Operation semantics

The following describes the semantics of operations defined in the XlaBuilder interface. Typically, these operations map one-to-one to operations defined in the RPC interface in xla_data.proto.

A note on nomenclature: the generalized data type XLA deals with is an N-dimensional array holding elements of some uniform type (such as 32-bit float). Throughout the documentation, array is used to denote an arbitrary-dimensional array. For convenience, special cases have more specific and familiar names; for example a vector is a 1-dimensional array and a matrix is a 2-dimensional array.

AfterAll

See also XlaBuilder::AfterAll.

AfterAll takes a variadic number of tokens and produces a single token. Tokens are primitive types which can be threaded between side-effecting operations to enforce ordering. AfterAll can be used as a join of tokens for ordering an operation after a set operations.

AfterAll(operands)

AllGather

See also XlaBuilder::AllGather.

Performs concatenation across replicas.

AllGather(operand, all_gather_dim, shard_count, replica_group_ids, channel_id)

• replica_groups is a list of replica groups between which the concatenation is performed (replica id for the current replica can be retrieved using ReplicaId). The order of replicas in each group determines the order in which their inputs are located in the result. replica_groups must either be empty (in which case all replicas belong to a single group, ordered from 0 to N - 1), or contain the same number of elements as the number of replicas. For example, replica_groups = {0, 2}, {1, 3} performs concatenation between the replicas 0 and 2, and 1 and 3.
• shard_count is the size of each replica group. We need this in cases where replica_groups are empty.
• channel_id is used for cross-module communication: only all-gather operations with the same channel_id can communicate to each other.

The output shape is the input shape with the all_gather_dim made shard_count times larger. For example, if there are two replicas and the operand has the value [1.0, 2.5] and [3.0, 5.25] respectively on the two replicas, then the output value from this op where all_gather_dim is 0 will be [1.0, 2.5, 3.0, 5.25] on both replicas.

AllReduce

See also XlaBuilder::AllReduce.

Performs a custom computation across replicas.

AllReduce(operand, computation, replica_group_ids, channel_id)

• When operand is a tuple of arrays, the all-reduce is performed on each element of the tuple.
• replica_groups is a list of replica groups between which the reduction is performed (replica id for the current replica can be retrieved using ReplicaId). replica_groups must either be empty (in which case all replicas belong to a single group), or contain the same number of elements as the number of replicas. For example, replica_groups = {0, 2}, {1, 3} performs reduction between the replicas 0 and 2, and 1 and 3.
• channel_id is used for cross-module communication: only all-reduce operations with the same channel_id can communicate to each other.

The output shape is the same as the input shape. For example, if there are two replicas and the operand has the value [1.0, 2.5] and [3.0, 5.25] respectively on the two replicas, then the output value from this op and summation computation will be [4.0, 7.75] on both replicas. If the input is a tuple, the output is a tuple as well.

Computing the result of AllReduce requires having one input from each replica, so if one replica executes an AllReduce node more times than another, then the former replica will wait forever. Since the replicas are all running the same program, there are not a lot of ways for that to happen, but it is possible when a while loop's condition depends on data from infeed and the data that is infed causes the while loop to iterate more times on one replica than another.

AllToAll

See also XlaBuilder::AllToAll.

AllToAll is a collective operation that sends data from all cores to all cores. It has two phases:

1. The scatter phase. On each core, the operand is split into split_count number of blocks along the split_dimensions, and the blocks are scattered to all cores, e.g., the ith block is sent to the ith core.
2. The gather phase. Each core concatenates the received blocks along the concat_dimension.

The participating cores can be configured by:

• replica_groups: each ReplicaGroup contains a list of replica ids participating in the computation (the replica id for the current replica can be retrieved using ReplicaId). AllToAll will be applied within subgroups in the specified order. For example, replica_groups = { {1,2,3}, {4,5,0} } means that an AllToAll will be applied within replicas {1, 2, 3}, and in the gather phase, and the received blocks will be concatenated in the same order of 1, 2, 3. Then, another AllToAll will be applied within replicas 4, 5, 0, and the concatenation order is also 4, 5, 0. If replica_groups is empty, all replicas belong to one group, in the concatenation order of their appearance.

Prerequisites:

• The dimension size of the operand on the split_dimension is divisible by split_count.
• The operand's shape is not tuple.

AllToAll(operand, split_dimension, concat_dimension, split_count, replica_groups)

Below shows an example of Alltoall.

XlaBuilder b("alltoall");
auto x = Parameter(&b, 0, ShapeUtil::MakeShape(F32, {4, 16}), "x");
AllToAll(x, /*split_dimension=*/1, /*concat_dimension=*/0, /*split_count=*/4);

In this example, there are 4 cores participating in the Alltoall. On each core, the operand is split into 4 parts along dimension 1, so each part has shape f32[4,4]. The 4 parts are scattered to all cores. Then each core concatenates the received parts along dimension 0, in the order of core 0-4. So the output on each core has shape f32[16,4].

BatchNormGrad

See also XlaBuilder::BatchNormGrad and the original batch normalization paper for a detailed description of the algorithm.

Calculates gradients of batch norm.

BatchNormGrad(operand, scale, mean, variance, grad_output, epsilon, feature_index)

For each feature in the feature dimension (feature_index is the index for the feature dimension in operand), the operation calculates the gradients with respect to operand, offset, and scale across all the other dimensions. The feature_index must be a valid index for the feature dimension in operand.

The three gradients are defined by the following formulas (assuming a 4-dimensional array as operand and with feature dimension index l, batch size m and spatial sizes w and h):

\[ \begin{split} c_l&= \frac{1}{mwh}\sum_{i=1}^m\sum_{j=1}^w\sum_{k=1}^h \left( \nabla y_{ijkl} \frac{x_{ijkl} - \mu_l}{\sigma^2_l+\epsilon} \right) \\\\ d_l&= \frac{1}{mwh}\sum_{i=1}^m\sum_{j=1}^w\sum_{k=1}^h \nabla y_{ijkl} \\\\ \nabla x_{ijkl} &= \frac{\gamma_{l} }{\sqrt{\sigma^2_{l}+\epsilon} } \left( \nabla y_{ijkl} - d_l - c_l (x_{ijkl} - \mu_{l}) \right) \\\\ \nabla \gamma_l &= \sum_{i=1}^m\sum_{j=1}^w\sum_{k=1}^h \left( \nabla y_{ijkl} \frac{x_{ijkl} - \mu_l}{\sqrt{\sigma^2_{l}+\epsilon} } \right) \\\\\ \nabla \beta_l &= \sum_{i=1}^m\sum_{j=1}^w\sum_{k=1}^h \nabla y_{ijkl} \end{split} \]

The inputs mean and variance represent moments values across batch and spatial dimensions.

The output type is a tuple of three handles:

BatchNormInference

See also XlaBuilder::BatchNormInference and the original batch normalization paper for a detailed description of the algorithm.

Normalizes an array across batch and spatial dimensions.

BatchNormInference(operand, scale, offset, mean, variance, epsilon, feature_index)

For each feature in the feature dimension (feature_index is the index for the feature dimension in operand), the operation calculates the mean and variance across all the other dimensions and uses the mean and variance to normalize each element in operand. The feature_index must be a valid index for the feature dimension in operand.

BatchNormInference is equivalent to calling BatchNormTraining without computing mean and variance for each batch. It uses the input mean and variance instead as estimated values. The purpose of this op is to reduce latency in inference, hence the name BatchNormInference.

The output is an n-dimensional, normalized array with the same shape as input operand.

BatchNormTraining

See also XlaBuilder::BatchNormTraining and the original batch normalization paper for a detailed description of the algorithm.

Normalizes an array across batch and spatial dimensions.

BatchNormTraining(operand, scale, offset, epsilon, feature_index)

For each feature in the feature dimension (feature_index is the index for the feature dimension in operand), the operation calculates the mean and variance across all the other dimensions and uses the mean and variance to normalize each element in operand. The feature_index must be a valid index for the feature dimension in operand.

The algorithm goes as follows for each batch in operand \(x\) that contains m elements with w and h as the size of spatial dimensions (assuming operand is a 4 dimensional array):

• Calculates batch mean \(\mu_l\) for each feature l in feature dimension: \(\mu_l=\frac{1}{mwh}\sum_{i=1}^m\sum_{j=1}^w\sum_{k=1}^h x_{ijkl}\)

• Calculates batch variance \(\sigma^2_l\): $\sigma^2l=\frac{1}{mwh}\sum{i=1}^m\sum{j=1}^w\sum{k=1}^h (x_{ijkl} - \mu_l)^2$

• Normalizes, scales and shifts: \(y_{ijkl}=\frac{\gamma_l(x_{ijkl}-\mu_l)}{\sqrt[2]{\sigma^2_l+\epsilon} }+\beta_l\)

The epsilon value, usually a small number, is added to avoid divide-by-zero errors.

The output type is a tuple of three XlaOps:

The batch_mean and batch_var are moments calculated across the batch and spatial dimensions using the formulas above.

BitcastConvertType

See also XlaBuilder::BitcastConvertType.

Similar to a tf.bitcast in TensorFlow, performs an element-wise bitcast operation from a data shape to a target shape. The input and output size must match: e.g. s32 elements become f32 elements via bitcast routine, and one s32 element will become four s8 elements. Bitcast is implemented as a low-level cast, so machines with different floating-point representations will give different results.

BitcastConvertType(operand, new_element_type)

The dimensions of the operand and the target shape must match, apart from the last dimension which will change by the ratio of the primitive size before and after the conversion.

The source and destination element types must not be tuples.

Bitcast-converting to primitive type of different width

BitcastConvert HLO instruction supports the case where the size of the output element type T' is not equal to the size of the input element T. As the whole operation is conceptually a bitcast and does not change the underlying bytes, the shape of the output element has to change. For B = sizeof(T), B' = sizeof(T'), there are two possible cases.

