LazyTensor
Summary
This section contains the full API documentation of the LazyTensor wrapper, which works identically on NumPy arrays and PyTorch tensors.

Symbolic wrapper for NumPy arrays and PyTorch tensors. 

Extension of the LazyTensor class for complex operations. 
Syntax
 class pykeops.common.lazy_tensor.GenericLazyTensor(x=None, axis=None)[source]
Symbolic wrapper for NumPy arrays and PyTorch tensors. This is the abstract class, end user should use
pykeops.numpy.LazyTensor
orpykeops.torch.LazyTensor
.LazyTensor
encode numerical arrays through the combination of a symbolic, mathematical formula and a list of small data arrays. They can be used to implement efficient algorithms on objects that are easy to define, but impossible to store in memory (e.g. the matrix of pairwise distances between two large point clouds).LazyTensor
may be created from standard NumPy arrays or PyTorch tensors, combined using simple mathematical operations and converted back to NumPy arrays or PyTorch tensors with efficient reduction routines, which outperform standard tensorized implementations by two orders of magnitude. __init__(x=None, axis=None)[source]
Creates a KeOps symbolic variable.
 Parameters:
x –
May be either:
A float, a list of floats, a NumPy float, a 0D or 1D NumPy array, a 0D or 1D PyTorch tensor, in which case the
LazyTensor
represents a constant vector of parameters, to be broadcasted on otherLazyTensor
.A 2D NumPy array or PyTorch tensor, in which case the
LazyTensor
represents a variable indexed by \(i\) if axis=0 or \(j\) if axis=1.A 3D+ NumPy array or PyTorch tensor with a dummy dimension (=1) at position 3 or 2, in which case the
LazyTensor
represents a variable indexed by \(i\) or \(j\), respectively. Dimensions before the last three will be handled as batch dimensions, that may support operator broadcasting.A tuple of 3 integers (ind,dim,cat), in which case the
LazyTensor
represents a symbolic variable that should be instantiated at calltime.An integer, in which case the
LazyTensor
represents an integer constant handled efficiently at compilation time.None, for internal use.
axis (int) – should be equal to 0 or 1 if x is a 2D tensor, and None otherwise.
Warning
A
LazyTensor
constructed from a NumPy array or a PyTorch tensor retains its dtype (float16, float32 or float64) and device properties (is it stored on the GPU?). Since KeOps does not support automatic type conversions and data transfers, please make sure not to mixLazyTensor
that come from different frameworks/devices or which are stored with different precisions.
 lt_constructor(x=None, axis=None)[source]
This method is specialized in
pykeops.numpy.LazyTensor
andpykeops.torch.LazyTensor
. It returns a new instance of a LazyTensor (numpy or pytorch).
 get_tools()[source]
This method is specialized in
pykeops.numpy.LazyTensor
andpykeops.torch.LazyTensor
. It populates the tools class.
 fixvariables()[source]
If needed, assigns final labels to each variable and pads their batch dimensions prior to a
Genred()
call.
 promote(other, props, is_complex=False)[source]
Creates a new
LazyTensor
whose None properties are set to those of self or other.
 join(other, is_complex=False)[source]
Merges the variables and attributes of two
LazyTensor
, with a compatibility check. This method concatenates tuples of variables, without paying attention to repetitions.
 unary(operation, dimres=None, opt_arg=None, opt_arg2=None, is_complex=None)[source]
Symbolically applies operation to self, with optional arguments if needed.
The optional argument dimres may be used to specify the dimension of the output result.
 binary(other, operation, is_operator=False, dimres=None, dimcheck='sameor1', opt_arg=None, opt_pos='last', rversion=False, is_complex=None)[source]
Symbolically applies operation to self, with optional arguments if needed.
 Keyword Arguments:
dimres () – May be used to specify the dimension of the output result.
is_operator () – May be used to specify if operation is an operator like
+
,
or a “genuine” function.dimcheck () – shall we check the input dimensions? Supported values are
"same"
,"sameor1"
,"vecand1"
or None.rversion () – shall we invert lhs and rhs of the binary op, e.g. as in __radd__, __rmut__, etc…
 ternary(other1, other2, operation, dimres=None, dimcheck='sameor1', opt_arg=None)[source]
Symbolically applies operation to self, with optional arguments if needed.
 Keyword Arguments:
dimres () – May be used to specify the dimension of the output result.
is_operator () – May be used to specify if operation is an operator like
+
,
or a “genuine” function.dimcheck () – shall we check the input dimensions? Supported values are
"same"
,"sameor1"
, or None.
 reduction(reduction_op, other=None, opt_arg=None, axis=None, dim=None, call=True, is_complex=None, **kwargs)[source]
Applies a reduction to a
LazyTensor
. This method is used internally by the LazyTensor class. :param reduction_op: the string identifier of the reduction, which will be passed to the KeOps routines. :type reduction_op: string Keyword Arguments:
other – May be used to specify some weights ; depends on the reduction.
opt_arg – typically, some integer needed by ArgKMin reductions ; depends on the reduction.
axis (integer) – The axis with respect to which the reduction should be performed. Supported values are nbatchdims and nbatchdims + 1, where nbatchdims is the number of “batch” dimensions before the last three (\(i\) indices, \(j\) indices, variables’ dimensions).
dim (integer) – alternative keyword for the axis argument.
