eigen/unsupported/Eigen/src/Tensor

Eigen Tensors

Tensors are multidimensional arrays of elements. Elements are typically scalars, but more complex types such as strings are also supported.

The Tensor module is part of Eigen's unsupported modules. While it is actively used in production (e.g. in TensorFlow), its API may change without notice.

To use the Tensor module, include the following header:

#include <unsupported/Eigen/Tensor>

Quick Start

#include <unsupported/Eigen/Tensor>
#include <iostream>

int main() {
  // Create a 3x4 matrix as a rank-2 tensor.
  Eigen::Tensor<float, 2> a(3, 4);
  a.setRandom();

  // Create another tensor and compute their element-wise sum.
  Eigen::Tensor<float, 2> b(3, 4);
  b.setConstant(1.0f);
  Eigen::Tensor<float, 2> c = a + b;

  // Reduce: compute the sum of all elements.
  Eigen::Tensor<float, 0> total = c.sum();
  std::cout << "Sum of all elements: " << total() << "\n";

  // Reshape and broadcast.
  Eigen::Tensor<float, 2> d = c.reshape(Eigen::array<Eigen::Index, 2>{{1, 12}})
                                .broadcast(Eigen::array<Eigen::Index, 2>{{3, 1}});
  std::cout << "d has shape: " << d.dimension(0) << " x " << d.dimension(1) << "\n";
  return 0;
}

Tensor Classes

You can manipulate a tensor with one of the following classes. They all are in the namespace ::Eigen.

Class Tensor<Scalar, NumIndices, Options, IndexType>

This is the class to use to create a tensor and allocate memory for it.

Template parameters:

Parameter	Description	Default
Scalar	Element type (e.g. float, int, std::string)	(required)
NumIndices	Rank (number of dimensions)	(required)
Options	ColMajor (0) or RowMajor	0 (ColMajor)
IndexType	Type used for indexing (e.g. int, long)	Eigen::DenseIndex

Tensors of this class are resizable. For example, if you assign a tensor of a different size to a Tensor, that tensor is resized to match its new value.

Constructor Tensor<Scalar, NumIndices>(size0, size1, ...)

Constructor for a Tensor. The constructor must be passed NumIndices integers indicating the sizes of the instance along each of the dimensions.

// Create a tensor of rank 3 of sizes 2, 3, 4.  This tensor owns
// memory to hold 24 floating point values (24 = 2 x 3 x 4).
Tensor<float, 3> t_3d(2, 3, 4);

// Resize t_3d by assigning a tensor of different sizes, but same rank.
t_3d = Tensor<float, 3>(3, 4, 3);

Constructor Tensor<Scalar, NumIndices>(size_array)

Constructor where the sizes for the constructor are specified as an array of values instead of an explicit list of parameters. The array type to use is Eigen::array<Eigen::Index, NumIndices>. The array can be constructed automatically from an initializer list.

// Create a tensor of strings of rank 2 with sizes 5, 7.
Tensor<string, 2> t_2d({5, 7});

Class TensorFixedSize<Scalar, Sizes<size0, size1, ...>, Options, IndexType>

Class to use for tensors of fixed size, where the size is known at compile time. Fixed sized tensors can provide very fast computations because all their dimensions are known by the compiler. FixedSize tensors are not resizable.

If the total number of elements in a fixed size tensor is small enough the tensor data is held onto the stack and does not cause heap allocation and free.

// Create a 4 x 3 tensor of floats.
TensorFixedSize<float, Sizes<4, 3>> t_4x3;

Class TensorMap<Tensor<Scalar, NumIndices, Options>>

This is the class to use to create a tensor on top of memory allocated and owned by another part of your code. It allows to view any piece of allocated memory as a Tensor. Instances of this class do not own the memory where the data are stored.

A TensorMap is not resizable because it does not own the memory where its data are stored.

An optional alignment template parameter controls whether Eigen can assume the data pointer is aligned: TensorMap<Tensor<float, 2>, Aligned>.

Constructor TensorMap<Tensor<Scalar, NumIndices>>(data, size0, size1, ...)

Constructor for a TensorMap. The constructor must be passed a pointer to the storage for the data, and NumIndices size attributes. The storage has to be large enough to hold all the data.

// Map a tensor of ints on top of stack-allocated storage.
int storage[128];  // 2 x 4 x 2 x 8 = 128
TensorMap<Tensor<int, 4>> t_4d(storage, 2, 4, 2, 8);

// The same storage can be viewed as a different tensor.
// You can also pass the sizes as an array.
TensorMap<Tensor<int, 2>> t_2d(storage, 16, 8);

// You can also map fixed-size tensors.  Here we get a 1d view of
// the 2d fixed-size tensor.
TensorFixedSize<float, Sizes<4, 3>> t_4x3;
TensorMap<Tensor<float, 1>> t_12(t_4x3.data(), 12);

Class TensorRef

See Assigning to a TensorRef.

Accessing Tensor Elements

Scalar tensor(index0, index1...)

Return the element at position (index0, index1...) in tensor tensor. You must pass as many parameters as the rank of tensor. The expression can be used as an l-value to set the value of the element at the specified position. The value returned is of the datatype of the tensor.

// Set the value of the element at position (0, 1, 0);
Tensor<float, 3> t_3d(2, 3, 4);
t_3d(0, 1, 0) = 12.0f;

// Initialize all elements to random values.
for (int i = 0; i < 2; ++i) {
  for (int j = 0; j < 3; ++j) {
    for (int k = 0; k < 4; ++k) {
      t_3d(i, j, k) = ...some random value...;
    }
  }
}

// Print elements of a tensor.
for (int i = 0; i < 2; ++i) {
  std::cout << t_3d(i, 0, 0);
}

TensorLayout

The tensor library supports 2 layouts: ColMajor (the default) and RowMajor.

The layout of a tensor is optionally specified as the third template parameter (Options). If not specified explicitly, column major is assumed.

Tensor<float, 3, ColMajor> col_major;  // equivalent to Tensor<float, 3>
TensorMap<Tensor<float, 3, RowMajor>> row_major(data, ...);

All the arguments to an expression must use the same layout. Attempting to mix different layouts will result in a compilation error.

It is possible to change the layout of a tensor or an expression using the swap_layout() method. Note that this will also reverse the order of the dimensions.

Tensor<float, 2, ColMajor> col_major(2, 4);
Tensor<float, 2, RowMajor> row_major(2, 4);

Tensor<float, 2> col_major_result = col_major;  // ok, layouts match
Tensor<float, 2> col_major_result = row_major;  // will not compile

// Simple layout swap
col_major_result = row_major.swap_layout();
eigen_assert(col_major_result.dimension(0) == 4);
eigen_assert(col_major_result.dimension(1) == 2);

// Swap the layout and preserve the order of the dimensions
array<int, 2> shuffle{{1, 0}};
col_major_result = row_major.swap_layout().shuffle(shuffle);
eigen_assert(col_major_result.dimension(0) == 2);
eigen_assert(col_major_result.dimension(1) == 4);

Tensor Operations

The Eigen Tensor library provides a vast library of operations on Tensors: numerical operations such as addition and multiplication, geometry operations such as slicing and shuffling, etc. These operations are available as methods of the Tensor classes, and in some cases as operator overloads. For example the following code computes the elementwise addition of two tensors:

Tensor<float, 3> t1(2, 3, 4);
t1.setRandom();
Tensor<float, 3> t2(2, 3, 4);
t2.setRandom();
// Set t3 to the element wise sum of t1 and t2
Tensor<float, 3> t3 = t1 + t2;

While the code above looks easy enough, it is important to understand that the expression t1 + t2 is not actually adding the values of the tensors. The expression instead constructs a "tensor operator" object of the class TensorCwiseBinaryOp<scalar_sum>, which has references to the tensors t1 and t2. This is a small C++ object that knows how to add t1 and t2. It is only when the value of the expression is assigned to the tensor t3 that the addition is actually performed. Technically, this happens through the overloading of operator= in the Tensor class.

This mechanism for computing tensor expressions allows for lazy evaluation and optimizations which are what make the tensor library very fast.

Of course, the tensor operators do nest, and the expression t1 + t2 * 0.3f is actually represented with the (approximate) tree of operators:

TensorCwiseBinaryOp<scalar_sum>(t1, TensorCwiseUnaryOp<scalar_mul>(t2, 0.3f))

Tensor Operations and C++ "auto"

Because Tensor operations create tensor operators, the C++ auto keyword does not have its intuitive meaning. Consider these 2 lines of code:

Tensor<float, 3> t3 = t1 + t2;
auto t4 = t1 + t2;

In the first line we allocate the tensor t3 and it will contain the result of the addition of t1 and t2. In the second line, t4 is actually the tree of tensor operators that will compute the addition of t1 and t2. In fact, t4 is not a tensor and you cannot get the values of its elements:

Tensor<float, 3> t3 = t1 + t2;
std::cout << t3(0, 0, 0);  // OK prints the value of t1(0, 0, 0) + t2(0, 0, 0)

auto t4 = t1 + t2;
std::cout << t4(0, 0, 0);  // Compilation error!

When you use auto you do not get a Tensor as a result but instead a non-evaluated expression. So only use auto to delay evaluation.

Unfortunately, there is no single underlying concrete type for holding non-evaluated expressions, hence you have to use auto in the case when you do want to hold non-evaluated expressions.

When you need the results of set of tensor computations you have to assign the result to a Tensor that will be capable of holding onto them. This can be either a normal Tensor, a TensorFixedSize, or a TensorMap on an existing piece of memory. All the following will work:

auto t4 = t1 + t2;

Tensor<float, 3> result = t4;  // Could also be: result(t4);
std::cout << result(0, 0, 0);

TensorMap<Tensor<float, 3>> result2(some_float_ptr, dim0, dim1, dim2);
result2 = t4;
std::cout << result2(0, 0, 0);

TensorFixedSize<float, Sizes<4, 4, 2>> result3 = t4;
std::cout << result3(0, 0, 0);

Until you need the results, you can keep the operation around, and even reuse it for additional operations. As long as you keep the expression as an operation, no computation is performed.

// One way to compute exp((t1 + t2) * 0.2f);
auto t3 = t1 + t2;
auto t4 = t3 * 0.2f;
auto t5 = t4.exp();
Tensor<float, 3> result = t5;

// Another way, exactly as efficient as the previous one:
Tensor<float, 3> result = ((t1 + t2) * 0.2f).exp();

Controlling When Expressions are Evaluated

There are several ways to control when expressions are evaluated:

• Assignment to a Tensor, TensorFixedSize, or TensorMap.
• Use of the eval() method.
• Assignment to a TensorRef.

Assigning to a Tensor, TensorFixedSize, or TensorMap.

The most common way to evaluate an expression is to assign it to a Tensor. In the example below, the auto declarations make the intermediate values "Operations", not Tensors, and do not cause the expressions to be evaluated. The assignment to the Tensor result causes the evaluation of all the operations.

auto t3 = t1 + t2;             // t3 is an Operation.
auto t4 = t3 * 0.2f;           // t4 is an Operation.
auto t5 = t4.exp();            // t5 is an Operation.
Tensor<float, 3> result = t5;  // The operations are evaluated.

