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// Copyright 2019 Jared Samet // // Licensed under the Apache License, Version 2.0 (the "License"); // you may not use this file except in compliance with the License. // You may obtain a copy of the License at // // http://www.apache.org/licenses/LICENSE-2.0 // // Unless required by applicable law or agreed to in writing, software // distributed under the License is distributed on an "AS IS" BASIS, // WITHOUT WARRANTIES OR CONDITIONS OF ANY KIND, either express or implied. // See the License for the specific language governing permissions and // limitations under the License. //! The `ndarray_einsum` crate implements the `einsum` function, originally //! implemented for numpy by Mark Wiebe and subsequently reimplemented for //! other tensor libraries such as Tensorflow and PyTorch. `einsum` (short for Einstein summation) //! implements general multidimensional tensor contraction. Many linear algebra operations //! and generalizations of those operations can be expressed as special cases of tensor //! contraction. Examples include matrix multiplication, matrix trace, vector dot product, //! tensor Hadamard [element-wise] product, axis permutation, outer product, batch //! matrix multiplication, bilinear transformations, and many more. //! //! Examples (deliberately similar to [numpy's documentation](https://docs.scipy.org/doc/numpy/reference/generated/numpy.einsum.html)): //! //! ``` //! # use ndarray_einsum_beta::*; //! # use ndarray::prelude::*; //! let a: Array2<f64> = Array::range(0., 25., 1.) //! .into_shape((5,5,)).unwrap(); //! let b: Array1<f64> = Array::range(0., 5., 1.); //! let c: Array2<f64> = Array::range(0., 6., 1.) //! .into_shape((2,3,)).unwrap(); //! let d: Array2<f64> = Array::range(0., 12., 1.) //! .into_shape((3,4,)).unwrap(); //! ``` //! //! Trace of a matrix //! ``` //! # use ndarray_einsum_beta::*; //! # use ndarray::prelude::*; //! # let a: Array2<f64> = Array::range(0., 25., 1.) //! # .into_shape((5,5,)).unwrap(); //! # let b: Array1<f64> = Array::range(0., 5., 1.); //! # let c: Array2<f64> = Array::range(0., 6., 1.) //! # .into_shape((2,3,)).unwrap(); //! # let d: Array2<f64> = Array::range(0., 12., 1.) //! # .into_shape((3,4,)).unwrap(); //! assert_eq!( //! einsum("ii", &[&a]).unwrap(), //! arr0(60.).into_dyn() //! ); //! assert_eq!( //! einsum("ii", &[&a]).unwrap(), //! arr0(a.diag().sum()).into_dyn() //! ); //! ``` //! //! Extract the diagonal //! ``` //! # use ndarray_einsum_beta::*; //! # use ndarray::prelude::*; //! # let a: Array2<f64> = Array::range(0., 25., 1.) //! # .into_shape((5,5,)).unwrap(); //! # let b: Array1<f64> = Array::range(0., 5., 1.); //! # let c: Array2<f64> = Array::range(0., 6., 1.) //! # .into_shape((2,3,)).unwrap(); //! # let d: Array2<f64> = Array::range(0., 12., 1.) //! # .into_shape((3,4,)).unwrap(); //! assert_eq!( //! einsum("ii->i", &[&a]).unwrap(), //! arr1(&[0., 6., 12., 18., 24.]).into_dyn() //! ); //! assert_eq!