call (True or False) – Should we actually perform the reduction on the current variables? If True, the returned object will be a NumPy array or a PyTorch tensor. Otherwise, we simply return a callable
LazyTensor
that may be used as apykeops.numpy.Genred
orpykeops.torch.Genred
function on arbitrary tensor data.backend (string) – Specifies the mapreduce scheme, as detailed in the documentation of the
Genred
module.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 IntTensors, None by default) – Ranges of integers that specify a blocksparse reduction scheme as detailed in the documentation of the
Genred
module. If None (default), we simply use a dense Kernel matrix as we loop over all indices \(i\in[0,M)\) and \(j\in[0,N)\).dtype_acc (string, default
"auto"
) –type for accumulator of reduction, before casting to dtype. It improves the accuracy of results in case of large sized data, but is slower. Default value “auto” will set this option to the value of dtype. The supported values are:
dtype_acc =
"float16"
: allowed only if dtype is “float16”.dtype_acc =
"float32"
: allowed only if dtype is “float16” or “float32”.dtype_acc =
"float64"
: allowed only if dtype is “float32” or “float64”..
use_double_acc (bool, default False) – same as setting dtype_acc=”float64” (only one of the two options can be set) If True, accumulate results of reduction in float64 variables, before casting to float32. This can only be set to True when data is in float32 or float64. It improves the accuracy of results in case of large sized data, but is slower.
sum_scheme (string, default
"auto"
) –method used to sum up results for reductions. This option may be changed only when reduction_op is one of: “Sum”, “MaxSumShiftExp”, “LogSumExp”, “Max_SumShiftExpWeight”, “LogSumExpWeight”, “SumSoftMaxWeight”. Default value “auto” will set this option to “block_red” for these reductions. Possible values are:
sum_scheme =
"direct_sum"
: direct summationsum_scheme =
"block_sum"
: use an intermediate accumulator in each block before accumulating in the output. This improves accuracy for large sized data.sum_scheme =
"kahan_scheme"
: use Kahan summation algorithm to compensate for roundoff errors. This improves accuracy for large sized data.
enable_chunks (bool, default True) – enable automatic selection of special “chunked” computation mode for accelerating reductions with formulas involving large dimension variables.
out (2d NumPy array or PyTorch Tensor, None by default) – The output numerical array, for inplace computation. If provided, the output array should all have the same
dtype
, be contiguous and be stored on the same device as the arguments. Moreover it should have the correct shape for the output.
 solve(other, var=None, call=True, **kwargs)[source]
Solves a positive definite linear system of the form
sum(self) = other
orsum(self*var) = other
, using a conjugate gradient solver. Parameters:
self (
LazyTensor
) – KeOps variable that encodes a symmetric positive definite matrix / linear operator.other (
LazyTensor
) – KeOps variable that encodes the second member of the equation.
 Keyword Arguments:
var (
LazyTensor
) – If var is None, solve will return the solution of theself * var = other
equation. Otherwise, if var is a KeOps symbolic variable, solve will assume that self defines an expression that is linear with respect to var and solve the equationself(var) = other
with respect to var.alpha (float, default=1e10) – Nonnegative ridge regularization parameter.
call (bool) – If True and if no other symbolic variable than var is contained in self, solve will return a tensor solution of our linear system. Otherwise solve will return a callable
LazyTensor
.backend (string) – Specifies the mapreduce scheme, as detailed in the documentation of the
Genred
module.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 IntTensors, None by default) – Ranges of integers that specify a blocksparse reduction scheme as detailed in the documentation of the
Genred
module. If None (default), we simply use a dense Kernel matrix as we loop over all indices \(i\in[0,M)\) and \(j\in[0,N)\).dtype_acc (string, default
"auto"
) –type for accumulator of reduction, before casting to dtype. It improves the accuracy of results in case of large sized data, but is slower. Default value “auto” will set this option to the value of dtype. The supported values are:
dtype_acc =
"float16"
: allowed only if dtype is “float16”.dtype_acc =
"float32"
: allowed only if dtype is “float16” or “float32”.dtype_acc =
"float64"
: allowed only if dtype is “float32” or “float64”..
use_double_acc (bool, default False) – same as setting dtype_acc=”float64” (only one of the two options can be set) If True, accumulate results of reduction in float64 variables, before casting to float32. This can only be set to True when data is in float32 or float64. It improves the accuracy of results in case of large sized data, but is slower.
sum_scheme (string, default
"auto"
) –method used to sum up results for reductions. This option may be changed only when reduction_op is one of: “Sum”, “MaxSumShiftExp”, “LogSumExp”, “Max_SumShiftExpWeight”, “LogSumExpWeight”, “SumSoftMaxWeight”. Default value “auto” will set this option to “block_red” for these reductions. Possible values are:
sum_scheme =
"direct_sum"
: direct summationsum_scheme =
"block_sum"
: use an intermediate accumulator in each block before accumulating in the output. This improves accuracy for large sized data.sum_scheme =
"kahan_scheme"
: use Kahan summation algorithm to compensate for roundoff errors. This improves accuracy for large sized data.
 enable_chunks (bool, default True): enable automatic selection of special “chunked” computation mode for accelerating reductions
with formulas involving large dimension variables.
Warning
Please note that no check of symmetry and definiteness will be performed prior to our conjugate gradient descent.
 __call__(*args, **kwargs)[source]
Executes a
Genred
orKernelSolve
call on the input data, as specified by self.formula .
 __add__(other)[source]
Broadcasted addition operator  a binary operation.
x + y
returns aLazyTensor
that encodes, symbolically, the addition ofx
andy
.
 __radd__(other)[source]
Broadcasted addition operator  a binary operation.
x + y
returns aLazyTensor
that encodes, symbolically, the addition ofx
andy
.
