Basic Tensor Functionality¶
Theano supports any kind of Python object, but its focus is support for symbolic matrix expressions. When you type,
>>> x = tt.fmatrix()
the x
is a TensorVariable
instance.
The tt.fmatrix
object itself is an instance of TensorType
.
Theano knows what type of variable x
is because x.type
points back to tt.fmatrix
.
This chapter explains the various ways of creating tensor variables,
the attributes and methods of TensorVariable
and TensorType
,
and various basic symbolic math and arithmetic that Theano supports for
tensor variables.
Creation¶
Theano provides a list of predefined tensor types that can be used
to create a tensor variables. Variables can be named to facilitate debugging,
and all of these constructors accept an optional name
argument.
For example, the following each produce a TensorVariable instance that stands
for a 0dimensional ndarray of integers with the name 'myvar'
:
>>> x = scalar('myvar', dtype='int32')
>>> x = iscalar('myvar')
>>> x = TensorType(dtype='int32', broadcastable=())('myvar')
Constructors with optional dtype¶
These are the simplest and oftenpreferred methods for creating symbolic
variables in your code. By default, they produce floatingpoint variables
(with dtype determined by config.floatX, see floatX
) so if you use
these constructors it is easy to switch your code between different levels of
floatingpoint precision.

theano.tensor.
scalar
(name=None, dtype=config.floatX)[source]¶ Return a Variable for a 0dimensional ndarray

theano.tensor.
vector
(name=None, dtype=config.floatX)[source]¶ Return a Variable for a 1dimensional ndarray

theano.tensor.
row
(name=None, dtype=config.floatX)[source]¶ Return a Variable for a 2dimensional ndarray in which the number of rows is guaranteed to be 1.

theano.tensor.
col
(name=None, dtype=config.floatX)[source]¶ Return a Variable for a 2dimensional ndarray in which the number of columns is guaranteed to be 1.

theano.tensor.
matrix
(name=None, dtype=config.floatX)[source]¶ Return a Variable for a 2dimensional ndarray

theano.tensor.
tensor3
(name=None, dtype=config.floatX)[source]¶ Return a Variable for a 3dimensional ndarray

theano.tensor.
tensor4
(name=None, dtype=config.floatX)[source]¶ Return a Variable for a 4dimensional ndarray

theano.tensor.
tensor5
(name=None, dtype=config.floatX)[source]¶ Return a Variable for a 5dimensional ndarray
All FullyTyped Constructors¶
The following TensorType instances are provided in the theano.tensor module.
They are all callable, and accept an optional name
argument. So for example:
from theano.tensor import *
x = dmatrix() # creates one Variable with no name
x = dmatrix('x') # creates one Variable with name 'x'
xyz = dmatrix('xyz') # creates one Variable with name 'xyz'
Constructor  dtype  ndim  shape  broadcastable 

bscalar  int8  0  ()  () 
bvector  int8  1  (?,)  (False,) 
brow  int8  2  (1,?)  (True, False) 
bcol  int8  2  (?,1)  (False, True) 
bmatrix  int8  2  (?,?)  (False, False) 
btensor3  int8  3  (?,?,?)  (False, False, False) 
btensor4  int8  4  (?,?,?,?)  (False, False, False, False) 
btensor5  int8  5  (?,?,?,?,?)  (False, False, False, False, False) 
btensor6  int8  6  (?,?,?,?,?,?)  (False,) * 6 
btensor7  int8  7  (?,?,?,?,?,?,?)  (False,) * 7 
wscalar  int16  0  ()  () 
wvector  int16  1  (?,)  (False,) 
wrow  int16  2  (1,?)  (True, False) 
wcol  int16  2  (?,1)  (False, True) 
wmatrix  int16  2  (?,?)  (False, False) 
wtensor3  int16  3  (?,?,?)  (False, False, False) 
wtensor4  int16  4  (?,?,?,?)  (False, False, False, False) 
wtensor5  int16  5  (?,?,?,?,?)  (False, False, False, False, False) 
wtensor6  int16  6  (?,?,?,?,?,?)  (False,) * 6 
wtensor7  int16  7  (?,?,?,?,?,?,?)  (False,) * 7 
iscalar  int32  0  ()  () 
ivector  int32  1  (?,)  (False,) 
irow  int32  2  (1,?)  (True, False) 
icol  int32  2  (?,1)  (False, True) 
imatrix  int32  2  (?,?)  (False, False) 
itensor3  int32  3  (?,?,?)  (False, False, False) 
itensor4  int32  4  (?,?,?,?)  (False, False, False, False) 
itensor5  int32  5  (?,?,?,?,?)  (False, False, False, False, False) 
itensor6  int32  6  (?,?,?,?,?,?)  (False,) * 6 
itensor7  int32  7  (?,?,?,?,?,?,?)  (False,) * 7 
lscalar  int64  0  ()  () 
lvector  int64  1  (?,)  (False,) 
lrow  int64  2  (1,?)  (True, False) 
lcol  int64  2  (?,1)  (False, True) 
lmatrix  int64  2  (?,?)  (False, False) 
ltensor3  int64  3  (?,?,?)  (False, False, False) 
ltensor4  int64  4  (?,?,?,?)  (False, False, False, False) 
ltensor5  int64  5  (?,?,?,?,?)  (False, False, False, False, False) 
ltensor6  int64  6  (?,?,?,?,?,?)  (False,) * 6 
ltensor7  int64  7  (?,?,?,?,?,?,?)  (False,) * 7 
dscalar  float64  0  ()  () 
dvector  float64  1  (?,)  (False,) 
drow  float64  2  (1,?)  (True, False) 
dcol  float64  2  (?,1)  (False, True) 
dmatrix  float64  2  (?,?)  (False, False) 
dtensor3  float64  3  (?,?,?)  (False, False, False) 
dtensor4  float64  4  (?,?,?,?)  (False, False, False, False) 
dtensor5  float64  5  (?,?,?,?,?)  (False, False, False, False, False) 
dtensor6  float64  6  (?,?,?,?,?,?)  (False,) * 6 
dtensor7  float64  7  (?,?,?,?,?,?,?)  (False,) * 7 
fscalar  float32  0  ()  () 
fvector  float32  1  (?,)  (False,) 
frow  float32  2  (1,?)  (True, False) 
fcol  float32  2  (?,1)  (False, True) 
fmatrix  float32  2  (?,?)  (False, False) 
ftensor3  float32  3  (?,?,?)  (False, False, False) 
ftensor4  float32  4  (?,?,?,?)  (False, False, False, False) 
ftensor5  float32  5  (?,?,?,?,?)  (False, False, False, False, False) 
ftensor6  float32  6  (?,?,?,?,?,?)  (False,) * 6 
ftensor7  float32  7  (?,?,?,?,?,?,?)  (False,) * 7 
cscalar  complex64  0  ()  () 
cvector  complex64  1  (?,)  (False,) 
crow  complex64  2  (1,?)  (True, False) 
ccol  complex64  2  (?,1)  (False, True) 
cmatrix  complex64  2  (?,?)  (False, False) 
ctensor3  complex64  3  (?,?,?)  (False, False, False) 
ctensor4  complex64  4  (?,?,?,?)  (False, False, False, False) 
ctensor5  complex64  5  (?,?,?,?,?)  (False, False, False, False, False) 
ctensor6  complex64  6  (?,?,?,?,?,?)  (False,) * 6 
ctensor7  complex64  7  (?,?,?,?,?,?,?)  (False,) * 7 
zscalar  complex128  0  ()  () 
zvector  complex128  1  (?,)  (False,) 
zrow  complex128  2  (1,?)  (True, False) 
zcol  complex128  2  (?,1)  (False, True) 
zmatrix  complex128  2  (?,?)  (False, False) 
ztensor3  complex128  3  (?,?,?)  (False, False, False) 
ztensor4  complex128  4  (?,?,?,?)  (False, False, False, False) 
ztensor5  complex128  5  (?,?,?,?,?)  (False, False, False, False, False) 
ztensor6  complex128  6  (?,?,?,?,?,?)  (False,) * 6 
ztensor7  complex128  7  (?,?,?,?,?,?,?)  (False,) * 7 
Plural Constructors¶
There are several constructors that can produce multiple variables at once. These are not frequently used in practice, but often used in tutorial examples to save space!

