text: // Copyright (c) 2022 PaddlePaddle Authors. All Rights Reserved.
  //
  // 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.
  
  #include "paddle/phi/kernels/norm_grad_kernel.h"
  #include "paddle/phi/kernels/funcs/eigen/eigen_function.h"
  #include "paddle/phi/kernels/funcs/math_function.h"
  
  #include "paddle/phi/kernels/funcs/eigen/common.h"
  
  #include "paddle/phi/backends/cpu/cpu_context.h"
  #include "paddle/phi/core/kernel_registry.h"
  
  #include "paddle/phi/kernels/funcs/common_shape.h"
  namespace phi {
  
  template <typename T, typename Context>
  void NormGradKernel(const Context& ctx,
                      const DenseTensor& out_grad,
                      const DenseTensor& x,
                      const DenseTensor& norm,
                      int axis,
                      float epsilon,
                      bool is_test,
                      DenseTensor* x_grad) {
    auto* in_x = &x;
    auto* in_dy = &out_grad;
    auto* in_norm = &norm;
    auto* out_dx = x_grad;
  
    ctx.template Alloc<T>(out_dx);
  
    auto xdim = in_x->dims();
    if (axis < 0) axis = xdim.size() + axis;
    int pre, n, post;
    funcs::GetPrePostNumel(xdim, axis, &pre, &n, &post);
  
    auto* place = ctx.eigen_device();
  
    auto x_e = paddle::framework::EigenVector<T>::Flatten(*in_x);
    auto dy_e = paddle::framework::EigenVector<T>::Flatten(*in_dy);
    auto norm_e = paddle::framework::EigenVector<T>::Flatten(*in_norm);
    auto dx_e = paddle::framework::EigenVector<T>::Flatten(*out_dx);
  
    Eigen::DSizes<int, 3> shape(pre, n, post);
    Eigen::DSizes<int, 3> rshape(pre, 1, post);
    auto x_r = x_e.reshape(shape);
    auto dy = dy_e.reshape(shape);
    auto norm_r = norm_e.reshape(rshape);
    auto dx = dx_e.reshape(shape);
  
    DenseTensor rsum;
    rsum.Resize({pre, post});
    ctx.template Alloc<T>(&rsum);
    auto sum = paddle::framework::EigenTensor<T, 2>::From(rsum);
  
    Eigen::DSizes<int, 1> rdim(1);
    Eigen::DSizes<int, 3> bcast(1, n, 1);
  
    // dx = ( dy/sqrt(sum(x*x)) ) * [1 - x*sum(x) / (sum(x*x) + e)]
    //    = [dy - dy * x * sum(x) / (sum(x*x) + e)] / sqrt(sum(x*x))
    //    = [dy - x * sum(x*dy) / (sum(x*x) + e)] / sqrt(sum(x*x))
    // 1. sum = sum(x*dy)
    sum.device(*place) = (x_r * dy).sum(rdim);
    // 2. dx = x * sum
    dx.device(*place) = sum.reshape(rshape).broadcast(bcast) * x_r;
    // 3. dx / (sum(x*x) + e)
    // where, norm.pow(2) = sum(x*x) + e, which is calculated in forward.
    dx.device(*place) = dx / norm_r.pow(2).broadcast(bcast);
    // 4. [dy - dx] / sqrt(sum(x*x))
    dx.device(*place) = (dy - dx) / norm_r.broadcast(bcast);
  }
  
  }  // namespace phi
  
  PT_REGISTER_KERNEL(
      norm_grad, CPU, ALL_LAYOUT, phi::NormGradKernel, float, double) {}
  