CaffeLoss - Lifted Struct Similarity Softmax Layer

Author： AIHGF
发布时间：May 21, 2018
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Categories：深度平台

Lifted Struct Similarity Softmax Layer, 聚类, 图像检索等特征学习论文: Deep Metric Learning via Lifted Structured Feature Embedding - CVPR2016 项目路径: Deep-Metric-Learning-CVPR16

<h2>1. 在 prototxt 中的定义</h2>

layer {
  name: "fc_embedding"
  type: "InnerProduct"
  bottom: "pool_ave"
  top: "fc_embedding"
  param {
    lr_mult: 1.0
    decay_mult: 1.0
  }
  param {
    lr_mult: 2.0
    decay_mult: 0.0
  }
  inner_product_param {
    num_output: 64  // feature dimension
    weight_filler {
      type: "xavier"
    }
    bias_filler {
      type: "constant"
      value: 0.0
    }
  }
}
####### LiftedStructSimilaritySoftmaxLoss #####
layer {
  name: "loss"
  type: "LiftedStructSimilaritySoftmaxLoss"
  bottom: "fc_embedding"
  bottom: "label"
  top: "loss"
  lifted_struct_sim_softmax_loss_param {
    margin: 1 // margin parameter alpha  
  }
}

<h2>2. caffe.proto 中的定义</h2>

message LayerParameter {
    optional LiftedStructSimilaritySoftmaxLossParameter lifted_struct_sim_softmax_loss_param = 148;
}

message LiftedStructSimilaritySoftmaxLossParameter {
  optional float margin = 1 [default = 1.0]; // margin parameter alpha
}

<h2>3. lifted_struct_similarity_softmax_layer.hpp</h2>

#ifndef CAFFE_LIFTED_STRUCT_SIMILARITY_LOSS_LAYER_HPP_
#define CAFFE_LIFTED_STRUCT_SIMILARITY_LOSS_LAYER_HPP_

#include <vector>

#include "caffe/blob.hpp"
#include "caffe/layer.hpp"
#include "caffe/proto/caffe.pb.h"

#include "caffe/layers/loss_layer.hpp"

namespace caffe {

template <typename Dtype>
class LiftedStructSimilaritySoftmaxLossLayer : public LossLayer<Dtype> {
 public:
  explicit LiftedStructSimilaritySoftmaxLossLayer(const LayerParameter& param)
      : LossLayer<Dtype>(param) {}
  virtual void LayerSetUp(const vector<Blob<Dtype>*>& bottom,
      const vector<Blob<Dtype>*>& top);

  virtual inline int ExactNumBottomBlobs() const { return 2; }
  virtual inline const char* type() const { return "LiftedStructSimilaritySoftmaxLoss"; }
  virtual inline bool AllowForceBackward(const int bottom_index) const {
    return bottom_index != 1;
  }

 protected:
  virtual void Forward_cpu(const vector<Blob<Dtype>*>& bottom,
      const vector<Blob<Dtype>*>& top);

  virtual void Backward_cpu(const vector<Blob<Dtype>*>& top,
      const vector<bool>& propagate_down, const vector<Blob<Dtype>*>& bottom);

  Blob<Dtype> dist_sq_;  // cached for backward pass 
  Blob<Dtype> dot_;  //
  Blob<Dtype> ones_;
  Blob<Dtype> blob_pos_diff_;
  Blob<Dtype> blob_neg_diff_;
  Blob<Dtype> loss_aug_inference_;
  Blob<Dtype> summer_vec_;
  Dtype num_constraints;
};

}  // namespace caffe

#endif  // CAFFE_LIFTED_STRUCT_SIMILARITY_LOSS_LAYER_HPP_

<h2>4. lifted_struct_similarity_softmax_layer.cpp</h2>

#include <algorithm>
#include <vector>

#include "caffe/layers/lifted_struct_similarity_softmax_layer.hpp"
#include "caffe/util/math_functions.hpp"

namespace caffe {

template <typename Dtype>
void LiftedStructSimilaritySoftmaxLossLayer<Dtype>::LayerSetUp(const vector<Blob<Dtype>>& bottom, const vector<Blob<Dtype>>& top) {
  LossLayer<Dtype>::LayerSetUp(bottom, top);
  CHECK_EQ(bottom[0]->height(), 1);
  CHECK_EQ(bottom[0]->width(), 1);
  CHECK_EQ(bottom[1]->channels(), 1);
  CHECK_EQ(bottom[1]->height(), 1);
  CHECK_EQ(bottom[1]->width(), 1);
  // List of member variables defined in /include/caffe/loss_layers.hpp;
  //   diff_, dist_sq_, summer_vec_, loss_aug_inference_;
  dist_sq_.Reshape(bottom[0]->num(), 1, 1, 1);
  dot_.Reshape(bottom[0]->num(), bottom[0]->num(), 1, 1);
  ones_.Reshape(bottom[0]->num(), 1, 1, 1);  // n by 1 vector of ones.
  for (int i=0; i < bottom[0]->num(); ++i){
    ones_.mutable_cpu_data()[i] = Dtype(1);
  }
  blob_pos_diff_.Reshape(bottom[0]->channels(), 1, 1, 1);
  blob_neg_diff_.Reshape(bottom[0]->channels(), 1, 1, 1);
} 

template <typename Dtype>
void LiftedStructSimilaritySoftmaxLossLayer<Dtype>::Forward_cpu(const vector<Blob<Dtype>>& bottom, const vector<Blob<Dtype>>& top) {

  const int channels = bottom[0]->channels(); // feature dims
  for (int i = 0; i < bottom[0]->num(); i++){
    dist_sq_.mutable_cpu_data()[i] = caffe_cpu_dot(channels, bottom[0]->cpu_data() + (ichannels), bottom[0]->cpu_data() + (ichannels));  // hat{x} in papaers
  }

  int M_ = bottom[0]->num(); // mini-batch 内样本数
  int N_ = bottom[0]->num();
  int K_ = bottom[0]->channels(); // 特征维度

  const Dtype* bottom_data1 = bottom[0]->cpu_data(); // 特征矩阵
  const Dtype* bottom_data2 = bottom[0]->cpu_data();