First, when B > B', the output shape gets a new minor-most dimension of size B/B'. For example:

  f16[10,2]{1,0} %output = f16[10,2]{1,0} bitcast-convert(f32[10]{0} %input)

The rule remains the same for effective scalars:

  f16[2]{0} %output = f16[2]{0} bitcast-convert(f32[] %input)

Alternatively, for B' > B the instruction requires the last logical dimension of the input shape to be equal to B'/B, and this dimension is dropped during the conversion:

  f32[10]{0} %output = f32[10]{0} bitcast-convert(f16[10,2]{1,0} %input)

Note that conversions between different bitwidths are not elementwise.

Broadcast

See also XlaBuilder::Broadcast.

Adds dimensions to an array by duplicating the data in the array.

Broadcast(operand, broadcast_sizes)

The new dimensions are inserted on the left, i.e. if broadcast_sizes has values {a0, ..., aN} and the operand shape has dimensions {b0, ..., bM} then the shape of the output has dimensions {a0, ..., aN, b0, ..., bM}.

The new dimensions index into copies of the operand, i.e.

output[i0, ..., iN, j0, ..., jM] = operand[j0, ..., jM]

For example, if operand is a scalar f32 with value 2.0f, and broadcast_sizes is {2, 3}, then the result will be an array with shape f32[2, 3] and all the values in the result will be 2.0f.

BroadcastInDim

See also XlaBuilder::BroadcastInDim.

Expands the size and rank of an array by duplicating the data in the array.

BroadcastInDim(operand, out_dim_size, broadcast_dimensions)

Similar to Broadcast, but allows adding dimensions anywhere and expanding existing dimensions with size 1.

The operand is broadcast to the shape described by out_dim_size. broadcast_dimensions maps the dimensions of operand to the dimensions of the target shape, i.e. the i'th dimension of the operand is mapped to the broadcast_dimension[i]'th dimension of the output shape. The dimensions of operand must have size 1 or be the same size as the dimension in the output shape they are mapped to. The remaining dimensions are filled with dimensions of size 1. Degenerate-dimension broadcasting then broadcasts along these degenerate dimensions to reach the output shape. The semantics are described in detail on the broadcasting page.

Call

See also XlaBuilder::Call.

Invokes a computation with the given arguments.

Call(computation, args...)

The arity and types of the args must match the parameters of the computation. It is allowed to have no args.

CompositeCall

See also XlaBuilder::CompositeCall.

Encapsulates an operation made up (composed) of other StableHLO operations, taking inputs and composite_attributes and producing results. The semantics of the op are implemented by the decomposition attribute. The composite op can be replaced with its decomposition without changing program semantics. In cases where inlining the decomposition does not provide the same op semantics, prefer using custom_call.

The version field (defaults to 0) is used to denote when a composite's semantics change.

This op is implemented as a kCall with attribute is_composite=true. The decomposition field is specified by the computation attribute. The frontend attributes store the remaining attributes prefixed with composite..

Example CompositeCall op:

f32[] call(f32[] %cst), to_apply=%computation, is_composite=true,
frontend_attributes = {
  composite.name="foo.bar",
  composite.attributes={n = 1 : i32, tensor = dense<1> : tensor<i32>},
  composite.version="1"
}

Call(computation, args..., name, composite_attributes, version)

Cholesky

See also XlaBuilder::Cholesky.

Computes the Cholesky decomposition of a batch of symmetric (Hermitian) positive definite matrices.

Cholesky(a, lower)

If lower is true, computes lower-triangular matrices l such that $a = l . l^T$. If lower is false, computes upper-triangular matrices u such that \(a = u^T . u\).

Input data is read only from the lower/upper triangle of a, depending on the value of lower. Values from the other triangle are ignored. Output data is returned in the same triangle; the values in the other triangle are implementation-defined and may be anything.

If the rank of a is greater than 2, a is treated as a batch of matrices, where all except the minor 2 dimensions are batch dimensions.

If a is not symmetric (Hermitian) positive definite, the result is implementation-defined.

Clamp

See also XlaBuilder::Clamp.

Clamps an operand to within the range between a minimum and maximum value.

Clamp(min, operand, max)

Given an operand and minimum and maximum values, returns the operand if it is in the range between the minimum and maximum, else returns the minimum value if the operand is below this range or the maximum value if the operand is above this range. That is, clamp(a, x, b) = min(max(a, x), b).

All three arrays must be the same shape. Alternatively, as a restricted form of broadcasting, min and/or max can be a scalar of type T.

Example with scalar min and max:

let operand: s32[3] = {-1, 5, 9};
let min: s32 = 0;
let max: s32 = 6;
==>
Clamp(min, operand, max) = s32[3]{0, 5, 6};

Collapse

See also XlaBuilder::Collapse and the tf.reshape operation.

Collapses dimensions of an array into one dimension.

Collapse(operand, dimensions)

Collapse replaces the given subset of the operand's dimensions by a single dimension. The input arguments are an arbitrary array of type T and a compile-time-constant vector of dimension indices. The dimension indices must be an in-order (low to high dimension numbers), consecutive subset of T's dimensions. Thus, {0, 1, 2}, {0, 1}, or {1, 2} are all valid dimension sets, but {1, 0} or {0, 2} are not. They are replaced by a single new dimension, in the same position in the dimension sequence as those they replace, with the new dimension size equal to the product of original dimension sizes. The lowest dimension number in dimensions is the slowest varying dimension (most major) in the loop nest which collapses these dimensions, and the highest dimension number is fastest varying (most minor). See the tf.reshape operator if more general collapse ordering is needed.

For example, let v be an array of 24 elements:

let v = f32[4x2x3] { { {10, 11, 12},  {15, 16, 17} },
{ {20, 21, 22},  {25, 26, 27} },
{ {30, 31, 32},  {35, 36, 37} },
{ {40, 41, 42},  {45, 46, 47} } };

// Collapse to a single dimension, leaving one dimension.
let v012 = Collapse(v, {0,1,2});
then v012 == f32[24] {10, 11, 12, 15, 16, 17,
20, 21, 22, 25, 26, 27,
30, 31, 32, 35, 36, 37,
40, 41, 42, 45, 46, 47};

// Collapse the two lower dimensions, leaving two dimensions.
let v01 = Collapse(v, {0,1});
then v01 == f32[4x6] { {10, 11, 12, 15, 16, 17},
{20, 21, 22, 25, 26, 27},
{30, 31, 32, 35, 36, 37},
{40, 41, 42, 45, 46, 47} };

// Collapse the two higher dimensions, leaving two dimensions.
let v12 = Collapse(v, {1,2});
then v12 == f32[8x3] { {10, 11, 12},
{15, 16, 17},
{20, 21, 22},
{25, 26, 27},
{30, 31, 32},
{35, 36, 37},
{40, 41, 42},
{45, 46, 47} };

CollectivePermute

See also XlaBuilder::CollectivePermute.

CollectivePermute is a collective operation that sends and receives data cross replicas.

CollectivePermute(operand, source_target_pairs)

Note that there are the following restrictions on the source_target_pair:

• Any two pairs should not have the same target replica id, and they should not have the same source replica id.
• If a replica id is not a target in any pair, then the output on that replica is a tensor consisting of 0(s) with the same shape as the input.

Concatenate

See also XlaBuilder::ConcatInDim.

Concatenate composes an array from multiple array operands. The array is of the same rank as each of the input array operands (which must be of the same rank as each other) and contains the arguments in the order that they were specified.

Concatenate(operands..., dimension)

With the exception of dimension all dimensions must be the same. This is because XLA does not support "ragged" arrays. Also note that rank-0 values cannot be concatenated (as it's impossible to name the dimension along which the concatenation occurs).

1-dimensional example:

Concat({ {2, 3}, {4, 5}, {6, 7} }, 0)
>>> {2, 3, 4, 5, 6, 7}

2-dimensional example:

let a = {
{1, 2},
{3, 4},
{5, 6},
};
let b = {
{7, 8},
};
Concat({a, b}, 0)
>>> {
{1, 2},
{3, 4},
{5, 6},
{7, 8},
}

Diagram:

Conditional

See also XlaBuilder::Conditional.

Conditional(pred, true_operand, true_computation, false_operand, false_computation)

Executes true_computation if pred is true, false_computation if pred is false, and returns the result.

The true_computation must take in a single argument of type \(T_0\) and will be invoked with true_operand which must be of the same type. The false_computation must take in a single argument of type \(T_1\) and will be invoked with false_operand which must be of the same type. The type of the returned value of true_computation and false_computation must be the same.

Note that only one of true_computation and false_computation will be executed depending on the value of pred.

Conditional(branch_index, branch_computations, branch_operands)

Executes branch_computations[branch_index], and returns the result. If branch_index is an S32 which is < 0 or >= N, then branch_computations[N-1] is executed as the default branch.

Each branch_computations[b] must take in a single argument of type \(T_b\) and will be invoked with branch_operands[b] which must be of the same type. The type of the returned value of each branch_computations[b] must be the same.

Note that only one of the branch_computations will be executed depending on the value of branch_index.

Conv (convolution)

See also XlaBuilder::Conv.

As ConvWithGeneralPadding, but the padding is specified in a short-hand way as either SAME or VALID. SAME padding pads the input (lhs) with zeroes so that the output has the same shape as the input when not taking striding into account. VALID padding simply means no padding.

ConvWithGeneralPadding (convolution)

See also XlaBuilder::ConvWithGeneralPadding.

Computes a convolution of the kind used in neural networks. Here, a convolution can be thought of as a n-dimensional window moving across a n-dimensional base area and a computation is performed for each possible position of the window.