iscalars, lscalars, fscalars, dscalars
Return one or more scalar variables.

ivectors, lvectors, fvectors, dvectors
Return one or more vector variables.

irows, lrows, frows, drows
Return one or more row variables.

icols, lcols, fcols, dcols
Return one or more col variables.

imatrices, lmatrices, fmatrices, dmatrices
Return one or more matrix variables.
Each of these plural constructors accepts an integer or several strings. If an integer is provided, the method will return that many Variables and if strings are provided, it will create one Variable for each string, using the string as the Variable’s name. For example:
from theano.tensor import *
x, y, z = dmatrices(3) # creates three matrix Variables with no names
x, y, z = dmatrices('x', 'y', 'z') # creates three matrix Variables named 'x', 'y' and 'z'
Custom tensor types¶
If you would like to construct a tensor variable with a nonstandard
broadcasting pattern, or a larger number of dimensions you’ll need to create
your own TensorType
instance. You create such an instance by passing
the dtype and broadcasting pattern to the constructor. For example, you
can create your own 8dimensional tensor type
>>> dtensor8 = TensorType('float64', (False,)*8)
>>> x = dtensor8()
>>> z = dtensor8('z')
You can also redefine some of the provided types and they will interact correctly:
>>> my_dmatrix = TensorType('float64', (False,)*2)
>>> x = my_dmatrix() # allocate a matrix variable
>>> my_dmatrix == dmatrix
True
See TensorType
for more information about creating new types of
Tensor.
Converting from Python Objects¶
Another way of creating a TensorVariable (a TensorSharedVariable to be
precise) is by calling shared()
x = shared(numpy.random.randn(3,4))
This will return a shared variable whose .value
is
a numpy ndarray. The number of dimensions and dtype of the Variable are
inferred from the ndarray argument. The argument to shared will not be
copied, and subsequent changes will be reflected in x.value
.
For additional information, see the shared()
documentation.
Finally, when you use a numpy ndarray or a Python number together with
TensorVariable
instances in arithmetic expressions, the result is a
TensorVariable
. What happens to the ndarray or the number?
Theano requires that the inputs to all expressions be Variable instances, so
Theano automatically wraps them in a TensorConstant
.
Note
Theano makes a copy of any ndarray that you use in an expression, so subsequent changes to that ndarray will not have any effect on the Theano expression.
For numpy ndarrays the dtype is given, but the broadcastable pattern must be
inferred. The TensorConstant is given a type with a matching dtype,
and a broadcastable pattern with a True
for every shape dimension that is 1.
For python numbers, the broadcastable pattern is ()
but the dtype must be
inferred. Python integers are stored in the smallest dtype that can hold
them, so small constants like 1
are stored in a bscalar
.
Likewise, Python floats are stored in an fscalar if fscalar suffices to hold
them perfectly, but a dscalar otherwise.
Note
When config.floatX==float32 (see config
), then Python floats
are stored instead as singleprecision floats.
For fine control of this rounding policy, see theano.tensor.basic.autocast_float.

theano.tensor.
as_tensor_variable
(x, name=None, ndim=None)[source]¶ Turn an argument x into a TensorVariable or TensorConstant.
Many tensor Ops run their arguments through this function as preprocessing. It passes through TensorVariable instances, and tries to wrap other objects into TensorConstant.
When x is a Python number, the dtype is inferred as described above.
When x is a list or tuple it is passed through numpy.asarray
If the ndim argument is not None, it must be an integer and the output will be broadcasted if necessary in order to have this many dimensions.
Return type: TensorVariable
orTensorConstant
TensorType and TensorVariable¶

class
theano.tensor.
TensorType
(Type)[source]¶ The Type class used to mark Variables that stand for numpy.ndarray values (numpy.memmap, which is a subclass of numpy.ndarray, is also allowed). Recalling to the tutorial, the purple box in the tutorial’s graphstructure figure is an instance of this class.

broadcastable
[source]¶ A tuple of True/False values, one for each dimension. True in position ‘i’ indicates that at evaluationtime, the ndarray will have size 1 in that ‘i’th dimension. Such a dimension is called a broadcastable dimension (see Broadcasting).
The broadcastable pattern indicates both the number of dimensions and whether a particular dimension must have length 1.
Here is a table mapping some broadcastable patterns to what they mean:
pattern interpretation [] scalar [True] 1D scalar (vector of length 1) [True, True] 2D scalar (1x1 matrix) [False] vector [False, False] matrix [False] * n nD tensor [True, False] row (1xN matrix) [False, True] column (Mx1 matrix) [False, True, False] A Mx1xP tensor (a) [True, False, False] A 1xNxP tensor (b) [False, False, False] A MxNxP tensor (pattern of a + b) For dimensions in which broadcasting is False, the length of this dimension can be 1 or more. For dimensions in which broadcasting is True, the length of this dimension must be 1.
When two arguments to an elementwise operation (like addition or subtraction) have a different number of dimensions, the broadcastable pattern is expanded to the left, by padding with
True
. For example, a vector’s pattern,[False]
, could be expanded to[True, False]
, and would behave like a row (1xN matrix). In the same way, a matrix ([False, False]
) would behave like a 1xNxP tensor ([True, False, False]
).If we wanted to create a type representing a matrix that would broadcast over the middle dimension of a 3dimensional tensor when adding them together, we would define it like this:
>>> middle_broadcaster = TensorType('complex64', [False, True, False])

ndim
[source]¶ The number of dimensions that a Variable’s value will have at evaluationtime. This must be known when we are building the expression graph.

dtype
[source]¶ A string indicating the numerical type of the ndarray for which a Variable of this Type is standing.
The dtype attribute of a TensorType instance can be any of the following strings.
dtype domain bits 'int8'
signed integer 8 'int16'
signed integer 16 'int32'
signed integer 32 'int64'
signed integer 64 'uint8'
unsigned integer 8 'uint16'
unsigned integer 16 'uint32'
unsigned integer 32 'uint64'
unsigned integer 64 'float32'
floating point 32 'float64'
floating point 64 'complex64'
complex 64 (two float32) 'complex128'
complex 128 (two float64)

__init__
(self, dtype, broadcastable)[source]¶ If you wish to use a type of tensor which is not already available (for example, a 5D tensor) you can build an appropriate type by instantiating
TensorType
.