  Dtype dot_scaler(-2.0);
  caffe_cpu_gemm<Dtype>(CblasNoTrans, CblasTrans, M_, N_, K_, dot_scaler, bottom_data1, bottom_data2, (Dtype)0., dot_.mutable_cpu_data());

  // add ||x_i||^2 to all elements in row i
  for (int i=0; i<N_; i++){
    caffe_axpy(N_, dist_sq_.cpu_data()[i], ones_.cpu_data(), dot_.mutable_cpu_data() + i*N_);
  }

  // add the norm vector to row i
  for (int i=0; i<N_; i++){
    caffe_axpy(N_, Dtype(1.0), dist_sq_.cpu_data(), dot_.mutable_cpu_data() + i*N_);
    caffe_abs(N_, dot_.mutable_cpu_data() + iN_, dot_.mutable_cpu_data() + iN_);
  }

  // construct pairwise label matrix
  vector<vector<bool> > label_mat(N_, vector<bool>(N_, false));
  for (int i=0; i<N_; i++){
    for (int j=0; j<N_; j++){
      label_mati = (bottom[1]->cpu_data()[i] == bottom[1]->cpu_data()[j]);
    }
  }

  Dtype margin = this->layer_param_.lifted_struct_sim_softmax_loss_param().margin();
  Dtype loss(0.0);
  num_constraints = Dtype(0.0); 
  const Dtype* bin = bottom[0]->cpu_data();
  Dtype* bout = bottom[0]->mutable_cpu_diff();

  // zero initialize bottom[0]->mutable_cpu_diff();
  for (int i=0; i<N_; i++){
    caffe_set(K_, Dtype(0.0), bout + i*K_);
  }

  // loop upper triangular matrix and look for positive anchors
  for (int i=0; i<N_; i++){
    for (int j=i+1; j<N_; j++){

      // found a positive pair @ anchor (i, j)
      if (label_mati){
        Dtype dist_pos = sqrt(dot_.cpu_data()[i*N_ + j] + 2e-10);

        caffe_sub(K_, bin + iK_, bin + jK_, blob_pos_diff_.mutable_cpu_data());

        // 1.count the number of negatives for this positive
        int num_negatives = 0;
        for (int k=0; k<N_; k++){
          if (!label_mati){
            num_negatives += 1;
          }
        }

        for (int k=0; k<N_; k++){
          if (!label_matj){
            num_negatives += 1;
          }
        }

        loss_aug_inference_.Reshape(num_negatives, 1, 1, 1);

        // vector of ones used to sum along channels
        summer_vec_.Reshape(num_negatives, 1, 1, 1);
        for (int ss = 0; ss < num_negatives; ++ss){
          summer_vec_.mutable_cpu_data()[ss] = Dtype(1);
        }

        // 2. compute loss augmented inference
        int neg_idx = 0;
        // mine negative (anchor i, neg k)
        for (int k=0; k<N_; k++){
          if (!label_mati){
            loss_aug_inference_.mutable_cpu_data()[neg_idx] = margin - sqrt(dot_.cpu_data()[i*N_ + k]);
            neg_idx++;
          }
        }

        // mine negative (anchor j, neg k)
        for (int k=0; k<N_; k++){
          if (!label_matj){
            loss_aug_inference_.mutable_cpu_data()[neg_idx] = margin - sqrt(dot_.cpu_data()[j*N_ + k]);
            neg_idx++;
          }
        }

        // compute softmax of loss aug inference vector;
        Dtype max_elem = *std::max_element(loss_aug_inference_.cpu_data(), loss_aug_inference_.cpu_data() + num_negatives);

        caffe_add_scalar(loss_aug_inference_.count(), Dtype(-1.0)*max_elem, loss_aug_inference_.mutable_cpu_data());
        caffe_exp(loss_aug_inference_.count(), loss_aug_inference_.mutable_cpu_data(), loss_aug_inference_.mutable_cpu_data());
        Dtype soft_maximum = log(caffe_cpu_dot(num_negatives, summer_vec_.cpu_data(), loss_aug_inference_.mutable_cpu_data())) + max_elem;

        Dtype this_loss = std::max(soft_maximum + dist_pos, Dtype(0.0));

        // squared hinge
        loss += this_loss * this_loss;
        num_constraints += Dtype(1.0);

        // 3. compute gradients
        Dtype sum_exp = caffe_cpu_dot(num_negatives, summer_vec_.cpu_data(), loss_aug_inference_.mutable_cpu_data());

        // update from positive distance dJ_dD_{ij}; update x_i, x_j
        Dtype scaler(0.0);


        scaler = Dtype(2.0)*this_loss / dist_pos;
        // update x_i
        caffe_axpy(K_, scaler  Dtype(1.0), blob_pos_diff_.cpu_data(), bout + iK_);
        // update x_j
        caffe_axpy(K_, scaler  Dtype(-1.0), blob_pos_diff_.cpu_data(), bout + jK_);

        // update from negative distance dJ_dD_{ik}; update x_i, x_k
        neg_idx = 0;
        Dtype dJ_dDik(0.0);
        for (int k=0; k<N_; k++){
          if (!label_mati){
            caffe_sub(K_, bin + iK_, bin + kK_, blob_neg_diff_.mutable_cpu_data());

            dJ_dDik = Dtype(2.0)this_loss  Dtype(-1.0)* loss_aug_inference_.cpu_data()[neg_idx] / sum_exp;
            neg_idx++;

            scaler = dJ_dDik / sqrt(dot_.cpu_data()[i*N_ + k]);

