# W3cubDocs

/TensorFlow Guide

# Operation Semantics

The following describes the semantics of operations defined in the ComputationBuilder 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.

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

Broadcast(operand, broadcast_sizes)

Arguments Type Semantics
operand ComputationDataHandle The array to duplicate
broadcast_sizes ArraySlice<int64> The sizes of the new dimensions

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.

## Call

See also ComputationBuilder::Call.

Invokes a computation with the given arguments.

Call(computation, args...)

Arguments Type Semantics
computation Computation computation of type T_0, T_1, ..., T_N -> S with N parameters of arbitrary type
args sequence of N ComputationDataHandles N arguments of arbitrary type

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

## Clamp

See also ComputationBuilder::Clamp.

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

Clamp(computation, args...)

Arguments Type Semantics
computation Computation computation of type T_0, T_1, ..., T_N -> S with N parameters of arbitrary type
operand ComputationDataHandle array of type T
min ComputationDataHandle array of type T
max ComputationDataHandle array of type T

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(x, a, b) = max(min(x, a), b).

All three arrays must be the same shape. Alternately, 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(operand, min, max) = s32[3]{0, 5, 6};


## Collapse

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

Collapses dimensions of an array into one dimension.

Collapse(operand, dimensions)

Arguments Type Semantics
operand ComputationDataHandle array of type T
dimensions int64 vector in-order, consecutive subset of T's 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 dimension, 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}};



## Concatenate

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)

Arguments Type Semantics
operands sequence of N ComputationDataHandle N arrays of type T with dimensions [L0, L1, ...]. Requires N >= 1.
dimension int64 A value in the interval [0, N) that names the dimension to be concatenated between the operands.

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:

## 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)

Arguments Type Semantics
operand ComputationDataHandle array of type T with dims D
new_element_type PrimitiveType type U

If the dimensions of the operand and the target shape do not match, or an invalid conversion is requested (e.g. to/from a tuple) an error will be produced.

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

Note: The precise float-to-int and visa-versa conversions are currently unspecified, but may become additional arguments to the convert operation in the future. Not all possible conversions have been implemented for all targets.
let a: s32[3] = {0, 1, 2};
let b: f32[3] = convert(a, f32);
then b == f32[3]{0.0, 1.0, 2.0}


## Conv (convolution)

See also ComputationBuilder::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.

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.

Arguments Type Semantics
lhs ComputationDataHandle rank n+2 array of inputs
rhs ComputationDataHandle rank n+2 array of kernel weights
window_strides ArraySlice<int64> n-d array of kernel strides
padding ArraySlice<pair<int64, int64>> n-d array of (low, high) padding
lhs_dilation ArraySlice<int64> n-d lhs dilation factor array
rhs_dilation ArraySlice<int64> n-d rhs dilation factor array

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 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 a 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 the tf.nn.atrous_conv2d. Dilation of the lhs is also called deconvolution.

The output shape has these dimensions, in this order:

• batch: Same size as batch on the input (lhs).
• z: Same size as output-z on the kernel (rhs).
• spatial_dims: One value for each valid placement of the convolutional window.

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;
}


## CrossReplicaSum

Computes a sum across replicas.

CrossReplicaSum(operand)

Arguments Type Semantics
operand ComputationDataHandle Array to sum across replicas.

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, 5.1) respectively on the two replicas, then the output value from this op will be (4.0, 7.6) on both replicas.

Computing the result of CrossReplicaSum requires having one input from each replica, so if one replica executes a CrossReplicaSum 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.

## CustomCall

Call a user-provided function within a computation.

CustomCall(target_name, args..., shape)

Arguments Type Semantics
target_name string Name of the function. A call instruction will be emitted which targets this symbol name.
args sequence of N ComputationDataHandles N arguments of arbitrary type, which will be passed to the function.
shape Shape Output shape of the function

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.

Note: The opaque nature of the user-provided function restricts optimization opportunities for the compiler. Try to express your computation in terms of native XLA ops whenever possible; only use CustomCall as a last resort.

## Dot

See also ComputationBuilder::Dot.

Dot(lhs, rhs)

Arguments Type Semantics
lhs ComputationDataHandle array of type T
rhs ComputationDataHandle array of type T

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

Input Output Semantics
vector [n] dot vector [n] scalar vector dot product
matrix [m x k] dot vector [k] vector [m] matrix-vector multiplication
matrix [m x k] dot matrix [k x n] matrix [m x n] matrix-matrix multiplication

The operation performs sum of products over the last dimension of lhs and the one-before-last 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.

## Element-wise binary arithmetic operations

See also ComputationBuilder::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), Rem (remainder), Max (maximum), Min (minimum), LogicalAnd (logical AND), or LogicalOr (logical OR).