TensorVariable¶

class
theano.tensor.
TensorVariable
(Variable, _tensor_py_operators)[source]¶ The result of symbolic operations typically have this type.
See
_tensor_py_operators
for most of the attributes and methods you’ll want to call.

class
theano.tensor.
TensorConstant
(Variable, _tensor_py_operators)[source]¶ Python and numpy numbers are wrapped in this type.
See
_tensor_py_operators
for most of the attributes and methods you’ll want to call.
This type is returned by
shared()
when the value to share is a numpy ndarray.See
_tensor_py_operators
for most of the attributes and methods you’ll want to call.
Shaping and Shuffling¶
To reorder the dimensions of a variable, to insert or remove broadcastable
dimensions, see _tensor_py_operators.dimshuffle()
.

theano.tensor.
reshape
(x, newshape, ndim=None)[source]¶ Parameters:  x (any TensorVariable (or compatible)) – variable to be reshaped
 newshape (lvector (or compatible)) – the new shape for x
 ndim – optional  the length that newshape’s value will have.
If this is
None
, then reshape() will infer it from newshape.
Return type: variable with x’s dtype, but ndim dimensions
Note
This function can infer the length of a symbolic newshape in some cases, but if it cannot and you do not provide the ndim, then this function will raise an Exception.

theano.tensor.
shape_padleft
(x, n_ones=1)[source]¶ Reshape x by left padding the shape with n_ones 1s. Note that all this new dimension will be broadcastable. To make them nonbroadcastable see the
unbroadcast()
.Parameters: x (any TensorVariable (or compatible)) – variable to be reshaped

theano.tensor.
shape_padright
(x, n_ones=1)[source]¶ Reshape x by right padding the shape with n_ones 1s. Note that all this new dimension will be broadcastable. To make them nonbroadcastable see the
unbroadcast()
.Parameters: x (any TensorVariable (or compatible)) – variable to be reshaped

theano.tensor.
shape_padaxis
(t, axis)[source]¶ Reshape t by inserting 1 at the dimension axis. Note that this new dimension will be broadcastable. To make it nonbroadcastable see the
unbroadcast()
.Parameters:  x (any TensorVariable (or compatible)) – variable to be reshaped
 axis (int) – axis where to add the new dimension to x
Example:
>>> tensor = theano.tensor.type.tensor3() >>> theano.tensor.shape_padaxis(tensor, axis=0) InplaceDimShuffle{x,0,1,2}.0 >>> theano.tensor.shape_padaxis(tensor, axis=1) InplaceDimShuffle{0,x,1,2}.0 >>> theano.tensor.shape_padaxis(tensor, axis=3) InplaceDimShuffle{0,1,2,x}.0 >>> theano.tensor.shape_padaxis(tensor, axis=1) InplaceDimShuffle{0,1,2,x}.0

theano.tensor.
unbroadcast
(x, *axes)[source]¶ Make the input impossible to broadcast in the specified axes.
For example, addbroadcast(x, 0) will make the first dimension of x broadcastable. When performing the function, if the length of x along that dimension is not 1, a ValueError will be raised.
We apply the opt here not to pollute the graph especially during the gpu optimization
Parameters:  x (tensor_like) – Input theano tensor.
 axis (an int or an iterable object such as list or tuple of int values) – The dimension along which the tensor x should be unbroadcastable. If the length of x along these dimensions is not 1, a ValueError will be raised.
Returns: A theano tensor, which is unbroadcastable along the specified dimensions.
Return type: tensor

theano.tensor.
addbroadcast
(x, *axes)[source]¶ Make the input broadcastable in the specified axes.
For example, addbroadcast(x, 0) will make the first dimension of x broadcastable. When performing the function, if the length of x along that dimension is not 1, a ValueError will be raised.
We apply the opt here not to pollute the graph especially during the gpu optimization
Parameters:  x (tensor_like) – Input theano tensor.
 axis (an int or an iterable object such as list or tuple of int values) – The dimension along which the tensor x should be broadcastable. If the length of x along these dimensions is not 1, a ValueError will be raised.
Returns: A theano tensor, which is broadcastable along the specified dimensions.
Return type: tensor

theano.tensor.
patternbroadcast
(x, broadcastable)[source]¶ Make the input adopt a specific broadcasting pattern.
Broadcastable must be iterable. For example, patternbroadcast(x, (True, False)) will make the first dimension of x broadcastable and the second dimension not broadcastable, so x will now be a row.
We apply the opt here not to pollute the graph especially during the gpu optimization.
Parameters:  x (tensor_like) – Input theano tensor.
 broadcastable (an iterable object such as list or tuple of bool values) – A set of boolean values indicating whether a dimension should be broadcastable or not. If the length of x along these dimensions is not 1, a ValueError will be raised.
Returns: A theano tensor, which is unbroadcastable along the specified dimensions.
Return type: tensor

theano.tensor.
flatten
(x, ndim=1)[source]¶ Similar to
reshape()
, but the shape is inferred from the shape of x.Parameters:  x (any TensorVariable (or compatible)) – variable to be flattened
 ndim (int) – the number of dimensions in the returned variable
Return type: variable with same dtype as x and ndim dimensions
Returns: variable with the same shape as x in the leading ndim1 dimensions, but with all remaining dimensions of x collapsed into the last dimension.
For example, if we flatten a tensor of shape (2, 3, 4, 5) with flatten(x, ndim=2), then we’ll have the same (21=1) leading dimensions (2,), and the remaining dimensions are collapsed. So the output in this example would have shape (2, 60).

theano.tensor.
tile
(x, reps, ndim=None)[source]¶ Construct an array by repeating the input x according to reps pattern.
Tiles its input according to reps. The length of reps is the number of dimension of x and contains the number of times to tile x in each dimension.
See: numpy.tile documentation for examples. See: theano.tensor.extra_ops.repeat
Note: Currently, reps must be a constant, x.ndim and len(reps) must be equal and, if specified, ndim must be equal to both.

theano.tensor.
roll
(x, shift, axis=None)[source]¶ Convenience function to roll TensorTypes along the given axis.
Syntax copies numpy.roll function.
Parameters:  x (tensor_like) – Input tensor.
 shift (int (symbolic or literal)) – The number of places by which elements are shifted.
 axis (int (symbolic or literal), optional) – The axis along which elements are shifted. By default, the array is flattened before shifting, after which the original shape is restored.
Returns: Output tensor, with the same shape as
x
.Return type: tensor
Creating Tensor¶

theano.tensor.
zeros_like
(x, dtype=None)[source]¶ Parameters:  x – tensor that has the same shape as output
 dtype – datatype, optional By default, it will be x.dtype.
Returns a tensor the shape of x filled with zeros of the type of dtype.

theano.tensor.
ones_like
(x)[source]¶ Parameters:  x – tensor that has the same shape as output
 dtype – datatype, optional By default, it will be x.dtype.
Returns a tensor the shape of x filled with ones of the type of dtype.

theano.tensor.
zeros
(shape, dtype=None)[source]¶ Parameters:  shape – a tuple/list of scalar with the shape information.
 dtype – the dtype of the new tensor. If None, will use floatX.
Returns a tensor filled with 0s of the provided shape.