 __sub__(other)[source]
Broadcasted subtraction operator  a binary operation.
x  y
returns aLazyTensor
that encodes, symbolically, the subtraction ofx
andy
.
 __rsub__(other)[source]
Broadcasted subtraction operator  a binary operation.
x  y
returns aLazyTensor
that encodes, symbolically, the subtraction ofx
andy
.
 __mul__(other)[source]
Broadcasted elementwise product  a binary operation.
x * y
returns aLazyTensor
that encodes, symbolically, the elementwise product ofx
andy
.
 __rmul__(other)[source]
Broadcasted elementwise product  a binary operation.
x * y
returns aLazyTensor
that encodes, symbolically, the elementwise product ofx
andy
.
 __truediv__(other)[source]
Broadcasted elementwise division  a binary operation.
x / y
returns aLazyTensor
that encodes, symbolically, the elementwise division ofx
byy
.
 __rtruediv__(other)[source]
Broadcasted elementwise division  a binary operation.
x / y
returns aLazyTensor
that encodes, symbolically, the elementwise division ofx
byy
.
 __or__(other)[source]
Euclidean scalar product  a binary operation.
(xy)
returns aLazyTensor
that encodes, symbolically, the scalar product ofx
andy
which are assumed to have the same shape.
 __ror__(other)[source]
Euclidean scalar product  a binary operation.
(xy)
returns aLazyTensor
that encodes, symbolically, the scalar product ofx
andy
which are assumed to have the same shape.
 __abs__()[source]
Elementwise absolute value  a unary operation.
abs(x)
returns aLazyTensor
that encodes, symbolically, the elementwise absolute value ofx
.
 abs()[source]
Elementwise absolute value  a unary operation.
x.abs()
returns aLazyTensor
that encodes, symbolically, the elementwise absolute value ofx
.
 __neg__()[source]
Elementwise minus  a unary operation.
x
returns aLazyTensor
that encodes, symbolically, the elementwise opposite ofx
.
 exp()[source]
Elementwise exponential  a unary operation.
x.exp()
returns aLazyTensor
that encodes, symbolically, the elementwise exponential ofx
.
 log()[source]
Elementwise logarithm  a unary operation.
x.log()
returns aLazyTensor
that encodes, symbolically, the elementwise logarithm ofx
.
 xlogx()[source]
Elementwise x*log(x) function  a unary operation.
x.xlogx()
returns aLazyTensor
that encodes, symbolically, the elementwisex
times logarithm ofx
(with value 0 at 0).
 cos()[source]
Elementwise cosine  a unary operation.
x.cos()
returns aLazyTensor
that encodes, symbolically, the elementwise cosine ofx
.
 sin()[source]
Elementwise sine  a unary operation.
x.sin()
returns aLazyTensor
that encodes, symbolically, the elementwise sine ofx
.
 sinxdivx()[source]
Elementwise sin(x)/x function  a unary operation.
x.sinxdivx()
returns aLazyTensor
that encodes, symbolically, the elementwise sinxdivx function ofx
.
 sinc()[source]
Elementwise sinc(x) = sin(pi x) / (pi x) function  a unary operation.
x.sinc()
returns aLazyTensor
that encodes, symbolically, the elementwise sinc function ofx
.
 asin()[source]
Elementwise arcsine  a unary operation.
x.asin()
returns aLazyTensor
that encodes, symbolically, the elementwise arcsine ofx
.
 acos()[source]
Elementwise arccosine  a unary operation.
x.acos()
returns aLazyTensor
that encodes, symbolically, the elementwise arccosine ofx
.
 atan()[source]
Elementwise arctangent  a unary operation.
x.atan()
returns aLazyTensor
that encodes, symbolically, the elementwise arctangent ofx
.
 atan2(other)[source]
Elementwise atan2  a binary operation.
y.atan2(x)
returns aLazyTensor
that encodes, symbolically, the elementwise atan2 ofx
andy
.
 sqrt()[source]
Elementwise square root  a unary operation.
x.sqrt()
returns aLazyTensor
that encodes, symbolically, the elementwise square root ofx
.
 rsqrt()[source]
Elementwise inverse square root  a unary operation.
x.rsqrt()
returns aLazyTensor
that encodes, symbolically, the elementwise inverse square root ofx
.
 __pow__(other)[source]
Broadcasted elementwise power operator  a binary operation.
x**y
returns aLazyTensor
that encodes, symbolically, the elementwise value ofx
to the powery
.Note
if y = 2,
x**y
relies on the"Square"
KeOps operation;if y = 0.5,
x**y
uses on the"Sqrt"
KeOps operation;if y = 0.5,
x**y
uses on the"Rsqrt"
KeOps operation.
 power(other)[source]
Broadcasted elementwise power operator  a binary operation.
pow(x,y)
is equivalent tox**y
.
 square()[source]
Elementwise square  a unary operation.
x.square()
is equivalent tox**2
and returns aLazyTensor
that encodes, symbolically, the elementwise square ofx
.
 sign()[source]
Elementwise sign in {1,0,+1}  a unary operation.
x.sign()
returns aLazyTensor
that encodes, symbolically, the elementwise sign ofx
.
 step()[source]
Elementwise step function  a unary operation.
x.step()
returns aLazyTensor
that encodes, symbolically, the elementwise sign ofx
.
 relu()[source]
Elementwise ReLU function  a unary operation.
x.relu()
returns aLazyTensor
that encodes, symbolically, the elementwise positive part ofx
.