            // update x_i
            caffe_axpy(K_, scaler  Dtype(1.0), blob_neg_diff_.cpu_data(), bout + iK_);
            // update x_k
            caffe_axpy(K_, scaler  Dtype(-1.0), blob_neg_diff_.cpu_data(), bout + kK_);
          }
        }

        // update from negative distance dJ_dD_{jk}; update x_j, x_k
        Dtype dJ_dDjk(0.0);
        for (int k=0; k<N_; k++){
          if (!label_matj){
            caffe_sub(K_, bin + jK_, bin + kK_, blob_neg_diff_.mutable_cpu_data());

            dJ_dDjk = Dtype(2.0)this_loss  Dtype(-1.0)*loss_aug_inference_.cpu_data()[neg_idx] / sum_exp;
            neg_idx++;

            scaler = dJ_dDjk / sqrt(dot_.cpu_data()[j*N_ + k]);

            // update x_j
            caffe_axpy(K_, scaler  Dtype(1.0), blob_neg_diff_.cpu_data(), bout + jK_);
            // update x_k
            caffe_axpy(K_, scaler  Dtype(-1.0), blob_neg_diff_.cpu_data(), bout + kK_);
          }
        }
      } // close this postive pair
    }
  }
  loss = loss / num_constraints / Dtype(2.0);
  top[0]->mutable_cpu_data()[0] = loss;
}

template <typename Dtype>
void LiftedStructSimilaritySoftmaxLossLayer<Dtype>::Backward_cpu(const vector<Blob<Dtype>*>& top,
    const vector<bool>& propagate_down, const vector<Blob<Dtype>*>& bottom) {

  const Dtype alpha = top[0]->cpu_diff()[0] / num_constraints / Dtype(2.0);

  int num = bottom[0]->num();
  int channels = bottom[0]->channels();
  for (int i = 0; i < num; i++){
    Dtype* bout = bottom[0]->mutable_cpu_diff();
    caffe_scal(channels, alpha, bout + (i*channels));
  }
}

#ifdef CPU_ONLY
STUB_GPU(LiftedStructSimilaritySoftmaxLossLayer);
#endif

INSTANTIATE_CLASS(LiftedStructSimilaritySoftmaxLossLayer);
REGISTER_LAYER_CLASS(LiftedStructSimilaritySoftmaxLoss);

}  // namespace caffe

<h2>5. lifted struct similarity softmax layer 实现细节分析</h2>

主要是分析理解论文中公式及源码分析.

const int channels = bottom[0]->channels(); // bottom[0] - channels = 64 dim
for (int i = 0; i < bottom[0]->num(); i++){ // i = 1:m
    dist_sq_.mutable_cpu_data()[i] = caffe_cpu_dot(channels, bottom[0]->cpu_data() + (ichannels), bottom[0]->cpu_data() + (ichannels)); // cpu 上的 dot 计算
}

dist_sq_[i]: ${ ||f(\mathbf{x} _i)|| _2^2 }$

dist_sq_: ${ \hat{\mathbf{x}} = [||f(\mathbf{x} _1)|| _2^2, ..., ||f(\mathbf{x} _m)|| _2^2]^T }$

int M_ = bottom[0]->num(); // M_ = m
int N_ = bottom[0]->num(); // N_ = m
int K_ = bottom[0]->channels(); // K_ = 64

const Dtype* bottom_data1 = bottom[0]->cpu_data(); // (m, 64, 1, 1)
const Dtype* bottom_data2 = bottom[0]->cpu_data(); // (m, 64, 1, 1)

Dtype dot_scaler(-2.0);
caffe_cpu_gemm<Dtype>(CblasNoTrans, CblasTrans, M_, N_, K_, dot_scaler, bottom_data1, bottom_data2, (Dtype)0., dot_.mutable_cpu_data());

void caffe_cpu_gemm(const CBLAS_TRANSPOSE TransA, const CBLAS_TRANSPOSE TransB, const int M, const int N, const int K, const Dtype alpha, const Dtype A, const Dtype B, const Dtype beta, Dtype* C); 功能： C=alpha*A*B+beta*C A,B,C 是输入矩阵（一维数组格式） CblasRowMajor :数据是行主序的（二维数据也是用一维数组储存的） TransA, TransB：是否要对A和B做转置操作（CblasTrans CblasNoTrans） M： A、C 的行数 N： B、C 的列数 K： A 的列数， B 的行数 lda ： A的列数（不做转置）行数（做转置） ldb： B的列数（不做转置）行数（做转置）

这里计算的是：
${ dot_ = -2.0 X X^T }$

// add ||x_i||^2 to all elements in row i
for (int i=0; i<N_; i++){ // N_ = 64
    caffe_axpy(N_, dist_sq_.cpu_data()[i], ones_.cpu_data(), dot_.mutable_cpu_data() + i*N_);
}

// add the norm vector to row i
for (int i=0; i<N_; i++){
    caffe_axpy(N_, Dtype(1.0), dist_sq_.cpu_data(), dot_.mutable_cpu_data() + i*N_); //这里可能出现负值
    caffe_abs(N_, dot_.mutable_cpu_data() + iN_, dot_.mutable_cpu_data() + iN_); // 加绝对值
}

这里计算的是：
${ D^2 = \hat{\mathbf{x}} \mathbf{1}^T + \mathbf{1} \hat{\mathbf{x}}^T- 2XX^T }$

${ D_{ij}^2 = ||f(\mathbf{x} _i)- \mathbf{x} _j||_2^2 }$

源码中的实现，${ D^2 }$ 可能出现负值，添加了一行绝对值 caffe_abs 处理.

// construct pairwise label matrix
vector<vector<bool> > label_mat(N_, vector<bool>(N_, false));
for (int i=0; i<N_; i++){
  for (int j=0; j<N_; j++){
    label_mati = (bottom[1]->cpu_data()[i] == bottom[1]->cpu_data()[j]);
  }
}

针对 mini-batch 内的正负样本建立矩阵，label_mat中相同 label 的位置值为 1，不同则为 0.

