include/ceres/internal/autodiff.h

// Ceres Solver - A fast non-linear least squares minimizer
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//
// Author: keir@google.com (Keir Mierle)
//
// Computation of the Jacobian matrix for vector-valued functions of multiple
// variables, using automatic differentiation based on the implementation of
// dual numbers in jet.h. Before reading the rest of this file, it is advisable
// to read jet.h's header comment in detail.
//
// The helper wrapper AutoDifferentiate() computes the Jacobian of
// functors with templated operator() taking this form:
//
//   struct F {
//     template<typename T>
//     bool operator()(const T *x, const T *y, ..., T *z) {
//       // Compute z[] based on x[], y[], ...
//       // return true if computation succeeded, false otherwise.
//     }
//   };
//
// All inputs and outputs may be vector-valued.
//
// To understand how Jets are used to compute the Jacobian, a
// picture may help. Consider a vector-valued function, F, returning 3
// dimensions and taking a vector-valued parameter of 4 dimensions:
//
//     y            x
//   [ * ]    F   [ * ]
//   [ * ]  <---  [ * ]
//   [ * ]        [ * ]
//                [ * ]
//
// Similar to the 2-parameter example for f described in jet.h, computing the
// Jacobian dy/dx is done by substituting a suitable Jet object for x and all
// intermediate steps of the computation of F. Since x has 4 dimensions, use
// a Jet<double, 4>.
//
// Before substituting a Jet object for x, the dual components are set
// appropriately for each dimension of x:
//
//          y                       x
//   [ * | * * * * ]    f   [ * | 1 0 0 0 ]   x0
//   [ * | * * * * ]  <---  [ * | 0 1 0 0 ]   x1
//   [ * | * * * * ]        [ * | 0 0 1 0 ]   x2
//         ---+---          [ * | 0 0 0 1 ]   x3
//            |                   ^ ^ ^ ^
//          dy/dx                 | | | +----- infinitesimal for x3
//                                | | +------- infinitesimal for x2
//                                | +--------- infinitesimal for x1
//                                +----------- infinitesimal for x0
//
// The reason to set the internal 4x4 submatrix to the identity is that we wish
// to take the derivative of y separately with respect to each dimension of x.
// Each column of the 4x4 identity is therefore for a single component of the
// independent variable x.
//
// Then the Jacobian of the mapping, dy/dx, is the 3x4 sub-matrix of the
// extended y vector, indicated in the above diagram.
//
// Functors with multiple parameters
// ---------------------------------
// In practice, it is often convenient to use a function f of two or more
// vector-valued parameters, for example, x[3] and z[6]. Unfortunately, the Jet
// framework is designed for a single-parameter vector-valued input. The wrapper
// in this file addresses this issue adding support for functions with one or
// more parameter vectors.
//
// To support multiple parameters, all the parameter vectors are concatenated
// into one and treated as a single parameter vector, except that since the
// functor expects different inputs, we need to construct the Jets as if they
// were part of a single parameter vector. The extended Jets are passed
// separately for each parameter.
//
// For example, consider a functor F taking two vector parameters, p[2] and
// q[3], and producing an output y[4]:
//
//   struct F {
//     template<typename T>
//     bool operator()(const T *p, const T *q, T *z) {
//       // ...
//     }
//   };
//
// In this case, the necessary Jet type is Jet<double, 5>. Here is a
// visualization of the Jet objects in this case:
//
//          Dual components for p ----+
//                                    |
//                                   -+-
//           y                 [ * | 1 0 | 0 0 0 ]    --- p[0]
//                             [ * | 0 1 | 0 0 0 ]    --- p[1]
//   [ * | . . | + + + ]         |
//   [ * | . . | + + + ]         v
//   [ * | . . | + + + ]  <--- F(p, q)
//   [ * | . . | + + + ]            ^
//         ^^^   ^^^^^              |
//        dy/dp  dy/dq            [ * | 0 0 | 1 0 0 ] --- q[0]
//                                [ * | 0 0 | 0 1 0 ] --- q[1]
//                                [ * | 0 0 | 0 0 1 ] --- q[2]
//                                            --+--
//                                              |
//          Dual components for q --------------+
//
// where the 4x2 submatrix (marked with ".") and 4x3 submatrix (marked with "+")
// of y in the above diagram are the derivatives of y with respect to p and q
// respectively. This is how autodiff works for functors taking multiple vector
// valued arguments (up to 6).
//
// Jacobian null pointers (nullptr)
// --------------------------------
// In general, the functions below will accept nullptr for all or some of the
// Jacobian parameters, meaning that those Jacobians will not be computed.

#ifndef CERES_PUBLIC_INTERNAL_AUTODIFF_H_
#define CERES_PUBLIC_INTERNAL_AUTODIFF_H_

#include <array>
#include <cstddef>
#include <utility>

#include "Eigen/Core"
#include "absl/log/check.h"
#include "ceres/internal/array_selector.h"
#include "ceres/internal/integer_sequence_algorithm.h"
#include "ceres/internal/variadic_evaluate.h"
#include "ceres/jet.h"
#include "ceres/types.h"

// If the number of parameters exceeds this values, the corresponding jets are
// placed on the heap. This will reduce performance by a factor of 2-5 on
// current compilers.
#ifndef CERES_AUTODIFF_MAX_PARAMETERS_ON_STACK
#define CERES_AUTODIFF_MAX_PARAMETERS_ON_STACK 50
#endif