 clamp(other1, other2)[source]
Elementwise Clamp function  a ternary operation.
x.clamp(a,b)
returns aLazyTensor
that encodes, symbolically, the elementwise clamping ofx
in(a,b)
. Broadcasting rules apply. a and b may be fixed integers or floats, or other LazyTensors.
 ifelse(other1, other2)[source]
Elementwise ifelse function  a ternary operation.
x.ifelse(a,b)
returns aLazyTensor
that encodes, symbolically,a
wherex >= 0
andb
wherex < 0
. Broadcasting rules apply. a and b may be fixed integers or floats, or other LazyTensors.
 mod(modulus, offset=0)[source]
Elementwise modulo with offset function  a ternary operation.
x.mod(a,b)
returns aLazyTensor
that encodes, symbolically, the elementwise modulo ofx
with modulusa
and offsetb
. By default b=0, so that x.mod(a) becomes equivalent to the NumPy function mod. Broadcasting rules apply. a and b are fixed integers or float.
 round(other=0)[source]
Elementwise rounding function  a unary operation.
x.round(d)
returns aLazyTensor
that encodes, symbolically, the elementwise rounding ofx
to d decimal places. d is int.
 sqnorm2()[source]
Squared Euclidean norm  a unary operation.
x.sqnorm2()
returns aLazyTensor
that encodes, symbolically, the squared Euclidean norm of a vectorx
.
 norm2()[source]
Euclidean norm  a unary operation.
x.norm2()
returns aLazyTensor
that encodes, symbolically, the Euclidean norm of a vectorx
.
 norm(dim)[source]
Euclidean norm  a unary operation.
x.norm(1)
is equivalent tox.norm2()
and returns aLazyTensor
that encodes, symbolically, the Euclidean norm of a vectorx
.
 normalize()[source]
Vector normalization  a unary operation.
x.normalize()
returns aLazyTensor
that encodes, symbolically, a vectorx
divided by its Euclidean norm.
 sqdist(other)[source]
Squared distance  a binary operation.
x.sqdist(y)
returns aLazyTensor
that encodes, symbolically, the squared Euclidean distance between two vectorsx
andy
.
 weightedsqnorm(other)[source]
Weighted squared norm of a LazyTensor
x
 a binary operation.x.weightedsqnorm(s)
returns aLazyTensor
that encodes, symbolically, the weighted squared Norm of a vectorx
with weights stored in the LazyTensors
 see the main reference page for details.
 weightedsqdist(g, s)[source]
Weighted squared distance.
x.weightedsqdist(y, s)
is equivalent to(x  y).weightedsqnorm(s)
.
 elem(i)[source]
Indexing of a vector  a unary operation.
x.elem(i)
returns aLazyTensor
that encodes, symbolically, the ith elementx[i]
of the vectorx
.
 extract(i, d)[source]
Range indexing  a unary operation.
x.extract(i, d)
returns aLazyTensor
that encodes, symbolically, the subvectorx[i:i+d]
of the vectorx
.
 __getitem__(key)[source]
Element or range indexing  a unary operation.
x[key]
redirects to theelem()
orextract()
methods, depending on thekey
argument. Supported values are:an integer
k
, in which casex[key]
redirects toelem(x,k)
,a tuple
..,:,:,k
withk
an integer, which is equivalent to the case above,a slice of the form
k:l
,k:
or:l
, withk
andl
two integers, in which casex[key]
redirects toextract(x,k,lk)
,a tuple of slices of the form
..,:,:,k:l
,..,:,:,k:
or..,:,:,:l
, withk
andl
two integers, which are equivalent to the case above.
 one_hot(D)[source]
Encodes a (rounded) scalar value as a onehot vector of dimension D.
x.one_hot(D)
returns aLazyTensor
that encodes, symbolically, a vector of length D whose round(x)th coordinate is equal to 1, and the other ones to zero.
 bspline(x, k=0)[source]
Vector of BSpline functions of order k for the knots (self), evaluated at x.
 Parameters:
x – a LazyTensor of dimension 1.
k – a nonnegative integer.
 concat(other)[source]
Concatenation of two
LazyTensor
 a binary operation.x.concat(y)
returns aLazyTensor
that encodes, symbolically, the concatenation ofx
andy
along their last dimension.
 static concatenate(tuple_of_lt, axis=1)[source]
Concatenation of a tuple of
GenericLazyTensor
.GenericLazyTensor.concatenate( (x_1, x_2, ..., x_n), 1)
returns aGenericLazyTensor
that encodes, symbolically, the concatenation ofx_1
,x_2
, …,x_n
along their last dimension. Note that axis should be equal to 1 or 2 (if thex_i
’s are 3D GenericLazyTensor): GenericLazyTensors only support concatenation and indexing operations with respect to the last dimension.
 static cat(tuple_of_lt, dim)[source]
Concatenation of a tuple of LazyTensors.
LazyTensor.cat( (x_1, x_2, ..., x_n), 1)
is a PyTorchfriendly alias forLazyTensor.concatenate( (x_1, x_2, ..., x_n), 1)
; just like indexing operations, it is only supported along the last dimension.
 matvecmult(other)[source]
Matrixvector product  a binary operation.
If
x._shape[1] == A*B
andy._shape[1] == B
,z = x.matvecmult(y)
returns aGenericLazyTensor
such thatz._shape[1] == A
which encodes, symbolically, the matrixvector product ofx
andy
along their last dimension. For details, please check the documentation of the KeOps operation"MatVecMult"
in the main reference page.
 vecmatmult(other)[source]
Vectormatrix product  a binary operation.