If you know the ranks and sizes of the Operation value you can assign the Operation to a TensorFixedSize instead of a Tensor, which is a bit more efficient.

// We know that the result is a 4x4x2 tensor!
TensorFixedSize<float, Sizes<4, 4, 2>> result = t5;

Similarly, assigning an expression to a TensorMap causes its evaluation. Like tensors of type TensorFixedSize, a TensorMap cannot be resized so they have to have the rank and sizes of the expression that are assigned to them.

Calling eval().

When you compute large composite expressions, you sometimes want to tell Eigen that an intermediate value in the expression tree is worth evaluating ahead of time. This is done by inserting a call to the eval() method of the expression Operation.

// The previous example could have been written:
Tensor<float, 3> result = ((t1 + t2) * 0.2f).exp();

// If you want to compute (t1 + t2) once ahead of time you can write:
Tensor<float, 3> result = ((t1 + t2).eval() * 0.2f).exp();

Semantically, calling eval() is equivalent to materializing the value of the expression in a temporary Tensor of the right size. The code above in effect does:

// .eval() knows the size!
TensorFixedSize<float, Sizes<4, 4, 2>> tmp = t1 + t2;
Tensor<float, 3> result = (tmp * 0.2f).exp();

Note that the return value of eval() is itself an Operation, so the following code does not do what you may think:

// Here t3 is an evaluation Operation.  t3 has not been evaluated yet.
auto t3 = (t1 + t2).eval();

// You can use t3 in another expression.  Still no evaluation.
auto t4 = (t3 * 0.2f).exp();

// The value is evaluated when you assign the Operation to a Tensor, using
// an intermediate tensor to represent t3.
Tensor<float, 3> result = t4;

While in the examples above calling eval() does not make a difference in performance, in other cases it can make a huge difference. In the expression below the broadcast() expression causes the X.maximum() expression to be evaluated many times:

Tensor<...> X ...;
Tensor<...> Y = ((X - X.maximum(depth_dim).reshape(dims2d).broadcast(bcast))
                 * beta).exp();

Inserting a call to eval() between the maximum() and reshape() calls guarantees that maximum() is only computed once and greatly speeds-up execution:

Tensor<...> Y =
  ((X - X.maximum(depth_dim).eval().reshape(dims2d).broadcast(bcast))
    * beta).exp();

In the other example below, the tensor Y is both used in the expression and its assignment. This is an aliasing problem and if the evaluation is not done in the right order Y will be updated incrementally during the evaluation resulting in bogus results:

 Tensor<...> Y ...;
 Y = Y / (Y.sum(depth_dim).reshape(dims2d).broadcast(bcast));

Inserting a call to eval() between the sum() and reshape() expressions ensures that the sum is computed before any updates to Y are done.

 Y = Y / (Y.sum(depth_dim).eval().reshape(dims2d).broadcast(bcast));

Note that an eval around the full right hand side expression is not needed because the generated has to compute the i-th value of the right hand side before assigning it to the left hand side.

However, if you were assigning the expression value to a shuffle of Y then you would need to force an eval for correctness by adding an eval() call for the right hand side:

 Y.shuffle(...) =
    (Y / (Y.sum(depth_dim).eval().reshape(dims2d).broadcast(bcast))).eval();

Assigning to a TensorRef.

If you need to access only a few elements from the value of an expression you can avoid materializing the value in a full tensor by using a TensorRef.

A TensorRef is a small wrapper class for any Eigen Operation. It provides overloads for the () operator that let you access individual values in the expression. TensorRef is convenient, because the Operation themselves do not provide a way to access individual elements.

// Create a TensorRef for the expression.  The expression is not
// evaluated yet.
TensorRef<Tensor<float, 3>> ref = ((t1 + t2) * 0.2f).exp();

// Use "ref" to access individual elements.  The expression is evaluated
// on the fly.
float at_0 = ref(0, 0, 0);
std::cout << ref(0, 1, 0);

Only use TensorRef when you need a subset of the values of the expression. TensorRef only computes the values you access. However note that if you are going to access all the values it will be much faster to materialize the results in a Tensor first.

In some cases, if the full Tensor result would be very large, you may save memory by accessing it as a TensorRef. But not always. So don't count on it.

Controlling How Expressions Are Evaluated

The tensor library provides several implementations of the various operations such as contractions and convolutions. The implementations are optimized for different environments: single threaded on CPU, multi threaded on CPU, or on a GPU using CUDA/HIP/SYCL.

You can choose which implementation to use with the device() call. If you do not choose an implementation explicitly the default implementation that uses a single thread on the CPU is used.

The default implementation has been optimized for modern CPUs, taking advantage of SSE, AVX, AVX-512, ARM NEON, SVE, RISC-V Vector (RVV), and other SIMD instruction sets. Note that you need to pass compiler-dependent flags to enable the use of these instructions (e.g. -mavx2, -march=native).

For example, the following code adds two tensors using the default single-threaded CPU implementation:

Tensor<float, 2> a(30, 40);
Tensor<float, 2> b(30, 40);
Tensor<float, 2> c = a + b;

To choose a different implementation you have to insert a device() call before the assignment of the result. For technical C++ reasons this requires that the Tensor for the result be declared on its own. This means that you have to know the size of the result.

Eigen::Tensor<float, 2> c(30, 40);
c.device(...) = a + b;

The call to device() must be the last call on the left of the operator=.

You must pass to the device() call an Eigen device object. There are presently four devices you can use: DefaultDevice, ThreadPoolDevice, GpuDevice, and SyclDevice.

Evaluating With the DefaultDevice

This is exactly the same as not inserting a device() call.

DefaultDevice my_device;
c.device(my_device) = a + b;

Evaluating with a Thread Pool

To use ThreadPoolDevice, you must define EIGEN_USE_THREADS before including the Tensor header:

#define EIGEN_USE_THREADS
#include <unsupported/Eigen/Tensor>

// Create the Eigen ThreadPool.
Eigen::ThreadPool pool(8 /* number of threads in pool */);

// Create the Eigen ThreadPoolDevice.
Eigen::ThreadPoolDevice my_device(&pool, 4 /* number of threads to use */);

// Now just use the device when evaluating expressions.
Eigen::Tensor<float, 2> c(30, 50);
c.device(my_device) = a.contract(b, dot_product_dims);

Evaluating On GPU

To use GpuDevice, you must define EIGEN_USE_GPU before including the Tensor header. GPU tensors require explicitly allocating device memory with CUDA or HIP APIs.

#define EIGEN_USE_GPU
#include <unsupported/Eigen/Tensor>

// Allocate data on GPU.
float* d_a;
float* d_b;
float* d_c;
cudaMalloc((void**)&d_a, 30 * 40 * sizeof(float));
cudaMalloc((void**)&d_b, 30 * 40 * sizeof(float));
cudaMalloc((void**)&d_c, 30 * 40 * sizeof(float));

// Copy host data to device.
cudaMemcpy(d_a, h_a, 30 * 40 * sizeof(float), cudaMemcpyHostToDevice);
cudaMemcpy(d_b, h_b, 30 * 40 * sizeof(float), cudaMemcpyHostToDevice);

// Create device maps.
Eigen::TensorMap<Eigen::Tensor<float, 2>> gpu_a(d_a, 30, 40);
Eigen::TensorMap<Eigen::Tensor<float, 2>> gpu_b(d_b, 30, 40);
Eigen::TensorMap<Eigen::Tensor<float, 2>> gpu_c(d_c, 30, 40);

// Create a GPU device and evaluate.
Eigen::GpuStreamDevice stream;
Eigen::GpuDevice gpu_device(&stream);
gpu_c.device(gpu_device) = gpu_a + gpu_b;

// Synchronize and copy back.
cudaStreamSynchronize(stream.stream());
cudaMemcpy(h_c, d_c, 30 * 40 * sizeof(float), cudaMemcpyDeviceToHost);

cudaFree(d_a);
cudaFree(d_b);
cudaFree(d_c);

For HIP, replace cuda* calls with the corresponding hip* calls.

Asynchronous Device Execution

You can pass a callback to the device() call that will be invoked when the computation completes. This is supported by ThreadPoolDevice and GpuDevice.

Eigen::Tensor<float, 2> c(30, 40);
auto done = []() { std::cout << "Computation complete!\n"; };
c.device(my_device, done) = a + b;
// The callback will be invoked when evaluation finishes.

API Reference

Datatypes

In the documentation of the tensor methods and Operation we mention datatypes that are tensor-type specific:

<Tensor-Type>::Dimensions

Acts like an array of Index. Has a size() method (inherited from std::array) and a static count member equal to the rank. Can be indexed like an array to access individual values. Used to represent the dimensions of a tensor. See dimensions().

<Tensor-Type>::Index

Acts like an int. Used for indexing tensors along their dimensions. See operator(), dimension(), and size().

<Tensor-Type>::Scalar

Represents the datatype of individual tensor elements. For example, for a Tensor<float>, Scalar is the type float. See setConstant().

(Operation)

We use this pseudo type to indicate that a tensor Operation is returned by a method. We indicate in the text the type and dimensions of the tensor that the Operation returns after evaluation.

The Operation will have to be evaluated, for example by assigning it to a Tensor, before you can access the values of the resulting tensor. You can also access the values through a TensorRef.

Built-in Tensor Methods

These are usual C++ methods that act on tensors immediately. They are not Operations which provide delayed evaluation of their results. Unless specified otherwise, all the methods listed below are available on all tensor classes: Tensor, TensorFixedSize, and TensorMap.

Metadata

int NumDimensions

Constant value indicating the number of dimensions of a Tensor. This is also known as the tensor rank.

Eigen::Tensor<float, 2> a(3, 4);
std::cout << "Dims " << a.NumDimensions;
// Dims 2

Dimensions dimensions()

Returns an array-like object representing the dimensions of the tensor. The actual type of the dimensions() result is <Tensor-Type>::Dimensions.

Eigen::Tensor<float, 2> a(3, 4);
const Eigen::Tensor<float, 2>::Dimensions& d = a.dimensions();
std::cout << "Dim size: " << d.size() << ", dim 0: " << d[0]
          << ", dim 1: " << d[1];
//  Dim size: 2, dim 0: 3, dim 1: 4

You can use auto to simplify the code:

const auto& d = a.dimensions();
std::cout << "Dim size: " << d.size() << ", dim 0: " << d[0]
        << ", dim 1: " << d[1];
// Dim size: 2, dim 0: 3, dim 1: 4

Index dimension(Index n)

Returns the n-th dimension of the tensor. The actual type of the dimension() result is <Tensor-Type>::Index, but you can always use it like an int.

