ndarray_einsum_beta 0.7.0

// 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())
}