正向传播计算：

// 在 label_mat 矩阵的上三角矩阵进行循环，寻找 positive anchors
for (int i=0; i<N_; i++){
  for (int j=i+1; j<N_; j++){

    // 如果 label_mat 值为1，则找到一个 positive pair @ anchor (i, j)
    if (label_mati){
      Dtype dist_pos = sqrt(dot_.cpu_data()[i*N_ + j] + 2e-10); // D_{ij} 

      caffe_sub(K_, bin + iK_, bin + jK_, blob_pos_diff_.mutable_cpu_data());
      // blob_pos_diff_ = [bin + iK] - [bin + jk]

dist_pos: ${ D_{ij} }$，如果两个样本足够相似，且它们的特征向量相似值很小，可能会出现 ${ D_{ij} = 0 }$. 这里添加了一个很小的数 2e-10.

void caffe_sub<float>(const int n, const float a, const float b, float* y) { vsSub(n, a, b, y); } y[i] = a[i] - b[i]

      // 1. 计算 positive 样本 i 的 negetives 样本数 
      int num_negatives = 0;
      for (int k=0; k<N_; k++){
        if (!label_mati){
          num_negatives += 1;
        }
      }

      // 计算 positive 样本 j 的 negetives 样本数 
      for (int k=0; k<N_; k++){
        if (!label_matj){
          num_negatives += 1;
        }
      }

      loss_aug_inference_.Reshape(num_negatives, 1, 1, 1);

      // vector of ones used to sum along channels
      summer_vec_.Reshape(num_negatives, 1, 1, 1);
      for (int ss = 0; ss < num_negatives; ++ss){
        summer_vec_.mutable_cpu_data()[ss] = Dtype(1);
      }

这里主要是统计 positive pair {i, j} 所对应的 negetives 样本总数.

      // 2. 计算 loss
      int neg_idx = 0;
      // mine negative (anchor i, neg k)
      for (int k=0; k<N_; k++){
        if (!label_mati){
          loss_aug_inference_.mutable_cpu_data()[neg_idx] = margin - sqrt(dot_.cpu_data()[i*N_ + k]); // margin - D_{i,k}
          neg_idx++;
        }
      }

      // mine negative (anchor j, neg k)
      for (int k=0; k<N_; k++){
        if (!label_matj){
          loss_aug_inference_.mutable_cpu_data()[neg_idx] = margin - sqrt(dot_.cpu_data()[j*N_ + k]); // margin - D_{j,k}
          neg_idx++;
        }
      }

      // compute softmax of loss aug inference vector;
      Dtype max_elem = *std::max_element(loss_aug_inference_.cpu_data(), loss_aug_inference_.cpu_data() + num_negatives);

      caffe_add_scalar(loss_aug_inference_.count(), Dtype(-1.0)*max_elem, loss_aug_inference_.mutable_cpu_data());
      caffe_exp(loss_aug_inference_.count(), loss_aug_inference_.mutable_cpu_data(), loss_aug_inference_.mutable_cpu_data());
      Dtype soft_maximum = log(caffe_cpu_dot(num_negatives, summer_vec_.cpu_data(), loss_aug_inference_.mutable_cpu_data())) + max_elem;

      // hinge the soft_maximum - S_ij (positive pair similarity)
      Dtype this_loss = std::max(soft_maximum + dist_pos, Dtype(0.0));

      // squared hinge
      loss += this_loss * this_loss; 
      num_constraints += Dtype(1.0);

这里主要进行的计算是：

positive i 与 negetive k 间的 ${ \alpha- D_{i, k} }$</p>
positive j 与 negetive k 间的 ${ \alpha- D_{j, k} }$</p>
采用 Log-Sum-Exp 的计算技巧</p>
<p>soft_maximum(sm): ${ log(\sum_{(i, k) \in \mathcal{N}} exp(\alpha- D_{i, k}) + \sum_{(j, l) \in \mathcal{N}} exp(\alpha- D_{j, l})) }$</p>
<p>this_loss: ${ \hat{J} = max(0, sm + D_{i, j}) }$</p>

<p>loss: ${ \hat{J} = max(0, sm + D_{i, j})^2 }$ 最终的loss是：

loss = loss / num_constraints / Dtype(2.0);

${ J = \frac{1}{2 \mathcal{P}} \sum_{(i,j) \in \mathcal{P}} max(0, sm + D_{i, j})^2 }$</p></li>
</ul>

<p>反向传播计算：

      // 3. 梯度计算
      Dtype sum_exp = caffe_cpu_dot(num_negatives, summer_vec_.cpu_data(), loss_aug_inference_.mutable_cpu_data());

      // update from positive distance dJ_dD_{ij}; update x_i, x_j
      Dtype scaler(0.0);

      scaler = Dtype(2.0)*this_loss / dist_pos;
      // update x_i
      caffe_axpy(K_, scaler  Dtype(1.0), blob_pos_diff_.cpu_data(), bout + iK_);
      // update x_j
      caffe_axpy(K_, scaler  Dtype(-1.0), blob_pos_diff_.cpu_data(), bout + jK_);

      // update from negative distance dJ_dD_{ik}; update x_i, x_k
      neg_idx = 0;
      Dtype dJ_dDik(0.0);
      for (int k=0; k<N_; k++){
        if (!label_mati){
          caffe_sub(K_, bin + iK_, bin + kK_, blob_neg_diff_.mutable_cpu_data());

          dJ_dDik = Dtype(2.0)this_loss  Dtype(-1.0)* loss_aug_inference_.cpu_data()[neg_idx] / sum_exp;
          neg_idx++;

          scaler = dJ_dDik / sqrt(dot_.cpu_data()[i*N_ + k]);