Eigen::Tensor<float, 2> a(3, 4);
int dim1 = a.dimension(1);
std::cout << "Dim 1: " << dim1;
// Dim 1: 4

Index size()

Returns the total number of elements in the tensor. This is the product of all the tensor dimensions. The actual type of the size() result is <Tensor-Type>::Index, but you can always use it like an int.

Eigen::Tensor<float, 2> a(3, 4);
std::cout << "Size: " << a.size();
/// Size: 12

Getting Dimensions From An Operation

A few operations provide dimensions() directly, e.g. TensorSlicingOp. Most operations defer calculating dimensions until the operation is being evaluated. If you need access to the dimensions of a deferred operation, you can wrap it in a TensorRef (see Assigning to a TensorRef above), which provides dimensions() and dimension() as above.

TensorRef can also wrap the plain Tensor types, so this is a useful idiom in templated contexts where the underlying object could be either a raw Tensor or some deferred operation (e.g. a slice of a Tensor). In this case, the template code can wrap the object in a TensorRef and reason about its dimensionality while remaining agnostic to the underlying type.

Constructors

Tensor

Creates a tensor of the specified size. The number of arguments must be equal to the rank of the tensor. The content of the tensor is not initialized.

Eigen::Tensor<float, 2> a(3, 4);
std::cout << "NumRows: " << a.dimension(0) << " NumCols: " << a.dimension(1) << endl;
// NumRows: 3 NumCols: 4

TensorFixedSize

Creates a tensor of the specified size. The number of arguments in the Sizes<> template parameter determines the rank of the tensor. The content of the tensor is not initialized.

Eigen::TensorFixedSize<float, Sizes<3, 4>> a;
std::cout << "Rank: " << a.rank() << endl;
// Rank: 2
std::cout << "NumRows: " << a.dimension(0)
          << " NumCols: " << a.dimension(1) << endl;
// NumRows: 3 NumCols: 4

TensorMap

Creates a tensor mapping an existing array of data. The data must not be freed until the TensorMap is discarded, and the size of the data must be large enough to accommodate the coefficients of the tensor.

float data[] = {0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11};
Eigen::TensorMap<Tensor<float, 2>> a(data, 3, 4);
std::cout << "NumRows: " << a.dimension(0) << " NumCols: " << a.dimension(1) << endl;
// NumRows: 3 NumCols: 4
std::cout << "a(1, 2): " << a(1, 2) << endl;
// a(1, 2): 7

Contents Initialization

When a new Tensor or a new TensorFixedSize are created, memory is allocated to hold all the tensor elements, but the memory is not initialized. Similarly, when a new TensorMap is created on top of non-initialized memory the memory its contents are not initialized.

You can use one of the methods below to initialize the tensor memory. These have an immediate effect on the tensor and return the tensor itself as a result. These are not tensor Operations which delay evaluation.

<Tensor-Type> setConstant(const Scalar& val)

Sets all elements of the tensor to the constant value val. Scalar is the type of data stored in the tensor. You can pass any value that is convertible to that type.

Returns the tensor itself in case you want to chain another call.

a.setConstant(12.3f);
std::cout << "Constant: " << endl << a << endl << endl;

// Constant:
// 12.3 12.3 12.3 12.3
// 12.3 12.3 12.3 12.3
// 12.3 12.3 12.3 12.3

Note that setConstant() can be used on any tensor where the element type has a copy constructor and an operator=():

Eigen::Tensor<string, 2> a(2, 3);
a.setConstant("yolo");
std::cout << "String tensor: " << endl << a << endl << endl;

// String tensor:
// yolo yolo yolo
// yolo yolo yolo

<Tensor-Type> setZero()

Fills the tensor with zeros. Equivalent to setConstant(Scalar(0)). Returns the tensor itself in case you want to chain another call.

a.setZero();
std::cout << "Zeros: " << endl << a << endl << endl;

// Zeros:
// 0 0 0 0
// 0 0 0 0
// 0 0 0 0

<Tensor-Type> setValues({..initializer_list})

Fills the tensor with explicit values specified in a std::initializer_list. The type of the initializer list depends on the type and rank of the tensor.

If the tensor has rank N, the initializer list must be nested N times. The most deeply nested lists must contains P scalars of the Tensor type where P is the size of the last dimension of the Tensor.

For example, for a TensorFixedSize<float, Sizes<2, 3>> the initializer list must contains 2 lists of 3 floats each.

setValues() returns the tensor itself in case you want to chain another call.

Eigen::Tensor<float, 2> a(2, 3);
a.setValues({{0.0f, 1.0f, 2.0f}, {3.0f, 4.0f, 5.0f}});
std::cout << "a" << endl << a << endl << endl;

// a
// 0 1 2
// 3 4 5

If a list is too short, the corresponding elements of the tensor will not be changed. This is valid at each level of nesting. For example the following code only sets the values of the first row of the tensor.

Eigen::Tensor<int, 2> a(2, 3);
a.setConstant(1000);
a.setValues({{10, 20, 30}});
std::cout << "a" << endl << a << endl << endl;
// a
// 10   20   30
// 1000 1000 1000

<Tensor-Type> setRandom()

Fills the tensor with random values. Returns the tensor itself in case you want to chain another call.

a.setRandom();
std::cout << "Random: " << endl << a << endl << endl;
// Random:
//   0.680375    0.59688  -0.329554    0.10794
//  -0.211234   0.823295   0.536459 -0.0452059
//   0.566198  -0.604897  -0.444451   0.257742

You can customize setRandom() by providing your own random number generator as a template argument:

a.setRandom<MyRandomGenerator>();

Here, MyRandomGenerator must be a struct with the following member functions, where Scalar and Index are the same as <Tensor-Type>::Scalar and <Tensor-Type>::Index.

See struct UniformRandomGenerator in TensorFunctors.h for an example.

// Custom number generator for use with setRandom().
struct MyRandomGenerator {
  // Default and copy constructors. Both are needed
  MyRandomGenerator() { }
  MyRandomGenerator(const MyRandomGenerator& ) { }

  // Return a random value to be used.  "element_location" is the
  // location of the entry to set in the tensor, it can typically
  // be ignored.
  Scalar operator()(Eigen::DenseIndex element_location,
                    Eigen::DenseIndex /*unused*/ = 0) const {
    return <randomly generated value of type T>;
  }

  // Same as above but generates several numbers at a time.
  typename internal::packet_traits<Scalar>::type packetOp(
      Eigen::DenseIndex packet_location, Eigen::DenseIndex /*unused*/ = 0) const {
    return <a packet of randomly generated values>;
  }
};

You can also use one of the 2 random number generators that are part of the tensor library:

• UniformRandomGenerator
• NormalRandomGenerator

Data Access

The Tensor, TensorFixedSize, and TensorRef classes provide the following accessors to access the tensor coefficients:

const Scalar& operator()(const array<Index, NumIndices>& indices)
const Scalar& operator()(Index firstIndex, IndexTypes... otherIndices)
Scalar& operator()(const array<Index, NumIndices>& indices)
Scalar& operator()(Index firstIndex, IndexTypes... otherIndices)

The number of indices must be equal to the rank of the tensor. Moreover, these accessors are not available on tensor expressions. In order to access the values of a tensor expression, the expression must either be evaluated or wrapped in a TensorRef.

Scalar* data() and const Scalar* data() const

Returns a pointer to the storage for the tensor. The pointer is const if the tensor was const. This allows direct access to the data. The layout of the data depends on the tensor layout: RowMajor or ColMajor.

This access is usually only needed for special cases, for example when mixing Eigen Tensor code with other libraries.

Scalar is the type of data stored in the tensor.

Eigen::Tensor<float, 2> a(3, 4);
float* a_data = a.data();
a_data[0] = 123.45f;
std::cout << "a(0, 0): " << a(0, 0);
// a(0, 0): 123.45

Tensor Operations

All the methods documented below return non evaluated tensor Operations. These can be chained: you can apply another Tensor Operation to the value returned by the method.

The chain of Operation is evaluated lazily, typically when it is assigned to a tensor. See Controlling When Expressions are Evaluated for more details about their evaluation.

(Operation) constant(const Scalar& val)

Returns a tensor of the same type and dimensions as the original tensor but where all elements have the value val.

This is useful, for example, when you want to add or subtract a constant from a tensor, or multiply every element of a tensor by a scalar. However, such operations can also be performed using operator overloads (see operator+).

Eigen::Tensor<float, 2> a(2, 3);
a.setConstant(1.0f);
Eigen::Tensor<float, 2> b = a + a.constant(2.0f);
Eigen::Tensor<float, 2> c = b * b.constant(0.2f);
std::cout << "a" << endl << a << endl << endl;
std::cout << "b" << endl << b << endl << endl;
std::cout << "c" << endl << c << endl << endl;
// a
// 1 1 1
// 1 1 1

// b
// 3 3 3
// 3 3 3

// c
// 0.6 0.6 0.6
// 0.6 0.6 0.6

(Operation) random()

Returns a tensor of the same type and dimensions as the current tensor but where all elements have random values.

This is for example useful to add random values to an existing tensor. The generation of random values can be customized in the same manner as for setRandom().

Eigen::Tensor<float, 2> a(2, 3);
a.setConstant(1.0f);
Eigen::Tensor<float, 2> b = a + a.random();
std::cout << "a\n" << a << "\n\n";
std::cout << "b\n" << b << "\n\n";

// a
// 1 1 1
// 1 1 1
// b
// 1.68038   1.5662  1.82329
// 0.788766  1.59688

Unary Element Wise Operations

All these operations take a single input tensor as argument and return a tensor of the same type and dimensions as the tensor to which they are applied. The requested operations are applied to each element independently.

(Operation) operator-()

Returns a tensor of the same type and dimensions as the original tensor containing the opposite values of the original tensor.

Eigen::Tensor<float, 2> a(2, 3);
a.setConstant(1.0f);
Eigen::Tensor<float, 2> b = -a;
std::cout << "a\n" << a << "\n\n";
std::cout << "b\n" << b << "\n\n";

// a
// 1 1 1
// 1 1 1
//
// b
// -1 -1 -1
// -1 -1 -1

(Operation) sqrt()

Returns a tensor containing the square roots of the original tensor.

(Operation) rsqrt()

Returns a tensor containing the inverse square roots (1/sqrt(x)) of the original tensor.

(Operation) square()

Returns a tensor containing the squares of the original tensor values.

(Operation) cube()

Returns a tensor containing the cubes (x^3) of the original tensor values.

(Operation) inverse()

Returns a tensor containing the inverse (1/x) of the original tensor values.

(Operation) exp()

Returns a tensor containing the exponential of the original tensor.

(Operation) expm1()

Returns a tensor containing exp(x) - 1 for each element. More accurate than exp(x) - 1 for small values of x.

(Operation) log()

Returns a tensor containing the natural logarithms of the original tensor.

(Operation) log1p()

Returns a tensor containing log(1 + x) for each element. More accurate than log(1 + x) for small values of x.

(Operation) log2()

Returns a tensor containing the base-2 logarithms of the original tensor.

(Operation) abs()

Returns a tensor containing the absolute values of the original tensor.