Let n be the number of spatial dimensions. The lhs argument is a rank n+2 array describing the base area. This is called the input, even though of course the rhs is also an input. In a neural network, these are the input activations. The n+2 dimensions are, in this order:

• batch: Each coordinate in this dimension represents an independent input for which convolution is carried out.
• z/depth/features: Each (y,x) position in the base area has a vector associated to it, which goes into this dimension.
• spatial_dims: Describes the n spatial dimensions that define the base area that the window moves across.

The rhs argument is a rank n+2 array describing the convolutional filter/kernel/window. The dimensions are, in this order:

• output-z: The z dimension of the output.
• input-z: The size of this dimension times feature_group_count should equal the size of the z dimension in lhs.
• spatial_dims: Describes the n spatial dimensions that define the n-d window that moves across the base area.

The window_strides argument specifies the stride of the convolutional window in the spatial dimensions. For example, if the stride in the first spatial dimension is 3, then the window can only be placed at coordinates where the first spatial index is divisible by 3.

The padding argument specifies the amount of zero padding to be applied to the base area. The amount of padding can be negative -- the absolute value of negative padding indicates the number of elements to remove from the specified dimension before doing the convolution. padding[0] specifies the padding for dimension y and padding[1] specifies the padding for dimension x. Each pair has the low padding as the first element and the high padding as the second element. The low padding is applied in the direction of lower indices while the high padding is applied in the direction of higher indices. For example, if padding[1] is (2,3) then there will be a padding by 2 zeroes on the left and by 3 zeroes on the right in the second spatial dimension. Using padding is equivalent to inserting those same zero values into the input (lhs) before doing the convolution.

The lhs_dilation and rhs_dilation arguments specify the dilation factor to be applied to the lhs and rhs, respectively, in each spatial dimension. If the dilation factor in a spatial dimension is d, then d-1 holes are implicitly placed between each of the entries in that dimension, increasing the size of the array. The holes are filled with a no-op value, which for convolution means zeroes.

Dilation of the rhs is also called atrous convolution. For more details, see tf.nn.atrous_conv2d. Dilation of the lhs is also called transposed convolution. For more details, see tf.nn.conv2d_transpose.

The feature_group_count argument (default value 1) can be used for grouped convolutions. feature_group_count needs to be a divisor of both the input and the output feature dimension. If feature_group_count is greater than 1, it means that conceptually the input and output feature dimension and the rhs output feature dimension are split evenly into many feature_group_count groups, each group consisting of a consecutive subsequence of features. The input feature dimension of rhs needs to be equal to the lhs input feature dimension divided by feature_group_count (so it already has the size of a group of input features). The i-th groups are used together to compute feature_group_count for many separate convolutions. The results of these convolutions are concatenated together in the output feature dimension.

For depthwise convolution the feature_group_count argument would be set to the input feature dimension, and the filter would be reshaped from [filter_height, filter_width, in_channels, channel_multiplier] to [filter_height, filter_width, 1, in_channels * channel_multiplier]. For more details, see tf.nn.depthwise_conv2d.

The batch_group_count (default value 1) argument can be used for grouped filters during backpropagation. batch_group_count needs to be a divisor of the size of the lhs (input) batch dimension. If batch_group_count is greater than 1, it means that the output batch dimension should be of size input batch / batch_group_count. The batch_group_count must be a divisor of the output feature size.

The output shape has these dimensions, in this order:

• batch: The size of this dimension times batch_group_count should equal the size of the batch dimension in lhs.
• z: Same size as output-z on the kernel (rhs).
• spatial_dims: One value for each valid placement of the convolutional window.

The figure above shows how the batch_group_count field works. Effectively, we slice each lhs batch into batch_group_count groups, and do the same for the output features. Then, for each of these groups we do pairwise convolutions and concatenate the output along the output feature dimension. The operational semantics of all the other dimensions (feature and spatial) remain the same.

The valid placements of the convolutional window are determined by the strides and the size of the base area after padding.

To describe what a convolution does, consider a 2d convolution, and pick some fixed batch, z, y, x coordinates in the output. Then (y,x) is a position of a corner of the window within the base area (e.g. the upper left corner, depending on how you interpret the spatial dimensions). We now have a 2d window, taken from the base area, where each 2d point is associated to a 1d vector, so we get a 3d box. From the convolutional kernel, since we fixed the output coordinate z, we also have a 3d box. The two boxes have the same dimensions, so we can take the sum of the element-wise products between the two boxes (similar to a dot product). That is the output value.

Note that if output-z is e.g., 5, then each position of the window produces 5 values in the output into the z dimension of the output. These values differ in what part of the convolutional kernel is used - there is a separate 3d box of values used for each output-z coordinate. So you could think of it as 5 separate convolutions with a different filter for each of them.

Here is pseudo-code for a 2d convolution with padding and striding:

for (b, oz, oy, ox) {  // output coordinates
  value = 0;
  for (iz, ky, kx) {  // kernel coordinates and input z
    iy = oy*stride_y + ky - pad_low_y;
    ix = ox*stride_x + kx - pad_low_x;
    if ((iy, ix) inside the base area considered without padding) {
      value += input(b, iz, iy, ix) * kernel(oz, iz, ky, kx);
    }
  }
  output(b, oz, oy, ox) = value;
}

ConvertElementType

See also XlaBuilder::ConvertElementType.

Similar to an element-wise static_cast in C++, performs an element-wise conversion operation from a data shape to a target shape. The dimensions must match, and the conversion is an element-wise one; e.g. s32 elements become f32 elements via an s32-to-f32 conversion routine.

ConvertElementType(operand, new_element_type)

The dimensions of the operand and the target shape must match. The source and destination element types must not be tuples.

A conversion such as T=s32 to U=f32 will perform a normalizing int-to-float conversion routine such as round-to-nearest-even.

let a: s32[3] = {0, 1, 2};
let b: f32[3] = convert(a, f32);
then b == f32[3]{0.0, 1.0, 2.0}

CrossReplicaSum

Performs AllReduce with a summation computation.

CustomCall

See also XlaBuilder::CustomCall.

Call a user-provided function within a computation.

CustomCall(target_name, args..., shape)

The function signature is the same, regardless of the arity or type of args:

extern "C" void target_name(void* out, void** in);

For example, if CustomCall is used as follows:

let x = f32[2] {1,2};
let y = f32[2x3] { {10, 20, 30}, {40, 50, 60} };

CustomCall("myfunc", {x, y}, f32[3x3])

Here is an example of an implementation of myfunc:

extern "C" void myfunc(void* out, void** in) {
  float (&x)[2] = *static_cast<float(*)[2]>(in[0]);
  float (&y)[2][3] = *static_cast<float(*)[2][3]>(in[1]);
  EXPECT_EQ(1, x[0]);
  EXPECT_EQ(2, x[1]);
  EXPECT_EQ(10, y[0][0]);
  EXPECT_EQ(20, y[0][1]);
  EXPECT_EQ(30, y[0][2]);
  EXPECT_EQ(40, y[1][0]);
  EXPECT_EQ(50, y[1][1]);
  EXPECT_EQ(60, y[1][2]);
  float (&z)[3][3] = *static_cast<float(*)[3][3]>(out);
  z[0][0] = x[1] + y[1][0];
  // ...
}

The user-provided function must not have side-effects and its execution must be idempotent.

Dot

See also XlaBuilder::Dot.

Dot(lhs, rhs)

The exact semantics of this operation depend on the ranks of the operands:

The operation performs sum of products over the second dimension of lhs (or the first if it has rank 1) and the first dimension of rhs. These are the "contracted" dimensions. The contracted dimensions of lhs and rhs must be of the same size. In practice, it can be used to perform dot products between vectors, vector/matrix multiplications or matrix/matrix multiplications.

DotGeneral

See also XlaBuilder::DotGeneral.

DotGeneral(lhs, rhs, dimension_numbers)

Similar to Dot, but allows contracting and batch dimension numbers to be specified for both the lhs and rhs.

DotGeneral performs the sum of products over contracting dimensions specified in dimension_numbers.

Associated contracting dimension numbers from the lhs and rhs do not need to be the same but must have the same dimension sizes.

Example with contracting dimension numbers:

lhs = { {1.0, 2.0, 3.0},
{4.0, 5.0, 6.0} }

rhs = { {1.0, 1.0, 1.0},
{2.0, 2.0, 2.0} }

DotDimensionNumbers dnums;
dnums.add_lhs_contracting_dimensions(1);
dnums.add_rhs_contracting_dimensions(1);

DotGeneral(lhs, rhs, dnums) -> { {6.0, 12.0},
{15.0, 30.0} }

Associated batch dimension numbers from the lhs and rhs must have the same dimension sizes.

Example with batch dimension numbers (batch size 2, 2x2 matrices):

lhs = { { {1.0, 2.0},
{3.0, 4.0} },
{ {5.0, 6.0},
{7.0, 8.0} } }

rhs = { { {1.0, 0.0},
{0.0, 1.0} },
{ {1.0, 0.0},
{0.0, 1.0} } }

DotDimensionNumbers dnums;
dnums.add_lhs_contracting_dimensions(2);
dnums.add_rhs_contracting_dimensions(1);
dnums.add_lhs_batch_dimensions(0);
dnums.add_rhs_batch_dimensions(0);

DotGeneral(lhs, rhs, dnums) -> { { {1.0, 2.0},
{3.0, 4.0} },
{ {5.0, 6.0},
{7.0, 8.0} } }

It follows that the resulting dimension number starts with the batch dimension, then the lhs non-contracting/non-batch dimension, and finally the rhs non-contracting/non-batch dimension.

DynamicSlice

See also XlaBuilder::DynamicSlice.

DynamicSlice extracts a sub-array from the input array at dynamic start_indices. The size of the slice in each dimension is passed in size_indices, which specify the end point of exclusive slice intervals in each dimension: [start, start + size). The shape of start_indices must be rank == 1, with dimension size equal to the rank of operand.