          // update x_i
          caffe_axpy(K_, scaler  Dtype(1.0), blob_neg_diff_.cpu_data(), bout + iK_);
          // update x_k
          caffe_axpy(K_, scaler  Dtype(-1.0), blob_neg_diff_.cpu_data(), bout + kK_);
        }
      }

      // update from negative distance dJ_dD_{jk}; update x_j, x_k
      Dtype dJ_dDjk(0.0);
      for (int k=0; k<N_; k++){
        if (!label_matj){
          caffe_sub(K_, bin + jK_, bin + kK_, blob_neg_diff_.mutable_cpu_data());

          dJ_dDjk = Dtype(2.0)this_loss  Dtype(-1.0)*loss_aug_inference_.cpu_data()[neg_idx] / sum_exp;
          neg_idx++;

          scaler = dJ_dDjk / sqrt(dot_.cpu_data()[j*N_ + k]);

          // update x_j
          caffe_axpy(K_, scaler  Dtype(1.0), blob_neg_diff_.cpu_data(), bout + jK_);
          // update x_k
          caffe_axpy(K_, scaler  Dtype(-1.0), blob_neg_diff_.cpu_data(), bout + kK_);
          }
        }
      } // close this postive pair

dist_pos: ${ D_{i,j} }$

第 7 行的 scaler: ${ \frac{\partial J}{\partial D_{i,j}} }$

Loss ${ J }$ 关于 ${ x_i }$ 的梯度计算：

${ \frac{\partial J} {\partial x_i} = \frac{\partial J}{\partial D_{i,j}} \cdot \frac{\partial D_{i.j}}{\partial x_i} = \frac{\partial J}{\partial D_{i,j}} \cdot (2*||f(\mathbf{x} _i)- f(\mathbf{x} _j)||_2) }$

Loss ${ J }$ 关于 ${ x_j }$ 的梯度计算：

${ \frac{\partial J} {\partial x_j} = \frac{\partial J}{\partial D_{i,j}} \cdot \frac{\partial D_{i.j}}{\partial x_j} = \frac{\partial J}{\partial D_{i,j}} \cdot (-2*||f(\mathbf{x} _i)- f(\mathbf{x} _j)||_2) }$

Loss ${ J }$ 关于 ${ D_{i,k} }$ 的梯度计算：

${ \frac{\partial J}{\partial D_{i,k}} = J_{i,j} \cdot \frac{-exp(\alpha- D_{i, k}) }{exp (J_{i,j}- D_{i,j}) } }$

反向传播计算流程：

<h2>6. Related</h2>

[1] - 论文阅读 - Deep Metric Learning via Lifted Structured Feature Embedding

Last modification：October 10th, 2018 at 04:31 pm

CaffeLoss - Lifted Struct Similarity Softmax Layer

AIHGF • 2018 年 05 月 21 日

Lifted Struct Similarity Softmax Layer, 聚类, 图像检索等特征学习论文: Deep Metric Learning via Lifted Structured Feature Embedding - CVPR2016 项目路径: Deep-Metric-Learning-CVPR16

<h2>1. 在 prototxt 中的定义</h2>

layer {
  name: "fc_embedding"
  type: "InnerProduct"
  bottom: "pool_ave"
  top: "fc_embedding"
  param {
    lr_mult: 1.0
    decay_mult: 1.0
  }
  param {
    lr_mult: 2.0
    decay_mult: 0.0
  }
  inner_product_param {
    num_output: 64  // feature dimension
    weight_filler {
      type: "xavier"
    }
    bias_filler {
      type: "constant"
      value: 0.0
    }
  }
}
####### LiftedStructSimilaritySoftmaxLoss #####
layer {
  name: "loss"
  type: "LiftedStructSimilaritySoftmaxLoss"
  bottom: "fc_embedding"
  bottom: "label"
  top: "loss"
  lifted_struct_sim_softmax_loss_param {
    margin: 1 // margin parameter alpha  
  }
}

<h2>2. caffe.proto 中的定义</h2>

message LayerParameter {
    optional LiftedStructSimilaritySoftmaxLossParameter lifted_struct_sim_softmax_loss_param = 148;
}

message LiftedStructSimilaritySoftmaxLossParameter {
  optional float margin = 1 [default = 1.0]; // margin parameter alpha
}

<h2>3. lifted_struct_similarity_softmax_layer.hpp</h2>

#ifndef CAFFE_LIFTED_STRUCT_SIMILARITY_LOSS_LAYER_HPP_
#define CAFFE_LIFTED_STRUCT_SIMILARITY_LOSS_LAYER_HPP_

#include <vector>

#include "caffe/blob.hpp"
#include "caffe/layer.hpp"
#include "caffe/proto/caffe.pb.h"

#include "caffe/layers/loss_layer.hpp"

namespace caffe {

template <typename Dtype>
class LiftedStructSimilaritySoftmaxLossLayer : public LossLayer<Dtype> {
 public:
  explicit LiftedStructSimilaritySoftmaxLossLayer(const LayerParameter& param)
      : LossLayer<Dtype>(param) {}
  virtual void LayerSetUp(const vector<Blob<Dtype>*>& bottom,
      const vector<Blob<Dtype>*>& top);

  virtual inline int ExactNumBottomBlobs() const { return 2; }
  virtual inline const char* type() const { return "LiftedStructSimilaritySoftmaxLoss"; }
  virtual inline bool AllowForceBackward(const int bottom_index) const {
    return bottom_index != 1;
  }

 protected:
  virtual void Forward_cpu(const vector<Blob<Dtype>*>& bottom,
      const vector<Blob<Dtype>*>& top);

  virtual void Backward_cpu(const vector<Blob<Dtype>*>& top,
      const vector<bool>& propagate_down, const vector<Blob<Dtype>*>& bottom);