theano.tensor.
ones
(shape, dtype=None)[source]¶ Parameters:  shape – a tuple/list of scalar with the shape information.
 dtype – the dtype of the new tensor. If None, will use floatX.
Returns a tensor filled with 1s of the provided shape.

theano.tensor.
fill
(a, b)[source]¶ Parameters:  a – tensor that has same shape as output
 b – theano scalar or value with which you want to fill the output
Create a matrix by filling the shape of a with b

theano.tensor.
alloc
(value, *shape)[source]¶ Parameters:  value – a value with which to fill the output
 shape – the dimensions of the returned array
Returns: an Ndimensional tensor initialized by value and having the specified shape.

theano.tensor.
eye
(n, m=None, k=0, dtype=theano.config.floatX)[source]¶ Parameters:  n – number of rows in output (value or theano scalar)
 m – number of columns in output (value or theano scalar)
 k – Index of the diagonal: 0 refers to the main diagonal, a positive value refers to an upper diagonal, and a negative value to a lower diagonal. It can be a theano scalar.
Returns: An array where all elements are equal to zero, except for the kth diagonal, whose values are equal to one.

theano.tensor.
identity_like
(x)[source]¶ Parameters: x – tensor Returns: A tensor of same shape as x that is filled with 0s everywhere except for the main diagonal, whose values are equal to one. The output will have same dtype as x.

theano.tensor.
stack
(tensors, axis=0)[source]¶ Stack tensors in sequence on given axis (default is 0).
Take a sequence of tensors and stack them on given axis to make a single tensor. The size in dimension axis of the result will be equal to the number of tensors passed.
Parameters:  tensors – a list or a tuple of one or more tensors of the same rank.
 axis – the axis along which the tensors will be stacked. Default value is 0.
Returns: A tensor such that rval[0] == tensors[0], rval[1] == tensors[1], etc.
Examples:
>>> a = theano.tensor.type.scalar() >>> b = theano.tensor.type.scalar() >>> c = theano.tensor.type.scalar() >>> x = theano.tensor.stack([a, b, c]) >>> x.ndim # x is a vector of length 3. 1 >>> a = theano.tensor.type.tensor4() >>> b = theano.tensor.type.tensor4() >>> c = theano.tensor.type.tensor4() >>> x = theano.tensor.stack([a, b, c]) >>> x.ndim # x is a 5d tensor. 5 >>> rval = x.eval(dict((t, np.zeros((2, 2, 2, 2))) for t in [a, b, c])) >>> rval.shape # 3 tensors are stacked on axis 0 (3, 2, 2, 2, 2)
We can also specify different axis than default value 0
>>> x = theano.tensor.stack([a, b, c], axis=3) >>> x.ndim 5 >>> rval = x.eval(dict((t, np.zeros((2, 2, 2, 2))) for t in [a, b, c])) >>> rval.shape # 3 tensors are stacked on axis 3 (2, 2, 2, 3, 2) >>> x = theano.tensor.stack([a, b, c], axis=2) >>> x.ndim 5 >>> rval = x.eval(dict((t, np.zeros((2, 2, 2, 2))) for t in [a, b, c])) >>> rval.shape # 3 tensors are stacked on axis 2 (2, 2, 2, 3, 2)

theano.tensor.
stack
(*tensors)[source]¶ Warning
The interface stack(*tensors) is deprecated! Use stack(tensors, axis=0) instead.
Stack tensors in sequence vertically (row wise).
Take a sequence of tensors and stack them vertically to make a single tensor.
Parameters: tensors – one or more tensors of the same rank Returns: A tensor such that rval[0] == tensors[0], rval[1] == tensors[1], etc. >>> x0 = tt.scalar() >>> x1 = tt.scalar() >>> x2 = tt.scalar() >>> x = tt.stack(x0, x1, x2) >>> x.ndim # x is a vector of length 3. 1

theano.tensor.
concatenate
(tensor_list, axis=0)[source]¶ Parameters:  tensor_list (a list or tuple of Tensors that all have the same shape in the axes not specified by the axis argument.) – one or more Tensors to be concatenated together into one.
 axis (literal or symbolic integer) – Tensors will be joined along this axis, so they may have different
shape[axis]
>>> x0 = tt.fmatrix() >>> x1 = tt.ftensor3() >>> x2 = tt.fvector() >>> x = tt.concatenate([x0, x1[0], tt.shape_padright(x2)], axis=1) >>> x.ndim 2

theano.tensor.
stacklists
(tensor_list)[source]¶ Parameters: tensor_list (an iterable that contains either tensors or other iterables of the same type as tensor_list (in other words, this is a tree whose leaves are tensors).) – tensors to be stacked together. Recursively stack lists of tensors to maintain similar structure.
This function can create a tensor from a shaped list of scalars:
>>> from theano.tensor import stacklists, scalars, matrices >>> from theano import function >>> a, b, c, d = scalars('abcd') >>> X = stacklists([[a, b], [c, d]]) >>> f = function([a, b, c, d], X) >>> f(1, 2, 3, 4) array([[ 1., 2.], [ 3., 4.]])
We can also stack arbitrarily shaped tensors. Here we stack matrices into a 2 by 2 grid:
>>> from numpy import ones >>> a, b, c, d = matrices('abcd') >>> X = stacklists([[a, b], [c, d]]) >>> f = function([a, b, c, d], X) >>> x = ones((4, 4), 'float32') >>> f(x, x, x, x).shape (2, 2, 4, 4)

theano.tensor.basic.
choose
(a, choices, out=None, mode='raise')[source]¶ Construct an array from an index array and a set of arrays to choose from.
First of all, if confused or uncertain, definitely look at the Examples  in its full generality, this function is less simple than it might seem from the following code description (below ndi = numpy.lib.index_tricks):
np.choose(a,c) == np.array([c[a[I]][I] for I in ndi.ndindex(a.shape)]).
But this omits some subtleties. Here is a fully general summary:
Given an
index
array (a) of integers and a sequence of n arrays (choices), a and each choice array are first broadcast, as necessary, to arrays of a common shape; calling these Ba and Bchoices[i], i = 0,…,n1 we have that, necessarily, Ba.shape == Bchoices[i].shape for each i. Then, a new array with shape Ba.shape is created as follows: if mode=raise (the default), then, first of all, each element of a (and thus Ba) must be in the range [0, n1]; now, suppose that i (in that range) is the value at the (j0, j1, …, jm) position in Ba  then the value at the same position in the new array is the value in Bchoices[i] at that same position;
 if mode=wrap, values in a (and thus Ba) may be any (signed) integer; modular arithmetic is used to map integers outside the range [0, n1] back into that range; and then the new array is constructed as above;
 if mode=clip, values in a (and thus Ba) may be any (signed) integer; negative integers are mapped to 0; values greater than n1 are mapped to n1; and then the new array is constructed as above.
Parameters:  a (int array) – This array must contain integers in [0, n1], where n is the number of choices, unless mode=wrap or mode=clip, in which cases any integers are permissible.
 choices (sequence of arrays) – Choice arrays. a and all of the choices must be broadcastable to
the same shape. If choices is itself an array (not recommended),
then its outermost dimension (i.e., the one corresponding to
choices.shape[0]) is taken as defining the
sequence
.  out (array, optional) – If provided, the result will be inserted into this array. It should be of the appropriate shape and dtype.
 mode ({
raise
(default),wrap
,clip
}, optional) – Specifies how indices outside [0, n1] will be treated:raise
: an exception is raisedwrap
: value becomes value mod nclip
: values < 0 are mapped to 0, values > n1 are mapped to n1
Returns: The merged result.
Return type: merged_array  array
Raises: ValueError  shape mismatch – If a and each choice array are not all broadcastable to the same shape.
Reductions¶