(Operation) sign()

Returns a tensor containing the sign (-1, 0, or +1) of each element.

(Operation) arg()

Returns a tensor containing the complex argument (phase angle) of the values of the original tensor.

(Operation) real()

Returns a tensor containing the real part of the complex values of the original tensor. The result has a real-valued scalar type.

(Operation) imag()

Returns a tensor containing the imaginary part of the complex values of the original tensor. The result has a real-valued scalar type.

(Operation) conjugate()

Returns a tensor containing the complex conjugate of each element. For real-valued tensors, this is a no-op.

(Operation) pow(Scalar exponent)

Returns a tensor containing the coefficients of the original tensor raised to the power of the exponent.

The type of the exponent, Scalar, is always the same as the type of the tensor coefficients. For example, only integer exponents can be used in conjunction with tensors of integer values.

You can use cast() to lift this restriction. For example this computes cubic roots of an int Tensor:

Eigen::Tensor<int, 2> a(2, 3);
a.setValues({{0, 1, 8}, {27, 64, 125}});
Eigen::Tensor<double, 2> b = a.cast<double>().pow(1.0 / 3.0);
std::cout << "a" << endl << a << endl << endl;
std::cout << "b" << endl << b << endl << endl;

// a
// 0   1   8
// 27  64 125
//
// b
// 0 1 2
// 3 4 5

(Operation) clip(Scalar min_val, Scalar max_val)

Returns a tensor with each element clamped to the range [min_val, max_val].

Eigen::Tensor<float, 1> a(5);
a.setValues({-2.0f, -0.5f, 0.0f, 0.5f, 2.0f});
Eigen::Tensor<float, 1> b = a.clip(-1.0f, 1.0f);
// b: -1 -0.5  0  0.5  1

Rounding Operations

(Operation) round()

Returns a tensor with each element rounded to the nearest integer.

(Operation) rint()

Returns a tensor with each element rounded to the nearest integer (using the current rounding mode).

(Operation) ceil()

Returns a tensor with each element rounded up to the nearest integer.

(Operation) floor()

Returns a tensor with each element rounded down to the nearest integer.

Predicates

(Operation) (isnan)()

Returns a bool tensor indicating which elements are NaN.

Eigen::Tensor<float, 1> a(3);
a.setValues({1.0f, std::numeric_limits<float>::quiet_NaN(), 3.0f});
Eigen::Tensor<bool, 1> b = a.isnan().cast<bool>();
// b: false true false

(Operation) (isinf)()

Returns a bool tensor indicating which elements are infinite.

(Operation) (isfinite)()

Returns a bool tensor indicating which elements are finite (not NaN or Inf).

Hyperbolic and Activation Functions

(Operation) tanh()

Returns a tensor containing the hyperbolic tangent of each element.

(Operation) sigmoid()

Returns a tensor containing the logistic sigmoid (1/(1+exp(-x))) of each element.

Error Functions

(Operation) erf()

Returns a tensor containing the error function of each element.

(Operation) erfc()

Returns a tensor containing the complementary error function (1 - erf(x)) of each element.

(Operation) ndtri()

Returns a tensor containing the inverse of the normal cumulative distribution function of each element.

Special Math Functions

These require including <unsupported/Eigen/SpecialFunctions> in addition to the Tensor header.

(Operation) lgamma()

Returns a tensor containing the log-gamma function of each element.

(Operation) digamma()

Returns a tensor containing the digamma (psi) function of each element.

(Operation) bessel_i0(), bessel_i0e(), bessel_i1(), bessel_i1e()

Modified Bessel functions of the first kind. The e variants are exponentially scaled.

(Operation) bessel_j0(), bessel_j1()

Bessel functions of the first kind.

(Operation) bessel_y0(), bessel_y1()

Bessel functions of the second kind.

(Operation) bessel_k0(), bessel_k0e(), bessel_k1(), bessel_k1e()

Modified Bessel functions of the second kind. The e variants are exponentially scaled.

(Operation) igamma(const OtherDerived& other)

Regularized lower incomplete gamma function. this is the parameter a and other is x.

(Operation) igammac(const OtherDerived& other)

Regularized upper incomplete gamma function (1 - igamma).

(Operation) zeta(const OtherDerived& other)

Riemann zeta function. this is x and other is q.

(Operation) polygamma(const OtherDerived& other)

Polygamma function. this is n and other is x.

Scalar Arithmetic

(Operation) operator*(Scalar s)

Multiplies every element of the input tensor by the scalar s:

Eigen::Tensor<int, 2> a(2, 3);
a.setValues({{1, 2, 3},
                {4, 5, 6}});
Eigen::Tensor<int,2> scaled_a = a * 2;

std::cout << "a\n" << a << "\n";
std::cout << "scaled_a\n" << scaled_a << "\n";

// a
// 1 2 3
// 4 5 6
//
// scaled_a
// 2  4  6
// 8 10 12

(Operation) operator+ (Scalar s)

Adds s to every element in the tensor.

(Operation) operator- (Scalar s)

Subtracts s from every element in the tensor.

(Operation) operator/ (Scalar s)

Divides every element in the tensor by s.

(Operation) operator% (Scalar s)

Computes the element-wise modulus (remainder) of each tensor element divided by s.

Only integer types are supported. For floating-point tensors, implement a unaryExpr using std::fmod.

(Operation) cwiseMax(Scalar threshold)

Returns a tensor where each element is the maximum of the original element and the scalar threshold.

Eigen::Tensor<int, 2> a(2, 3);
a.setValues({{0, 100, 200}, {300, 400, 500}});

Eigen::Tensor<int, 2> b(2, 3);
b.setValues({{-1, -2, 300}, {-4, 555, -6}});

Eigen::Tensor<int, 2> c = a.cwiseMax(b);

std::cout << "a\n" << a << "\n"
            << "b\n" << b << "\n"
            << "c\n" << c << "\n";

// a
//   0 100 200
// 300 400 500

// b
// -1  -2 300
// -4 555  -6

// c
//   0 100 300
// 300 555 500

(Operation) cwiseMin(Scalar threshold)

Returns a tensor where each element is the minimum of the original element and the scalar threshold.

Eigen::Tensor<int, 2> a(2, 2);
a.setValues({{0, 100}, {300, -900}});

Eigen::Tensor<int, 2> b(2, 2);
b.setValues({{-1, -2}, {400, 555}});

Eigen::Tensor<int, 2> c = a.cwiseMin(b);

std::cout << "a\n" << a << "\n"
          << "b\n" << b << "\n"
          << "c\n" << c << "\n";

// a
//   0  100
// 300 -900

// b
//  -1  -2
// 400 555

// c
//  -1   -2
// 300 -900

NaN Propagation for cwiseMax and cwiseMin

The cwiseMax and cwiseMin operations accept an optional template parameter controlling NaN propagation:

• cwiseMax<Eigen::PropagateNaN>(other) — if either operand is NaN, the result is NaN.
• cwiseMax<Eigen::PropagateNumbers>(other) — NaN is treated as missing; the non-NaN value wins.
• cwiseMax(other) — default behavior (fast; may or may not propagate NaN, depends on platform).

Eigen::Tensor<float, 1> a(3), b(3);
a.setValues({1.0f, NAN, 3.0f});
b.setValues({2.0f, 2.0f, NAN});

Eigen::Tensor<float, 1> c = a.cwiseMax<Eigen::PropagateNaN>(b);
// c: 2.0, NaN, NaN

Eigen::Tensor<float, 1> d = a.cwiseMax<Eigen::PropagateNumbers>(b);
// d: 2.0, 2.0, 3.0

(Operation) unaryExpr(const CustomUnaryOp& func)

Applies a user defined function to each element in the tensor. Supports lambdas or functor structs with an operator().

Using lambda:

Eigen::Tensor<float, 2> a(2, 3);
a.setValues({{0, -.5, -1}, {.5, 1.5, 2.0}});
auto my_func = [](float el){ return std::abs(el + 0.5f);};
Eigen::Tensor<float, 2> b = a.unaryExpr(my_func);
std::cout << "a\n" << a << "\n"
        << "b\n" << b << "\n";
=>
a
    0  -0.5   -1
0.5   1.5    2
b
0.5     0  0.5
    1     2  2.5

Using a functor to normalize and clamp values to [-1.0, 1.0]:

template<typename Scalar>
struct NormalizedClamp {
NormalizedClamp(Scalar lo, Scalar hi) : _lo(lo), _hi(hi) {}
Scalar operator()(Scalar x) const {
    if (x < _lo) return Scalar(0);
    if (x > _hi) return Scalar(1);
    return (x - _lo) / (_hi - _lo);
}
Scalar _lo, _hi;
};

Eigen::Tensor<float, 2> c = a.unaryExpr(NormalizedClamp<float>(-1.0f, 1.0f));
std::cout << "c\n" << c << "\n";

// c
// 0.5    0.25    0
// 0.75   1       1

Bitwise and Boolean Unary Operations

(Operation) operator~()

Bitwise NOT of each element (integer types only).

(Operation) operator!()

Boolean NOT of each element.

Binary Element Wise Operations

These operations take two input tensors as arguments. The 2 input tensors should be of the same type and dimensions. The result is a tensor of the same dimensions as the tensors to which they are applied, and unless otherwise specified it is also of the same type. The requested operations are applied to each pair of elements independently.

(Operation) operator+(const OtherDerived& other)

Returns a tensor of the same type and dimensions as the input tensors containing the coefficient wise sums of the inputs.

(Operation) operator-(const OtherDerived& other)

Returns a tensor of the same type and dimensions as the input tensors containing the coefficient wise differences of the inputs.

(Operation) operator*(const OtherDerived& other)

Returns a tensor of the same type and dimensions as the input tensors containing the coefficient wise products of the inputs.

(Operation) operator/(const OtherDerived& other)

Returns a tensor of the same type and dimensions as the input tensors containing the coefficient wise quotients of the inputs.

This operator is not supported for integer types.

(Operation) cwiseMax(const OtherDerived& other)

Returns a tensor of the same type and dimensions as the input tensors containing the coefficient wise maximums of the inputs.

(Operation) cwiseMin(const OtherDerived& other)

Returns a tensor of the same type and dimensions as the input tensors containing the coefficient wise minimums of the inputs.

(Operation) binaryExpr(const OtherDerived& other, const CustomBinaryOp& func)

Applies a custom binary functor element-wise to two tensors.

Eigen::Tensor<float, 2> a(2, 3), b(2, 3);
a.setRandom(); b.setRandom();
auto my_op = [](float x, float y) { return x * x + y * y; };
Eigen::Tensor<float, 2> c = a.binaryExpr(b, my_op);

(Operation) Logical operators

The following boolean operators are supported:

• operator&&(const OtherDerived& other)
• operator||(const OtherDerived& other)
• operator<(const OtherDerived& other)
• operator<=(const OtherDerived& other)
• operator>(const OtherDerived& other)
• operator>=(const OtherDerived& other)
• operator==(const OtherDerived& other)
• operator!=(const OtherDerived& other)

as well as bitwise operators:

• operator&(const OtherDerived& other)
• operator|(const OtherDerived& other)
• operator^(const OtherDerived& other)

The resulting tensor retains the input scalar type.

