# Reduction

TensorFlow provides several operations that you can use to perform common math computations that reduce various dimensions of a tensor.

### tf.reduce_sum(input_tensor, axis=None, keep_dims=False, name=None, reduction_indices=None)

Computes the sum of elements across dimensions of a tensor.

Reduces input_tensor along the dimensions given in axis. Unless keep_dims is true, the rank of the tensor is reduced by 1 for each entry in axis. If keep_dims is true, the reduced dimensions are retained with length 1.

If axis has no entries, all dimensions are reduced, and a tensor with a single element is returned.

For example:

# 'x' is [[1, 1, 1]
#         [1, 1, 1]]
tf.reduce_sum(x) ==> 6
tf.reduce_sum(x, 0) ==> [2, 2, 2]
tf.reduce_sum(x, 1) ==> [3, 3]
tf.reduce_sum(x, 1, keep_dims=True) ==> [[3], [3]]
tf.reduce_sum(x, [0, 1]) ==> 6

##### Args:
• input_tensor: The tensor to reduce. Should have numeric type.
• axis: The dimensions to reduce. If None (the default), reduces all dimensions.
• keep_dims: If true, retains reduced dimensions with length 1.
• name: A name for the operation (optional).
• reduction_indices: The old (deprecated) name for axis.
##### Returns:

The reduced tensor.

@compatibility(numpy) Equivalent to np.sum @end_compatibility

### tf.reduce_prod(input_tensor, axis=None, keep_dims=False, name=None, reduction_indices=None)

Computes the product of elements across dimensions of a tensor.

Reduces input_tensor along the dimensions given in axis. Unless keep_dims is true, the rank of the tensor is reduced by 1 for each entry in axis. If keep_dims is true, the reduced dimensions are retained with length 1.

If axis has no entries, all dimensions are reduced, and a tensor with a single element is returned.

##### Args:
• input_tensor: The tensor to reduce. Should have numeric type.
• axis: The dimensions to reduce. If None (the default), reduces all dimensions.
• keep_dims: If true, retains reduced dimensions with length 1.
• name: A name for the operation (optional).
• reduction_indices: The old (deprecated) name for axis.
##### Returns:

The reduced tensor.

@compatibility(numpy) Equivalent to np.prod @end_compatibility

### tf.reduce_min(input_tensor, axis=None, keep_dims=False, name=None, reduction_indices=None)

Computes the minimum of elements across dimensions of a tensor.

Reduces input_tensor along the dimensions given in axis. Unless keep_dims is true, the rank of the tensor is reduced by 1 for each entry in axis. If keep_dims is true, the reduced dimensions are retained with length 1.

If axis has no entries, all dimensions are reduced, and a tensor with a single element is returned.

##### Args:
• input_tensor: The tensor to reduce. Should have numeric type.
• axis: The dimensions to reduce. If None (the default), reduces all dimensions.
• keep_dims: If true, retains reduced dimensions with length 1.
• name: A name for the operation (optional).
• reduction_indices: The old (deprecated) name for axis.
##### Returns:

The reduced tensor.

@compatibility(numpy) Equivalent to np.min @end_compatibility

### tf.reduce_max(input_tensor, axis=None, keep_dims=False, name=None, reduction_indices=None)

Computes the maximum of elements across dimensions of a tensor.

Reduces input_tensor along the dimensions given in axis. Unless keep_dims is true, the rank of the tensor is reduced by 1 for each entry in axis. If keep_dims is true, the reduced dimensions are retained with length 1.

If axis has no entries, all dimensions are reduced, and a tensor with a single element is returned.

##### Args:
• input_tensor: The tensor to reduce. Should have numeric type.
• axis: The dimensions to reduce. If None (the default), reduces all dimensions.
• keep_dims: If true, retains reduced dimensions with length 1.
• name: A name for the operation (optional).
• reduction_indices: The old (deprecated) name for axis.
##### Returns:

The reduced tensor.

@compatibility(numpy) Equivalent to np.max @end_compatibility

### tf.reduce_mean(input_tensor, axis=None, keep_dims=False, name=None, reduction_indices=None)

Computes the mean of elements across dimensions of a tensor.

Reduces input_tensor along the dimensions given in axis. Unless keep_dims is true, the rank of the tensor is reduced by 1 for each entry in axis. If keep_dims is true, the reduced dimensions are retained with length 1.

If axis has no entries, all dimensions are reduced, and a tensor with a single element is returned.