( //! einsum("ii->i", &[&a]).unwrap(), //! a.diag().into_dyn() //! ); //! //! ``` //! //! Sum over an axis //! ``` //! # use ndarray_einsum_beta::*; //! # use ndarray::prelude::*; //! # let a: Array2<f64> = Array::range(0., 25., 1.) //! # .into_shape((5,5,)).unwrap(); //! # let b: Array1<f64> = Array::range(0., 5., 1.); //! # let c: Array2<f64> = Array::range(0., 6., 1.) //! # .into_shape((2,3,)).unwrap(); //! # let d: Array2<f64> = Array::range(0., 12., 1.) //! # .into_shape((3,4,)).unwrap(); //! assert_eq!( //! einsum("ij->i", &[&a]).unwrap(), //! arr1(&[10., 35., 60., 85., 110.]).into_dyn() //! ); //! assert_eq!( //! einsum("ij->i", &[&a]).unwrap(), //! a.sum_axis(Axis(1)).into_dyn() //! ); //! //! ``` //! //! Compute matrix transpose //! ``` //! # use ndarray_einsum_beta::*; //! # use ndarray::prelude::*; //! # let a: Array2<f64> = Array::range(0., 25., 1.) //! # .into_shape((5,5,)).unwrap(); //! # let b: Array1<f64> = Array::range(0., 5., 1.); //! # let c: Array2<f64> = Array::range(0., 6., 1.) //! # .into_shape((2,3,)).unwrap(); //! # let d: Array2<f64> = Array::range(0., 12., 1.) //! # .into_shape((3,4,)).unwrap(); //! assert_eq!( //! einsum("ji", &[&c]).unwrap(), //! c.t().into_dyn() //! ); //! assert_eq!( //! einsum("ji", &[&c]).unwrap(), //! arr2(&[[0., 3.], [1., 4.], [2., 5.]]).into_dyn() //! ); //! assert_eq!( //! einsum("ji", &[&c]).unwrap(), //! einsum("ij->ji", &[&c]).unwrap() //! ); //! //! ``` //! //! Multiply two matrices //! ``` //! # use ndarray_einsum_beta::*; //! # use ndarray::prelude::*; //! # let a: Array2<f64> = Array::range(0., 25., 1.) //! # .into_shape((5,5,)).unwrap(); //! # let b: Array1<f64> = Array::range(0., 5., 1.); //! # let c: Array2<f64> = Array::range(0., 6., 1.) //! # .into_shape((2,3,)).unwrap(); //! # let d: Array2<f64> = Array::range(0., 12., 1.) //! # .into_shape((3,4,)).unwrap(); //! assert_eq!( //! einsum("ij,jk->ik", &[&c, &d]).unwrap(), //! c.dot(&d).into_dyn() //! ); //! ``` //! //! Compute the path separately from the result //! ``` //! # use ndarray_einsum_beta::*; //! # use ndarray::prelude::*; //! # let a: Array2<f64> = Array::range(0., 25., 1.) //! # .into_shape((5,5,)).unwrap(); //! # let b: Array1<f64> = Array::range(0., 5., 1.); //! # let c: Array2<f64> = Array::range(0., 6., 1.) //! # .into_shape((2,3,)).unwrap(); //! # let d: Array2<f64> = Array::range(0., 12., 1.) //! # .into_shape((3,4,)).unwrap(); //! let path = einsum_path( //! "ij,jk->ik", //! &[&c, &d], //! OptimizationMethod::Naive //! ).unwrap(); //! assert_eq!( //! path.contract_operands(&[&c, &d]), //! c.dot(&d).into_dyn() //! ); //! ``` use ndarray::prelude::*; use ndarray::{Data, IxDyn, LinalgScalar}; mod validation; pub use validation::{ validate, validate_and_optimize_order, validate_and_size, Contraction, SizedContraction, }; mod optimizers; pub use optimizers::{generate_optimized_order, ContractionOrder, OptimizationMethod}; mod contractors; pub use contractors::{EinsumPath, EinsumPathSteps}; use contractors::{PairContractor, TensordotGeneral}; /// This trait is implemented for all `ArrayBase` variants and is parameterized by the data type. /// /// It's here so `einsum` and the other functions accepting a list of operands /// can take a slice `&[&dyn ArrayLike<A>]` where