Arguments Type Semantics
lhs ComputationDataHandle left-hand-side operand: array of type T
rhs ComputationDataHandle right-hand-side operand: array of type T

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.

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 dimension to equalize the shapes of both operands. The semantics are described in detail on the broadcasting page.

## Element-wise comparison operations

See also ComputationBuilder::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), Le (less-than).

Arguments Type Semantics
lhs ComputationDataHandle left-hand-side operand: array of type T
rhs ComputationDataHandle right-hand-side operand: array of type T

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

ComputationBuilder supports these element-wise unary functions:

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

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

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

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

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

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).

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

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

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

$$\text{sgn}(x) = \begin{cases} -1 & x < 0\\ 0 & x = 0\\ 1 & x > 0 \end{cases}$$

using the comparison operator of the element type of operand.

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

Arguments Type Semantics
operand ComputationDataHandle The operand to the function

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).

## BatchNormTraining

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

Warning: Not implemented on GPU backend yet.

Normalizes an array across batch and spatial dimensions.

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

Arguments Type Semantics
operand ComputationDataHandle n dimensional array to be normalized
scale ComputationDataHandle 1 dimensional array ($$\gamma$$)
offset ComputationDataHandle 1 dimensional array ($$\beta\ ) epsilon float Epsilon value (\(\epsilon$$)
feature_index int64 Index to feature dimension in operand

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 use the mean and variance to normalize each element in operand. If an invalid feature_index is passed, an error is produced.

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 an 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 $$\sigma^2_l$$: $$\sigma^2_l=\frac{1}{mwh}\sum_{i=1}^m\sum_{j=1}^w\sum_{k=1}^h (x_{ijkl} - \mu_l)^2$$

• 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 ComputationDataHandles:

Outputs Type Semantics
output ComputationDataHandle n dimensional array with the same shape as input operand (y)
batch_mean ComputationDataHandle 1 dimensional array ($$\mu$$)
batch_var ComputationDataHandle 1 dimensional array ($$\sigma^2$$)

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

## BatchNormInference

Warning: Not implemented yet.

Normalizes an array across batch and spatial dimensions.

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

Arguments Type Semantics
operand ComputationDataHandle n dimensional array to be normalized
scale ComputationDataHandle 1 dimensional array
offset ComputationDataHandle 1 dimensional array
mean ComputationDataHandle 1 dimensional array
variance ComputationDataHandle 1 dimensional array
epsilon float Epsilon value
feature_index int64 Index to feature dimension in operand

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 use the mean and variance to normalize each element in operand. If an invalid feature_index is passed, an error is produced.

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 a n dimensional, normalized array with the same shape as input operand.

Warning: Not implemented yet.

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

Arguments Type Semantics
operand ComputationDataHandle n dimensional array to be normalized (x)
scale ComputationDataHandle 1 dimensional array ($$\gamma$$)
mean ComputationDataHandle 1 dimensional array ($$\mu$$)
variance ComputationDataHandle 1 dimensional array ($$\sigma^2$$)
grad_output ComputationDataHandle Gradients passed to BatchNormTraining ($$\nabla y$$)
epsilon float Epsilon value ($$\epsilon$$)
feature_index int64 Index to feature dimension in operand

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. If an invalid feature_index is passed, an error is produced.

The three gradients are defined by the following formulas:

$$\nabla x = \nabla y * \gamma * \sqrt{\sigma^2+\epsilon}$$

$$\nabla \gamma = sum(\nabla y * (x - \mu) * \sqrt{\sigma^2 + \epsilon})$$

$$\nabla \beta = sum(\nabla y)$$

The inputs mean and variance represents moments value across batch and spatial dimensions.

The output type is a tuple of three ComputationDataHandles:

Outputs Type Semantics
grad_operand ComputationDataHandle gradient with respect to input
operand grad_offset ComputationDataHandle
grad_scale ComputationDataHandle gradient with respect to input scale

## 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 ComputationBuilder::Infeed.

Infeed(shape)

Argument Type Semantics
shape Shape Shape of the data read from the Infeed interface. The layout field of the shape must be set to match the layout of the data sent to the device; otherwise its behavior is undefined.

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 ComputationDataHandle 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. The compiler issues an error if there isn't a total order.

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 nop and proceeds without reading any data from the Infeed of the device.

Note: We plan to allow multiple Infeed operations without a total order, in which case the compiler will provide information about how the Infeed operations are serialized in the compiled program.

## Map

See also ComputationBuilder::Map.