  Blob<Dtype> dist_sq_;  // cached for backward pass 
  Blob<Dtype> dot_;  //
  Blob<Dtype> ones_;
  Blob<Dtype> blob_pos_diff_;
  Blob<Dtype> blob_neg_diff_;
  Blob<Dtype> loss_aug_inference_;
  Blob<Dtype> summer_vec_;
  Dtype num_constraints;
};

}  // namespace caffe

#endif  // CAFFE_LIFTED_STRUCT_SIMILARITY_LOSS_LAYER_HPP_

<h2>4. lifted_struct_similarity_softmax_layer.cpp</h2>

#include <algorithm>
#include <vector>

#include "caffe/layers/lifted_struct_similarity_softmax_layer.hpp"
#include "caffe/util/math_functions.hpp"

namespace caffe {

template <typename Dtype>
void LiftedStructSimilaritySoftmaxLossLayer<Dtype>::LayerSetUp(const vector<Blob<Dtype>>& bottom, const vector<Blob<Dtype>>& top) {
  LossLayer<Dtype>::LayerSetUp(bottom, top);
  CHECK_EQ(bottom[0]->height(), 1);
  CHECK_EQ(bottom[0]->width(), 1);
  CHECK_EQ(bottom[1]->channels(), 1);
  CHECK_EQ(bottom[1]->height(), 1);
  CHECK_EQ(bottom[1]->width(), 1);
  // List of member variables defined in /include/caffe/loss_layers.hpp;
  //   diff_, dist_sq_, summer_vec_, loss_aug_inference_;
  dist_sq_.Reshape(bottom[0]->num(), 1, 1, 1);
  dot_.Reshape(bottom[0]->num(), bottom[0]->num(), 1, 1);
  ones_.Reshape(bottom[0]->num(), 1, 1, 1);  // n by 1 vector of ones.
  for (int i=0; i < bottom[0]->num(); ++i){
    ones_.mutable_cpu_data()[i] = Dtype(1);
  }
  blob_pos_diff_.Reshape(bottom[0]->channels(), 1, 1, 1);
  blob_neg_diff_.Reshape(bottom[0]->channels(), 1, 1, 1);
} 

template <typename Dtype>
void LiftedStructSimilaritySoftmaxLossLayer<Dtype>::Forward_cpu(const vector<Blob<Dtype>>& bottom, const vector<Blob<Dtype>>& top) {

  const int channels = bottom[0]->channels(); // feature dims
  for (int i = 0; i < bottom[0]->num(); i++){
    dist_sq_.mutable_cpu_data()[i] = caffe_cpu_dot(channels, bottom[0]->cpu_data() + (ichannels), bottom[0]->cpu_data() + (ichannels));  // hat{x} in papaers
  }

  int M_ = bottom[0]->num(); // mini-batch 内样本数
  int N_ = bottom[0]->num();
  int K_ = bottom[0]->channels(); // 特征维度

  const Dtype* bottom_data1 = bottom[0]->cpu_data(); // 特征矩阵
  const Dtype* bottom_data2 = bottom[0]->cpu_data();

  Dtype dot_scaler(-2.0);
  caffe_cpu_gemm<Dtype>(CblasNoTrans, CblasTrans, M_, N_, K_, dot_scaler, bottom_data1, bottom_data2, (Dtype)0., dot_.mutable_cpu_data());

  // add ||x_i||^2 to all elements in row i
  for (int i=0; i<N_; i++){
    caffe_axpy(N_, dist_sq_.cpu_data()[i], ones_.cpu_data(), dot_.mutable_cpu_data() + i*N_);
  }

  // add the norm vector to row i
  for (int i=0; i<N_; i++){
    caffe_axpy(N_, Dtype(1.0), dist_sq_.cpu_data(), dot_.mutable_cpu_data() + i*N_);
    caffe_abs(N_, dot_.mutable_cpu_data() + iN_, dot_.mutable_cpu_data() + iN_);
  }

  // construct pairwise label matrix
  vector<vector<bool> > label_mat(N_, vector<bool>(N_, false));
  for (int i=0; i<N_; i++){
    for (int j=0; j<N_; j++){
      label_mati = (bottom[1]->cpu_data()[i] == bottom[1]->cpu_data()[j]);
    }
  }

  Dtype margin = this->layer_param_.lifted_struct_sim_softmax_loss_param().margin();
  Dtype loss(0.0);
  num_constraints = Dtype(0.0); 
  const Dtype* bin = bottom[0]->cpu_data();
  Dtype* bout = bottom[0]->mutable_cpu_diff();

  // zero initialize bottom[0]->mutable_cpu_diff();
  for (int i=0; i<N_; i++){
    caffe_set(K_, Dtype(0.0), bout + i*K_);
  }

  // loop upper triangular matrix and look for positive anchors
  for (int i=0; i<N_; i++){
    for (int j=i+1; j<N_; j++){

      // found a positive pair @ anchor (i, j)
      if (label_mati){
        Dtype dist_pos = sqrt(dot_.cpu_data()[i*N_ + j] + 2e-10);

        caffe_sub(K_, bin + iK_, bin + jK_, blob_pos_diff_.mutable_cpu_data());

        // 1.count the number of negatives for this positive
        int num_negatives = 0;
        for (int k=0; k<N_; k++){
          if (!label_mati){
            num_negatives += 1;
          }
        }

        for (int k=0; k<N_; k++){
          if (!label_matj){
            num_negatives += 1;
          }
        }

        loss_aug_inference_.Reshape(num_negatives, 1, 1, 1);

        // vector of ones used to sum along channels
        summer_vec_.Reshape(num_negatives, 1, 1, 1);
        for (int ss = 0; ss < num_negatives; ++ss){
          summer_vec_.mutable_cpu_data()[ss] = Dtype(1);
        }