theano.tensor.
max
(x, axis=None, keepdims=False)[source]¶ Parameter: x  symbolic Tensor (or compatible) Parameter: axis  axis or axes along which to compute the maximum Parameter: keepdims  (boolean) If this is set to True, the axes which are reduced are left in the result as dimensions with size one. With this option, the result will broadcast correctly against the original tensor. Returns: maximum of x along axis  axis can be:
 None  in which case the maximum is computed along all axes (like numpy)
 an int  computed along this axis
 a list of ints  computed along these axes

theano.tensor.
argmax
(x, axis=None, keepdims=False)[source]¶ Parameter: x  symbolic Tensor (or compatible) Parameter: axis  axis along which to compute the index of the maximum Parameter: keepdims  (boolean) If this is set to True, the axis which is reduced is left in the result as a dimension with size one. With this option, the result will broadcast correctly against the original tensor. Returns: the index of the maximum value along a given axis  if axis=None, Theano 0.5rc1 or later: argmax over the flattened tensor (like numpy)
 older: then axis is assumed to be ndim(x)1

theano.tensor.
max_and_argmax
(x, axis=None, keepdims=False)[source]¶ Parameter: x  symbolic Tensor (or compatible) Parameter: axis  axis along which to compute the maximum and its index Parameter: keepdims  (boolean) If this is set to True, the axis which is reduced is left in the result as a dimension with size one. With this option, the result will broadcast correctly against the original tensor. Returns: the maximum value along a given axis and its index.  if axis=None, Theano 0.5rc1 or later: max_and_argmax over the flattened tensor (like numpy)
 older: then axis is assumed to be ndim(x)1

theano.tensor.
min
(x, axis=None, keepdims=False)[source]¶ Parameter: x  symbolic Tensor (or compatible) Parameter: axis  axis or axes along which to compute the minimum Parameter: keepdims  (boolean) If this is set to True, the axes which are reduced are left in the result as dimensions with size one. With this option, the result will broadcast correctly against the original tensor. Returns: minimum of x along axis  axis can be:
 None  in which case the minimum is computed along all axes (like numpy)
 an int  computed along this axis
 a list of ints  computed along these axes

theano.tensor.
argmin
(x, axis=None, keepdims=False)[source]¶ Parameter: x  symbolic Tensor (or compatible) Parameter: axis  axis along which to compute the index of the minimum Parameter: keepdims  (boolean) If this is set to True, the axes which are reduced are left in the result as dimensions with size one. With this option, the result will broadcast correctly against the original tensor. Returns: the index of the minimum value along a given axis  if axis=None, Theano 0.5rc1 or later: argmin over the flattened tensor (like numpy)
 older: then axis is assumed to be ndim(x)1

theano.tensor.
sum
(x, axis=None, dtype=None, keepdims=False, acc_dtype=None)[source]¶ Parameter: x  symbolic Tensor (or compatible)
Parameter: axis  axis or axes along which to compute the sum
Parameter: dtype  The dtype of the returned tensor. If None, then we use the default dtype which is the same as the input tensor’s dtype except when:
 the input dtype is a signed integer of precision < 64 bit, in which case we use int64
 the input dtype is an unsigned integer of precision < 64 bit, in which case we use uint64
This default dtype does _not_ depend on the value of “acc_dtype”.
Parameter: keepdims  (boolean) If this is set to True, the axes which are reduced are left in the result as dimensions with size one. With this option, the result will broadcast correctly against the original tensor.
Parameter: acc_dtype  The dtype of the internal accumulator. If None (default), we use the dtype in the list below, or the input dtype if its precision is higher:
 for int dtypes, we use at least int64;
 for uint dtypes, we use at least uint64;
 for float dtypes, we use at least float64;
 for complex dtypes, we use at least complex128.
Returns: sum of x along axis
 axis can be:
 None  in which case the sum is computed along all axes (like numpy)
 an int  computed along this axis
 a list of ints  computed along these axes

theano.tensor.
prod
(x, axis=None, dtype=None, keepdims=False, acc_dtype=None, no_zeros_in_input=False)[source]¶ Parameter: x  symbolic Tensor (or compatible)
Parameter: axis  axis or axes along which to compute the product
Parameter: dtype  The dtype of the returned tensor. If None, then we use the default dtype which is the same as the input tensor’s dtype except when:
 the input dtype is a signed integer of precision < 64 bit, in which case we use int64
 the input dtype is an unsigned integer of precision < 64 bit, in which case we use uint64
This default dtype does _not_ depend on the value of “acc_dtype”.
Parameter: keepdims  (boolean) If this is set to True, the axes which are reduced are left in the result as dimensions with size one. With this option, the result will broadcast correctly against the original tensor.
Parameter: acc_dtype  The dtype of the internal accumulator. If None (default), we use the dtype in the list below, or the input dtype if its precision is higher:
 for int dtypes, we use at least int64;
 for uint dtypes, we use at least uint64;
 for float dtypes, we use at least float64;
 for complex dtypes, we use at least complex128.
Parameter: no_zeros_in_input  The grad of prod is complicated as we need to handle 3 different cases: without zeros in the input reduced group, with 1 zero or with more zeros.
This could slow you down, but more importantly, we currently don’t support the second derivative of the 3 cases. So you cannot take the second derivative of the default prod().
To remove the handling of the special cases of 0 and so get some small speed up and allow second derivative set
no_zeros_in_inputs
toTrue
. It defaults toFalse
.It is the user responsibility to make sure there are no zeros in the inputs. If there are, the grad will be wrong.
Returns: product of every term in x along axis
 axis can be:
 None  in which case the sum is computed along all axes (like numpy)
 an int  computed along this axis
 a list of ints  computed along these axes

theano.tensor.
mean
(x, axis=None, dtype=None, keepdims=False, acc_dtype=None)[source]¶ Parameter: x  symbolic Tensor (or compatible) Parameter: axis  axis or axes along which to compute the mean Parameter: dtype  The dtype to cast the result of the inner summation into. For instance, by default, a sum of a float32 tensor will be done in float64 (acc_dtype would be float64 by default), but that result will be casted back in float32. Parameter: keepdims  (boolean) If this is set to True, the axes which are reduced are left in the result as dimensions with size one. With this option, the result will broadcast correctly against the original tensor. Parameter: acc_dtype  The dtype of the internal accumulator of the inner summation. This will not necessarily be the dtype of the output (in particular if it is a discrete (int/uint) dtype, the output will be in a float type). If None, then we use the same rules as sum()
.Returns: mean value of x along axis  axis can be:
 None  in which case the mean is computed along all axes (like numpy)
 an int  computed along this axis
 a list of ints  computed along these axes