If
x._shape[1] == A
andy._shape[1] == A*B
,z = x.vecmatmult(y)
returns aGenericLazyTensor
such thatz._shape[1] == B
which encodes, symbolically, the vectormatrix product ofx
andy
along their last dimension. For details, please check the documentation of the KeOps operation"VecMacMult"
in the main reference page.
 tensorprod(other)[source]
Tensor product of vectors  a binary operation.
If
x._shape[1] == A
andy._shape[1] == B
,z = x.tensorprod(y)
returns aGenericLazyTensor
such thatz._shape[1] == A*B
which encodes, symbolically, the tensor product ofx
andy
along their last dimension. For details, please check the documentation of the KeOps operation"TensorProd"
in the main reference page.
 keops_tensordot(other, dimfa, dimfb, contfa, contfb, *args)[source]
Tensor dot product (on KeOps internal dimensions)  a binary operation.
 Parameters:
other – a
LazyTensor
dimfa – tuple/list of int
dimfb – tuple/list of int
contfa – tuple/list of int listing contraction dimension of a (could be empty)
contfb – tuple/list of int listing contraction dimension of b (could be empty)
args – a tuple/list of int containing the graph of a permutation of the output
 Returns:
a
LazyTensor
 keops_kron(other, dimfself, dimfother)[source]
Kronecker product (on KeOps internal dimensions)  a binary operation.
If
self._shape[1] == d0 * d1 * ... * dN
andother._shape[1] == D0 * D1 * ... * DN
,z = x.keops_kron(y, [d0, d1, ..., dN], [D0, D1, ..., DN])
returns aGenericLazyTensor
of shapez._shape[1] == d0 * D0 * d1 * D1 * ... * dN * DN
which encodes, symbolically, the (flattened version of) Kronecker product ofself
andother
along their internal dimension. Parameters:
other – a
LazyTensor
dimfself – tuple/list of int listing the dimensions of a (prod(dimfself) == self._shape[1]) and len(dimfself) == len(dimfother)
dimfother – tuple/list of int listing the dimensions of b (prod(dimfother) == y._shape[1]) and len(dimfself) == len(dimfother)
 Returns:
a
LazyTensor
of internal shape prod(dimfself) * prod(dimfother)>>> import numpy as np >>> from pykeops.numpy import LazyTensor >>> >>> dimfb = [3, 3, 2, 1] >>> dimfa = [3, 1, 3, 2] >>> >>> M, N, axis = 11, 150, 1 >>> >>> x = np.random.rand(M, np.array(dimfa).prod()) >>> y = np.random.rand(N, np.array(dimfb).prod()) >>> >>> gamma_py = np.zeros((M, N, np.array(dimfa).prod() * np.array(dimfb).prod())) >>> for i in range(M): >>> for j in range(N): >>> gamma_py[i, j, :] = np.kron(x[i, :].reshape(*dimfa), y[j, :].reshape(*dimfb)).flatten() >>> >>> gamma_py = gamma_py.sum(axis=axis) >>> >>> X = LazyTensor(x[:, None, :]) >>> Y = LazyTensor(y[None, :, :]) >>> >>> gamma_keops = (X.keops_kron(Y, dimfa, dimfb)).sum(axis=axis) >>> >>> print(np.allclose(gamma_keops, gamma_py, atol=1e6))
 grad(other, gradin)[source]
Symbolic gradient operation.
z = x.grad(v,e)
returns aLazyTensor
which encodes, symbolically, the gradient (more precisely, the adjoint of the differential operator) ofx
, with respect to variablev
, and applied toe
. For details, please check the documentation of the KeOps operation"Grad"
in the main reference page.
 sum(axis=1, dim=None, **kwargs)[source]
Summation unary operation, or Sum reduction.
sum(axis, dim, **kwargs)
will:if axis or dim = 0, return the sum reduction of self over the “i” indexes.
if axis or dim = 1, return the sum reduction of self over the “j” indexes.
if axis or dim = 2, return a new
LazyTensor
object representing the sum of the values of the vector self,
 Keyword Arguments:
axis (integer) – reduction dimension, which should be equal to the number of batch dimensions plus 0 (= reduction over \(i\)), 1 (= reduction over \(j\)) or 2 (i.e. 1, sum along the dimension of the vector variable).
dim (integer) – alternative keyword for the axis parameter.
**kwargs – optional parameters that are passed to the
reduction()
method.
 sum_reduction(axis=None, dim=None, **kwargs)[source]
Sum reduction.
sum_reduction(axis, dim, **kwargs)
will return the sum reduction of self. Keyword Arguments:
axis (integer) – reduction dimension, which should be equal to the number of batch dimensions plus 0 (= reduction over \(i\)), or 1 (= reduction over \(j\)).
dim (integer) – alternative keyword for the axis parameter.
**kwargs – optional parameters that are passed to the
reduction()
method.
 logsumexp(axis=None, dim=None, weight=None, **kwargs)[source]
LogSumExp reduction.
logsumexp(axis, dim, weight, **kwargs)
will:if axis or dim = 0, return the “logsumexp” reduction of self over the “i” indexes.
if axis or dim = 1, return the “logsumexp” reduction of self over the “j” indexes.