DynamicSlice(operand, start_indices, size_indices)

The effective slice indices are computed by applying the following transformation for each index i in [1, N) before performing the slice:

start_indices[i] = clamp(start_indices[i], 0, operand.dimension_size[i] - size_indices[i])

This ensures that the extracted slice is always in-bounds with respect to the operand array. If the slice is in-bounds before the transformation is applied, the transformation has no effect.

1-dimensional example:

let a = {0.0, 1.0, 2.0, 3.0, 4.0}
let s = {2}

DynamicSlice(a, s, {2}) produces:
{2.0, 3.0}

2-dimensional example:

let b =
{ {0.0,  1.0,  2.0},
{3.0,  4.0,  5.0},
{6.0,  7.0,  8.0},
{9.0, 10.0, 11.0} }
let s = {2, 1}

DynamicSlice(b, s, {2, 2}) produces:
{ { 7.0,  8.0},
{10.0, 11.0} }

DynamicUpdateSlice

See also XlaBuilder::DynamicUpdateSlice.

DynamicUpdateSlice generates a result which is the value of the input array operand, with a slice update overwritten at start_indices. The shape of update determines the shape of the sub-array of the result which is updated. The shape of start_indices must be rank == 1, with dimension size equal to the rank of operand.

DynamicUpdateSlice(operand, update, start_indices)

The effective slice indices are computed by applying the following transformation for each index i in [1, N) before performing the slice:

start_indices[i] = clamp(start_indices[i], 0, operand.dimension_size[i] - update.dimension_size[i])

This ensures that the updated slice is always in-bounds with respect to the operand array. If the slice is in-bounds before the transformation is applied, the transformation has no effect.

1-dimensional example:

let a = {0.0, 1.0, 2.0, 3.0, 4.0}
let u = {5.0, 6.0}
let s = {2}

DynamicUpdateSlice(a, u, s) produces:
{0.0, 1.0, 5.0, 6.0, 4.0}

2-dimensional example:

let b =
{ {0.0,  1.0,  2.0},
{3.0,  4.0,  5.0},
{6.0,  7.0,  8.0},
{9.0, 10.0, 11.0} }
let u =
{ {12.0,  13.0},
{14.0,  15.0},
{16.0,  17.0} }

let s = {1, 1}

DynamicUpdateSlice(b, u, s) produces:
{ {0.0,  1.0,  2.0},
{3.0, 12.0, 13.0},
{6.0, 14.0, 15.0},
{9.0, 16.0, 17.0} }

Element-wise binary arithmetic operations

See also XlaBuilder::Add.

A set of element-wise binary arithmetic operations is supported.

Op(lhs, rhs)

Where Op is one of Add (addition), Sub(subtraction), Mul (multiplication), Div (division), Pow (power), Rem (remainder), Max (maximum), Min (minimum), And (logical AND), Or (logical OR), Xor (logical XOR), ShiftLeft (Left Shift), ShiftRightArithmetic (arithmetic Right Shift), ShiftRightLogical (logical Right Shift), Atan2 (2-argument arctangent), or Complex (combines real and imaginary parts into a complex number)

The arguments' shapes have to be either similar or compatible. See the broadcasting documentation about what it means for shapes to be compatible. The result of an operation has a shape which is the result of broadcasting the two input arrays. In this variant, operations between arrays of different ranks are not supported, unless one of the operands is a scalar.

When Op is Rem, the sign of the result is taken from the dividend, and the absolute value of the result is always less than the divisor's absolute value.

Integer division overflow (signed/unsigned division/remainder by zero or signed division/remainder of INT_SMIN with -1) produces an implementation defined value.

An alternative variant with different-rank broadcasting support exists for these operations:

Op(lhs, rhs, broadcast_dimensions)

Where Op is the same as above. This variant of the operation should be used for arithmetic operations between arrays of different ranks (such as adding a matrix to a vector).

The additional broadcast_dimensions operand is a slice of integers used to expand the rank of the lower-rank operand up to the rank of the higher-rank operand. broadcast_dimensions maps the dimensions of the lower-rank shape to the dimensions of the higher-rank shape. The unmapped dimensions of the expanded shape are filled with dimensions of size one. Degenerate-dimension broadcasting then broadcasts the shapes along these degenerate dimensions to equalize the shapes of both operands. The semantics are described in detail on the broadcasting page.

Element-wise comparison operations

See also XlaBuilder::Eq.

A set of standard element-wise binary comparison operations is supported. Note that standard IEEE 754 floating-point comparison semantics apply when comparing floating-point types.

Op(lhs, rhs)

Where Op is one of Eq (equal-to), Ne (not equal-to), Ge (greater-or-equal-than), Gt (greater-than), Le (less-or-equal-than), Lt (less-than). Another set of operators, EqTotalOrder, NeTotalOrder, GeTotalOrder, GtTotalOrder, LeTotalOrder, and LtTotalOrder, provide the same functionalities, except that they additionally support a total order over the floating point numbers, by enforcing -NaN < -Inf < -Finite < -0 < +0 < +Finite < +Inf < +NaN.

The arguments' shapes have to be either similar or compatible. See the broadcasting documentation about what it means for shapes to be compatible. The result of an operation has a shape which is the result of broadcasting the two input arrays with the element type PRED. In this variant, operations between arrays of different ranks are not supported, unless one of the operands is a scalar.

An alternative variant with different-rank broadcasting support exists for these operations:

Op(lhs, rhs, broadcast_dimensions)

Where Op is the same as above. This variant of the operation should be used for comparison operations between arrays of different ranks (such as adding a matrix to a vector).

The additional broadcast_dimensions operand is a slice of integers specifying the dimensions to use for broadcasting the operands. The semantics are described in detail on the broadcasting page.

Element-wise unary functions

XlaBuilder supports these element-wise unary functions:

Abs(operand) Element-wise abs x -> |x|.

Cbrt(operand) Element-wise cubic root operation x -> cbrt(x).

Ceil(operand) Element-wise ceil x -> ⌈x⌉.

Clz(operand) Element-wise count leading zeros.

Cos(operand) Element-wise cosine x -> cos(x).

Erf(operand) Element-wise error function x -> erf(x) where

\(\text{erf}(x) = \frac{2}{\sqrt{\pi} }\int_0^x e^{-t^2} \, dt\).

Exp(operand) Element-wise natural exponential x -> e^x.

Expm1(operand) Element-wise natural exponential minus one x -> e^x - 1.

Floor(operand) Element-wise floor x -> ⌊x⌋.

Imag(operand) Element-wise imaginary part of a complex (or real) shape. x -> imag(x). If the operand is a floating point type, returns 0.

IsFinite(operand) Tests whether each element of operand is finite, i.e., is not positive or negative infinity, and is not NaN. Returns an array of PRED values with the same shape as the input, where each element is true if and only if the corresponding input element is finite.

Log(operand) Element-wise natural logarithm x -> ln(x).

Log1p(operand) Element-wise shifted natural logarithm x -> ln(1+x).

Logistic(operand) Element-wise logistic function computation x -> logistic(x).

Neg(operand) Element-wise negation x -> -x.

Not(operand) Element-wise logical not x -> !(x).

PopulationCount(operand) Computes the number of bits set in each element of operand.

Real(operand) Element-wise real part of a complex (or real) shape. x -> real(x). If the operand is a floating point type, returns the same value.

Round(operand) Element-wise rounding, ties away from zero.

RoundNearestEven(operand) Element-wise rounding, ties to nearest even.

Rsqrt(operand) Element-wise reciprocal of square root operation x -> 1.0 / sqrt(x).

Sign(operand) Element-wise sign operation x -> sgn(x) where

\[\text{sgn}(x) = \begin{cases} -1 & x < 0\\ -0 & x = -0\\ NaN & x = NaN\\ +0 & x = +0\\ 1 & x > 0 \end{cases}\]

using the comparison operator of the element type of operand.

Sin(operand) Element-wise sine x -> sin(x).

Sqrt(operand) Element-wise square root operation x -> sqrt(x).

Tan(operand) Element-wise tangent x -> tan(x).

Tanh(operand) Element-wise hyperbolic tangent x -> tanh(x).

The function is applied to each element in the operand array, resulting in an array with the same shape. It is allowed for operand to be a scalar (rank 0).

Fft

The XLA FFT operation implements the forward and inverse Fourier Transforms for real and complex inputs/outputs. Multidimensional FFTs on up to 3 axes are supported.

See also XlaBuilder::Fft.

Multidimensional FFT

When more than 1 fft_length is provided, this is equivalent to applying a cascade of FFT operations to each of the innermost axes. Note that for the real->complex and complex->real cases, the innermost axis transform is (effectively) performed first (RFFT; last for IRFFT), which is why the innermost axis is the one which changes size. Other axis transforms will then be complex->complex.

Implementation details

CPU FFT is backed by Eigen's TensorFFT. GPU FFT uses cuFFT.

Gather

The XLA gather operation stitches together several slices (each slice at a potentially different runtime offset) of an input array.

General Semantics

See also XlaBuilder::Gather. For a more intuitive description, see the "Informal Description" section below.

gather(operand, start_indices, offset_dims, collapsed_slice_dims, slice_sizes, start_index_map)

For convenience, we label dimensions in the output array not in offset_dims as batch_dims.

The output is an array of rank batch_dims.size + offset_dims.size.

The operand.rank must equal the sum of offset_dims.size and collapsed_slice_dims.size. Also, slice_sizes.size has to be equal to operand.rank.

If index_vector_dim is equal to start_indices.rank we implicitly consider start_indices to have a trailing 1 dimension (i.e. if start_indices was of shape [6,7] and index_vector_dim is 2 then we implicitly consider the shape of start_indices to be [6,7,1]).

The bounds for the output array along dimension i is computed as follows:

1. If i is present in batch_dims (i.e. is equal to batch_dims[k] for some k) then we pick the corresponding dimension bounds out of start_indices.shape, skipping index_vector_dim (i.e. pick start_indices.shape.dims[k] if k < index_vector_dim and start_indices.shape.dims[k+1] otherwise).

