# `tensor.elemwise` – Tensor Elemwise¶

class `theano.tensor.elemwise.``All`(axis=None)[source]

Applies logical and to all the values of a tensor along the specified axis(es).

class `theano.tensor.elemwise.``Any`(axis=None)[source]

Applies bitwise or to all the values of a tensor along the specified axis(es).

class `theano.tensor.elemwise.``CAReduce`(scalar_op, axis=None)[source]

CAReduce = Commutative Associative Reduce Reduces a scalar operation along the specified axis(es). (The scalar op should be both commutative and assocative)

The output will have the same shape as the input minus the reduced dimensions. It will contain the variable of accumulating all values over the reduced dimensions using the specified scalar op.

Parameters: scalar_op – A binary scalar op with only one output. It must be commutative and associative. axis – The dimension along which we want to reduce List of dimensions that we want to reduce If None, all dimensions are reduced

Note

```CAReduce(add)      # sum (ie, acts like the numpy sum operation)
CAReduce(mul)      # product
CAReduce(maximum)  # max
CAReduce(minimum)  # min
CAReduce(or_)      # any # not lazy
CAReduce(and_)     # all # not lazy
CAReduce(xor)      # a bit at 1 tell that there was an odd number of
# bit at that position that where 1. 0 it was an
# even number ...
```

In order to (eventually) optimize memory usage patterns, CAReduce makes zero guarantees on the order in which it iterates over the dimensions and the elements of the array(s). Therefore, to ensure consistent variables, the scalar operation represented by the reduction must be both commutative and associative (eg add, multiply, maximum, binary or/and/xor - but not subtract, divide or power).

class `theano.tensor.elemwise.``CAReduceDtype`(scalar_op, axis=None, dtype=None, acc_dtype=None)[source]

Reduces a scalar operation along the specified axis(es).

This subclass of CAReduce accepts an additional “dtype” parameter, that specifies which dtype the output should be.

It also accepts an optional “acc_dtype”, which specify the dtype that will be used for the accumulation.

So, the accumulation will be done into a tensor of dtype “acc_dtype”, then it will be casted into “dtype” and returned.

If no dtype is provided, one will be inferred so as not to lose too much precision.

Parameters: scalar_op – A binary scalar op with only one output. It must be commutative and associative. axis – the dimension along which we want to reduce list of dimensions that we want to reduce if None, all dimensions are reduced 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”. This behavior is similar in spirit to that of numpy (except numpy uses the default machine integer while we always use 64 bit integers to avoid platform-dependent behavior). 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.
class `theano.tensor.elemwise.``DimShuffle`(input_broadcastable, new_order, inplace=True)[source]

Allows to reorder the dimensions of a tensor or insert or remove broadcastable dimensions.

In the following examples, ‘x’ means that we insert a broadcastable dimension and a numerical index represents the dimension of the same rank in the tensor passed to perform.

Parameters: input_broadcastable – The expected broadcastable pattern of the input new_order – A list representing the relationship between the input’s dimensions and the output’s dimensions. Each element of the list can either be an index or ‘x’. Indices must be encoded as python integers, not theano symbolic integers. inplace (bool, optional) – If True (default), the output will be a view of the input.

Note

If j = new_order[i] is an index, the output’s ith dimension will be the input’s jth dimension. If new_order[i] is x, the output’s ith dimension will be 1 and Broadcast operations will be allowed to do broadcasting over that dimension.

If input.broadcastable[i] == False then i must be found in new_order. Broadcastable dimensions, on the other hand, can be discarded.

Note

```DimShuffle((False, False, False), ['x', 2, 'x', 0, 1])
```

This op will only work on 3d tensors with no broadcastable dimensions. The first dimension will be broadcastable, then we will have the third dimension of the input tensor as the second of the resulting tensor, etc. If the tensor has shape (20, 30, 40), the resulting tensor will have dimensions (1, 40, 1, 20, 30). (AxBxC tensor is mapped to 1xCx1xAxB tensor)

```DimShuffle((True, False), [1])
```

This op will only work on 2d tensors with the first dimension broadcastable. The second dimension of the input tensor will be the first dimension of the resulting tensor. If the tensor has shape (1, 20), the resulting tensor will have shape (20, ).

Example

```DimShuffle((), ['x'])  # make a 0d (scalar) into a 1d vector
DimShuffle((False, False), [0, 1])  # identity
DimShuffle((False, False), [1, 0])  # inverts the 1st and 2nd dimensions
DimShuffle((False,), ['x', 0])  # make a row out of a 1d vector
# (N to 1xN)
DimShuffle((False,), [0, 'x'])  # make a column out of a 1d vector
# (N to Nx1)
DimShuffle((False, False, False), [2, 0, 1])  # AxBxC to CxAxB
DimShuffle((False, False), [0, 'x', 1])  # AxB to Ax1xB
DimShuffle((False, False), [1, 'x', 0])  # AxB to Bx1xA
```

The reordering of the dimensions can be done with the numpy.transpose function. Adding, subtracting dimensions can be done with reshape.

class `theano.tensor.elemwise.``Elemwise`(scalar_op, inplace_pattern=None, name=None, nfunc_spec=None, openmp=None)[source]

Generalizes a scalar op to tensors.