For details, please check the documentation of the KeOps reductions
LogSumExp
andLogSumExpWeight
in the main reference page. Keyword Arguments:
axis (integer) – reduction dimension, which should be equal to the number of batch dimensions plus 0 (= reduction over \(i\)), or 1 (= reduction over \(j\)).
dim (integer) – alternative keyword for the axis parameter.
weight (
LazyTensor
) – optional object that specifies scalar or vectorvalued weights in the logsumexp operation**kwargs – optional parameters that are passed to the
reduction()
method.
 logsumexp_reduction(**kwargs)[source]
LogSumExp reduction. Redirects to
logsumexp()
method.
 sumsoftmaxweight(weight, axis=None, dim=None, **kwargs)[source]
Sum of weighted SoftMax reduction.
sumsoftmaxweight(weight, axis, dim, **kwargs)
will:if axis or dim = 0, return the “sum of weighted SoftMax” reduction of self over the “i” indexes.
if axis or dim = 1, return the “sum of weighted SoftMax” reduction of self over the “j” indexes.
For details, please check the documentation of the KeOps reduction
SumSoftMaxWeight
in the main reference page. Keyword Arguments:
weight (
LazyTensor
) – object that specifies scalar or vectorvalued weights.axis (integer) – reduction dimension, which should be equal to the number of batch dimensions plus 0 (= reduction over \(i\)), or 1 (= reduction over \(j\)).
dim (integer) – alternative keyword for the axis parameter.
**kwargs – optional parameters that are passed to the
reduction()
method.
 sumsoftmaxweight_reduction(**kwargs)[source]
Sum of weighted SoftMax reduction. Redirects to
sumsoftmaxweight()
method.
 min(axis=1, dim=None, **kwargs)[source]
Minimum unary operation, or Min reduction.
min(axis, dim, **kwargs)
will:if axis or dim = 0, return the min reduction of self over the “i” indexes.
if axis or dim = 1, return the min reduction of self over the “j” indexes.
if axis or dim = 2, return a new
LazyTensor
object representing the min of the values of the vector self,
 Keyword Arguments:
axis (integer) – reduction dimension, which should be equal to the number of batch dimensions plus 0 (= reduction over \(i\)), 1 (= reduction over \(j\)) or 2 (i.e. 1, min along the dimension of the vector variable).
dim (integer) – alternative keyword for the axis parameter.
**kwargs – optional parameters that are passed to the
reduction()
method.
 min_reduction(axis=None, dim=None, **kwargs)[source]
Min reduction.
min_reduction(axis, dim, **kwargs)
will return the min reduction of self. Keyword Arguments:
axis (integer) – reduction dimension, which should be equal to the number of batch dimensions plus 0 (= reduction over \(i\)), or 1 (= reduction over \(j\)).
dim (integer) – alternative keyword for the axis parameter.
**kwargs – optional parameters that are passed to the
reduction()
method.
 argmin(axis=1, dim=None, **kwargs)[source]
ArgMin unary operation, or ArgMin reduction.
argmin(axis, dim, **kwargs)
will:if axis or dim = 0, return the argmin reduction of self over the “i” indexes.
if axis or dim = 1, return the argmin reduction of self over the “j” indexes.
if axis or dim = 2, return a new
LazyTensor
object representing the argmin of the values of the vector self,
 Keyword Arguments:
axis (integer) – reduction dimension, which should be equal to the number of batch dimensions plus 0 (= reduction over \(i\)), 1 (= reduction over \(j\)) or 2 (i.e. 1, argmin along the dimension of the vector variable).
dim (integer) – alternative keyword for the axis parameter.
**kwargs – optional parameters that are passed to the
reduction()
method.
 argmin_reduction(axis=None, dim=None, **kwargs)[source]
ArgMin reduction.
argmin_reduction(axis, dim, **kwargs)
will return the argmin reduction of self. Keyword Arguments:
axis (integer) – reduction dimension, which should be equal to the number of batch dimensions plus 0 (= reduction over \(i\)), or 1 (= reduction over \(j\)).
dim (integer) – alternative keyword for the axis parameter.
**kwargs – optional parameters that are passed to the
reduction()
method.
 min_argmin(axis=None, dim=None, **kwargs)[source]
MinArgMin reduction.
min_argmin(axis, dim, **kwargs)
will:if axis or dim = 0, return the minimal values and its indices of self over the “i” indexes.
if axis or dim = 1, return the minimal values and its indices of self over the “j” indexes.
 Keyword Arguments:
axis (integer) – reduction dimension, which should be equal to the number of batch dimensions plus 0 (= reduction over \(i\)), or 1 (= reduction over \(j\)).
dim (integer) – alternative keyword for the axis parameter.
**kwargs – optional parameters that are passed to the
reduction()
method.
 min_argmin_reduction(**kwargs)[source]
MinArgMin reduction. Redirects to
min_argmin()
method.
 max(axis=1, dim=None, **kwargs)[source]
Miaximum unary operation, or Max reduction.
max(axis, dim, **kwargs)
will:if axis or dim = 0, return the max reduction of self over the “i” indexes.
if axis or dim = 1, return the max reduction of self over the “j” indexes.
if axis or dim = 2, return a new
LazyTensor
object representing the max of the values of the vector self,
 Keyword Arguments:
axis (integer) – reduction dimension, which should be equal to the number of batch dimensions plus 0 (= reduction over \(i\)), 1 (= reduction over \(j\)) or 2 (i.e. 1, max along the dimension of the vector variable).
dim (integer) – alternative keyword for the axis parameter.
**kwargs – optional parameters that are passed to the
reduction()
method.
 max_reduction(axis=None, dim=None, **kwargs)[source]
Max reduction.
max_reduction(axis, dim, **kwargs)
will return the max reduction of self. Keyword Arguments:
axis (integer) – reduction dimension, which should be equal to the number of batch dimensions plus 0 (= reduction over \(i\)), or 1 (= reduction over \(j\)).
dim (integer) – alternative keyword for the axis parameter.