2. If i is present in offset_dims (i.e. equal to offset_dims[k] for some k) then we pick the corresponding bound out of slice_sizes after accounting for collapsed_slice_dims (i.e. we pick adjusted_slice_sizes[k] where adjusted_slice_sizes is slice_sizes with the bounds at indices collapsed_slice_dims removed).

Formally, the operand index In corresponding to a given output index Out is calculated as follows:

1. Let G = { Out[k] for k in batch_dims }. Use G to slice out a vector S such that S[i] = start_indices[Combine(G, i)] where Combine(A, b) inserts b at position index_vector_dim into A. Note that this is well defined even if G is empty: If G is empty then S = start_indices.

2. Create a starting index, Sin, into operand using S by scattering S using start_index_map. More precisely:

   1. Sin[start_index_map[k]] = S[k] if k < start_index_map.size.

   2. Sin[_] = 0 otherwise.

3. Create an index Oin into operand by scattering the indices at the offset dimensions in Out according to the collapsed_slice_dims set. More precisely:

   1. Oin[remapped_offset_dims(k)] = Out[offset_dims[k]] if k < offset_dims.size (remapped_offset_dims is defined below).

   2. Oin[_] = 0 otherwise.

4. In is Oin + Sin where + is element-wise addition.

remapped_offset_dims is a monotonic function with domain [0, offset_dims.size) and range [0, operand.rank) \ collapsed_slice_dims. So if, e.g., offset_dims.size is 4, operand.rank is 6 and collapsed_slice_dims is {0, 2} then remapped_offset_dims is {0→1, 1→3, 2→4, 3→5}.

If indices_are_sorted is set to true then XLA can assume that start_indices are sorted (in ascending order, after scattering its values according to start_index_map) by the user. If they are not then the semantics are implementation defined.

Informal Description and Examples

Informally, every index Out in the output array corresponds to an element E in the operand array, computed as follows:

• We use the batch dimensions in Out to look up a starting index from start_indices.

• We use start_index_map to map the starting index (whose size may be less than operand.rank) to a "full" starting index into the operand.

• We dynamic-slice out a slice with size slice_sizes using the full starting index.

• We reshape the slice by collapsing the collapsed_slice_dims dimensions. Since all collapsed slice dimensions must have a bound of 1, this reshape is always legal.

• We use the offset dimensions in Out to index into this slice to get the input element, E, corresponding to output index Out.

index_vector_dim is set to start_indices.rank - 1 in all of the examples that follow. More interesting values for index_vector_dim do not change the operation fundamentally, but make the visual representation more cumbersome.

To get an intuition on how all of the above fits together, let's look at an example that gathers 5 slices of shape [8,6] from a [16,11] array. The position of a slice into the [16,11] array can be represented as an index vector of shape S64[2], so the set of 5 positions can be represented as a S64[5,2] array.

The behavior of the gather operation can then be depicted as an index transformation that takes [G,O0,O1], an index in the output shape, and maps it to an element in the input array in the following way:

We first select an (X,Y) vector from the gather indices array using G. The element in the output array at index [G,O0,O1] is then the element in the input array at index [X+O0,Y+O1].

slice_sizes is [8,6], which decides the range of O₀ and O₁, and this in turn decides the bounds of the slice.

This gather operation acts as a batch dynamic slice with G as the batch dimension.

The gather indices may be multidimensional. For instance, a more general version of the example above using a "gather indices" array of shape [4,5,2] would translate indices like this:

Again, this acts as a batch dynamic slice G0 and G1 as the batch dimensions. The slice size is still [8,6].

The gather operation in XLA generalizes the informal semantics outlined above in the following ways:

1. We can configure which dimensions in the output shape are the offset dimensions (dimensions containing O0, O1 in the last example). The output batch dimensions (dimensions containing G0, G1 in the last example) are defined to be the output dimensions that are not offset dimensions.

2. The number of output offset dimensions explicitly present in the output shape may be smaller than the input rank. These "missing" dimensions, which are listed explicitly as collapsed_slice_dims, must have a slice size of 1. Since they have a slice size of 1 the only valid index for them is 0 and eliding them does not introduce ambiguity.

3. The slice extracted from the "Gather Indices" array ((X, Y) in the last example) may have fewer elements than the input array rank, and an explicit mapping dictates how the index should be expanded to have the same rank as the input.

As a final example, we use (2) and (3) to implement tf.gather_nd:

G0 and G1 are used to slice out a starting index from the gather indices array as usual, except the starting index has only one element, X. Similarly, there is only one output offset index with the value O0. However, before being used as indices into the input array, these are expanded in accordance to "Gather Index Mapping" (start_index_map in the formal description) and "Offset Mapping" (remapped_offset_dims in the formal description) into [X,0] and [0,O0] respectively, adding up to [X,O0]. In other words, the output index [G0,G1,O0] maps to the input index [GatherIndices[G0,G1,0],O0] which gives us the semantics for tf.gather_nd.

slice_sizes for this case is [1,11]. Intuitively this means that every index X in the gather indices array picks an entire row and the result is the concatenation of all these rows.

GetDimensionSize

See also XlaBuilder::GetDimensionSize.

Returns the size of the given dimension of the operand. The operand must be array shaped.

GetDimensionSize(operand, dimension)

SetDimensionSize

See also XlaBuilder::SetDimensionSize.

Sets the dynamic size of XlaOp's given dimension. The operand must be array shaped.

SetDimensionSize(operand, size, dimension)

Pass through the operand as result, with dynamic dimension tracked by the compiler.

Padded values will be ignored by downstream reduction ops.

let v: f32[10] = f32[10]{1, 2, 3, 4, 5, 6, 7, 8, 9, 10};
let five: s32 = 5;
let six: s32 = 6;

// Setting dynamic dimension size doesn't change the upper bound of the static
// shape.
let padded_v_five: f32[10] = set_dimension_size(v, five, /*dimension=*/0);
let padded_v_six: f32[10] = set_dimension_size(v, six, /*dimension=*/0);

// sum == 1 + 2 + 3 + 4 + 5
let sum:f32[] = reduce_sum(padded_v_five);
// product == 1 * 2 * 3 * 4 * 5
let product:f32[] = reduce_product(padded_v_five);

// Changing padding size will yield different result.
// sum == 1 + 2 + 3 + 4 + 5 + 6
let sum:f32[] = reduce_sum(padded_v_six);

GetTupleElement

See also XlaBuilder::GetTupleElement.

Indexes into a tuple with a compile-time-constant value.

The value must be a compile-time-constant so that shape inference can determine the type of the resulting value.

This is analogous to std::get<int N>(t) in C++. Conceptually:

let v: f32[10] = f32[10]{0, 1, 2, 3, 4, 5, 6, 7, 8, 9};
let s: s32 = 5;
let t: (f32[10], s32) = tuple(v, s);
let element_1: s32 = gettupleelement(t, 1);  // Inferred shape matches s32.

See also tf.tuple.

Infeed

See also XlaBuilder::Infeed.

Infeed(shape)

Reads a single data item from the implicit Infeed streaming interface of the device, interpreting the data as the given shape and its layout, and returns a XlaOp of the data. Multiple Infeed operations are allowed in a computation, but there must be a total order among the Infeed operations. For example, two Infeeds in the code below have a total order since there is a dependency between the while loops.

result1 = while (condition, init = init_value) {
  Infeed(shape)
}

result2 = while (condition, init = result1) {
  Infeed(shape)
}

Nested tuple shapes are not supported. For an empty tuple shape, the Infeed operation is effectively a no-op and proceeds without reading any data from the Infeed of the device.

Iota

See also XlaBuilder::Iota.

Iota(shape, iota_dimension)

Builds a constant literal on device rather than a potentially large host transfer. Creates an array that has specified shape and holds values starting at zero and incrementing by one along the specified dimension. For floating-point types, the produced array is equivalent to ConvertElementType(Iota(...)) where the Iota is of integral type and the conversion is to the floating-point type.

For example, Iota(s32[4, 8], 0) returns

  [[0, 0, 0, 0, 0, 0, 0, 0 ],
   [1, 1, 1, 1, 1, 1, 1, 1 ],
   [2, 2, 2, 2, 2, 2, 2, 2 ],
   [3, 3, 3, 3, 3, 3, 3, 3 ]]

Iota(s32[4, 8], 1) returns

  [[0, 1, 2, 3, 4, 5, 6, 7 ],
   [0, 1, 2, 3, 4, 5, 6, 7 ],
   [0, 1, 2, 3, 4, 5, 6, 7 ],
   [0, 1, 2, 3, 4, 5, 6, 7 ]]

Map

See also XlaBuilder::Map.

Map(operands..., computation)

Applies a scalar function over the given operands arrays, producing an array of the same dimensions where each element is the result of the mapped function applied to the corresponding elements in the input arrays.

The mapped function is an arbitrary computation with the restriction that it has N inputs of scalar type T and a single output with type S. The output has the same dimensions as the operands except that the element type T is replaced with S.

For example: Map(op1, op2, op3, computation, par1) maps elem_out <- computation(elem1, elem2, elem3, par1) at each (multi-dimensional) index in the input arrays to produce the output array.

OptimizationBarrier

Blocks any optimization pass from moving computations across the barrier.

Ensures that all inputs are evaluated before any operators that depend on the barrier's outputs.

Pad

See also XlaBuilder::Pad.

Pad(operand, padding_value, padding_config)

Expands the given operand array by padding around the array as well as between the elements of the array with the given padding_value. padding_config specifies the amount of edge padding and the interior padding for each dimension.