All the inputs must have the same number of dimensions. When the Op is performed, for each dimension, each input’s size for that dimension must be the same. As a special case, it can also be 1 but only if the input’s broadcastable flag is True for that dimension. In that case, the tensor is (virtually) replicated along that dimension to match the size of the others.

The dtypes of the outputs mirror those of the scalar Op that is being generalized to tensors. In particular, if the calculations for an output are done inplace on an input, the output type must be the same as the corresponding input type (see the doc of scalar.ScalarOp to get help about controlling the output type)

Parameters: scalar_op – An instance of a subclass of scalar.ScalarOp which works uniquely on scalars. inplace_pattern – A dictionary that maps the index of an output to the index of an input so the output is calculated inplace using the input’s storage. (Just like destroymap, but without the lists.) nfunc_spec – Either None or a tuple of three elements, (nfunc_name, nin, nout) such that getattr(numpy, nfunc_name) implements this operation, takes nin inputs and nout outputs. Note that nin cannot always be inferred from the scalar op’s own nin field because that value is sometimes 0 (meaning a variable number of inputs), whereas the numpy function may not have varargs.

Note

Elemwise(add) represents + on tensors (x + y)
Elemwise(add, {0 : 0}) represents the += operation (x += y)
Elemwise(add, {0 : 1}) represents += on the second argument (y += x)
Elemwise(mul)(rand(10, 5), rand(1, 5)) the second input is completed along the first dimension to match the first input
Elemwise(true_div)(rand(10, 5), rand(10, 1)) same but along the second dimension
Elemwise(int_div)(rand(1, 5), rand(10, 1)) the output has size (10, 5)
Elemwise(log)(rand(3, 4, 5))
`get_output_info`(dim_shuffle, *inputs)[source]

Return the outputs dtype and broadcastable pattern and the dimshuffled niputs.

`make_node`(*inputs)[source]

If the inputs have different number of dimensions, their shape is left-completed to the greatest number of dimensions with 1s using DimShuffle.

`python_constant_folding`(node)[source]

Return True if we do not want to compile c code when doing constant folding of this node.

class `theano.tensor.elemwise.``Prod`(axis=None, dtype=None, acc_dtype=None, no_zeros_in_input=False)[source]

Multiplies all the values of a tensor along the specified axis(es).

Equivalent to CAReduce(scalar.prod, axis = axis), with the difference that this defines the gradient of prod wrt its tensor input.

`L_op`(inp, out, grads)[source]

The grad of this Op could be very easy, if it is was not for the case where zeros are present in a given “group” (ie. elements reduced together to form the product).

If no zeros are found in the elements of the product, then the partial derivative of the product relative to one of the elements (one of the inputs) is simply the product of the other elements. That’s easy to see from the chain rule.

Now the trick (with no zeros) is to take the overall product, then for every original element, the partial derivative is given by this product divided by the element itself (which equals the product of the other terms). This is easy to do by broadcasting the original product.

(Note that we also need to broadcast-multiply by the “incoming gradient”, ie. the gradient of the cost relative to the output/product).

With zeros, things get more complicated. For a given group, we have 3 cases:

• No zeros in the group. Use previous trick.

• If only one zero is present, then the gradient for that element is

non-zero, but is zero for all others.

• If more than one zero is present, then all the derivatives are zero.

For the last two cases (with 1 or more zeros), we can’t use the division trick, as this gives divisions by 0.

Implementing that case-by-case logic is not as trivial, so a bunch of hacks are piled down here to do it. Notably, for the “only one zero” case, there’s a special Op that computes the product of the elements in the group, minus the zero (see ProdWithoutZero). The trick is then to use the division trick for groups with no zero, to use the ProdWithoutZeros op where there’s only one zero, and to output a derivative of zero for any element part of a group with more than one zero.

I do this by first counting the number of zeros in each group (see the “T.eq()” bits), then taking this or that behavior (see T.switch) based on the result of this count.

class `theano.tensor.elemwise.``Sum`(axis=None, dtype=None, acc_dtype=None)[source]

Sums all the values of a tensor along the specified axis(es).

Equivalent to CAReduceDtype(scalar.add, axis=axis, dtype=dtype), with the difference that this defines the gradient of sum wrt its tensor input.

Parameters: axis – Axis(es) along which the tensor should be summed (use None to sum over all axes, and a list or tuple to sum along more than one axis). dtype – The dtype of the internal accumulator and 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 value does not depend on the value of “acc_dtype”. 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.