theano.tensor.
var
(x, axis=None, keepdims=False)[source]¶ Parameter: x  symbolic Tensor (or compatible) Parameter: axis  axis or axes along which to compute the variance Parameter: keepdims  (boolean) If this is set to True, the axes which are reduced are left in the result as dimensions with size one. With this option, the result will broadcast correctly against the original tensor. Returns: variance of x along axis  axis can be:
 None  in which case the variance is computed along all axes (like numpy)
 an int  computed along this axis
 a list of ints  computed along these axes

theano.tensor.
std
(x, axis=None, keepdims=False)[source]¶ Parameter: x  symbolic Tensor (or compatible) Parameter: axis  axis or axes along which to compute the standard deviation Parameter: keepdims  (boolean) If this is set to True, the axes which are reduced are left in the result as dimensions with size one. With this option, the result will broadcast correctly against the original tensor. Returns: variance of x along axis  axis can be:
 None  in which case the standard deviation is computed along all axes (like numpy)
 an int  computed along this axis
 a list of ints  computed along these axes

theano.tensor.
all
(x, axis=None, keepdims=False)[source]¶ Parameter: x  symbolic Tensor (or compatible) Parameter: axis  axis or axes along which to apply ‘bitwise and’ Parameter: keepdims  (boolean) If this is set to True, the axes which are reduced are left in the result as dimensions with size one. With this option, the result will broadcast correctly against the original tensor. Returns: bitwise and of x along axis  axis can be:
 None  in which case the ‘bitwise and’ is computed along all axes (like numpy)
 an int  computed along this axis
 a list of ints  computed along these axes

theano.tensor.
any
(x, axis=None, keepdims=False)[source]¶ Parameter: x  symbolic Tensor (or compatible) Parameter: axis  axis or axes along which to apply bitwise or Parameter: keepdims  (boolean) If this is set to True, the axes which are reduced are left in the result as dimensions with size one. With this option, the result will broadcast correctly against the original tensor. Returns: bitwise or of x along axis  axis can be:
 None  in which case the ‘bitwise or’ is computed along all axes (like numpy)
 an int  computed along this axis
 a list of ints  computed along these axes

theano.tensor.
ptp
(x, axis=None)[source]¶ Range of values (maximum  minimum) along an axis. The name of the function comes from the acronym for peak to peak.
Parameter: x Input tensor. Parameter: axis Axis along which to find the peaks. By default, flatten the array. Returns: A new array holding the result.
Indexing¶
Like NumPy, Theano distinguishes between basic and advanced indexing. Theano fully supports basic indexing (see NumPy’s indexing) and integer advanced indexing. Since version 0.10.0 Theano also supports boolean indexing with boolean NumPy arrays or Theano tensors.
Indexassignment is not supported. If you want to do something like a[5]
= b
or a[5]+=b
, see theano.tensor.subtensor.set_subtensor()
and
theano.tensor.subtensor.inc_subtensor()
below.

theano.tensor.subtensor.
set_subtensor
(x, y, inplace=False, tolerate_inplace_aliasing=False)[source]¶ Return x with the given subtensor overwritten by y.
Parameters:  x – Symbolic variable for the lvalue of = operation.
 y – Symbolic variable for the rvalue of = operation.
 tolerate_inplace_aliasing – See inc_subtensor for documentation.
Examples
To replicate the numpy expression “r[10:] = 5”, type
>>> r = ivector() >>> new_r = set_subtensor(r[10:], 5)

theano.tensor.subtensor.
inc_subtensor
(x, y, inplace=False, set_instead_of_inc=False, tolerate_inplace_aliasing=False)[source]¶ Return x with the given subtensor incremented by y.
Parameters:  x – The symbolic result of a Subtensor operation.
 y – The amount by which to increment the subtensor in question.
 inplace – Don’t use. Theano will do it when possible.
 set_instead_of_inc – If True, do a set_subtensor instead.
 tolerate_inplace_aliasing – Allow x and y to be views of a single underlying array even while working inplace. For correct results, x and y must not be overlapping views; if they overlap, the result of this Op will generally be incorrect. This value has no effect if inplace=False.
Examples
To replicate the numpy expression “r[10:] += 5”, type
>>> r = ivector() >>> new_r = inc_subtensor(r[10:], 5)
Operator Support¶
Many Python operators are supported.
>>> a, b = tt.itensor3(), tt.itensor3() # example inputs
Arithmetic¶
>>> a + 3 # tt.add(a, 3) > itensor3
>>> 3  a # tt.sub(3, a)
>>> a * 3.5 # tt.mul(a, 3.5) > ftensor3 or dtensor3 (depending on casting)
>>> 2.2 / a # tt.truediv(2.2, a)
>>> 2.2 // a # tt.intdiv(2.2, a)
>>> 2.2**a # tt.pow(2.2, a)
>>> b % a # tt.mod(b, a)
Bitwise¶
>>> a & b # tt.and_(a,b) bitwise and (alias tt.bitwise_and)
>>> a ^ 1 # tt.xor(a,1) bitwise xor (alias tt.bitwise_xor)
>>> a  b # tt.or_(a,b) bitwise or (alias tt.bitwise_or)
>>> ~a # tt.invert(a) bitwise invert (alias tt.bitwise_not)
Inplace¶
Inplace operators are not supported. Theano’s graphoptimizations
will determine which intermediate values to use for inplace
computations. If you would like to update the value of a
shared variable, consider using the updates
argument to
theano.function()
.
Elementwise¶
Casting¶

theano.tensor.
cast
(x, dtype)[source]¶ Cast any tensor x to a Tensor of the same shape, but with a different numerical type dtype.
This is not a reinterpret cast, but a coercion cast, similar to
numpy.asarray(x, dtype=dtype)
.import theano.tensor as tt x = tt.matrix() x_as_int = tt.cast(x, 'int32')
Attempting to casting a complex value to a real value is ambiguous and will raise an exception. Use real(), imag(), abs(), or angle().
Comparisons¶
 The six usual equality and inequality operators share the same interface.
Parameter: a  symbolic Tensor (or compatible) Parameter: b  symbolic Tensor (or compatible) Return type: symbolic Tensor Returns: a symbolic tensor representing the application of the logical elementwise operator. Note
Theano has no boolean dtype. Instead, all boolean tensors are represented in
'int8'
.Here is an example with the lessthan operator.
import theano.tensor as tt x,y = tt.dmatrices('x','y') z = tt.le(x,y)

theano.tensor.
lt
(a, b)[source]¶ Returns a symbolic
'int8'
tensor representing the result of logical lessthan (a<b).Also available using syntax
a < b

theano.tensor.
gt
(a, b)[source]¶ Returns a symbolic
'int8'
tensor representing the result of logical greaterthan (a>b).Also available using syntax
a > b

theano.tensor.
le
(a, b)[source]¶ Returns a variable representing the result of logical less than or equal (a<=b).
Also available using syntax
a <= b

theano.tensor.
ge
(a, b)[source]¶ Returns a variable representing the result of logical greater or equal than (a>=b).
Also available using syntax
a >= b

theano.tensor.
eq
(a, b)[source]¶ Returns a variable representing the result of logical equality (a==b).