Scalar comparison variants are also available (e.g. a < 0.5f).

Selection (select(const ThenDerived& thenTensor, const ElseDerived& elseTensor)

Selection is a coefficient-wise ternary operator that is the tensor equivalent to the if-then-else operation.

    Tensor<bool, 3> if = ...;
    Tensor<float, 3> then = ...;
    Tensor<float, 3> else = ...;
    Tensor<float, 3> result = if.select(then, else);

The 3 arguments must be of the same dimensions, which will also be the dimension of the result. The 'if' tensor must be of type boolean, the 'then' and the 'else' tensor must be of the same type, which will also be the type of the result.

Each coefficient in the result is equal to the corresponding coefficient in the 'then' tensor if the corresponding value in the 'if' tensor is true. If not, the resulting coefficient will come from the 'else' tensor.

Contraction

Tensor contractions are a generalization of the matrix product to the multidimensional case.

// Create 2 matrices using tensors of rank 2
Eigen::Tensor<int, 2> a(2, 3);
a.setValues({{1, 2, 3}, {6, 5, 4}});
Eigen::Tensor<int, 2> b(3, 2);
b.setValues({{1, 2}, {4, 5}, {5, 6}});

// Compute the traditional matrix product
Eigen::array<Eigen::IndexPair<int>, 1> product_dims = { Eigen::IndexPair<int>(1, 0) };
Eigen::Tensor<int, 2> AB = a.contract(b, product_dims);

// Compute the product of the transpose of the matrices
Eigen::array<Eigen::IndexPair<int>, 1> transposed_product_dims = { Eigen::IndexPair<int>(0, 1) };
Eigen::Tensor<int, 2> AtBt = a.contract(b, transposed_product_dims);

// Contraction to scalar value using a double contraction.
// First coordinate of both tensors are contracted as well as both second coordinates, i.e., this computes the sum of the squares of the elements.
Eigen::array<Eigen::IndexPair<int>, 2> double_contraction_product_dims = { Eigen::IndexPair<int>(0, 0), Eigen::IndexPair<int>(1, 1) };
Eigen::Tensor<int, 0> AdoubleContractedA = a.contract(a, double_contraction_product_dims);

// Extracting the scalar value of the tensor contraction for further usage
int value = AdoubleContractedA(0);

Reduction Operations

A Reduction operation returns a tensor with fewer dimensions than the original tensor. The values in the returned tensor are computed by applying a reduction operator to slices of values from the original tensor. You specify the dimensions along which the slices are made.

The Eigen Tensor library provides a set of predefined reduction operators such as maximum() and sum() and lets you define additional operators by implementing a few methods from a reductor template.

Reduction Dimensions

All reduction operations take a single parameter of type <TensorType>::``Dimensions which can always be specified as an array of ints. These are called the "reduction dimensions." The values are the indices of the dimensions of the input tensor over which the reduction is done. The parameter can have at most as many element as the rank of the input tensor; each element must be less than the tensor rank, as it indicates one of the dimensions to reduce.

Each dimension of the input tensor should occur at most once in the reduction dimensions as the implementation does not remove duplicates.

The order of the values in the reduction dimensions does not affect the results, but the code may execute faster if you list the dimensions in increasing order.

Example: Reduction along one dimension.

// Create a tensor of 2 dimensions
Eigen::Tensor<int, 2> a(2, 3);
a.setValues({{1, 2, 3}, {6, 5, 4}});
// Reduce it along the second dimension (1)...
Eigen::array<int, 1> dims{1 /* dimension to reduce */};
// ...using the "maximum" operator.
// The result is a tensor with one dimension.  The size of
// that dimension is the same as the first (non-reduced) dimension of a.
Eigen::Tensor<int, 1> b = a.maximum(dims);
std::cout << "a" << endl << a << endl << endl;
std::cout << "b" << endl << b << endl << endl;

// a
// 1 2 3
// 6 5 4

// b
// 3
// 6

Example: Reduction along two dimensions.

Eigen::Tensor<float, 3, Eigen::ColMajor> a(2, 3, 4);
a.setValues({{{0.0f, 1.0f, 2.0f, 3.0f},
                {7.0f, 6.0f, 5.0f, 4.0f},
                {8.0f, 9.0f, 10.0f, 11.0f}},
                {{12.0f, 13.0f, 14.0f, 15.0f},
                {19.0f, 18.0f, 17.0f, 16.0f},
                {20.0f, 21.0f, 22.0f, 23.0f}}});
// The tensor a has 3 dimensions.  We reduce along the
// first 2, resulting in a tensor with a single dimension
// of size 4 (the last dimension of a.)
// Note that we pass the array of reduction dimensions
// directly to the maximum() call.
Eigen::Tensor<float, 1, Eigen::ColMajor> b =
    a.maximum(Eigen::array<int, 2>{0, 1});
std::cout << "b" << endl << b << endl << endl;

// b
// 20
// 21
// 22
// 23

Reduction along all dimensions

As a special case, if you pass no parameter to a reduction operation the original tensor is reduced along all its dimensions. The result is a scalar, represented as a zero-dimension tensor.

Eigen::Tensor<float, 3> a(2, 3, 4);
a.setValues({{{0.0f, 1.0f, 2.0f, 3.0f},
              {7.0f, 6.0f, 5.0f, 4.0f},
              {8.0f, 9.0f, 10.0f, 11.0f}},
              {{12.0f, 13.0f, 14.0f, 15.0f},
              {19.0f, 18.0f, 17.0f, 16.0f},
              {20.0f, 21.0f, 22.0f, 23.0f}}});
// Reduce along all dimensions using the sum() operator.
Eigen::Tensor<float, 0> b = a.sum();
std::cout << "b\n" << b;

// b
// 276

You can extract the scalar directly by casting the expression and extract the first and only coefficient:

float sum = static_cast<Eigen::Tensor<float, 0>>(a.sum())();

(Operation) sum(const Dimensions& reduction_dims)

(Operation) sum()

Reduce a tensor using the sum() operator. The resulting values are the sum of the reduced values.

(Operation) mean(const Dimensions& reduction_dims)

(Operation) mean()

Reduce a tensor using the mean() operator. The resulting values are the mean of the reduced values.

(Operation) maximum(const Dimensions& reduction_dims)

(Operation) maximum()

Reduce a tensor using the maximum() operator. The resulting values are the largest of the reduced values.

(Operation) minimum(const Dimensions& reduction_dims)

(Operation) minimum()

Reduce a tensor using the minimum() operator. The resulting values are the smallest of the reduced values.

NaN Propagation for maximum and minimum

Like cwiseMax and cwiseMin, the maximum and minimum reductions accept an optional NaN propagation template parameter:

// If any element along the reduction is NaN, the result is NaN.
Eigen::Tensor<float, 1> b = a.maximum<Eigen::PropagateNaN>(dims);

// NaN values are ignored during reduction.
Eigen::Tensor<float, 1> c = a.maximum<Eigen::PropagateNumbers>(dims);

(Operation) prod(const Dimensions& reduction_dims)

(Operation) prod()

Reduce a tensor using the prod() operator. The resulting values are the product of the reduced values.

(Operation) all(const Dimensions& reduction_dims)

(Operation) all()

Reduce a tensor using the all() operator. Casts tensor to bool and then checks whether all elements are true. Runs through all elements rather than short-circuiting, so may be significantly inefficient.

(Operation) any(const Dimensions& reduction_dims)

(Operation) any()

Reduce a tensor using the any() operator. Casts tensor to bool and then checks whether any element is true. Runs through all elements rather than short-circuiting, so may be significantly inefficient.

(Operation) argmax(const Dimensions& reduction_dim)

(Operation) argmax()

Reduce a tensor using the argmax() operator.

The resulting values are the indices of the largest elements along the specified dimension.

Only a single reduction_dim is supported.

If multiple elements share the maximum value, the one with the lowest index is returned.

Eigen::Tensor<float, 2> a(2, 3);
a.setValues({{1, 4, 8}, {3, 4, 2}});

Eigen::Tensor<Eigen::Index, 1> argmax_dim0 = a.argmax(0);

std::cout << "a:\n" << a << "\n";
for (int i = 0; i < argmax_dim0.size(); ++i) {
    std::cout << "argmax along dim 0 at index " << i << " = " << argmax_dim0(i) << "\n";
}

// a:
// 1 4 8
// 3 4 2
// argmax along dim 0 at index 0 = 1
// argmax along dim 0 at index 1 = 0
// argmax along dim 0 at index 2 = 0

To compute the index of the global maximum, use the overload without arguments (which flattens the tensor).

Eigen::Tensor<Eigen::Index, 0> argmax_flat = a.argmax();
std::cout << "Flat argmax index: " << argmax_flat();

// Flat argmax index: 4

(Operation) argmin(const Dimensions& reduction_dim)

(Operation) argmin()

See argmax.

(Operation) reduce(const Dimensions& reduction_dims, const Reducer& reducer)

Reduce a tensor using a user-defined reduction operator. See SumReducer in TensorFunctors.h for information on how to implement a reduction operator.

Trace

A Trace operation returns a tensor with fewer dimensions than the original tensor. It returns a tensor whose elements are the sum of the elements of the original tensor along the main diagonal for a list of specified dimensions, the "trace dimensions". Similar to the Reduction Dimensions, the trace dimensions are passed as an input parameter to the operation, are of type <TensorType>::``Dimensions , and have the same requirements when passed as an input parameter. In addition, the trace dimensions must have the same size.

Example: Trace along 2 dimensions.

// Create a tensor of 3 dimensions
Eigen::Tensor<int, 3> a(2, 2, 3);
a.setValues({{{1, 2, 3}, {4, 5, 6}}, {{7, 8, 9}, {10, 11, 12}}});
// Specify the dimensions along which the trace will be computed.
// In this example, the trace can only be computed along the dimensions
// with indices 0 and 1
Eigen::array<int, 2> dims{0, 1};
// The output tensor contains all but the trace dimensions.
Tensor<int, 1> a_trace = a.trace(dims);
std::cout << "a_trace:" << endl;
std::cout << a_trace << endl;

// a_trace:
// 11
// 13
// 15

(Operation) trace(const Dimensions& new_dims)

(Operation) trace()

As a special case, if no parameter is passed to the operation, trace is computed along all dimensions of the input tensor.

Example: Trace along all dimensions.

// Create a tensor of 3 dimensions, with all dimensions having the same size.
Eigen::Tensor<int, 3> a(3, 3, 3);
a.setValues({{{1, 2, 3}, {4, 5, 6}, {7, 8, 9}},
            {{10, 11, 12}, {13, 14, 15}, {16, 17, 18}},
            {{19, 20, 21}, {22, 23, 24}, {25, 26, 27}}});
// Result is a zero dimension tensor
Tensor<int, 0> a_trace = a.trace();
std::cout<<"a_trace:"<<endl;
std::cout<<a_trace<<endl;

// a_trace:
// 42

Scan Operations

A Scan operation returns a tensor with the same dimensions as the original tensor. The operation performs an inclusive scan along the specified axis, which means it computes a running total along the axis for a given reduction operation. If the reduction operation corresponds to summation, then this computes the prefix sum of the tensor along the given axis.