**kwargs – optional parameters that are passed to the
reduction()
method.
 argmax(axis=1, dim=None, **kwargs)[source]
ArgMax unary operation, or ArgMax reduction.
argmax(axis, dim, **kwargs)
will:if axis or dim = 0, return the argmax reduction of self over the “i” indexes.
if axis or dim = 1, return the argmax reduction of self over the “j” indexes.
if axis or dim = 2, return a new
LazyTensor
object representing the argmax of the values of the vector self,
 Keyword Arguments:
axis (integer) – reduction dimension, which should be equal to the number of batch dimensions plus 0 (= reduction over \(i\)), 1 (= reduction over \(j\)) or 2 (i.e. 1, argmax along the dimension of the vector variable).
dim (integer) – alternative keyword for the axis parameter.
**kwargs – optional parameters that are passed to the
reduction()
method.
 argmax_reduction(axis=None, dim=None, **kwargs)[source]
ArgMax reduction.
argmax_reduction(axis, dim, **kwargs)
will return the argmax reduction of self. Keyword Arguments:
axis (integer) – reduction dimension, which should be equal to the number of batch dimensions plus 0 (= reduction over \(i\)), or 1 (= reduction over \(j\)).
dim (integer) – alternative keyword for the axis parameter.
**kwargs – optional parameters that are passed to the
reduction()
method.
 max_argmax(axis=None, dim=None, **kwargs)[source]
MaxArgMax reduction.
max_argmax(axis, dim, **kwargs)
will:if axis or dim = 0, return the maximal values and its indices of self over the “i” indexes.
if axis or dim = 1, return the maximal values and its indices of self over the “j” indexes.
 Keyword Arguments:
axis (integer) – reduction dimension, which should be equal to the number of batch dimensions plus 0 (= reduction over \(i\)), or 1 (= reduction over \(j\)).
dim (integer) – alternative keyword for the axis parameter.
**kwargs – optional parameters that are passed to the
reduction()
method.
 max_argmax_reduction(**kwargs)[source]
MaxArgMax reduction. Redirects to
max_argmax()
method.
 Kmin(K, axis=None, dim=None, **kwargs)[source]
KMin reduction.
Kmin(K, axis, dim, **kwargs)
will:if axis or dim = 0, return the K minimal values of self over the “i” indexes.
if axis or dim = 1, return the K minimal values of self over the “j” indexes.
 Keyword Arguments:
K (integer) – number of minimal values required
axis (integer) – reduction dimension, which should be equal to the number of batch dimensions plus 0 (= reduction over \(i\)), or 1 (= reduction over \(j\)).
dim (integer) – alternative keyword for the axis parameter.
**kwargs – optional parameters that are passed to the
reduction()
method.
 argKmin(K, axis=None, dim=None, **kwargs)[source]
argKmin reduction.
argKmin(K, axis, dim, **kwargs)
will:if axis or dim = 0, return the indices of the K minimal values of self over the “i” indexes.
if axis or dim = 1, return the indices of the K minimal values of self over the “j” indexes.
 Keyword Arguments:
K (integer) – number of minimal values required
axis (integer) – reduction dimension, which should be equal to the number of batch dimensions plus 0 (= reduction over \(i\)), or 1 (= reduction over \(j\)).
dim (integer) – alternative keyword for the axis parameter.
**kwargs – optional parameters that are passed to the
reduction()
method.
 Kmin_argKmin(K, axis=None, dim=None, **kwargs)[source]
KMinargKmin reduction.
Kmin_argKmin(K, axis, dim, **kwargs)
will:if axis or dim = 0, return the K minimal values and its indices of self over the “i” indexes.
if axis or dim = 1, return the K minimal values and its indices of self over the “j” indexes.
 Keyword Arguments:
K (integer) – number of minimal values required
axis (integer) – reduction dimension, which should be equal to the number of batch dimensions plus 0 (= reduction over \(i\)), or 1 (= reduction over \(j\)).
dim (integer) – alternative keyword for the axis parameter.
**kwargs – optional parameters that are passed to the
reduction()
method.
 Kmin_argKmin_reduction(**kwargs)[source]
Kmin_argKmin reduction. Redirects to
Kmin_argKmin()
method.
 __matmul__(v, **kwargs)[source]
Matrixvector or Matrixmatrix product, supporting batch dimensions.
If
K
is aLazyTensor
whose trailing dimensionK._shape[1]
is equal to 1, we can understand it as a linear operator and apply it to arbitrary NumPy arrays or PyTorch Tensors. Assuming thatv
is a 1D (resp. ND) tensor such thatK.shape[1] == v.shape[1]
(resp.v.shape[2]
),K @ v
denotes the matrixvector (resp. matrixmatrix) product between the two objects, encoded as a vanilla NumPy or PyTorch 1D (resp. ND) tensor.Example
>>> x, y = torch.randn(1000, 3), torch.randn(2000, 3) >>> x_i, y_j = LazyTensor( x[:,None,:] ), LazyTensor( y[None,:,:] ) >>> K = ( ((x_i  y_j)**2).sum(2) ).exp() # Symbolic (1000,2000,1) Gaussian kernel matrix >>> v = torch.rand(2000, 2) >>> print( (K @ v).shape ) ... torch.Size([1000, 2])
 t()[source]
Matrix transposition, permuting the axes of \(i\) and \(j\)variables.