theano.tensor.
neq
(a, b)[source]¶ Returns a variable representing the result of logical inequality (a!=b).

theano.tensor.
isnan
(a)[source]¶ Returns a variable representing the comparison of
a
elements with nan.This is equivalent to
numpy.isnan
.

theano.tensor.
isinf
(a)[source]¶ Returns a variable representing the comparison of
a
elements with inf or inf.This is equivalent to
numpy.isinf
.

theano.tensor.
isclose
(a, b, rtol=1e05, atol=1e08, equal_nan=False)[source]¶ Returns a symbolic
'int8'
tensor representing where two tensors are equal within a tolerance.The tolerance values are positive, typically very small numbers. The relative difference (rtol * abs(b)) and the absolute difference atol are added together to compare against the absolute difference between a and b.
For finite values, isclose uses the following equation to test whether two floating point values are equivalent:
a  b <= (atol + rtol * b)
For infinite values, isclose checks if both values are the same signed inf value.
If equal_nan is True, isclose considers NaN values in the same position to be close. Otherwise, NaN values are not considered close.
This is equivalent to
numpy.isclose
.
Condition¶

theano.tensor.
switch
(cond, ift, iff)[source]¶  Returns a variable representing a switch between ift (iftrue) and iff (iffalse)
based on the condition cond. This is the theano equivalent of numpy.where.
Parameter: cond  symbolic Tensor (or compatible) Parameter: ift  symbolic Tensor (or compatible) Parameter: iff  symbolic Tensor (or compatible) Return type: symbolic Tensor
import theano.tensor as tt a,b = tt.dmatrices('a','b') x,y = tt.dmatrices('x','y') z = tt.switch(tt.lt(a,b), x, y)

theano.tensor.
clip
(x, min, max)[source]¶ Return a variable representing x, but with all elements greater than max clipped to max and all elements less than min clipped to min.
Normal broadcasting rules apply to each of x, min, and max.
Note that there is no warning for inputs that are the wrong way round (min > max), and that results in this case may differ from
numpy.clip
.
Bitwise¶
 The bitwise operators possess this interface:
Parameter: a  symbolic Tensor of integer type. Parameter: b  symbolic Tensor of integer type. Note
The bitwise operators must have an integer type as input.
The bitwise not (invert) takes only one parameter.
Return type: symbolic Tensor with corresponding dtype.
Here is an example using the bitwise and_
via the &
operator:
import theano.tensor as tt
x,y = tt.imatrices('x','y')
z = x & y
Mathematical¶

theano.tensor.
abs_
(a)[source]¶ Returns a variable representing the absolute of a, ie
a
.Note
Can also be accessed with
abs(a)
.

theano.tensor.
angle
(a)[source]¶ Returns a variable representing angular component of complexvalued Tensor a.

theano.tensor.
maximum
(a, b)[source]¶ Returns a variable representing the maximum element by element of a and b

theano.tensor.
minimum
(a, b)[source]¶ Returns a variable representing the minimum element by element of a and b

theano.tensor.
inv
(a)[source]¶ Returns a variable representing the inverse of a, ie 1.0/a. Also called reciprocal.

theano.tensor.
log
(a), log2(a), log10(a)[source]¶ Returns a variable representing the base e, 2 or 10 logarithm of a.

theano.tensor.
ceil
(a)[source]¶ Returns a variable representing the ceiling of a (for example ceil(2.1) is 3).

theano.tensor.
floor
(a)[source]¶ Returns a variable representing the floor of a (for example floor(2.9) is 2).

theano.tensor.
round
(a, mode='half_away_from_zero')[source]¶ Returns a variable representing the rounding of a in the same dtype as a. Implemented rounding mode are half_away_from_zero and half_to_even.

theano.tensor.
iround
(a, mode='half_away_from_zero')[source]¶ Short hand for cast(round(a, mode),’int64’).

theano.tensor.
cos
(a), sin(a), tan(a)[source]¶ Returns a variable representing the trigonometric functions of a (cosine, sine and tangent).

theano.tensor.
cosh
(a), sinh(a), tanh(a)[source]¶ Returns a variable representing the hyperbolic trigonometric functions of a (hyperbolic cosine, sine and tangent).

theano.tensor.
erf
(a), erfc(a)[source]¶ Returns a variable representing the error function or the complementary error function. wikipedia

theano.tensor.
erfinv
(a), erfcinv(a)[source]¶ Returns a variable representing the inverse error function or the inverse complementary error function. wikipedia

theano.tensor.
gammaln
(a)[source]¶ Returns a variable representing the logarithm of the gamma function.

theano.tensor.
psi
(a)[source]¶ Returns a variable representing the derivative of the logarithm of the gamma function (also called the digamma function).

theano.tensor.
chi2sf
(a, df)[source]¶ Returns a variable representing the survival function (1cdf — sometimes more accurate).
C code is provided in the Theano_lgpl repository. This makes it faster.
You can find more information about Broadcasting in the Broadcasting tutorial.
Linear Algebra¶

theano.tensor.
dot
(X, Y)[source]¶  For 2D arrays it is equivalent to matrix multiplication, and for 1D arrays to inner product of vectors (without complex conjugation). For N dimensions it is a sum product over the last axis of a and the secondtolast of b:
Parameters:  X (symbolic tensor) – left term
 Y (symbolic tensor) – right term
Return type: symbolic matrix or vector
Returns: the inner product of X and Y.

theano.tensor.
outer
(X, Y)[source]¶ Parameters:  X (symbolic vector) – left term
 Y (symbolic vector) – right term
Return type: symbolic matrix
Returns: vectorvector outer product

theano.tensor.
tensordot
(a, b, axes=2)[source]¶ Given two tensors a and b,tensordot computes a generalized dot product over the provided axes. Theano’s implementation reduces all expressions to matrix or vector dot products and is based on code from Tijmen Tieleman’s gnumpy (http://www.cs.toronto.edu/~tijmen/gnumpy.html).
Parameters:  a (symbolic tensor) – the first tensor variable
 b (symbolic tensor) – the second tensor variable
 axes (int or arraylike of length 2) –
an integer or array. If an integer, the number of axes to sum over. If an array, it must have two array elements containing the axes to sum over in each tensor.
Note that the default value of 2 is not guaranteed to work for all values of a and b, and an error will be raised if that is the case. The reason for keeping the default is to maintain the same signature as numpy’s tensordot function (and np.tensordot raises analogous errors for noncompatible inputs).
If an integer i, it is converted to an array containing the last i dimensions of the first tensor and the first i dimensions of the second tensor:
axes = [range(a.ndim  i, b.ndim), range(i)]If an array, its two elements must contain compatible axes of the two tensors. For example, [[1, 2], [2, 0]] means sum over the 2nd and 3rd axes of a and the 3rd and 1st axes of b. (Remember axes are zeroindexed!) The 2nd axis of a and the 3rd axis of b must have the same shape; the same is true for the 3rd axis of a and the 1st axis of b.
Returns: a tensor with shape equal to the concatenation of a’s shape (less any dimensions that were summed over) and b’s shape (less any dimensions that were summed over).
Return type: symbolic tensor
It may be helpful to consider an example to see what tensordot does. Theano’s implementation is identical to NumPy’s. Here a has shape (2, 3, 4) and b has shape (5, 6, 4, 3). The axes to sum over are [[1, 2], [3, 2]] – note that a.shape[1] == b.shape[3] and a.shape[2] == b.shape[2]; these axes are compatible. The resulting tensor will have shape (2, 5, 6) – the dimensions that are not being summed:
import numpy as np a = np.random.random((2,3,4)) b = np.random.random((5,6,4,3)) #tensordot c = np.tensordot(a, b, [[1,2],[3,2]]) #loop replicating tensordot a0, a1, a2 = a.shape b0, b1, _, _ = b.shape cloop = np.zeros((a0,b0,b1)) #loop over nonsummed indices  these exist #in the tensor product. for i in range(a0): for j in range(b0): for k in range(b1): #loop over summed indices  these don't exist #in the tensor product. for l in range(a1): for m in range(a2): cloop[i,j,k] += a[i,l,m] * b[j,k,m,l] assert np.allclose(c, cloop)
This specific implementation avoids a loop by transposing a and b such that the summed axes of a are last and the summed axes of b are first. The resulting arrays are reshaped to 2 dimensions (or left as vectors, if appropriate) and a matrix or vector dot product is taken. The result is reshaped back to the required output dimensions.
In an extreme case, no axes may be specified. The resulting tensor will have shape equal to the concatenation of the shapes of a and b:
>>> c = np.tensordot(a, b, 0) >>> a.shape (2, 3, 4) >>> b.shape (5, 6, 4, 3) >>> print(c.shape) (2, 3, 4, 5, 6, 4, 3)
Note: See the documentation of numpy.tensordot for more examples.