Example: Cumulative sum along the second dimension

// Create a tensor of 2 dimensions
Eigen::Tensor<int, 2> a(2, 3);
a.setValues({{1, 2, 3}, {4, 5, 6}});
// Scan it along the second dimension (1) using summation
Eigen::Tensor<int, 2> b = a.cumsum(1);
// The result is a tensor with the same size as the input
std::cout << "a" << endl << a << endl << endl;
std::cout << "b" << endl << b << endl << endl;

// a
// 1 2 3
// 4 5 6

// b
// 1  3  6
// 4  9 15

(Operation) cumsum(const Index& axis, bool exclusive = false)

Perform a scan by summing consecutive entries.

When exclusive is true, element i contains the sum of all elements before index i (exclusive prefix sum). The first element along the axis is 0.

Eigen::Tensor<int, 1> a(4);
a.setValues({1, 2, 3, 4});

Eigen::Tensor<int, 1> inclusive = a.cumsum(0);           // 1, 3, 6, 10
Eigen::Tensor<int, 1> exclusive = a.cumsum(0, true);     // 0, 1, 3, 6

(Operation) cumprod(const Index& axis, bool exclusive = false)

Perform a scan by multiplying consecutive entries.

When exclusive is true, element i contains the product of all elements before index i. The first element along the axis is 1.

Eigen::Tensor<int, 1> a(4);
a.setValues({1, 2, 3, 4});

Eigen::Tensor<int, 1> inclusive = a.cumprod(0);           // 1, 2, 6, 24
Eigen::Tensor<int, 1> exclusive = a.cumprod(0, true);     // 1, 1, 2, 6

Convolutions

(Operation) convolve(const Kernel& kernel, const Dimensions& dims)

Returns a tensor that is the output of the convolution of the input tensor with the kernel, along the specified dimensions of the input tensor. The dimension size for dimensions of the output tensor which were part of the convolution will be reduced by the formula:

output_dim_size = input_dim_size - kernel_dim_size + 1 // (requires: input_dim_size >= kernel_dim_size).

The dimension sizes for dimensions that were not part of the convolution will remain the same. Performance of the convolution can depend on the length of the stride(s) of the input tensor dimension(s) along which the convolution is computed (the first dimension has the shortest stride for ColMajor, whereas RowMajor's shortest stride is for the last dimension).

// Compute convolution along the second and third dimension.
Tensor<float, 4, DataLayout> input(3, 3, 7, 11);
Tensor<float, 2, DataLayout> kernel(2, 2);
Tensor<float, 4, DataLayout> output(3, 2, 6, 11);
input.setRandom();
kernel.setRandom();

Eigen::array<ptrdiff_t, 2> dims{1, 2};  // Specify second and third dimension for convolution.
output = input.convolve(kernel, dims);

for (int i = 0; i < 3; ++i) {
  for (int j = 0; j < 2; ++j) {
    for (int k = 0; k < 6; ++k) {
      for (int l = 0; l < 11; ++l) {
        const float result = output(i,j,k,l);
        const float expected = input(i,j+0,k+0,l) * kernel(0,0) +
                               input(i,j+1,k+0,l) * kernel(1,0) +
                               input(i,j+0,k+1,l) * kernel(0,1) +
                               input(i,j+1,k+1,l) * kernel(1,1);
        VERIFY_IS_APPROX(result, expected);
      }
    }
  }
}

FFT (Fast Fourier Transform)

(Operation) fft<FFTResultType, FFTDirection>(const FFTDims& dims)

Computes the Fast Fourier Transform of the input tensor along the specified dimensions.

Template parameters:

Parameter	Values	Description
FFTResultType	RealPart, ImagPart, BothParts	Which part(s) of the result to return
FFTDirection	FFT_FORWARD, FFT_REVERSE	Forward or inverse transform

When FFTResultType is BothParts, the output scalar type is std::complex<Scalar>. When RealPart or ImagPart, the output retains the real scalar type.

// Forward FFT of a 2D tensor along both dimensions.
Eigen::Tensor<float, 2> input(8, 16);
input.setRandom();
Eigen::array<int, 2> fft_dims{{0, 1}};

// Get the full complex result.
Eigen::Tensor<std::complex<float>, 2> complex_result =
    input.fft<Eigen::BothParts, Eigen::FFT_FORWARD>(fft_dims);

// Get only the real part.
Eigen::Tensor<float, 2> real_result =
    input.fft<Eigen::RealPart, Eigen::FFT_FORWARD>(fft_dims);

// Inverse FFT to recover the original signal.
Eigen::Tensor<float, 2> recovered =
    complex_result.fft<Eigen::RealPart, Eigen::FFT_REVERSE>(fft_dims);

The FFT uses the Cooley-Tukey algorithm for power-of-2 sizes and falls back to the Bluestein algorithm for arbitrary sizes.

Geometrical Operations

These operations return a Tensor with different dimensions than the original Tensor. They can be used to access slices of tensors, see them with different dimensions, or pad tensors with additional data.

(Operation) reshape(const Dimensions& new_dims)

Returns a view of the input tensor that has been reshaped to the specified new dimensions.

The argument new_dims is an array of Index values.

The rank of the resulting tensor is equal to the number of elements in new_dims.

The product of all the sizes in the new dimension array must be equal to the number of elements in the input tensor.

// Increase the rank of the input tensor by introducing a new dimension
// of size 1.
Tensor<float, 2> input(7, 11);
array<int, 3> three_dims{{7, 11, 1}};
Tensor<float, 3> result = input.reshape(three_dims);

// Decrease the rank of the input tensor by merging 2 dimensions;
array<int, 1> one_dim{{7 * 11}};
Tensor<float, 1> result = input.reshape(one_dim);

This operation does not move any data in the input tensor, so the resulting contents of a reshaped Tensor depend on the data layout of the original Tensor.

For example this is what happens when you reshape() a 2D ColMajor tensor to one dimension:

Eigen::Tensor<float, 2, Eigen::ColMajor> a(2, 3);
a.setValues({{0.0f, 100.0f, 200.0f}, {300.0f, 400.0f, 500.0f}});
Eigen::array<Eigen::DenseIndex, 1> one_dim{3 * 2};
Eigen::Tensor<float, 1, Eigen::ColMajor> b = a.reshape(one_dim);
std::cout << "b" << endl << b << endl;

// b
//   0
// 300
// 100
// 400
// 200
// 500

This is what happens when the 2D Tensor is RowMajor:

Eigen::Tensor<float, 2, Eigen::RowMajor> a(2, 3);
a.setValues({{0.0f, 100.0f, 200.0f}, {300.0f, 400.0f, 500.0f}});
Eigen::array<Eigen::DenseIndex, 1> one_dim{3 * 2};
Eigen::Tensor<float, 1, Eigen::RowMajor> b = a.reshape(one_dim);
std::cout << "b" << endl << b << endl;

// b
//   0
// 100
// 200
// 300
// 400
// 500

The reshape operation is a lvalue. In other words, it can be used on the left side of the assignment operator.

The previous example can be rewritten as follow:

Eigen::Tensor<float, 2, Eigen::ColMajor> a(2, 3);
a.setValues({{0.0f, 100.0f, 200.0f}, {300.0f, 400.0f, 500.0f}});
Eigen::array<Eigen::DenseIndex, 2> two_dim{2, 3};
Eigen::Tensor<float, 1, Eigen::ColMajor> b(6);
b.reshape(two_dim) = a;
std::cout << "b" << endl << b << endl;

// b
//   0
// 300
// 100
// 400
// 200
// 500

Note that "b" itself was not reshaped but that instead the assignment is done to the reshape view of b.

(Operation) shuffle(const Shuffle& shuffle)

Returns a view of the input tensor whose dimensions have been reordered according to the specified permutation.

The argument shuffle is an array of Index values:

• Its size is the rank of the input tensor.
• It must contain a permutation of [0, 1, ..., rank - 1].
• The i-th dimension of the output tensor corresponds to the size of the dimension at position shuffle[i] in the input tensor. For example:

// Shuffle all dimensions to the left by 1.
Tensor<float, 3> input(20, 30, 50);
// ... set some values in input.
Tensor<float, 3> output = input.shuffle({1, 2, 0});

eigen_assert(output.dimension(0) == 30);
eigen_assert(output.dimension(1) == 50);
eigen_assert(output.dimension(2) == 20);

// Indices into the output tensor are shuffled accordingly to formulate
// indices into the input tensor.
eigen_assert(output(3, 7, 11) == input(11, 3, 7));

// In general:
eigen_assert(output(..., indices[shuffle[i]], ...) ==
             input(..., indices[i], ...));

The shuffle operation results in a lvalue, which means that it can be assigned to. In other words, it can be used on the left side of the assignment operator.

Let's rewrite the previous example to take advantage of this feature:

// Shuffle all dimensions to the left by 1.
Tensor<float, 3> input(20, 30, 50);
input.setRandom();
Tensor<float, 3> output(30, 50, 20);
output.shuffle({2, 0, 1}) = input;

(Operation) stride(const Strides& strides)

Returns a view of the input tensor that strides (skips stride-1 elements) along each of the dimensions.

The argument strides is an array of Index values:

• Its size is the rank of the input tensor.
• Must be >= 1

The dimensions of the resulting tensor are ceil(input_dimensions[i] / strides[i]).

For example this is what happens when you stride() a 2D tensor:

Eigen::Tensor<int, 2> a(4, 3);
a.setValues({{0, 100, 200},
             {300, 400, 500},
             {600, 700, 800},
             {900, 1000, 1100}});
Eigen::array<Eigen::DenseIndex, 2> strides{3, 2};
Eigen::Tensor<int, 2> b = a.stride(strides);
std::cout << "b" << endl << b << endl;
// b
//    0   200
//  900  1100

It is possible to assign a tensor to a stride:

Tensor<float, 3> input(20, 30, 50);
input.setRandom();
Tensor<float, 3> output(40, 90, 200);
output.stride({2, 3, 4}) = input;

(Operation) slice(const StartIndices& offsets, const Sizes& extents)

Returns a sub-tensor of the given tensor. For each dimension i, the slice is made of the coefficients stored between offset[i] and offset[i] + extents[i] in the input tensor.