For example:

# 'x' is [[1., 1.]
#         [2., 2.]]
tf.reduce_mean(x) ==> 1.5
tf.reduce_mean(x, 0) ==> [1.5, 1.5]
tf.reduce_mean(x, 1) ==> [1.,  2.]

##### Args:
• input_tensor: The tensor to reduce. Should have numeric type.
• axis: The dimensions to reduce. If None (the default), reduces all dimensions.
• keep_dims: If true, retains reduced dimensions with length 1.
• name: A name for the operation (optional).
• reduction_indices: The old (deprecated) name for axis.
##### Returns:

The reduced tensor.

@compatibility(numpy) Equivalent to np.mean @end_compatibility

### tf.reduce_all(input_tensor, axis=None, keep_dims=False, name=None, reduction_indices=None)

Computes the "logical and" of elements across dimensions of a tensor.

Reduces input_tensor along the dimensions given in axis. Unless keep_dims is true, the rank of the tensor is reduced by 1 for each entry in axis. If keep_dims is true, the reduced dimensions are retained with length 1.

If axis has no entries, all dimensions are reduced, and a tensor with a single element is returned.

For example:

# 'x' is [[True,  True]
#         [False, False]]
tf.reduce_all(x) ==> False
tf.reduce_all(x, 0) ==> [False, False]
tf.reduce_all(x, 1) ==> [True, False]

##### Args:
• input_tensor: The boolean tensor to reduce.
• axis: The dimensions to reduce. If None (the default), reduces all dimensions.
• keep_dims: If true, retains reduced dimensions with length 1.
• name: A name for the operation (optional).
• reduction_indices: The old (deprecated) name for axis.
##### Returns:

The reduced tensor.

@compatibility(numpy) Equivalent to np.all @end_compatibility

### tf.reduce_any(input_tensor, axis=None, keep_dims=False, name=None, reduction_indices=None)

Computes the "logical or" of elements across dimensions of a tensor.

Reduces input_tensor along the dimensions given in axis. Unless keep_dims is true, the rank of the tensor is reduced by 1 for each entry in axis. If keep_dims is true, the reduced dimensions are retained with length 1.

If axis has no entries, all dimensions are reduced, and a tensor with a single element is returned.

For example:

# 'x' is [[True,  True]
#         [False, False]]
tf.reduce_any(x) ==> True
tf.reduce_any(x, 0) ==> [True, True]
tf.reduce_any(x, 1) ==> [True, False]

##### Args:
• input_tensor: The boolean tensor to reduce.
• axis: The dimensions to reduce. If None (the default), reduces all dimensions.
• keep_dims: If true, retains reduced dimensions with length 1.
• name: A name for the operation (optional).
• reduction_indices: The old (deprecated) name for axis.
##### Returns:

The reduced tensor.

@compatibility(numpy) Equivalent to np.any @end_compatibility

### tf.reduce_logsumexp(input_tensor, axis=None, keep_dims=False, name=None, reduction_indices=None)

Computes log(sum(exp(elements across dimensions of a tensor))).

Reduces input_tensor along the dimensions given in axis. Unless keep_dims is true, the rank of the tensor is reduced by 1 for each entry in axis. If keep_dims is true, the reduced dimensions are retained with length 1.

If axis has no entries, all dimensions are reduced, and a tensor with a single element is returned.

This function is more numerically stable than log(sum(exp(input))). It avoids overflows caused by taking the exp of large inputs and underflows caused by taking the log of small inputs.

For example:

# 'x' is [[0, 0, 0]]
#         [0, 0, 0]]
tf.reduce_logsumexp(x) ==> log(6)
tf.reduce_logsumexp(x, 0) ==> [log(2), log(2), log(2)]
tf.reduce_logsumexp(x, 1) ==> [log(3), log(3)]
tf.reduce_logsumexp(x, 1, keep_dims=True) ==> [[log(3)], [log(3)]]
tf.reduce_logsumexp(x, [0, 1]) ==> log(6)

##### Args:
• input_tensor: The tensor to reduce. Should have numeric type.
• axis: The dimensions to reduce. If None (the default), reduces all dimensions.
• keep_dims: If true, retains reduced dimensions with length 1.
• name: A name for the operation (optional).
• reduction_indices: The old (deprecated) name for axis.
##### Returns:

The reduced tensor.

### tf.count_nonzero(input_tensor, axis=None, keep_dims=False, dtype=tf.int64, name=None, reduction_indices=None)

Computes number of nonzero elements across dimensions of a tensor.