#ifndef CERES_AUTODIFF_MAX_RESIDUALS_ON_STACK
#define CERES_AUTODIFF_MAX_RESIDUALS_ON_STACK 20
#endif

namespace ceres::internal {

// Extends src by a 1st order perturbation for every dimension and puts it in
// dst. The size of src is N. Since this is also used for perturbations in
// blocked arrays, offset is used to shift which part of the jet the
// perturbation occurs. This is used to set up the extended x augmented by an
// identity matrix. The JetT type should be a Jet type, and T should be a
// numeric type (e.g. double). For example,
//
//             0   1 2   3 4 5   6 7 8
//   dst[0]  [ * | . . | 1 0 0 | . . . ]
//   dst[1]  [ * | . . | 0 1 0 | . . . ]
//   dst[2]  [ * | . . | 0 0 1 | . . . ]
//
// is what would get put in dst if N was 3, offset was 3, and the jet type JetT
// was 8-dimensional.
template <int N, int Offset, typename T, typename JetT>
inline void Make1stOrderPerturbation(const T* src, JetT* dst) {
  DCHECK(src);
  DCHECK(dst);
  for (int i = 0; i < N; ++i) {
    dst[i] = JetT(src[i], i + Offset);
  }
}

// Internal non-recursive implementation of Make1stOrderPerturbations.
template <int... Ns,
          int... Offsets,
          std::size_t... BlockIdx,
          typename T,
          typename JetT>
inline void Make1stOrderPerturbationsImpl(
    T const* const* parameters,
    JetT* x,
    std::integer_sequence<int, Ns...>,
    std::integer_sequence<int, Offsets...>,
    std::index_sequence<BlockIdx...>) {
  ((Make1stOrderPerturbation<Ns, Offsets>(parameters[BlockIdx], x + Offsets)),
   ...);
}

// Takes the 0th order part of src, assumed to be a Jet type, and puts it in
// dst. This is used to pick out the "vector" part of the extended y.
template <typename JetT, typename T>
inline void Take0thOrderPart(int M, const JetT* src, T* dst) {
  DCHECK(src);
  for (int i = 0; i < M; ++i) {
    dst[i] = src[i].a;
  }
}

// Takes N 1st order parts, starting at index N0, and puts them in the M x N
// matrix 'dst'. This is used to pick out the "matrix" parts of the extended y.
template <int N0, int N, typename JetT, typename T>
inline void Take1stOrderPart(const int M, const JetT* src, T* dst) {
  DCHECK(src);
  DCHECK(dst);
  for (int i = 0; i < M; ++i) {
    Eigen::Map<Eigen::Matrix<T, N, 1>>(dst + N * i, N) =
        src[i].v.template segment<N>(N0);
  }
}

// Internal non-recursive implementation of Take1stOrderParts.
template <int... Ns,
          int... Offsets,
          std::size_t... BlockIdx,
          typename JetT,
          typename T>
inline void Take1stOrderPartsImpl(int num_outputs,
                                  JetT* output,
                                  T** jacobians,
                                  std::integer_sequence<int, Ns...>,
                                  std::integer_sequence<int, Offsets...>,
                                  std::index_sequence<BlockIdx...>) {
  ((void)(jacobians[BlockIdx] != nullptr &&
          (Take1stOrderPart<Offsets, Ns>(
               num_outputs, output, jacobians[BlockIdx]),
           true)),
   ...);
}

template <int kNumResiduals,
          typename ParameterDims,
          typename Functor,
          typename T>
inline bool AutoDifferentiate(const Functor& functor,
                              T const* const* parameters,
                              int dynamic_num_outputs,
                              T* function_value,
                              T** jacobians) {
  using JetT = Jet<T, ParameterDims::kNumParameters>;
  using Parameters = typename ParameterDims::Parameters;

  if (kNumResiduals != DYNAMIC) {
    DCHECK_EQ(kNumResiduals, dynamic_num_outputs);
  }

  ArraySelector<JetT,
                ParameterDims::kNumParameters,
                CERES_AUTODIFF_MAX_PARAMETERS_ON_STACK>
      parameters_as_jets(ParameterDims::kNumParameters);

  // Pointers to the beginning of each parameter block
  std::array<JetT*, ParameterDims::kNumParameterBlocks> unpacked_parameters =
      ParameterDims::GetUnpackedParameters(parameters_as_jets.data());

  // If the number of residuals is fixed, we use the template argument as the
  // number of outputs. Otherwise we use the num_outputs parameter. Note: The
  // ?-operator here is compile-time evaluated, therefore num_outputs is also
  // a compile-time constant for functors with fixed residuals.
  const int num_outputs =
      kNumResiduals == DYNAMIC ? dynamic_num_outputs : kNumResiduals;
  DCHECK_GT(num_outputs, 0);

  ArraySelector<JetT, kNumResiduals, CERES_AUTODIFF_MAX_RESIDUALS_ON_STACK>
      residuals_as_jets(num_outputs);

  for (int i = 0; i < num_outputs; ++i) {
    residuals_as_jets[i].a = kImpossibleValue;
    residuals_as_jets[i].v.setConstant(kImpossibleValue);
  }

  using Offsets = ExclusiveScan<Parameters>;
  Make1stOrderPerturbationsImpl(
      parameters,
      parameters_as_jets.data(),
      Parameters{},
      Offsets{},
      std::make_index_sequence<ParameterDims::kNumParameterBlocks>{});

  if (!VariadicEvaluate<ParameterDims>(
          functor, unpacked_parameters.data(), residuals_as_jets.data())) {
    return false;
  }

  Take0thOrderPart(num_outputs, residuals_as_jets.data(), function_value);
  Take1stOrderPartsImpl(
      num_outputs,
      residuals_as_jets.data(),
      jacobians,
      Parameters{},
      Offsets{},
      std::make_index_sequence<ParameterDims::kNumParameterBlocks>{});

  return true;
}

}  // namespace ceres::internal

#endif  // CERES_PUBLIC_INTERNAL_AUTODIFF_H_