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# Sparse Tensors

Note: Functions taking Tensor arguments can also take anything accepted by tf.convert_to_tensor.

## Sparse Tensor Representation

Tensorflow supports a SparseTensor representation for data that is sparse in multiple dimensions. Contrast this representation with IndexedSlices, which is efficient for representing tensors that are sparse in their first dimension, and dense along all other dimensions.

### class tf.SparseTensor

Represents a sparse tensor.

Tensorflow represents a sparse tensor as three separate dense tensors: indices, values, and dense_shape. In Python, the three tensors are collected into a SparseTensor class for ease of use. If you have separate indices, values, and dense_shape tensors, wrap them in a SparseTensor object before passing to the Ops below.

Concretely, the sparse tensor SparseTensor(values, indices, dense_shape) is

• indices: A 2-D int64 tensor of shape [N, ndims].
• values: A 1-D tensor of any type and shape [N].
• dense_shape: A 1-D int64 tensor of shape [ndims].

where N and ndims are the number of values, and number of dimensions in the SparseTensor respectively.

The corresponding dense tensor satisfies

dense.shape = dense_shape
dense[tuple(indices[i])] = values[i]

By convention, indices should be sorted in row-major order (or equivalently lexigraphic order on the tuples indices[i]). This is not enforced when SparseTensor objects are constructed, but most Ops assume correct ordering. If the ordering is wrong, it can be fixed by calling sparse_reorder on the misordered SparseTensor.

Example: The sparse tensor

  SparseTensor(values=[1, 2], indices=[[0, 0], [1, 2]], shape=[3, 4])

represents the dense tensor

  [[1, 0, 0, 0]
[0, 0, 2, 0]
[0, 0, 0, 0]]

#### tf.SparseTensor.__init__(indices, values, shape)

Creates a SparseTensor.

##### Args:
• indices: A 2-D int64 tensor of shape [N, ndims].
• values: A 1-D tensor of any type and shape [N].
• dense_shape: A 1-D int64 tensor of shape [ndims].
##### Returns:

A SparseTensor

#### tf.SparseTensor.indices

The indices of non-zero values in the represented dense tensor.

##### Returns:

A 2-D Tensor of int64 with shape [N, ndims], where N is the number of non-zero values in the tensor, and ndims is the rank.

#### tf.SparseTensor.values

The non-zero values in the represented dense tensor.

##### Returns:

A 1-D Tensor of any data type.

#### tf.SparseTensor.dtype

The DType of elements in this tensor.

#### tf.SparseTensor.shape

A 1-D Tensor of int64 representing the shape of the dense tensor.

#### tf.SparseTensor.graph

The Graph that contains the index, value, and shape tensors.

### class tf.SparseTensorValue

SparseTensorValue(indices, values, shape)

#### tf.SparseTensorValue.indices

Alias for field number 0

#### tf.SparseTensorValue.shape

Alias for field number 2

#### tf.SparseTensorValue.values

Alias for field number 1

## Sparse to Dense Conversion

### tf.sparse_to_dense(sparse_indices, output_shape, sparse_values, default_value, name=None)

Converts a sparse representation into a dense tensor.

Builds an array dense with shape output_shape such that

# If sparse_indices is scalar
dense[i] = (i == sparse_indices ? sparse_values : default_value)

# If sparse_indices is a vector, then for each i
dense[sparse_indices[i]] = sparse_values[i]

# If sparse_indices is an n by d matrix, then for each i in [0, n)
dense[sparse_indices[i][0], ..., sparse_indices[i][d-1]] = sparse_values[i]

All other values in dense are set to default_value. If sparse_values is a scalar, all sparse indices are set to this single value.

##### Args:
• sparse_indices: A Tensor. Must be one of the following types: int32, int64. 0-D, 1-D, or 2-D. sparse_indices[i] contains the complete index where sparse_values[i] will be placed.
• output_shape: A Tensor. Must have the same type as sparse_indices. 1-D. Shape of the dense output tensor.
• sparse_values: A Tensor. 1-D. Values corresponding to each row of sparse_indices, or a scalar value to be used for all sparse indices.
• default_value: A Tensor. Must have the same type as sparse_values. Scalar value to set for indices not specified in sparse_indices.
• name: A name for the operation (optional).
##### Returns:

A Tensor. Has the same type as sparse_values. Dense output tensor of shape output_shape.

### tf.sparse_tensor_to_dense(sp_input, default_value, name=None)

Converts a SparseTensor into a dense tensor.

This op is a convenience wrapper around sparse_to_dense for SparseTensors.

For example, if sp_input has shape [3, 5] and non-empty string values:

[0, 1]: a
[0, 3]: b
[2, 0]: c

and default_value is x, then the output will be a dense [3, 5] string tensor with values:

[[x a x b x]
[x x x x x]
[c x x x x]]
##### Args:
• sp_input: The input SparseTensor.
• default_value: Scalar value to set for indices not specified in sp_input.
• name: A name prefix for the returned tensors (optional).
##### Returns:

A dense tensor with shape sp_input.shape and values specified by the non-empty values in sp_input. Indices not in sp_input are assigned default_value.

##### Raises:
• TypeError: If sp_input is not a SparseTensor.

• -

### tf.sparse_to_indicator(sp_input, vocab_size, name=None)

Converts a SparseTensor of ids into a dense bool indicator tensor.

The last dimension of sp_input is discarded and replaced with the values of sp_input. If sp_input.shape = [D0, D1, ..., Dn, K], then output.shape = [D0, D1, ..., Dn, vocab_size], where

output[d_0, d_1, ..., d_n, sp_input[d_0, d_1, ..., d_n, k]] = True

and False elsewhere in output.