        // 2. compute loss augmented inference
        int neg_idx = 0;
        // mine negative (anchor i, neg k)
        for (int k=0; k<N_; k++){
          if (!label_mati){
            loss_aug_inference_.mutable_cpu_data()[neg_idx] = margin - sqrt(dot_.cpu_data()[i*N_ + k]);
            neg_idx++;
          }
        }

        // mine negative (anchor j, neg k)
        for (int k=0; k<N_; k++){
          if (!label_matj){
            loss_aug_inference_.mutable_cpu_data()[neg_idx] = margin - sqrt(dot_.cpu_data()[j*N_ + k]);
            neg_idx++;
          }
        }

        // compute softmax of loss aug inference vector;
        Dtype max_elem = *std::max_element(loss_aug_inference_.cpu_data(), loss_aug_inference_.cpu_data() + num_negatives);

        caffe_add_scalar(loss_aug_inference_.count(), Dtype(-1.0)*max_elem, loss_aug_inference_.mutable_cpu_data());
        caffe_exp(loss_aug_inference_.count(), loss_aug_inference_.mutable_cpu_data(), loss_aug_inference_.mutable_cpu_data());
        Dtype soft_maximum = log(caffe_cpu_dot(num_negatives, summer_vec_.cpu_data(), loss_aug_inference_.mutable_cpu_data())) + max_elem;

        Dtype this_loss = std::max(soft_maximum + dist_pos, Dtype(0.0));

        // squared hinge
        loss += this_loss * this_loss;
        num_constraints += Dtype(1.0);

        // 3. compute gradients
        Dtype sum_exp = caffe_cpu_dot(num_negatives, summer_vec_.cpu_data(), loss_aug_inference_.mutable_cpu_data());

        // update from positive distance dJ_dD_{ij}; update x_i, x_j
        Dtype scaler(0.0);


        scaler = Dtype(2.0)*this_loss / dist_pos;
        // update x_i
        caffe_axpy(K_, scaler  Dtype(1.0), blob_pos_diff_.cpu_data(), bout + iK_);
        // update x_j
        caffe_axpy(K_, scaler  Dtype(-1.0), blob_pos_diff_.cpu_data(), bout + jK_);

        // update from negative distance dJ_dD_{ik}; update x_i, x_k
        neg_idx = 0;
        Dtype dJ_dDik(0.0);
        for (int k=0; k<N_; k++){
          if (!label_mati){
            caffe_sub(K_, bin + iK_, bin + kK_, blob_neg_diff_.mutable_cpu_data());

            dJ_dDik = Dtype(2.0)this_loss  Dtype(-1.0)* loss_aug_inference_.cpu_data()[neg_idx] / sum_exp;
            neg_idx++;

            scaler = dJ_dDik / sqrt(dot_.cpu_data()[i*N_ + k]);

            // update x_i
            caffe_axpy(K_, scaler  Dtype(1.0), blob_neg_diff_.cpu_data(), bout + iK_);
            // update x_k
            caffe_axpy(K_, scaler  Dtype(-1.0), blob_neg_diff_.cpu_data(), bout + kK_);
          }
        }

        // update from negative distance dJ_dD_{jk}; update x_j, x_k
        Dtype dJ_dDjk(0.0);
        for (int k=0; k<N_; k++){
          if (!label_matj){
            caffe_sub(K_, bin + jK_, bin + kK_, blob_neg_diff_.mutable_cpu_data());

            dJ_dDjk = Dtype(2.0)this_loss  Dtype(-1.0)*loss_aug_inference_.cpu_data()[neg_idx] / sum_exp;
            neg_idx++;

            scaler = dJ_dDjk / sqrt(dot_.cpu_data()[j*N_ + k]);

            // update x_j
            caffe_axpy(K_, scaler  Dtype(1.0), blob_neg_diff_.cpu_data(), bout + jK_);
            // update x_k
            caffe_axpy(K_, scaler  Dtype(-1.0), blob_neg_diff_.cpu_data(), bout + kK_);
          }
        }
      } // close this postive pair
    }
  }
  loss = loss / num_constraints / Dtype(2.0);
  top[0]->mutable_cpu_data()[0] = loss;
}

template <typename Dtype>
void LiftedStructSimilaritySoftmaxLossLayer<Dtype>::Backward_cpu(const vector<Blob<Dtype>*>& top,
    const vector<bool>& propagate_down, const vector<Blob<Dtype>*>& bottom) {

  const Dtype alpha = top[0]->cpu_diff()[0] / num_constraints / Dtype(2.0);

  int num = bottom[0]->num();
  int channels = bottom[0]->channels();
  for (int i = 0; i < num; i++){
    Dtype* bout = bottom[0]->mutable_cpu_diff();
    caffe_scal(channels, alpha, bout + (i*channels));
  }
}

#ifdef CPU_ONLY
STUB_GPU(LiftedStructSimilaritySoftmaxLossLayer);
#endif

INSTANTIATE_CLASS(LiftedStructSimilaritySoftmaxLossLayer);
REGISTER_LAYER_CLASS(LiftedStructSimilaritySoftmaxLoss);

}  // namespace caffe

<h2>5. lifted struct similarity softmax layer 实现细节分析</h2>

主要是分析理解论文中公式及源码分析.

const int channels = bottom[0]->channels(); // bottom[0] - channels = 64 dim
for (int i = 0; i < bottom[0]->num(); i++){ // i = 1:m
    dist_sq_.mutable_cpu_data()[i] = caffe_cpu_dot(channels, bottom[0]->cpu_data() + (ichannels), bottom[0]->cpu_data() + (ichannels)); // cpu 上的 dot 计算
}

dist_sq_[i]: ${ ||f(\mathbf{x} _i)|| _2^2 }$

dist_sq_: ${ \hat{\mathbf{x}} = [||f(\mathbf{x} _1)|| _2^2, ..., ||f(\mathbf{x} _m)|| _2^2]^T }$

int M_ = bottom[0]->num(); // M_ = m
int N_ = bottom[0]->num(); // N_ = m
int K_ = bottom[0]->channels(); // K_ = 64

const Dtype* bottom_data1 = bottom[0]->cpu_data(); // (m, 64, 1, 1)
const Dtype* bottom_data2 = bottom[0]->cpu_data(); // (m, 64, 1, 1)