PaddingConfig is a repeated field of PaddingConfigDimension, which contains three fields for each dimension: edge_padding_low, edge_padding_high, and interior_padding.

edge_padding_low and edge_padding_high specify the amount of padding added at the low-end (next to index 0) and the high-end (next to the highest index) of each dimension respectively. The amount of edge padding can be negative -- the absolute value of negative padding indicates the number of elements to remove from the specified dimension.

interior_padding specifies the amount of padding added between any two elements in each dimension; it may not be negative. Interior padding occurs logically before edge padding, so in the case of negative edge padding, elements are removed from the interior-padded operand.

This operation is a no-op if the edge padding pairs are all (0, 0) and the interior padding values are all 0. The figure below shows examples of different edge_padding and interior_padding values for a two-dimensional array.

Recv

See also XlaBuilder::Recv.

Recv(shape, channel_handle)

Receives data of the given shape from a Send instruction in another computation that shares the same channel handle. Returns a XlaOp for the received data.

The client API of Recv operation represents synchronous communication. However, the instruction is internally decomposed into 2 HLO instructions (Recv and RecvDone) to enable asynchronous data transfers. See also HloInstruction::CreateRecv and HloInstruction::CreateRecvDone.

Recv(const Shape& shape, int64 channel_id)

Allocates resources required to receive data from a Send instruction with the same channel_id. Returns a context for the allocated resources, which is used by a following RecvDone instruction to wait for the completion of the data transfer. The context is a tuple of {receive buffer (shape), request identifier (U32)} and it can only be used by a RecvDone instruction.

RecvDone(HloInstruction context)

Given a context created by a Recv instruction, waits for the data transfer to complete and returns the received data.

Reduce

See also XlaBuilder::Reduce.

Applies a reduction function to one or more arrays in parallel.

Reduce(operands..., init_values..., computation, dimensions)

Where:

• N is required to be greater or equal to 1.
• The computation has to be "roughly" associative (see below).
• All input arrays must have the same dimensions.
• All initial values have to form an identity under computation.
• If N = 1, Collate(T) is T.
• If N > 1, Collate(T_0, ..., T_{N-1}) is a tuple of N elements of type T.

This operation reduces one or more dimensions of each input array into scalars. The rank of each returned array is rank(operand) - len(dimensions). The output of the op is Collate(Q_0, ..., Q_N) where Q_i is an array of type T_i, the dimensions of which are described below.

Different backends are allowed to reassociate the reduction computation. This can lead to numerical differences, as some reduction functions like addition are not associative for floats. However, if the range of the data is limited, floating-point addition is close enough to being associative for most practical uses.

Examples

When reducing across one dimension in a single 1D array with values [10, 11, 12, 13], with reduction function f (this is computation) then that could be computed as

f(10, f(11, f(12, f(init_value, 13)))

but there are also many other possibilities, e.g.

f(init_value, f(f(10, f(init_value, 11)), f(f(init_value, 12), f(init_value, 13))))

The following is a rough pseudo-code example of how reduction could be implemented, using summation as the reduction computation with an initial value of 0.

result_shape <- remove all dims in dimensions from operand_shape

# Iterate over all elements in result_shape. The number of r's here is equal
# to the rank of the result
for r0 in range(result_shape[0]), r1 in range(result_shape[1]), ...:
  # Initialize this result element
  result[r0, r1...] <- 0

  # Iterate over all the reduction dimensions
  for d0 in range(dimensions[0]), d1 in range(dimensions[1]), ...:
    # Increment the result element with the value of the operand's element.
    # The index of the operand's element is constructed from all ri's and di's
    # in the right order (by construction ri's and di's together index over the
    # whole operand shape).
    result[r0, r1...] += operand[ri... di]

Here's an example of reducing a 2D array (matrix). The shape has rank 2, dimension 0 of size 2 and dimension 1 of size 3:

Results of reducing dimensions 0 or 1 with an "add" function:

Note that both reduction results are 1D arrays. The diagram shows one as column and another as row just for visual convenience.

For a more complex example, here is a 3D array. Its rank is 3, dimension 0 of size 4, dimension 1 of size 2 and dimension 2 of size 3. For simplicity, the values 1 to 6 are replicated across dimension 0.

Similarly to the 2D example, we can reduce just one dimension. If we reduce dimension 0, for example, we get a rank-2 array where all values across dimension 0 were folded into a scalar:

|  4   8  12 |
| 16  20  24 |

If we reduce dimension 2, we also get a rank-2 array where all values across dimension 2 were folded into a scalar:

| 6  15 |
| 6  15 |
| 6  15 |
| 6  15 |

Note that the relative order between the remaining dimensions in the input is preserved in the output, but some dimensions may get assigned new numbers (since the rank changes).

We can also reduce multiple dimensions. Add-reducing dimensions 0 and 1 produces the 1D array [20, 28, 36].

Reducing the 3D array over all its dimensions produces the scalar 84.

Variadic Reduce

When N > 1, reduce function application is slightly more complex, as it is applied simultaneously to all inputs. The operands are supplied to the computation in the following order:

• Running reduced value for the first operand
• ...
• Running reduced value for the N'th operand
• Input value for the first operand
• ...
• Input value for the N'th operand

For example, consider the following reduction function, which can be used to compute the max and the argmax of a 1-D array in parallel:

f: (Float, Int, Float, Int) -> Float, Int
f(max, argmax, value, index):
  if value >= max:
    return (value, index)
  else:
    return (max, argmax)

For 1-D Input arrays V = Float[N], K = Int[N], and init values I_V = Float, I_K = Int, the result f_(N-1) of reducing across the only input dimension is equivalent to the following recursive application:

f_0 = f(I_V, I_K, V_0, K_0)
f_1 = f(f_0.first, f_0.second, V_1, K_1)
...
f_(N-1) = f(f_(N-2).first, f_(N-2).second, V_(N-1), K_(N-1))

Applying this reduction to an array of values, and an array of sequential indices (i.e. iota), will co-iterate over the arrays, and return a tuple containing the maximal value and the matching index.

ReducePrecision

See also XlaBuilder::ReducePrecision.

Models the effect of converting floating-point values to a lower-precision format (such as IEEE-FP16) and back to the original format. The number of exponent and mantissa bits in the lower-precision format can be specified arbitrarily, although all bit sizes may not be supported on all hardware implementations.

ReducePrecision(operand, mantissa_bits, exponent_bits)

The result is an array of type T. The input values are rounded to the nearest value representable with the given number of mantissa bits (using "ties to even" semantics), and any values that exceed the range specified by the number of exponent bits are clamped to positive or negative infinity. NaN values are retained, although they may be converted to canonical NaN values.

The lower-precision format must have at least one exponent bit (in order to distinguish a zero value from an infinity, since both have a zero mantissa), and must have a non-negative number of mantissa bits. The number of exponent or mantissa bits may exceed the corresponding value for type T; the corresponding portion of the conversion is then simply a no-op.

ReduceScatter

See also XlaBuilder::ReduceScatter.

ReduceScatter is a collective operation that effectively does an AllReduce and then scatters the result by splitting it into shard_count blocks along the scatter_dimension and replica i in the replica group receives the ith shard.

ReduceScatter(operand, computation, scatter_dim, shard_count, replica_group_ids, channel_id)

• When operand is a tuple of arrays, the reduce-scatter is performed on each element of the tuple.
• replica_groups is a list of replica groups between which the reduction is performed (replica id for the current replica can be retrieved using ReplicaId). The order of replicas in each group determines the order in which the all-reduce result will be scattered. replica_groups must either be empty (in which case all replicas belong to a single group), or contain the same number of elements as the number of replicas. When there are more than one replica groups, they all must be of the same size. For example, replica_groups = {0, 2}, {1, 3} performs reduction between the replicas 0 and 2, and 1 and 3 and then scatters the result.
• shard_count is the size of each replica group. We need this in cases where replica_groups are empty. If replica_groups is not empty, shard_count must be equal to the size of each replica group.
• channel_id is used for cross-module communication: only reduce-scatter operations with the same channel_id can communicate with each other.

The output shape is the input shape with the scatter_dimension made shard_count times smaller. For example, if there are two replicas and the operand has the value [1.0, 2.25] and [3.0, 5.25] respectively on the two replicas, then the output value from this op where scatter_dim is 0 will be [4.0] for the first replica and [7.5] for the second replica.

ReduceWindow

See also XlaBuilder::ReduceWindow.

Applies a reduction function to all elements in each window of a sequence of N multi-dimensional arrays, producing a single or a tuple of N multi-dimensional arrays as output. Each output array has the same number of elements as the number of valid positions of the window. A pooling layer can be expressed as a ReduceWindow. Similar to Reduce, the applied computation is always passed the init_values on the left-hand side.

ReduceWindow(operands..., init_values..., computation, window_dimensions, window_strides, padding)

Where:

• N is required to be greater or equal to 1.
• All input arrays must have the same dimensions.
• If N = 1, Collate(T) is T.
• If N > 1, Collate(T_0, ..., T_{N-1}) is a tuple of N elements of type (T0,...T{N-1}).

Below code and figure shows an example of using ReduceWindow. Input is a matrix of size [4x6] and both window_dimensions and window_stride_dimensions are [2x3].

// Create a computation for the reduction (maximum).
XlaComputation max;
{
  XlaBuilder builder(client_, "max");
  auto y = builder.Parameter(0, ShapeUtil::MakeShape(F32, {}), "y");
  auto x = builder.Parameter(1, ShapeUtil::MakeShape(F32, {}), "x");
  builder.Max(y, x);
  max = builder.Build().value();
}

// Create a ReduceWindow computation with the max reduction computation.
XlaBuilder builder(client_, "reduce_window_2x3");
auto shape = ShapeUtil::MakeShape(F32, {4, 6});
auto input = builder.Parameter(0, shape, "input");
builder.ReduceWindow(
    input,
    /*init_val=*/builder.ConstantLiteral(LiteralUtil::MinValue(F32)),
    *max,
    /*window_dimensions=*/{2, 3},
    /*window_stride_dimensions=*/{2, 3},
    Padding::kValid);

Stride of 1 in a dimension specifies that the position of a window in the dimension is 1 element away from its adjacent window. In order to specify that no windows overlap with each other, window_stride_dimensions should be equal to window_dimensions. The figure below illustrates the use of two different stride values. Padding is applied to each dimension of the input and the calculations are the same as though the input came in with the dimensions it has after padding.

