Fastor - C++11/14/17 中的轻量级高性能SIMD优化张量代数框架

Fastor - C++11/14/17 中的轻量级高性能SIMD优化张量代数框架

Fastor

Fastor is a stack-based high performance tensor (fixed multi-dimensional array) library for modern C++.

Fastor offers:

High-level interface for manipulating multi-dimensional arrays in C++ that look and feel native to scientific programmersBare metal performance for small matrix/tensor multiplications, contractions and tensor factorisations [LU, QR etc]. Refer to benchmarks to see how Fastor delivers performance on par with MKL JIT's dedicated APICompile time operation minimisation such as graph optimisation and nearly symbolic tensor algebraic manipulation to reduce the complexity of evaluation of BLAS or non-BLAS type expressions by orders of magnitudeExplicit and configurable SIMD vectorisation supporting all numeric data types float32, float64, complex float32 and complex float64 as well as integral typesOptional JIT backend using Intel's MKL-JIT and LIBXSMM for performance portable codeAbility to wrap existing data and operate on them using Fastor's highly optimised kernelsLight weight header-only library with no external dependencies offering fast compilation timesWell-tested on most compilers including GCC, Clang, Intel's ICC and MSVC

Documentation

Documenation can be found under the Wiki pages.

High-level interface

Fastor provides a high level interface for tensor algebra. To get a glimpse, consider the following

Tensor scalar = 3.5; // A scalarTensor vector3 = {1,2,3}; // A vectorTensor matrix{{1,2},{3,4},{5,6}}; // A second order tensorTensor tensor_3; // A third order tensor with dimension 3x3x3tensor_3.arange(0); // fill tensor with sequentially ascending numberstensor_3(0,2,1); // index a tensortensor_3(all,last,seq(0,2)); // slice a tensor tensor_3[:,-1,:2]tensor_3.rank(); // get rank of tensor, 3 in this caseTensor t_12; // A 12th order tensor

Tensor contraction

Einstein summation as well as summing over multiple (i.e. more than two) indices are supported. As a complete example consider

#include using namespace Fastor;enum {I,J,K,L,M,N};int main() { // An example of Einstein summation Tensor A; Tensor B; // fill A and B A.random(); B.random(); auto C = einsum,Index>(A,B); // An example of summing over three indices Tensor D; D.random(); auto E = inner(D); // An example of tensor permutation Tensor F; F.random(); auto G = permutation>(F); // Output the results print("Our big tensors:",C,E,G); return 0;}

You can compile this by providing the following flags to your compiler -std=c++14 -O3 -march=native -DNDEBUG.

Tensor views: A powerful indexing, slicing and broadcasting mechanism

Fastor provides powerful tensor views for block indexing, slicing and broadcasting familiar to scientific programmers. Consider the following examples

Tensor A, B;A.random(); B.random();Tensor C; Tensor D;// Dynamic views -> seq(first,last,step)C = A(seq(0,2),seq(0,2),seq(0,last,2)); // C = A[0:2,0:2,0::2]D = B(all,all,0) + A(all,all,last); // D = B[:,:,0] + A[:,:,-1]A(2,all,3) = 5.0; // A[2,:,3] = 5.0// Static views -> fseqC = A(fseq<0,2>(),fseq<0,2>(),fseq<0,last,2>()); // C = A[0:2,0:2,0::2]D = B(all, all, fix<0>) + A(all, all, fix()); // D = B[:,:,0] + A[:,:,-1]A(2,all,3) = 5.0; // A[2,:,3] = 5.0// Overlapping is also allowed without having undefined/aliasing behaviourA(seq(2,last),all,all).noalias() += A(seq(0,last-2),all,all); // A[2::,:,:] += A[::-2,:,:]// Note that in case of perfect overlapping noalias is not requiredA(seq(0,last-2),all,all) += A(seq(0,last-2),all,all); // A[::2,:,:] += A[::2,:,:]// If instead of a tensor view, one needs an actual tensor the iseq could be used// iseqC = A(iseq<0,2>(),iseq<0,2>(),iseq<0,last,2>()); // C = A[0:2,0:2,0::2]// Note that iseq returns an immediate tensor rather than a tensor view and hence cannot appear// on the left hand side, for instanceA(iseq<0,2>(),iseq<0,2>(),iseq<0,last,2>()) = 2; // Will not compile, as left operand is an rvalue// One can also index a tensor with another tensor(s)Tensor E; E.fill(2);Tensor it = {0,1,3,6,8};Tensor t_it; t_it.arange();E(it,0) = 2;E(it,seq(0,last,3)) /= -1000.;E(all,it) += E(all,it) * 15.;E(t_it) -= 42 + E;// Masked and filtered views are also supportedTensor F;Tensor mask = {{true,false},{false,true}};F(mask) += 10;