Eigen::Tensor<int, 2> a(4, 3);
a.setValues({{0, 100, 200}, {300, 400, 500},
             {600, 700, 800}, {900, 1000, 1100}});
Eigen::array<Eigen::Index, 2> offsets = {1, 0};
Eigen::array<Eigen::Index, 2> extents = {2, 2};
Eigen::Tensor<int, 2> slice = a.slice(offsets, extents);
std::cout << "a" << endl << a << endl;
// a
//    0   100   200
//  300   400   500
//  600   700   800
//  900  1000  1100

std::cout << "slice" << endl << slice << endl;
// slice
//  300   400
//  600   700

(Operation) stridedSlice(const StartIndices& start, const StopIndices& stop, const Strides& strides)

Returns a sub-tensor by selecting elements using start, stop (exclusive), and strides for each dimension.

This is similar to slicing in Python using [start:stop:step].

Eigen::Tensor<int, 2> a(4, 6);
a.setValues({{  0,  10,  20,  30,  40,   50},
             {100, 110, 120, 130, 140,  150},
             {200, 210, 220, 230, 240,  250},
             {300, 310, 320, 330, 340,  350}});

Eigen::array<Eigen::Index, 2> start = {1, 1};
Eigen::array<Eigen::Index, 2> stop  = {4, 6}; // Stop is exclusive
Eigen::array<Eigen::Index, 2> strides = {2, 2};

Eigen::Tensor<int, 2> sub = a.stridedSlice(start, stop, strides);

std::cout << "a\n" << a << "\n";
std::cout << "sub\n" << sub << "\n";

// a
//   0  10  20  30  40  50
// 100 110 120 130 140 150
// 200 210 220 230 240 250
// 300 310 320 330 340 350

// sub
// 110 130 150
// 310 330 350

It is also possible to assign to a strided slice:

Eigen::Tensor<int, 2> b(sub.dimensions());
b.setConstant(-1);
a.stridedSlice(start, stop, strides) = b;
std::cout << "modified a\n" << a << "\n";


// modified a
//   0  10  20  30  40  50
// 100  -1 120  -1 140  -1
// 200 210 220 230 240 250
// 300  -1 320  -1 340  -1

(Operation) chip(const Index offset, const Index dim)

A chip is a special kind of slice. It is the subtensor at the given offset in the dimension dim.

The returned tensor has one fewer dimension than the input tensor: the dimension dim is removed.

For example, a matrix chip would be either a row or a column of the input matrix:

Eigen::Tensor<int, 2> a(4, 3);
a.setValues({{0, 100, 200}, {300, 400, 500},
             {600, 700, 800}, {900, 1000, 1100}});
Eigen::Tensor<int, 1> row_3 = a.chip(2, 0);
Eigen::Tensor<int, 1> col_2 = a.chip(1, 1);
std::cout << "a\n" << a << "\n";

// a
//    0   100   200
//  300   400   500
//  600   700   800
//  900  1000  1100

std::cout << "row_3\n" << row_3 << "\n";
// row_3
//    600   700   800

std::cout << "col_2\n" << col_2 << "\n";
// col_2
//   100   400   700    1000

It is possible to assign values to a tensor chip since the chip operation is a lvalue. For example:

Eigen::Tensor<int, 1> a(3);
a.setValues({{100, 200, 300}});
Eigen::Tensor<int, 2> b(2, 3);
b.setZero();
b.chip(0, 0) = a;
std::cout << "a\n" << a << "\n";
std::cout << "b\n" << b << "\n";

// a
// 100
// 200
// 300

// b
//   100   200   300
//     0     0     0

The dimension can also be passed as a template parameter:

b.chip<0>(1) = a;  // Equivalent to b.chip(1,0) = a;

Note that only one dimension can be chipped at a time. To chip off multiple dimensions, you can chain calls

Eigen::Tensor<int, 3> a(2, 3, 4);
Eigen::Tensor<int, 1> b = b.chip<2>(0) // Now has shape [2,3]
                           .chip<1>(0); // Now has shape [2]

Be careful in which order you chip, as each operation affects the shape of the intermediate result. For example:

// AVOID THIS
Eigen::Tensor<int, 1> c = b.chip<1>(0) // Now has shape [2,4]
                           .chip<1>(0); // Now has shape [2]

In general, it's more intuitive to chip from the outermost dimension first.

(Operation) reverse(const ReverseDimensions& reverse)

Returns a view of the input tensor that reverses the order of the coefficients along a subset of the dimensions. The argument reverse is an array of boolean values that indicates whether or not the order of the coefficients should be reversed along each of the dimensions. This operation preserves the dimensions of the input tensor.

For example this is what happens when you reverse() the first dimension of a 2D tensor:

Eigen::Tensor<int, 2> a(4, 3);
a.setValues({{0, 100, 200}, {300, 400, 500},
            {600, 700, 800}, {900, 1000, 1100}});
Eigen::array<bool, 2> reverse{true, false};
Eigen::Tensor<int, 2> b = a.reverse(reverse);
std::cout << "a\n" << a << "\n";
std::cout << "b\n" << b << "\n";

// a
//    0   100   200
//  300   400   500
//  600   700   800
//  900  1000  1100
// b
//  900  1000  1100
//  600   700   800
//  300   400   500
//    0   100   200

(Operation) roll(const Rolls& shifts)

Returns a tensor with the elements circularly shifted (like bit rotation) along one or more dimensions.

For each dimension i, the content is shifted by shifts[i] positions:

• A positive shift of +s moves each value to a lower index by s.
• A negative shift of -s moves each value to a higher index by s.

Eigen::Tensor<int, 2> a(3, 4);
a.setValues({{ 1,  2,  3,  4},
             { 5,  6,  7,  8},
             { 9, 10, 11, 12}});

Eigen::array<Eigen::Index, 2> shifts = {1, -2};

Eigen::Tensor<int, 2> rolled = a.roll(shifts);

std::cout << "a\n" << a << "\n";
std::cout << "rolled\n" << rolled << "\n";

// a
// 1  2  3  4
// 5  6  7  8
// 9 10 11 12
//
// rolled
// 7  8  5  6
// 11 12  9 10
// 3  4  1  2

(Operation) broadcast(const Broadcast& broadcast)

Returns a view of the input tensor in which the input is replicated one to many times. The broadcast argument specifies how many copies of the input tensor need to be made in each of the dimensions.

Eigen::Tensor<int, 2> a(2, 3);
a.setValues({{0, 100, 200}, {300, 400, 500}});
Eigen::array<int, 2> bcast{3, 2};
Eigen::Tensor<int, 2> b = a.broadcast(bcast);
std::cout << "a" << endl << a << endl << "b" << endl << b << endl;
// a
//    0   100   200
//  300   400   500
// b
//    0   100   200    0   100   200
//  300   400   500  300   400   500
//    0   100   200    0   100   200
//  300   400   500  300   400   500
//    0   100   200    0   100   200
//  300   400   500  300   400   500

Note: Broadcasting does not increase rank. To broadcast into higher dimensions, you must first reshape the tensor with singleton (1) dimensions:

Eigen::Tensor<int, 2> a(2, 3);
a.setValues({{0, 100, 200}, {300, 400, 500}});

Eigen::array<Eigen::Index, 3> new_shape = {1, 2, 3}; //Reshape to [1, 2, 3]
Eigen::array<int, 3> bcast = {4, 1, 1}; // Broadcast to [4, 2, 3]
Eigen::Tensor<int, 3> b = a.reshape(new_shape).broadcast(bcast);

std::cout << "b dimensions: " << b.dimensions() << "\n";
std::cout << b << "\n";

(Operation) concatenate(const OtherDerived& other, Axis axis)

Returns a view of two tensors joined along a specified axis. The dimensions of the two tensors must match on all axes except the concatenation axis. The resulting tensor has the same rank as the inputs.

Eigen::Tensor<int, 2> a(2, 3);
a.setValues({{0, 100, 200}, {300, 400, 500}});

Eigen::Tensor<int, 2> b(2, 3);
b.setValues({{-1, -2, -3}, {-4, -5, -6}});

// Concatenate along dimension 0: resulting shape is [4, 3]
Eigen::Tensor<int, 2> c = a.concatenate(b, 0);

// Concatenate along dimension 1: resulting shape is [2, 6]
Eigen::Tensor<int, 2> d = a.concatenate(b, 1);

std::cout << "a\n" << a << "\n"
          << "b\n" << b << "\n"
          << "c (concatenated along dim 0)\n" << c << "\n"
          << "d (concatenated along dim 1)\n" << d << "\n";
// a
//   0 100 200
// 300 400 500
// b
// -1 -2 -3
// -4 -5 -6
// c (concatenated along dim 0)
//   0 100 200
// 300 400 500
//  -1  -2  -3
//  -4  -5  -6
// d (concatenated along dim 1)
//   0 100 200  -1  -2  -3
// 300 400 500  -4  -5  -6

(Operation) pad(const PaddingDimensions& padding)

Returns a view of the input tensor in which the input is padded with zeros.

An optional second argument specifies the padding value (default is zero): a.pad(paddings, 42) pads with the value 42.

Eigen::Tensor<int, 2> a(2, 3);
a.setValues({{0, 100, 200}, {300, 400, 500}});
Eigen::array<pair<int, int>, 2> paddings;
paddings[0] = make_pair(0, 1);
paddings[1] = make_pair(2, 3);
Eigen::Tensor<int, 2> b = a.pad(paddings);
std::cout << "a" << endl << a << endl << "b" << endl << b << endl;
// a
//    0   100   200
//  300   400   500
// b
//    0     0     0    0
//    0     0     0    0
//    0   100   200    0
//  300   400   500    0
//    0     0     0    0
//    0     0     0    0
//    0     0     0    0

(Operation) inflate(const Strides& strides)

Returns a tensor with zeros inserted between the elements of the input tensor along each dimension. The strides array specifies the inflation factor for each dimension: a stride of s inserts s-1 zeros between consecutive elements in that dimension. A stride of 1 leaves the dimension unchanged.

The output dimension sizes are (input_dim - 1) * stride + 1.

Eigen::Tensor<float, 2> a(2, 3);
a.setValues({{1, 2, 3}, {4, 5, 6}});
Eigen::array<Eigen::Index, 2> strides{{2, 3}};
Eigen::Tensor<float, 2> b = a.inflate(strides);
std::cout << "b dimensions: " << b.dimension(0) << " x " << b.dimension(1) << "\n";
std::cout << "b\n" << b << "\n";

// b dimensions: 3 x 7
// b
// 1 0 0 2 0 0 3
// 0 0 0 0 0 0 0
// 4 0 0 5 0 0 6

This is the adjoint of the stride() operation and is useful for implementing transposed convolutions (deconvolutions).

(Operation) extract_patches(const PatchDims& patch_dims)

Returns a tensor of coefficient patches extracted from the input tensor, where each patch is of dimension specified by patch_dims. The returned tensor has one greater dimension than the input tensor, which is used to index each patch. The patch index in the output tensor depends on the data layout of the input tensor: the patch index is the last dimension ColMajor layout, and the first dimension in RowMajor layout.