Map(operands..., computation)

Arguments Type Semantics
operands sequence of N ComputationDataHandles N arrays of types T_0..T_{N-1}
computation Computation computation of type T_0, T_1, ..., T_{N + M -1} -> S with N parameters of type T and M of arbitrary type
dimensions int64 array array of map dimensions
static_operands sequence of M ComputationDataHandles M arrays of arbitrary type

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 with static_operands given as additional input to computation.

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.

See also ComputationBuilder::Pad.

Pad(operand, padding_value, padding_config)

Arguments Type Semantics
operand ComputationDataHandle array of type T
padding_value ComputationDataHandle scalar of type T to fill in the added padding
padding_config PaddingConfig padding amount on both edges (low, high) and between the elements of each dimension

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 specifies 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. 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. Figure below shows examples of different edge_padding and interior_padding values for a two dimensional array.

## Reduce

See also ComputationBuilder::Reduce.

Applies a reduction function to an array.

Reduce(operand, init_value, computation, dimensions)

Arguments Type Semantics
operand ComputationDataHandle array of type T
init_value ComputationDataHandle scalar of type T
computation Computation computation of type T, T -> T
dimensions int64 array unordered array of dimensions to reduce

Conceptually, this operation reduces one or more dimensions in the input array into scalars. The rank of the result array is rank(operand) - len(dimensions). init_value is the initial value used for every reduction and may also be inserted anywhere during computation if the back-end chooses to do so. So in most cases init_value should be an identity of the reduction function (for example, 0 for addition).

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.

Some reduction functions like addition are not strictly 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. It is possible to conceive of some completely non-associative reductions, however, and these will produce incorrect or unpredictable results in XLA reductions.

As an example, when reducing across the one dimension in a 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(13, init_value))))

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.

## 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)

Arguments Type Semantics
operand ComputationDataHandle array of floating-point type T.
exponent_bits int32 number of exponent bits in lower-precision format
mantissa_bits int32 number of mantissa bits in lower-precision format

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.

## ReduceWindow

Applies a reduction function to all elements in each window of the input multi-dimensional array, producing an output multi-dimensional array with the same number of elements as the number of valid positions of the window. A pooling layer can be expressed as a ReduceWindow.

ReduceWindow(operand, init_value, computation, window_dimensions, window_strides, padding)

Arguments Type Semantics
operand ComputationDataHandle N dimensional array containing elements of type T. This is the base area on which the window is placed.
init_value ComputationDataHandle Starting value for the reduction. See Reduce for details.
computation Computation Reduction function of type T, T -> T, to apply to all elements in each window
window_dimensions ArraySlice<int64> array of integers for window dimension values
window_strides ArraySlice<int64> array of integers for window stride values
padding Padding padding type for window (Padding\:\:kSame or Padding\:\:kValid)

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).
Computation max;
{
ComputationBuilder 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().ConsumeValueOrDie();
}

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


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.

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.

## Reshape

See also ComputationBuilder::Reshape and the Collapse operation.

Reshapes the dimensions of an array into a new configuration.

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

Arguments Type Semantics
operand ComputationDataHandle array of type T
dimensions int64 vector order in which dimensions are collapsed
new_sizes int64 vector vector of sizes of new dimensions

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 ComputationBuilder::Rev.

Rev(operand, dimensions)

Arguments Type Semantics
operand ComputationDataHandle array of type T
dimensions ArraySlice<int64> dimensions to reverse

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.

## RngBernoulli

Constructs an output of a given shape with random numbers generated following the Bernoulli distribution. The parameter needs to be a scalar valued F32 operand while the output shape needs to have elemental type U32.

RngBernoulli(mean, shape)

Arguments Type Semantics
mean ComputationDataHandle Scalar of type F32 specifying mean of generated numbers
shape Shape Output shape of type U32

## 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 elemental type F32. The parameters furthermore have to be scalar valued.

RngNormal(mean, sigma, shape)

Arguments Type Semantics
mu ComputationDataHandle Scalar of type F32 specifying mean of generated numbers
sigma ComputationDataHandle Scalar of type F32 specifying standard deviation of generated numbers
shape Shape Output shape of type F32

## 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
shape may be either F32, S32 or U32, but the types have to be consistent.
Furthermore, the parameters need to be scalar valued. If $$b <= a$$ the result
is implementation-defined.

RngUniform(a, b, shape)

Arguments Type Semantics
a ComputationDataHandle Scalar of type T specifying lower limit of interval
b ComputationDataHandle Scalar of type T specifying upper limit of interval
shape Shape Output shape of type T

## 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)

Arguments Type Semantics
operand ComputationDataHandle array of type T over which the windows slide
select Computation binary computation of type T, T -> PRED, to apply to all elements in each window; returns true if the first parameter is selected and returns false if the second parameter is selected
window_dimensions ArraySlice<int64> array of integers for window dimension values
window_strides ArraySlice<int64> array of integers for window stride values
padding Padding padding type for window (Padding\:\:kSame or Padding\:\:kValid)
source ComputationDataHandle array of type T with the values to scatter
init_value ComputationDataHandle scalar value of type T for the initial value of the output array
scatter Computation binary computation of type T, T -> T, to apply each scatter source element with its destination element

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.