For example, if sp_input.shape = [2, 3, 4] with non-empty values:

[0, 0, 0]: 0
[0, 1, 0]: 10
[1, 0, 3]: 103
[1, 1, 2]: 112
[1, 1, 3]: 113
[1, 2, 1]: 121

and vocab_size = 200, then the output will be a [2, 3, 200] dense bool tensor with False everywhere except at positions

(0, 0, 0), (0, 1, 10), (1, 0, 103), (1, 1, 112), (1, 1, 113), (1, 2, 121).

This op is useful for converting SparseTensors into dense formats for compatibility with ops that expect dense tensors.

The input SparseTensor must be in row-major order.

##### Args:
• sp_input: A SparseTensor of type int32 or int64.
• vocab_size: The new size of the last dimension, with all(0 <= sp_input.values < vocab_size).
• name: A name prefix for the returned tensors (optional)
##### Returns:

A dense bool indicator tensor representing the indices with specified value.

##### Raises:
• TypeError: If sp_input is not a SparseTensor.

## Manipulation

### tf.sparse_concat(concat_dim, sp_inputs, name=None)

Concatenates a list of SparseTensor along the specified dimension.

Concatenation is with respect to the dense versions of each sparse input. It is assumed that each inputs is a SparseTensor whose elements are ordered along increasing dimension number.

All inputs' shapes must match, except for the concat dimension. The indices, values, and shapes lists must have the same length.

The output shape is identical to the inputs', except along the concat dimension, where it is the sum of the inputs' sizes along that dimension.

The output elements will be resorted to preserve the sort order along increasing dimension number.

This op runs in O(M log M) time, where M is the total number of non-empty values across all inputs. This is due to the need for an internal sort in order to concatenate efficiently across an arbitrary dimension.

For example, if concat_dim = 1 and the inputs are

sp_inputs[0]: shape = [2, 3]
[0, 2]: "a"
[1, 0]: "b"
[1, 1]: "c"

sp_inputs[1]: shape = [2, 4]
[0, 1]: "d"
[0, 2]: "e"

then the output will be

shape = [2, 7]
[0, 2]: "a"
[0, 4]: "d"
[0, 5]: "e"
[1, 0]: "b"
[1, 1]: "c"

Graphically this is equivalent to doing

[    a] concat [  d e  ] = [    a   d e  ]
[b c  ]        [       ]   [b c          ]
##### Args:
• concat_dim: Dimension to concatenate along.
• sp_inputs: List of SparseTensor to concatenate.
• name: A name prefix for the returned tensors (optional).
##### Returns:

A SparseTensor with the concatenated output.

##### Raises:
• TypeError: If sp_inputs is not a list of SparseTensor.

• -

### tf.sparse_reorder(sp_input, name=None)

Reorders a SparseTensor into the canonical, row-major ordering.

Note that by convention, all sparse ops preserve the canonical ordering along increasing dimension number. The only time ordering can be violated is during manual manipulation of the indices and values to add entries.

Reordering does not affect the shape of the SparseTensor.

For example, if sp_input has shape [4, 5] and indices / values:

[0, 3]: b
[0, 1]: a
[3, 1]: d
[2, 0]: c

then the output will be a SparseTensor of shape [4, 5] and indices / values:

[0, 1]: a
[0, 3]: b
[2, 0]: c
[3, 1]: d
##### Args:
• sp_input: The input SparseTensor.
• name: A name prefix for the returned tensors (optional)
##### Returns:

A SparseTensor with the same shape and non-empty values, but in canonical ordering.

##### Raises:
• TypeError: If sp_input is not a SparseTensor.

• -

### tf.sparse_retain(sp_input, to_retain)

Retains specified non-empty values within a SparseTensor.

For example, if sp_input has shape [4, 5] and 4 non-empty string values:

[0, 1]: a
[0, 3]: b
[2, 0]: c
[3, 1]: d

and to_retain = [True, False, False, True], then the output will be a SparseTensor of shape [4, 5] with 2 non-empty values:

[0, 1]: a
[3, 1]: d
##### Args:
• sp_input: The input SparseTensor with N non-empty elements.
• to_retain: A bool vector of length N with M true values.
##### Returns:

A SparseTensor with the same shape as the input and M non-empty elements corresponding to the true positions in to_retain.

##### Raises:
• TypeError: If sp_input is not a SparseTensor.

• -

### tf.sparse_fill_empty_rows(sp_input, default_value, name=None)

Fills empty rows in the input 2-D SparseTensor with a default value.

This op adds entries with the specified default_value at index [row, 0] for any row in the input that does not already have a value.

For example, suppose sp_input has shape [5, 6] and non-empty values:

[0, 1]: a
[0, 3]: b
[2, 0]: c
[3, 1]: d

Rows 1 and 4 are empty, so the output will be of shape [5, 6] with values:

[0, 1]: a
[0, 3]: b
[1, 0]: default_value
[2, 0]: c
[3, 1]: d
[4, 0]: default_value

Note that the input may have empty columns at the end, with no effect on this op.

The output SparseTensor will be in row-major order and will have the same shape as the input.

This op also returns an indicator vector such that

empty_row_indicator[i] = True iff row i was an empty row.
##### Args:
• sp_input: A SparseTensor with shape [N, M].
• default_value: The value to fill for empty rows, with the same type as sp_input.
• name: A name prefix for the returned tensors (optional)
##### Returns:
• sp_ordered_output: A SparseTensor with shape [N, M], and with all empty rows filled in with default_value.
• empty_row_indicator: A bool vector of length N indicating whether each input row was empty.
##### Raises:
• TypeError: If sp_input is not a SparseTensor.