For instance, if
K
is a LazyTensor of shape(B,M,N,D)
,K.t()
returns a symbolic copy ofK
whose axes 1 and 2 have been switched with each other:K.t().shape == (B,N,M,D)
.Example
>>> x, y = torch.randn(1000, 3), torch.randn(2000, 3) >>> x_i, y_j = LazyTensor( x[:,None,:] ), LazyTensor( y[None,:,:] ) >>> K = ( (( x_i  y_j )**2).sum(2) ).exp() # Symbolic (1000,2000) Gaussian kernel matrix >>> K_ = ( ((x[:,None,:]  y[None,:,:])**2).sum(2) ).exp() # Explicit (1000,2000) Gaussian kernel matrix >>> w = torch.rand(1000, 2) >>> print( (K.t() @ w  K_.t() @ w).abs().mean() ) ... tensor(1.7185e05)
 matvec(v)[source]
Alias for the matrixvector product, added for compatibility with
scipy.sparse.linalg
.If
K
is aLazyTensor
whose trailing dimensionK._shape[1]
is equal to 1, we can understand it as a linear operator and wrap it into ascipy.sparse.linalg.LinearOperator
object, thus getting access to robust solvers and spectral routines.Example
>>> import numpy as np >>> x = np.random.randn(1000,3) >>> x_i, x_j = LazyTensor( x[:,None,:] ), LazyTensor( x[None,:,:] ) >>> K_xx = ( ((x_i  x_j)**2).sum(2) ).exp() # Symbolic (1000,1000) Gaussian kernel matrix >>> from scipy.sparse.linalg import eigsh, aslinearoperator >>> eigenvalues, eigenvectors = eigsh( aslinearoperator( K_xx ), k=5 ) >>> print(eigenvalues) ... [ 35.5074527 59.01096445 61.35075268 69.34038814 123.77540277] >>> print( eigenvectors.shape) ... (1000, 5)
 rmatvec(v)[source]
Alias for the transposed matrixvector product, added for compatibility with
scipy.sparse.linalg
.See
matvec()
for further reference.
 real2complex()[source]
Elementwise “real 2 complex” operation  a unary operation.
x.real2complex()
returns aComplexLazyTensor
that encodes, symbolically, the same tensor asx
, but seen as complexvalued (with zero imaginary part for each coefficient)
 class pykeops.common.lazy_tensor.ComplexGenericLazyTensor(x=None, axis=None)[source]
Extension of the LazyTensor class for complex operations.
 property real[source]
Elementwise real part of complex  a unary operation.
z.real
returns aLazyTensor
that encodes, symbolically, the elementwise real part ofz
.
 property imag[source]
Elementwise imaginary part of complex  a unary operation.
z.imag
returns aLazyTensor
that encodes, symbolically, the elementwise imaginary part ofz
.
 angle()[source]
Elementwise angle (or argument) of complex  a unary operation.
z.angle()
returns aLazyTensor
that encodes, symbolically, the elementwise angle ofz
.
 conj()[source]
Elementwise complex conjugate  a unary operation.
z.conj()
returns aComplexLazyTensor
that encodes, symbolically, the elementwise complex conjugate ofz
.
 sum(axis=1, dim=None, **kwargs)[source]
Summation unary operation, or Sum reduction.
sum(axis, dim, **kwargs)
will:if axis or dim = 0, return the sum reduction of self over the “i” indexes.
if axis or dim = 1, return the sum reduction of self over the “j” indexes.
if axis or dim = 2, return a new
LazyTensor
object representing the sum of the values of the vector self,
 Keyword Arguments:
axis (integer) – reduction dimension, which should be equal to the number of batch dimensions plus 0 (= reduction over \(i\)), 1 (= reduction over \(j\)) or 2 (i.e. 1, sum along the dimension of the vector variable).
dim (integer) – alternative keyword for the axis parameter.
**kwargs – optional parameters that are passed to the
reduction()
method.
 __abs__()[source]
Elementwise absolute value (or modulus) of complex  a unary operation.
z.abs()
returns aLazyTensor
that encodes, symbolically, the elementwise absolute value ofz
.
 exp()[source]
Elementwise complex exponential  a unary operation.
z.exp()
returns aComplexLazyTensor
that encodes, symbolically, the elementwise complex exponential ofz
.
 real2complex()[source]
Elementwise “real 2 complex” operation  a unary operation.
x.real2complex()
returns aComplexLazyTensor
that encodes, symbolically, the same tensor asx
, but seen as complexvalued (with zero imaginary part for each coefficient)
 imag2complex()[source]
Elementwise “imag 2 complex” operation  a unary operation.
x.real2complex()
returns aComplexLazyTensor
that encodes, symbolically, the multiplication of1j
withx
.
 exp1j()[source]
Elementwise “complex exponential of 1j x” operation  a unary operation.
x.exp1j()
returns aComplexLazyTensor
that encodes, symbolically, the complex exponential of1j*x
.
 keops_tensordot(*args)[source]
Tensor dot product (on KeOps internal dimensions)  a binary operation.
 Parameters:
other – a
LazyTensor
dimfa – tuple/list of int
dimfb – tuple/list of int
contfa – tuple/list of int listing contraction dimension of a (could be empty)
contfb – tuple/list of int listing contraction dimension of b (could be empty)
args – a tuple/list of int containing the graph of a permutation of the output
 Returns:
a
LazyTensor
 __call__(*args, **kwargs)[source]
Executes a
Genred
orKernelSolve
call on the input data, as specified by self.formula .