theano.tensor.
batched_dot
(X, Y)[source]¶ Parameters:  x – A Tensor with sizes e.g.: for 3D (dim1, dim3, dim2)
 y – A Tensor with sizes e.g.: for 3D (dim1, dim2, dim4)
This function computes the dot product between the two tensors, by iterating over the first dimension using scan. Returns a tensor of size e.g. if it is 3D: (dim1, dim3, dim4) Example:
>>> first = tt.tensor3('first') >>> second = tt.tensor3('second') >>> result = batched_dot(first, second)
Note: This is a subset of numpy.einsum, but we do not provide it for now. But numpy einsum is slower than dot or tensordot: http://mail.scipy.org/pipermail/numpydiscussion/2012October/064259.html
Parameters:  X (symbolic tensor) – left term
 Y (symbolic tensor) – right term
Returns: tensor of products

theano.tensor.
batched_tensordot
(X, Y, axes=2)[source]¶ Parameters:  x – A Tensor with sizes e.g.: for 3D (dim1, dim3, dim2)
 y – A Tensor with sizes e.g.: for 3D (dim1, dim2, dim4)
 axes (int or arraylike of length 2) –
an integer or array. If an integer, the number of axes to sum over. If an array, it must have two array elements containing the axes to sum over in each tensor.
If an integer i, it is converted to an array containing the last i dimensions of the first tensor and the first i dimensions of the second tensor (excluding the first (batch) dimension):
axes = [range(a.ndim  i, b.ndim), range(1,i+1)]
If an array, its two elements must contain compatible axes of the two tensors. For example, [[1, 2], [2, 4]] means sum over the 2nd and 3rd axes of a and the 3rd and 5th axes of b. (Remember axes are zeroindexed!) The 2nd axis of a and the 3rd axis of b must have the same shape; the same is true for the 3rd axis of a and the 5th axis of b.
Returns: a tensor with shape equal to the concatenation of a’s shape (less any dimensions that were summed over) and b’s shape (less first dimension and any dimensions that were summed over).
Return type: tensor of tensordots
A hybrid of batch_dot and tensordot, this function computes the tensordot product between the two tensors, by iterating over the first dimension using scan to perform a sequence of tensordots.
Note: See tensordot()
andbatched_dot()
for supplementary documentation.

theano.tensor.
mgrid
()[source]¶ Returns: an instance which returns a dense (or fleshed out) meshgrid when indexed, so that each returned argument has the same shape. The dimensions and number of the output arrays are equal to the number of indexing dimensions. If the step length is not a complex number, then the stop is not inclusive. Example:
>>> a = tt.mgrid[0:5, 0:3] >>> a[0].eval() array([[0, 0, 0], [1, 1, 1], [2, 2, 2], [3, 3, 3], [4, 4, 4]]) >>> a[1].eval() array([[0, 1, 2], [0, 1, 2], [0, 1, 2], [0, 1, 2], [0, 1, 2]])

theano.tensor.
ogrid
()[source]¶ Returns: an instance which returns an open (i.e. not fleshed out) meshgrid when indexed, so that only one dimension of each returned array is greater than 1. The dimension and number of the output arrays are equal to the number of indexing dimensions. If the step length is not a complex number, then the stop is not inclusive. Example:
>>> b = tt.ogrid[0:5, 0:3] >>> b[0].eval() array([[0], [1], [2], [3], [4]]) >>> b[1].eval() array([[0, 1, 2]])
Gradient / Differentiation¶
Driver for gradient calculations.

theano.gradient.
grad
(cost, wrt, consider_constant=None, disconnected_inputs='raise', add_names=True, known_grads=None, return_disconnected='zero', null_gradients='raise')[source] Return symbolic gradients of one cost with respect to one or more variables.
For more information about how automatic differentiation works in Theano, see
gradient
. For information on how to implement the gradient of a certain Op, seegrad()
.Parameters:  cost (
Variable
scalar (0dimensional) tensor variable orNone
) – Value that we are differentiating (that we want the gradient of). May be None if known_grads is provided.  wrt (
Variable
or list of Variables) – Term[s] with respect to which we want gradients  consider_constant (list of variables) – Expressions not to backpropagate through
 disconnected_inputs ({'ignore', 'warn', 'raise'}) –
Defines the behaviour if some of the variables in wrt are not part of the computational graph computing cost (or if all links are nondifferentiable). The possible values are:
 ’ignore’: considers that the gradient on these parameters is zero.
 ’warn’: consider the gradient zero, and print a warning.
 ’raise’: raise DisconnectedInputError.
 add_names (bool) – If True, variables generated by grad will be named (d<cost.name>/d<wrt.name>) provided that both cost and wrt have names
 known_grads (OrderedDict, optional) – A ordered dictionary mapping variables to their gradients. This is useful in the case where you know the gradient on some variables but do not know the original cost.
 return_disconnected ({'zero', 'None', 'Disconnected'}) –
 ‘zero’ : If wrt[i] is disconnected, return value i will be wrt[i].zeros_like()
 ’None’ : If wrt[i] is disconnected, return value i will be None
 ’Disconnected’ : returns variables of type DisconnectedType
 null_gradients ({'raise', 'return'}) –
Defines the behaviour if some of the variables in wrt have a null gradient. The possibles values are:
 ’raise’ : raise a NullTypeGradError exception
 ’return’ : return the null gradients
Returns: Symbolic expression of gradient of cost with respect to each of the wrt terms. If an element of wrt is not differentiable with respect to the output, then a zero variable is returned.
Return type: variable or list/tuple of variables (matches wrt)
 cost (
See the gradient page for complete documentation of the gradient module.