the elements of the slice can have /// different numbers of dimensions and can be a mixture of `Array` and `ArrayView`. pub trait ArrayLike<A> { fn into_dyn_view(&self) -> ArrayView<A, IxDyn>; } impl<A, S, D> ArrayLike<A> for ArrayBase<S, D> where S: Data<Elem = A>, D: Dimension, { fn into_dyn_view(&self) -> ArrayView<A, IxDyn> { self.view().into_dyn() } } /// Wrapper around [SizedContraction::contract_operands](struct.SizedContraction.html#method.contract_operands). pub fn einsum_sc<A: LinalgScalar>( sized_contraction: &SizedContraction, operands: &[&dyn ArrayLike<A>], ) -> ArrayD<A> { sized_contraction.contract_operands(operands) } /// Create a [SizedContraction](struct.SizedContraction.html), optimize the contraction order, and compile the result into an [EinsumPath](struct.EinsumPath.html). pub fn einsum_path<A>( input_string: &str, operands: &[&dyn ArrayLike<A>], optimization_strategy: OptimizationMethod, ) -> Result<EinsumPath<A>, &'static str> { let contraction_order = validate_and_optimize_order(input_string, operands, optimization_strategy)?; Ok(EinsumPath::from_path(&contraction_order)) } /// Performs all steps of the process in one function: parse the string, compile the execution plan, and execute the contraction. pub fn einsum<A: LinalgScalar>( input_string: &str, operands: &[&dyn ArrayLike<A>], ) -> Result<ArrayD<A>, &'static str> { let sized_contraction = validate_and_size(input_string, operands)?; Ok(einsum_sc(&sized_contraction, operands)) } /// Compute tensor dot product between two tensors. /// /// Similar to [the numpy function of the same name](https://docs.scipy.org/doc/numpy/reference/generated/numpy.tensordot.html). /// Easiest to explain by showing the `einsum` equivalents: /// /// ``` /// # use ndarray::prelude::*; /// # use ndarray_einsum_beta::*; /// let m1 = Array::range(0., (3*4*5*6) as f64, 1.) /// .into_shape((3,4,5,6,)) /// .unwrap(); /// let m2 = Array::range(0., (4*5*6*7) as f64, 1.) /// .into_shape((4,5,6,7)) /// .unwrap(); /// assert_eq!( /// einsum( /// "ijkl,jklm->im", /// &[&m1, &m2] /// ).unwrap(), /// tensordot( /// &m1, /// &m2, /// &[Axis(1), Axis(2), Axis(3)], /// &[Axis(0), Axis(1), Axis(2)] /// ) /// ); /// /// assert_eq!( /// einsum( /// "abic,dief->abcdef", /// &[&m1, &m2] /// ).unwrap(), /// tensordot( /// &m1, /// &m2, /// &[Axis(2)], /// &[Axis(1)] /// ) /// ); /// ``` pub fn tensordot<A, S, S2, D, E>( lhs: &ArrayBase<S, D>, rhs: &ArrayBase<S2, E>, lhs_axes: &[Axis], rhs_axes: &[Axis], ) -> ArrayD<A> where A: ndarray::LinalgScalar, S: Data<Elem = A>, S2: Data<Elem = A>, D: Dimension, E: Dimension, { assert_eq!(lhs_axes.len(), rhs_axes.len()); let lhs_axes_copy: Vec<_> = lhs_axes.iter().map(|x| x.index()).collect(); let rhs_axes_copy: Vec<_> = rhs_axes.iter().map(|x| x.index()).collect(); let output_order: Vec<usize> = (0..(lhs.ndim() + rhs.ndim() - 2 * (lhs_axes.len()))).collect(); let tensordotter = TensordotGeneral::from_shapes_and_axis_numbers( &lhs.shape(), &rhs.shape(), &lhs_axes_copy, &rhs_axes_copy, &output_order, ); tensordotter.contract_pair(&lhs.view().into_dyn(), &rhs.view().into_dyn()) }