For example, given the following input tensor:

Eigen::Tensor<float, 2, DataLayout> tensor(3,4);
tensor.setValues({{0.0f, 1.0f, 2.0f, 3.0f},
                  {4.0f, 5.0f, 6.0f, 7.0f},
                  {8.0f, 9.0f, 10.0f, 11.0f}});

std::cout << "tensor: " << endl << tensor << endl;

// tensor:
//  0   1   2   3
//  4   5   6   7
//  8   9  10  11

Six 2x2 patches can be extracted and indexed using the following code:

Eigen::Tensor<float, 3, DataLayout> patch;
Eigen::array<ptrdiff_t, 2> patch_dims;
patch_dims[0] = 2;
patch_dims[1] = 2;
patch = tensor.extract_patches(patch_dims);
for (int k = 0; k < 6; ++k) {
  std::cout << "patch index: " << k << endl;
  for (int i = 0; i < 2; ++i) {
    for (int j = 0; j < 2; ++j) {
      if (DataLayout == ColMajor) {
        std::cout << patch(i, j, k) << " ";
      } else {
        std::cout << patch(k, i, j) << " ";
      }
    }
    std::cout << endl;
  }
}

This code results in the following output when the data layout is ColMajor:

patch index: 0
0 1
4 5
patch index: 1
4 5
8 9
patch index: 2
1 2
5 6
patch index: 3
5 6
9 10
patch index: 4
2 3
6 7
patch index: 5
6 7
10 11

This code results in the following output when the data layout is RowMajor:

NOTE: the set of patches is the same as in ColMajor, but are indexed differently

patch index: 0
0 1
4 5
patch index: 1
1 2
5 6
patch index: 2
2 3
6 7
patch index: 3
4 5
8 9
patch index: 4
5 6
9 10
patch index: 5
6 7
10 11

(Operation) extract_image_patches(const Index patch_rows, const Index patch_cols, const Index row_stride, const Index col_stride, const PaddingType padding_type)

Returns a tensor of coefficient image patches extracted from the input tensor, which is expected to have dimensions ordered as follows (depending on the data layout of the input tensor, and the number of additional dimensions 'N'):

• ColMajor
  • 1st dimension: channels (of size d)
  • 2nd dimension: rows (of size r)
  • 3rd dimension: columns (of size c)
  • 4th-Nth dimension: time (for video) or batch (for bulk processing).

• RowMajor (reverse order of ColMajor)
  • 1st-Nth dimension: time (for video) or batch (for bulk processing).
  • N+1'th dimension: columns (of size c)
  • N+2'th dimension: rows (of size r)
  • N+3'th dimension: channels (of size d)

The returned tensor has one greater dimension than the input tensor, which is used to index each patch. The patch index in the output tensor depends on the data layout of the input tensor: the patch index is the 4'th dimension in ColMajor layout, and the 4'th from the last dimension in RowMajor layout.

For example, given the following input tensor with the following dimension sizes:

• depth: 2
• rows: 3
• columns: 5
• batch: 7

Tensor<float, 4> tensor(2,3,5,7);
Tensor<float, 4, RowMajor> tensor_row_major = tensor.swap_layout();

2x2 image patches can be extracted and indexed using the following code:

2D patch: ColMajor (patch indexed by second-to-last dimension)

Tensor<float, 5> twod_patch;
twod_patch = tensor.extract_image_patches<2, 2>();
// twod_patch.dimension(0) == 2
// twod_patch.dimension(1) == 2
// twod_patch.dimension(2) == 2
// twod_patch.dimension(3) == 3*5
// twod_patch.dimension(4) == 7

2D patch: RowMajor (patch indexed by the second dimension)

Tensor<float, 5, RowMajor> twod_patch_row_major;
twod_patch_row_major = tensor_row_major.extract_image_patches<2, 2>();
// twod_patch_row_major.dimension(0) == 7
// twod_patch_row_major.dimension(1) == 3*5
// twod_patch_row_major.dimension(2) == 2
// twod_patch_row_major.dimension(3) == 2
// twod_patch_row_major.dimension(4) == 2

Generation and Custom Operations

(Operation) generate(const Generator& generator)

Returns a tensor whose values are computed by the given generator functor based on element coordinates. The generator must define operator() taking an array<Index, NumDims> of coordinates and returning a Scalar.

// Generator that produces the linear index of each element.
template <typename Index, int NumDims>
struct LinearIndexGenerator {
  Eigen::array<Index, NumDims> dims_;
  LinearIndexGenerator(const Eigen::array<Index, NumDims>& dims) : dims_(dims) {}

  float operator()(const Eigen::array<Index, NumDims>& coords) const {
    float idx = 0;
    float stride = 1;
    for (int i = 0; i < NumDims; ++i) {
      idx += coords[i] * stride;
      stride *= dims_[i];
    }
    return idx;
  }
};

Eigen::Tensor<float, 2> t(3, 4);
Eigen::array<Eigen::Index, 2> dims{{3, 4}};
Eigen::Tensor<float, 2> result = t.generate(LinearIndexGenerator<Eigen::Index, 2>(dims));

(Operation) customOp(const CustomUnaryFunc& func)

Applies a custom operation that can produce output with different dimensions than the input. Unlike unaryExpr() which is element-wise, customOp() gives full control over how the output is computed.

The functor must implement:

• dimensions(const InputType& input) — returns the output dimensions.
• eval(const InputType& input, OutputType& output, const Device& device) — computes the result.

struct RowSumOp {
  // Output is a 1D tensor with size equal to the number of rows.
  template <typename Input>
  Eigen::DSizes<Eigen::Index, 1> dimensions(const Input& input) const {
    return Eigen::DSizes<Eigen::Index, 1>(input.dimension(0));
  }

  template <typename Input, typename Output, typename Device>
  void eval(const Input& input, Output& output, const Device& device) const {
    Eigen::array<Eigen::Index, 1> reduce_dims{{1}};
    output.device(device) = input.sum(reduce_dims);
  }
};

Eigen::Tensor<float, 2> a(3, 4);
a.setRandom();
Eigen::Tensor<float, 1> row_sums = a.customOp(RowSumOp());

A binary variant is also available:

Eigen::Tensor<float, 2> result = a.customOp(b, MyBinaryCustomOp());

(Operation) nullaryExpr(const CustomNullaryOp& func)

Creates a tensor from a custom nullary functor. The functor is called for each element position.

Special Operations

(Operation) cast<T>()

Returns a tensor of type T with the same dimensions as the original tensor. The returned tensor contains the values of the original tensor converted to type T.

Eigen::Tensor<float, 2> a(2, 3);
Eigen::Tensor<int, 2> b = a.cast<int>();

This can be useful for example if you need to do element-wise division of Tensors of integers. This is not currently supported by the Tensor library but you can easily cast the tensors to floats to do the division:

Eigen::Tensor<int, 2> a(2, 3);
a.setValues({{0, 1, 2}, {3, 4, 5}});
Eigen::Tensor<int, 2> b =
    (a.cast<float>() / a.constant(2).cast<float>()).cast<int>();
std::cout << "a\n" << a << "\n";
std::cout << "b\n" << b << "\n";

// a
// 0 1 2
// 3 4 5
//
// b
// 0 0 1
// 1 2 2

(Operation) eval()

See Calling eval().

Tensor Printing

Tensors can be printed into a stream object (e.g. std::cout) using different formatting options.

Eigen::Tensor<float, 3> tensor3d(4, 3, 2);
tensor3d.setValues( {{{1, 2},
                      {3, 4},
                      {5, 6}},
                     {{7, 8},
                      {9, 10},
                      {11, 12}},
                     {{13, 14},
                      {15, 16},
                      {17, 18}},
                     {{19, 20},
                      {21, 22},
                      {23, 24}}} );
std::cout << tensor3d.format(Eigen::TensorIOFormat::Plain()) << "\n";
//  1  2
//  3  4
//  5  6
//
//  7  8
//  9 10
// 11 12
//
// 13 14
// 15 16
// 17 18
//
// 19 20
// 21 22
// 23 24

In the example, we used the predefined format Eigen::TensorIOFormat::Plain. Here is the list of all predefined formats from which you can choose:

• Eigen::TensorIOFormat::Plain() for a plain output without braces. Different submatrices are separated by a blank line.
• Eigen::TensorIOFormat::Numpy() for numpy-like output.
• Eigen::TensorIOFormat::Native() for a c++ like output which can be directly copy-pasted to setValues().
• Eigen::TensorIOFormat::Legacy() for a backwards compatible printing of tensors.

If you send the tensor directly to the stream the default format is called which is Eigen::TensorIOFormat::Plain().

You can define your own format by explicitly providing a Eigen::TensorIOFormat class instance. Here, you can specify:

• The overall prefix and suffix with std::string tenPrefix and std::string tenSuffix
• The prefix, separator and suffix for each new element, row, matrix, 3d subtensor, ... with std::vector<std::string> prefix, std::vector<std::string> separator and std::vector<std::string> suffix. Note that the first entry in each of the vectors refer to the last dimension of the tensor, e.g. separator[0] will be printed between adjacent elements, separator[1] will be printed between adjacent matrices, ...
• char fill: character which will be placed if the elements are aligned.
• int precision
• int flags: an OR-ed combination of flags, the default value is 0, the only currently available flag is Eigen::DontAlignCols which allows to disable the alignment of columns, resulting in faster code.

Interop with Eigen Matrix and Vector Types

Tensor data can be wrapped as an Eigen Map<Matrix>, and vice versa, Eigen dense matrix/vector data can be wrapped as a TensorMap. This is a zero-copy operation that simply reinterprets the underlying memory.

Wrapping a Tensor as a Matrix

Eigen::Tensor<float, 2> tensor(3, 4);
tensor.setRandom();

// View the tensor's data as an Eigen Matrix (no copy).
Eigen::Map<Eigen::MatrixXf> matrix(tensor.data(), 3, 4);
std::cout << "Matrix view:\n" << matrix << "\n";

// Modifications through the map are reflected in the tensor.
matrix(0, 0) = 42.0f;
assert(tensor(0, 0) == 42.0f);

Wrapping a Matrix as a Tensor

Eigen::MatrixXf matrix(3, 4);
matrix.setRandom();

// View the matrix's data as a rank-2 Tensor (no copy).
Eigen::TensorMap<Eigen::Tensor<float, 2>> tensor(matrix.data(), 3, 4);
std::cout << "Tensor view:\n" << tensor << "\n";

// You can also reshape to a different rank.
Eigen::TensorMap<Eigen::Tensor<float, 1>> flat(matrix.data(), 12);

Important: Both the Map and TensorMap are non-owning views. The underlying data must remain valid for the lifetime of the view. Also note that the default storage order of Eigen matrices is ColMajor, which matches the default Tensor layout.

Representation of scalar values

Scalar values are often represented by tensors of size 1 and rank 0.

For example Tensor<T, N>::maximum() returns a Tensor<T, 0>.

Similarly, the inner product of 2 1d tensors (through contractions) returns a 0d tensor.

The scalar value can be extracted as explained in Reduction along all dimensions.

Limitations

• The number of tensor dimensions is currently limited to 250.
• On GPUs only floating point values are properly tested and optimized for.
• C++14 or later is required to use the Tensor module.