For a non-trivial padding example, consider computing reduce-window minimum (initial value is MAX_FLOAT) with dimension 3 and stride 2 over the input array [10000, 1000, 100, 10, 1]. Padding kValid computes minimums over two valid windows: [10000, 1000, 100] and [100, 10, 1], resulting in the output [100, 1]. Padding kSame first pads the array so that the shape after the reduce-window would be the same as input for stride one by adding initial elements on both sides, getting [MAX_VALUE, 10000, 1000, 100, 10, 1, MAX_VALUE]. Running reduce-window over the padded array operates on three windows [MAX_VALUE, 10000, 1000], [1000, 100, 10], [10, 1, MAX_VALUE], and yields [1000, 10, 1].

The evaluation order of the reduction function is arbitrary and may be non-deterministic. Therefore, the reduction function should not be overly sensitive to reassociation. See the discussion about associativity in the context of Reduce for more details.

ReplicaId

See also XlaBuilder::ReplicaId.

Returns the unique ID (U32 scalar) of the replica.

ReplicaId()

The unique ID of each replica is an unsigned integer in the interval [0, N), where N is the number of replicas. Since all the replicas are running the same program, a ReplicaId() call in the program will return a different value on each replica.

Reshape

See also XlaBuilder::Reshape and the Collapse operation.

Reshapes the dimensions of an array into a new configuration.

Reshape(operand, new_sizes) Reshape(operand, dimensions, new_sizes)

Conceptually, reshape first flattens an array into a one-dimensional vector of data values, and then refines this vector into a new shape. The input arguments are an arbitrary array of type T, a compile-time-constant vector of dimension indices, and a compile-time-constant vector of dimension sizes for the result. The values in the dimension vector, if given, must be a permutation of all of T's dimensions; the default if not given is {0, ..., rank - 1}. The order of the dimensions in dimensions is from slowest-varying dimension (most major) to fastest-varying dimension (most minor) in the loop nest which collapses the input array into a single dimension. The new_sizes vector determines the size of the output array. The value at index 0 in new_sizes is the size of dimension 0, the value at index 1 is the size of dimension 1, and so on. The product of the new_size dimensions must equal the product of the operand's dimension sizes. When refining the collapsed array into the multidimensional array defined by new_sizes, the dimensions in new_sizes are ordered from slowest varying (most major) and to fastest varying (most minor).

For example, let v be an array of 24 elements:

let v = f32[4x2x3] { { {10, 11, 12}, {15, 16, 17} },
                    { {20, 21, 22}, {25, 26, 27} },
                    { {30, 31, 32}, {35, 36, 37} },
                    { {40, 41, 42}, {45, 46, 47} } };

In-order collapse:
let v012_24 = Reshape(v, {0,1,2}, {24});
then v012_24 == f32[24] {10, 11, 12, 15, 16, 17, 20, 21, 22, 25, 26, 27,
                         30, 31, 32, 35, 36, 37, 40, 41, 42, 45, 46, 47};

let v012_83 = Reshape(v, {0,1,2}, {8,3});
then v012_83 == f32[8x3] { {10, 11, 12}, {15, 16, 17},
                          {20, 21, 22}, {25, 26, 27},
                          {30, 31, 32}, {35, 36, 37},
                          {40, 41, 42}, {45, 46, 47} };

Out-of-order collapse:
let v021_24 = Reshape(v, {1,2,0}, {24});
then v012_24 == f32[24]  {10, 20, 30, 40, 11, 21, 31, 41, 12, 22, 32, 42,
                          15, 25, 35, 45, 16, 26, 36, 46, 17, 27, 37, 47};

let v021_83 = Reshape(v, {1,2,0}, {8,3});
then v021_83 == f32[8x3] { {10, 20, 30}, {40, 11, 21},
                          {31, 41, 12}, {22, 32, 42},
                          {15, 25, 35}, {45, 16, 26},
                          {36, 46, 17}, {27, 37, 47} };


let v021_262 = Reshape(v, {1,2,0}, {2,6,2});
then v021_262 == f32[2x6x2] { { {10, 20}, {30, 40},
                              {11, 21}, {31, 41},
                              {12, 22}, {32, 42} },
                             { {15, 25}, {35, 45},
                              {16, 26}, {36, 46},
                              {17, 27}, {37, 47} } };

As a special case, reshape can transform a single-element array to a scalar and vice versa. For example,

Reshape(f32[1x1] { {5} }, {0,1}, {}) == 5;
Reshape(5, {}, {1,1}) == f32[1x1] { {5} };

Rev (reverse)

See also XlaBuilder::Rev.

Rev(operand, dimensions)

Reverses the order of elements in the operand array along the specified dimensions, generating an output array of the same shape. Each element of the operand array at a multidimensional index is stored into the output array at a transformed index. The multidimensional index is transformed by reversing the index in each dimension to be reversed (i.e., if a dimension of size N is one of the reversing dimensions, its index i is transformed into N - 1 - i).

One use for the Rev operation is to reverse the convolution weight array along the two window dimensions during the gradient computation in neural networks.

RngNormal

See also XlaBuilder::RngNormal.

Constructs an output of a given shape with random numbers generated following the \(N(\mu, \sigma)\) normal distribution. The parameters \(\mu\) and \(\sigma\), and output shape have to have a floating point elemental type. The parameters furthermore have to be scalar valued.

RngNormal(mu, sigma, shape)

RngUniform

See also XlaBuilder::RngUniform.

Constructs an output of a given shape with random numbers generated following the uniform distribution over the interval \([a,b)\). The parameters and output element type have to be a boolean type, an integral type or a floating point types, and the types have to be consistent. The CPU and GPU backends currently only support F64, F32, F16, BF16, S64, U64, S32 and U32. Furthermore, the parameters need to be scalar valued. If \(b <= a\) the result is implementation-defined.

RngUniform(a, b, shape)

RngBitGenerator

Generates an output with a given shape filled with uniform random bits using the specified algorithm (or backend default) and returns an updated state (with the same shape as initial state) and the generated random data.

Initial state is the initial state of the current random number generation. It and the required shape and valid values are dependent on the algorithm used.

The output is guaranteed to be a deterministic function of the initial state but it is not guaranteed to be deterministic between backends and different compiler versions.

RngBitGenerator(algorithm, key, shape)

Available values for algorithm:

• rng_default: Backend specific algorithm with backend specific shape requirements.

• rng_three_fry: ThreeFry counter-based PRNG algorithm. The initial_state shape is u64[2] with arbitrary values. Salmon et al. SC 2011. Parallel random numbers: as easy as 1, 2, 3.

• rng_philox: Philox algorithm to generate random numbers in parallel. The initial_state shape is u64[3] with arbitrary values. Salmon et al. SC 2011. Parallel random numbers: as easy as 1, 2, 3.

Scatter

The XLA scatter operation generates a sequence of results which are the values of the input array operands, with several slices (at indices specified by scatter_indices) updated with the sequence of values in updates using update_computation.

See also XlaBuilder::Scatter.

scatter(operands..., scatter_indices, updates..., update_computation, index_vector_dim, update_window_dims, inserted_window_dims, scatter_dims_to_operand_dims)

Where:

• N is required to be greater or equal to 1.
• operands[0], ..., operands[N-1] must all have the same dimensions.
• updates[0], ..., updates[N-1] must all have the same dimensions.
• If N = 1, Collate(T) is T.
• If N > 1, Collate(T_0, ..., T_N) is a tuple of N elements of type T.

If index_vector_dim is equal to scatter_indices.rank we implicitly consider scatter_indices to have a trailing 1 dimension.

We define update_scatter_dims of type ArraySlice<int64> as the set of dimensions in updates shape that are not in update_window_dims, in ascending order.

The arguments of scatter should follow these constraints:

• Each updates array must be of rank update_window_dims.size + scatter_indices.rank - 1.

• Bounds of dimension i in each updates array must conform to the following:

  • If i is present in update_window_dims (i.e. equal to update_window_dims[k] for some k), then the bound of dimension i in updates must not exceed the corresponding bound of operand after accounting for the inserted_window_dims (i.e. adjusted_window_bounds[k], where adjusted_window_bounds contains the bounds of operand with the bounds at indices inserted_window_dims removed).
  • If i is present in update_scatter_dims (i.e. equal to update_scatter_dims[k] for some k), then the bound of dimension i in updates must be equal to the corresponding bound of scatter_indices, skipping index_vector_dim (i.e. scatter_indices.shape.dims[k], if k < index_vector_dim and scatter_indices.shape.dims[k+1] otherwise).

• update_window_dims must be in ascending order, not have any repeating dimension numbers, and be in the range [0, updates.rank).

• inserted_window_dims must be in ascending order, not have any repeating dimension numbers, and be in the range [0, operand.rank).

• operand.rank must equal the sum of update_window_dims.size and inserted_window_dims.size.

• scatter_dims_to_operand_dims.size must be equal to scatter_indices.shape.dims[index_vector_dim], and its values must be in the range [0, operand.rank).