All possible combination of slicing and broadcasting is possible. For instance, one complex slicing and broadcasting example is given below

A(all,all) -= log(B(all,all,0)) + abs(B(all,all,1)) + sin(C(all,0,all,0)) - 102. - cos(B(all,all,0));

SIMD optimised linear algebra kernels for fixed size tensors

All basic linear algebra subroutines for small matrices/tensors (where the overhead of calling vendor/optimised BLAS is typically high) are fully SIMD vectorised and efficiently implemented. Note that Fastor exposes two functionally equivalent interfaces for linear algebra functions, the more verbose names such as matmul, determinant, inverse etc that evaluate immediately and the less verbose ones (%, det, inv) that evaluate lazy

Tensor A,B;// fill A and Bauto mulab = matmul(A,B); // matrix matrix multiplication [or equivalently A % B]auto norma = norm(A); // Frobenious norm of Aauto detb = determinant(B); // determinant of B [or equivalently det(B)]auto inva = inverse(A); // inverse of A [or equivalently inv(A)]auto cofb = cofactor(B); // cofactor of B [or equivalently cof(B)]lu(A, L, U); // LU decomposition of A in to L and Uqr(A, Q, R); // QR decomposition of A in to Q and R

Boolean tensor algebra

A set of boolean operations are available that whenever possible are performed at compile time

isuniform(A); // does the tensor expression span equally in all dimensions - generalisation of square matricesisorthogonal(A); // is the tensor expression orthogonalisequal(A,B,tol); // Are two tensor expressions equal within a tolerancedoesbelongtoSL3(A); // does the tensor expression belong to the special linear 3D groupdoesbelongtoSO3(A); // does the tensor expression belong to the special orthogonal 3D groupissymmetric(A); // is the tensor expression symmetric in the axis_1 x axis_3 planeisdeviatoric(A); // is the tensor expression deviatoric [trace free]isvolumetric(A); // is the tensor expression volumetric [A = 1/3*trace(A) * I]all_of(A < B); // Are all elements in A less than Bany_of(A >= B); // is any element in A greater than or equal to the corresponding element in Bnone_of(A == B); // is no element in A and B equal

Interfacing with C arrays and external buffers

Alernatively Fastor can be used as a pure wrapper over existing buffer. You can wrap C arrays or map any external piece of memory as Fastor tensors and operate on them just like you would on Fastor's tensors without making any copies, using the Fastor::TensorMap feature. For instance

double c_array[4] = {1,2,3,4};// Map to a Fastor vectorTensorMap tn1(c_array);// Map to a Fastor matrix of 2x2TensorMap tn2(c_array);// You can now operate on these. This will also modify c_arraytn1 += 1;tn2(0,1) = 5;

Basic expression templates

Expression templates are archetypal of array/tensor libraries in C++ as they provide a means for lazy evaluation of arbitrary chained operations. Consider the following expression

Tensor tn1 ,tn2, tn3;tn1.random(); tn2.random(); tn3.random();auto tn4 = 2*tn1+sqrt(tn2-tn3);

Here tn4 is not another tensor but rather an expression that is not yet evaluated. The expression is evaluated if you explicitly assign it to another tensor or call the free function evaluate on the expression

Tensor tn5 = tn4;// orauto tn6 = evaluate(tn4);

this mechanism helps chain the operations to avoid the need for intermediate memory allocations. Various re-structuring of the expression before evaluation is possible depending on the chosen policy.