Reduces input_tensor along the dimensions given in axis. Unless keep_dims is true, the rank of the tensor is reduced by 1 for each entry in axis. If keep_dims is true, the reduced dimensions are retained with length 1.

If axis has no entries, all dimensions are reduced, and a tensor with a single element is returned.

NOTE Floating point comparison to zero is done by exact floating point equality check. Small values are not rounded to zero for purposes of the nonzero check.

For example:

# 'x' is [[0, 1, 0]
#         [1, 1, 0]]
tf.count_nonzero(x) ==> 3
tf.count_nonzero(x, 0) ==> [1, 2, 0]
tf.count_nonzero(x, 1) ==> [1, 2]
tf.count_nonzero(x, 1, keep_dims=True) ==> [[1], [2]]
tf.count_nonzero(x, [0, 1]) ==> 3

##### Args:
• input_tensor: The tensor to reduce. Should be of numeric type, or bool.
• axis: The dimensions to reduce. If None (the default), reduces all dimensions.
• keep_dims: If true, retains reduced dimensions with length 1.
• dtype: The output dtype; defaults to tf.int64.
• name: A name for the operation (optional).
• reduction_indices: The old (deprecated) name for axis.
##### Returns:

The reduced tensor (number of nonzero values).

### tf.accumulate_n(inputs, shape=None, tensor_dtype=None, name=None)

Returns the element-wise sum of a list of tensors.

Optionally, pass shape and tensor_dtype for shape and type checking, otherwise, these are inferred.

NOTE: This operation is not differentiable and cannot be used if inputs depend on trainable variables. Please use tf.add_n for such cases.

For example:

# tensor 'a' is [[1, 2], [3, 4]]
# tensor b is [[5, 0], [0, 6]]
tf.accumulate_n([a, b, a]) ==> [[7, 4], [6, 14]]

# Explicitly pass shape and type
tf.accumulate_n([a, b, a], shape=[2, 2], tensor_dtype=tf.int32)
==> [[7, 4], [6, 14]]

##### Args:
• inputs: A list of Tensor objects, each with same shape and type.
• shape: Shape of elements of inputs.
• tensor_dtype: The type of inputs.
• name: A name for the operation (optional).
##### Returns:

A Tensor of same shape and type as the elements of inputs.

##### Raises:
• ValueError: If inputs don't all have same shape and dtype or the shape cannot be inferred.

### tf.einsum(equation, *inputs)

A generalized contraction between tensors of arbitrary dimension.

This function returns a tensor whose elements are defined by equation, which is written in a shorthand form inspired by the Einstein summation convention. As an example, consider multiplying two matrices A and B to form a matrix C. The elements of C are given by:

  C[i,k] = sum_j A[i,j] * B[j,k]


The corresponding equation is:

  ij,jk->ik


In general, the equation is obtained from the more familiar element-wise equation by 1. removing variable names, brackets, and commas, 2. replacing "*" with ",", 3. dropping summation signs, and 4. moving the output to the right, and replacing "=" with "->".

Many common operations can be expressed in this way. For example:

# Matrix multiplication

einsum('ij,jk->ik', m0, m1) # output[i,k] = sum_j m0[i,j] * m1[j, k]

# Dot product

einsum('i,i->', u, v) # output = sum_i u[i]*v[i]

# Outer product

einsum('i,j->ij', u, v) # output[i,j] = u[i]*v[j]

# Transpose

einsum('ij->ji', m) # output[j,i] = m[i,j]

# Batch matrix multiplication

einsum('aij,ajk->aik', s, t) # out[a,i,k] = sum_j s[a,i,j] * t[a, j, k]

This function behaves like numpy.einsum, but does not support: Ellipses (subscripts like ij...,jk...->ik...) Subscripts where an axis appears more than once for a single input (e.g. ijj,k->ik). * Subscripts that are summed across multiple inputs (e.g., ij,ij,jk->ik).

##### Args:
• equation: a str describing the contraction, in the same format as numpy.einsum.
• inputs: the inputs to contract (each one a Tensor), whose shapes should be consistent with equation.
##### Returns:

The contracted Tensor, with shape determined by equation.

##### Raises:
• ValueError: If
• the format of equation is incorrect,
• the number of inputs implied by equation does not match len(inputs),
• an axis appears in the output subscripts but not in any of the inputs,
• the number of dimensions of an input differs from the number of indices in its subscript, or
• the input shapes are inconsistent along a particular axis.