}^J_k,\operatorname{end}^J_k)\) in \([0,N]\) that specify our Nc blocks along the axis 1 of “\(j\) variables”.slices_j
, (Nc,) integer array  consecutive slice indices \([\operatorname{end}^S_1, ..., \operatorname{end}^S_{N_c}]\) that specify Nc ranges \([\operatorname{start}^S_k,\operatorname{end}^S_k)\) inredranges_i
, with \(\operatorname{start}^S_k = \operatorname{end}^S_{k1}\). The first 0 is implicit, meaning that \(\operatorname{start}^S_0 = 0\), and we typically expect thatslices_j[1] == len(redrange_i)
.redranges_i
, (Ncc,2) integer array  slice indices \([\operatorname{start}^I_l,\operatorname{end}^I_l)\) in \([0,M]\) that specify reduction ranges along the axis 0 of “\(i\) variables”.
If axis = 0, these integer arrays allow us to say that
for k in range(Nc)
, the output values for indicesj in range( ranges_j[k,0], ranges_j[k,1] )
should be computed using a MapReduce scheme over indicesi in Union( range( redranges_i[l, 0], redranges_i[l, 1] ))
forl in range( slices_j[k1], slices_j[k] )
.
 Returns
The output of the reduction, a 2dtensor with \(M\) or \(N\) lines (if axis = 1 or axis = 0, respectively) and a number of columns that is inferred from the formula.
 Return type
(M,D) or (N,D) array