For a given index U in each updates array, the corresponding index I in the corresponding operands array into which this update has to be applied is computed as follows:

1. Let G = { U[k] for k in update_scatter_dims }. Use G to look up an index vector S in the scatter_indices array such that S[i] = scatter_indices[Combine(G, i)] where Combine(A, b) inserts b at positions index_vector_dim into A.
2. Create an index Sin into operand using S by scattering S using the scatter_dims_to_operand_dims map. More formally:
   1. Sin[scatter_dims_to_operand_dims[k]] = S[k] if k < scatter_dims_to_operand_dims.size.
   2. Sin[_] = 0 otherwise.
3. Create an index Win into each operands array by scattering the indices at update_window_dims in U according to inserted_window_dims. More formally:
   1. Win[window_dims_to_operand_dims(k)] = U[k] if k is in update_window_dims, where window_dims_to_operand_dims is the monotonic function with domain [0, update_window_dims.size) and range [0, operand.rank) \ inserted_window_dims. (For example, if update_window_dims.size is 4, operand.rank is 6, and inserted_window_dims is {0, 2} then window_dims_to_operand_dims is {0→1, 1→3, 2→4, 3→5}).
   2. Win[_] = 0 otherwise.
4. I is Win + Sin where + is element-wise addition.

In summary, the scatter operation can be defined as follows.

• Initialize output with operands, i.e. for all indices J, for all indices O in the operands[J] array:
  output[J][O] = operands[J][O]
• For every index U in the updates[J] array and the corresponding index O in the operand[J] array, if O is a valid index for output:
  (output[0][O], ..., output[N-1][O]) =update_computation(output[0][O], ..., ,output[N-1][O],updates[0][U], ...,updates[N-1][U])

The order in which updates are applied is non-deterministic. So, when multiple indices in updates refer to the same index in operands, the corresponding value in output will be non-deterministic.

Note that the first parameter that is passed into the update_computation will always be the current value from the output array and the second parameter will always be the value from the updates array. This is important specifically for cases when the update_computation is not commutative.

If indices_are_sorted is set to true then XLA can assume that scatter_indices are sorted (in ascending order, after scattering its values according to scatter_dims_to_operand_dims) by the user. If they are not then the semantics are implementation defined.

If unique_indices is set to true then XLA can assume that all elements scattered to are unique. So XLA could use non-atomic operations. If unique_indices is set to true and the indices being scattered to are not unique then the semantics is implementation defined.

Informally, the scatter op can be viewed as an inverse of the gather op, i.e. the scatter op updates the elements in the input that are extracted by the corresponding gather op.

For a detailed informal description and examples, refer to the "Informal Description" section under Gather.

Select

See also XlaBuilder::Select.

Constructs an output array from elements of two input arrays, based on the values of a predicate array.

Select(pred, on_true, on_false)

The arrays on_true and on_false must have the same shape. This is also the shape of the output array. The array pred must have the same dimensionality as on_true and on_false, with the PRED element type.

For each element P of pred, the corresponding element of the output array is taken from on_true if the value of P is true, and from on_false if the value of P is false. As a restricted form of broadcasting, pred can be a scalar of type PRED. In this case, the output array is taken wholly from on_true if pred is true, and from on_false if pred is false.

Example with non-scalar pred:

let pred: PRED[4] = {true, false, false, true};
let v1: s32[4] = {1, 2, 3, 4};
let v2: s32[4] = {100, 200, 300, 400};
==>
Select(pred, v1, v2) = s32[4]{1, 200, 300, 4};

Example with scalar pred:

let pred: PRED = true;
let v1: s32[4] = {1, 2, 3, 4};
let v2: s32[4] = {100, 200, 300, 400};
==>
Select(pred, v1, v2) = s32[4]{1, 2, 3, 4};

Selections between tuples are supported. Tuples are considered to be scalar types for this purpose. If on_true and on_false are tuples (which must have the same shape!) then pred has to be a scalar of type PRED.

SelectAndScatter

See also XlaBuilder::SelectAndScatter.

This operation can be considered as a composite operation that first computes ReduceWindow on the operand array to select an element from each window, and then scatters the source array to the indices of the selected elements to construct an output array with the same shape as the operand array. The binary select function is used to select an element from each window by applying it across each window, and it is called with the property that the first parameter's index vector is lexicographically less than the second parameter's index vector. The select function returns true if the first parameter is selected and returns false if the second parameter is selected, and the function must hold transitivity (i.e., if select(a, b) and select(b, c) are true, then select(a, c) is also true) so that the selected element does not depend on the order of the elements traversed for a given window.

The function scatter is applied at each selected index in the output array. It takes two scalar parameters:

1. Current value at the selected index in the output array
2. The scatter value from source that applies to the selected index

It combines the two parameters and returns a scalar value that's used to update the value at the selected index in the output array. Initially, all indices of the output array are set to init_value.

The output array has the same shape as the operand array and the source array must have the same shape as the result of applying a ReduceWindow operation on the operand array. SelectAndScatter can be used to backpropagate the gradient values for a pooling layer in a neural network.

SelectAndScatter(operand, select, window_dimensions, window_strides, padding, source, init_value, scatter)

The figure below shows examples of using SelectAndScatter, with the select function computing the maximal value among its parameters. Note that when the windows overlap, as in the figure (2) below, an index of the operand array may be selected multiple times by different windows. In the figure, the element of value 9 is selected by both of the top windows (blue and red) and the binary addition scatter function produces the output element of value 8 (2 + 6).

The evaluation order of the scatter function is arbitrary and may be non-deterministic. Therefore, the scatter function should not be overly sensitive to reassociation. See the discussion about associativity in the context of Reduce for more details.

Send

See also XlaBuilder::Send.

Send(operand, channel_handle)

Sends the given operand data to a Recv instruction in another computation that shares the same channel handle. Does not return any data.

Similar to the Recv operation, the client API of Send operation represents synchronous communication, and is internally decomposed into 2 HLO instructions (Send and SendDone) to enable asynchronous data transfers. See also HloInstruction::CreateSend and HloInstruction::CreateSendDone.

Send(HloInstruction operand, int64 channel_id)

Initiates an asynchronous transfer of the operand to the resources allocated by the Recv instruction with the same channel id. Returns a context, which is used by a following SendDone instruction to wait for the completion of the data transfer. The context is a tuple of {operand (shape), request identifier (U32)} and it can only be used by a SendDone instruction.

SendDone(HloInstruction context)

Given a context created by a Send instruction, waits for the data transfer to complete. The instruction does not return any data.

Scheduling of channel instructions

The execution order of the 4 instructions for each channel (Recv, RecvDone, Send, SendDone) is as below.

• Recv happens before Send
• Send happens before RecvDone
• Recv happens before RecvDone
• Send happens before SendDone

When the backend compilers generate a linear schedule for each computation that communicates via channel instructions, there must not be cycles across the computations. For example, below schedules lead to deadlocks.

Slice

See also XlaBuilder::Slice.

Slicing extracts a sub-array from the input array. The sub-array is of the same rank as the input and contains the values inside a bounding box within the input array where the dimensions and indices of the bounding box are given as arguments to the slice operation.

Slice(operand, start_indices, limit_indices, strides)

1-dimensional example:

let a = {0.0, 1.0, 2.0, 3.0, 4.0}
Slice(a, {2}, {4}) produces:
  {2.0, 3.0}

2-dimensional example:

let b =
 { {0.0,  1.0,  2.0},
   {3.0,  4.0,  5.0},
   {6.0,  7.0,  8.0},
   {9.0, 10.0, 11.0} }

Slice(b, {2, 1}, {4, 3}) produces:
  { { 7.0,  8.0},
    {10.0, 11.0} }

Sort

See also XlaBuilder::Sort.

Sort(operands, comparator, dimension, is_stable)

If only one operand is provided:

• If the operand is a rank-1 tensor (an array), the result is a sorted array. If you want to sort the array into ascending order, the comparator should perform a less-than comparison. Formally, after the array is sorted, it holds for all index positions i, j with i < j that either comparator(value[i], value[j]) = comparator(value[j], value[i]) = false or comparator(value[i], value[j]) = true.

• If the operand has higher rank, the operand is sorted along the provided dimension. For example, for a rank-2 tensor (a matrix), a dimension value of 0 will independently sort every column, and a dimension value of 1 will independently sort each row. If no dimension number is provided, then the last dimension is chosen by default. For the dimension which is sorted, the same sorting order applies as in the rank-1 case.

If n > 1 operands are provided:

• All n operands must be tensors with the same dimensions. The element types of the tensors may be different.

• All operands are sorted together, not individually. Conceptually the operands are treated as a tuple. When checking whether the elements of each operand at index positions i and j need to be swapped, the comparator is called with 2 * n scalar parameters, where parameter 2 * k corresponds to the value at position i from the k-th operand, and parameter 2 * k + 1 corresponds to the value at position j from the k-th operand. Usually, the comparator would thus compare parameters 2 * k and 2 * k + 1 with each other and possibly use other parameter pairs as tie breakers.

• The result is a tuple that consists of the operands in sorted order (along the provided dimension, as above). The i-th operand of the tuple corresponds to the i-th operand of Sort.

For example, if there are three operands operand0 = [3, 1], operand1 = [42, 50], operand2 = [-3.0, 1.1], and the comparator compares only the values of operand0 with less-than, then the output of the sort is the tuple ([1, 3], [50, 42], [1.1, -3.0]).

If is_stable is set to true, the sort is guaranteed to be stable, that is, if there are elements which are considered to be equal by the comparator, the relative order of the equal values is preserved. Two elements e1 and e2 are equal if and only if comparator(e1, e2) = comparator(e2, e1) = false. By default, is_stable is set to false.

Transpose

See also the tf.reshape operation.

Transpose(operand)

Permutes the operand dimensions with the given permutation, so ∀ i . 0 ≤ i < rank ⇒ input_dimensions[permutation[i]] = output_dimensions[i].

This is the same as Reshape(operand, permutation, Permute(permutation, operand.shape.dimensions)).

TriangularSolve

See also XlaBuilder::TriangularSolve.

Solves systems of linear equations with lower or upper triangular coefficient matrices by forward- or back-substitution. Broadcasting along leading dimensions, this routine solves one of the matrix systems op(a) * x = b, or x * op(a) = b, for the variable x, given a and b, where op(a) is either op(a) = a, or op(a) = Transpose(a), or op(a) = Conj(Transpose(a)).