Smart expression templates

Aside from basic expression templates, by employing further template metaprogrommaing techniques Fastor can mathematically transform expressions and/or apply compile time graph optimisation to find optimal contraction indices of complex tensor networks, for instance. This gives Fastor the ability to re-structure or completely re-write an expression and simplify it rather symbolically. As an example, consider the expression trace(matmul(transpose(A),B)) which is O(n^3) in computational complexity. Fastor can determine this to be inefficient and will statically dispatch the call to an equivalent but much more efficient routine, in this case A_ij*B_ij or inner(A,B) which is O(n^2). Further examples of such mathematical transformation include (but certainly not exclusive to)

det(inv(A)); // transformed to 1/det(A), O(n^3) reduction in computationtrans(cof(A)); // transformed to adj(A), O(n^2) reduction in memory accesstrans(adj(A)); // transformed to cof(A), O(n^2) reduction in memory accessA % B % b; // transformed to A % (B % b), O(n) reduction in computation [% is the operator matrix multiplication]// and many more

These expressions are not treated as special cases but rather the Einstein indicial notation of the whole expression is constructed under the hood and by simply simplifying/collapsing the indices one obtains the most efficient form that an expression can be evaluated. The expression is then sent to an optimised kernel for evaluation. Note that there are situations that the user may write a complex chain of operations in the most verbose/obvious way perhaps for readibility purposes, but Fastor delays the evaluation of the expression and checks if an equivalent but efficient expression can be computed.

Operation minimisation for tensor networks

For tensor networks comprising of many higher rank tensors, a full generalisation of the above mathematical transformation can be performed through a constructive graph search optimisation. This typically involves finding the most optimal pattern of tensor contraction by studying the indices of contraction wherein tensor pairs are multiplied, summed over and factorised out in all possible combinations in order to come up with a cost model. Once again, knowing the dimensions of the tensor and the contraction pattern, Fastor performs this operation minimisation step at compile time and further checks the SIMD vectorisability of the tensor contraction loop nest (i.e. full/partial/strided vectorisation). In a nutshell, it not only minimises the the number of floating point operations but also generates the most optimal vectorisable loop nest for attaining theoretical peak for those remaining FLOPs. The following figures show the benefit of operation minimisation (FLOP optimal) over a single expression evaluation (Memory-optimal - as temporaries are not created) approach (for instance, NumPy's einsum uses the single expression evaluation technique where the whole expression in einsum is computed without being broken up in to smaller computations) in contracting a three-tensor-network fitting in L1, L2 and L3 caches, respectively

The x-axis shows the number FLOPS saved/reduced over single expression evaluation technique. Certainly, the bigger the size of tensors the more reduction in FLOPs is necessary to compensate for the temporaries created during by-pair (FLOP optimal) evaluation.

Specialised tensors

A set of specialised tensors are available that provide optimised tensor algebraic computations, for instance SingleValueTensor or IdentityTensor. Some of the computations performed on these tensors have almost zero cost no matter how big the tensor is. These tensors work in the exact same way as the Tensor class and can be assigned to one another. Consider for example the einsum between two SingleValueTensors. A SingleValueTensor is a tensor of any dimension and size whose elements are all the same (a matrix of ones for instance).

SingleValueTensor a(3.51);SingleValueTensor b(2.76);auto c = einsum,Index<0,2>>(a,b);

This will incur almost no runtime cost. As where if the tensors were of type Tensor then a heavy computation would ensue.

Tested Compilers

Fastor has been tested against the following compilers (on Ubuntu 14.04/16.04/18.04, macOS 10.13+ and Windows 10)

GCC 4.8, GCC 4.9, GCC 5.1, GCC 5.2, GCC 5.3, GCC 5.4, GCC 6.2, GCC 7.3, GCC 8, GCC 9.1, GCC 9.2, GCC 9.3, GCC 10.1Clang 3.6, Clang 3.7, Clang 3.8, Clang 3.9, Clang 5, Clang 7, Clang 8, Clang 10.0.0Intel 16.0.1, Intel 16.0.2, Intel 16.0.3, Intel 17.0.1, Intel 18.2, Intel 19.3MSVC 2019

References

For academic purposes, Fastor can be cited as

@Article{Poya2017, author="Poya, Roman and Gil, Antonio J. and Ortigosa, Rogelio", title = "A high performance data parallel tensor contraction framework: Application to coupled electro-mechanics", journal = "Computer Physics Communications", year="2017", doi = "http://dx.doi.org/10.1016/j.cpc.2017.02.016", url = "http://sciencedirect.com/science/article/pii/S0010465517300681"}

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