## Select

See also ComputationBuilder::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)

Arguments Type Semantics
pred ComputationDataHandle array of type PRED
on_true ComputationDataHandle array of type T
on_false ComputationDataHandle array of type T

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.

## Slice

See also ComputationBuilder::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)

Arguments Type Semantics
operand ComputationDataHandle N dimensional array of type T
start_indices ArraySlice<int64> List of N integers containing the starting indices of the slice for each dimension. Values must be greater than or equal to zero.
limit_indices ArraySlice<int64> List of N integers containing the ending indices (exclusive) for the slice for each dimension. Each value must be strictly greater than the respective start_indices value for the dimension and less than or equal to the size of the dimension.

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} }


## 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. Note: handling of out-of-bounds slice indices (generated by incorrect runtime calculation of 'start_indices') is currently implementation-defined. Currently, slice indices are computed modulo input dimension sizes to prevent out-of-bound array accesses, but this behavior may change in future implementations.

DynamicSlice(operand, start_indices, size_indices)

Arguments Type Semantics
operand ComputationDataHandle N dimensional array of type T
start_indices ComputationDataHandle Rank 1 array of N integers containing the starting indices of the slice for each dimension. Value must be greater than or equal to zero.
size_indices ArraySlice<int64> List of N integers containing the slice size for each dimension. Each value must be strictly greater than zero, and start + size must be less than or equal to the size of the dimension to avoid wrapping modulo dimension size.

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 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. Note: handling of out-of-bounds slice indices (generated by incorrect runtime calculation of 'start_indices') is currently implementation-defined. Currently, slice indices are computed modulo update dimension sizes to prevent out-of-bound array accesses, but this behavior may change in future implementations.

DynamicUpdateSlice(operand, update, start_indices)

Arguments Type Semantics
operand ComputationDataHandle N dimensional array of type T
update ComputationDataHandle N dimensional array of type T containing the slice update. Each dimension of update shape must be strictly greater than zero, and start + update must be less than operand size for each dimension to avoid generating out-of-bounds update indices.
start_indices ComputationDataHandle Rank 1 array of N integers containing the starting indices of the slice for each dimension. Value must be greater than or equal to zero.

1-dimensional example:

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

{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}

{ {0.0,  1.0,  2.0},
{3.0, 12.0, 13.0},
{6.0, 14.0, 15.0},
{9.0, 16.0, 17.0} }


## Sort

See also ComputationBuilder::Sort.

Sorts the elements in the operand.

Sort(operand)

Arguments Type Semantics
operand ComputationDataHandle The operand to sort

## Transpose

See also the tf.reshape operation.

Transpose(operand)

Arguments Type Semantics
operand ComputationDataHandle The operand to transpose.
permutation ArraySlice<int64> How to permute the dimensions.

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)).

## Tuple

See also ComputationBuilder::Tuple.

A tuple containing a variable number of data handles, each of which has its own shape.

This is analogous to std::tuple 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);


Tuples can be deconstructed (accessed) via the GetTupleElement operation.

## While

See also ComputationBuilder::While.

While(condition, body, init)

Arguments Type Semantics
condition Computation Computation of type T -> PRED which defines the termination condition of the loop.
body Computation Computation of type T -> T which defines the body of the loop.
init T Initial value for the parameter of condition and body.

Sequentially executes the body until the condition fails. This is similar to a typical while loop in many other languages except for the differences and restrictions listed below.

• A While node returns a value of type T, which is the result from the last execution of the body.
• The shape of the type T is statically determined and must be the same across all iterations.
• While nodes are not allowed to be nested. (This restriction may be lifted in the future on some targets.)

The T parameters of the computations are initialized with the init value in the first iteration and are automatically updated to the new result from body in each subsequent iteration.

One main use case of the While node is to implement the repeated execution of training in neural networks. Simplified pseudocode is shown below with a graph that represents the computation. The code can be found in while_test.cc. The type T in this example is a Tuple consisting of an int32 for the iteration count and a vector[10] for the accumulator. For 1000 iterations, the loop keeps adding a constant vector to the accumulator.

// Pseudocode for the computation.
init = {0, zero_vector[10]} // Tuple of int32 and float[10].
result = init;
while (result(0) < 1000) {
iteration = result(0) + 1;
new_vector = result(1) + constant_vector[10];
result = {iteration, new_vector};
}