Dtype dot_scaler(-2.0);
caffe_cpu_gemm<Dtype>(CblasNoTrans, CblasTrans, M_, N_, K_, dot_scaler, bottom_data1, bottom_data2, (Dtype)0., dot_.mutable_cpu_data());

void caffe_cpu_gemm(const CBLAS_TRANSPOSE TransA, const CBLAS_TRANSPOSE TransB, const int M, const int N, const int K, const Dtype alpha, const Dtype A, const Dtype B, const Dtype beta, Dtype* C); 功能： C=alpha*A*B+beta*C A,B,C 是输入矩阵（一维数组格式） CblasRowMajor :数据是行主序的（二维数据也是用一维数组储存的） TransA, TransB：是否要对A和B做转置操作（CblasTrans CblasNoTrans） M： A、C 的行数 N： B、C 的列数 K： A 的列数， B 的行数 lda ： A的列数（不做转置）行数（做转置） ldb： B的列数（不做转置）行数（做转置）

这里计算的是：
${ dot_ = -2.0 X X^T }$

// add ||x_i||^2 to all elements in row i
for (int i=0; i<N_; i++){ // N_ = 64
    caffe_axpy(N_, dist_sq_.cpu_data()[i], ones_.cpu_data(), dot_.mutable_cpu_data() + i*N_);
}

// add the norm vector to row i
for (int i=0; i<N_; i++){
    caffe_axpy(N_, Dtype(1.0), dist_sq_.cpu_data(), dot_.mutable_cpu_data() + i*N_); //这里可能出现负值
    caffe_abs(N_, dot_.mutable_cpu_data() + iN_, dot_.mutable_cpu_data() + iN_); // 加绝对值
}

这里计算的是：
${ D^2 = \hat{\mathbf{x}} \mathbf{1}^T + \mathbf{1} \hat{\mathbf{x}}^T- 2XX^T }$

${ D_{ij}^2 = ||f(\mathbf{x} _i)- \mathbf{x} _j||_2^2 }$

源码中的实现，${ D^2 }$ 可能出现负值，添加了一行绝对值 caffe_abs 处理.

// construct pairwise label matrix
vector<vector<bool> > label_mat(N_, vector<bool>(N_, false));
for (int i=0; i<N_; i++){
  for (int j=0; j<N_; j++){
    label_mati = (bottom[1]->cpu_data()[i] == bottom[1]->cpu_data()[j]);
  }
}

针对 mini-batch 内的正负样本建立矩阵，label_mat中相同 label 的位置值为 1，不同则为 0.

正向传播计算：

// 在 label_mat 矩阵的上三角矩阵进行循环，寻找 positive anchors
for (int i=0; i<N_; i++){
  for (int j=i+1; j<N_; j++){

    // 如果 label_mat 值为1，则找到一个 positive pair @ anchor (i, j)
    if (label_mati){
      Dtype dist_pos = sqrt(dot_.cpu_data()[i*N_ + j] + 2e-10); // D_{ij} 

      caffe_sub(K_, bin + iK_, bin + jK_, blob_pos_diff_.mutable_cpu_data());
      // blob_pos_diff_ = [bin + iK] - [bin + jk]

dist_pos: ${ D_{ij} }$，如果两个样本足够相似，且它们的特征向量相似值很小，可能会出现 ${ D_{ij} = 0 }$. 这里添加了一个很小的数 2e-10.

void caffe_sub<float>(const int n, const float a, const float b, float* y) { vsSub(n, a, b, y); } y[i] = a[i] - b[i]

      // 1. 计算 positive 样本 i 的 negetives 样本数 
      int num_negatives = 0;
      for (int k=0; k<N_; k++){
        if (!label_mati){
          num_negatives += 1;
        }
      }

      // 计算 positive 样本 j 的 negetives 样本数 
      for (int k=0; k<N_; k++){
        if (!label_matj){
          num_negatives += 1;
        }
      }

      loss_aug_inference_.Reshape(num_negatives, 1, 1, 1);

      // vector of ones used to sum along channels
      summer_vec_.Reshape(num_negatives, 1, 1, 1);
      for (int ss = 0; ss < num_negatives; ++ss){
        summer_vec_.mutable_cpu_data()[ss] = Dtype(1);
      }

这里主要是统计 positive pair {i, j} 所对应的 negetives 样本总数.

      // 2. 计算 loss
      int neg_idx = 0;
      // mine negative (anchor i, neg k)
      for (int k=0; k<N_; k++){
        if (!label_mati){
          loss_aug_inference_.mutable_cpu_data()[neg_idx] = margin - sqrt(dot_.cpu_data()[i*N_ + k]); // margin - D_{i,k}
          neg_idx++;
        }
      }

      // mine negative (anchor j, neg k)
      for (int k=0; k<N_; k++){
        if (!label_matj){
          loss_aug_inference_.mutable_cpu_data()[neg_idx] = margin - sqrt(dot_.cpu_data()[j*N_ + k]); // margin - D_{j,k}
          neg_idx++;
        }
      }

      // compute softmax of loss aug inference vector;
      Dtype max_elem = *std::max_element(loss_aug_inference_.cpu_data(), loss_aug_inference_.cpu_data() + num_negatives);

      caffe_add_scalar(loss_aug_inference_.count(), Dtype(-1.0)*max_elem, loss_aug_inference_.mutable_cpu_data());
      caffe_exp(loss_aug_inference_.count(), loss_aug_inference_.mutable_cpu_data(), loss_aug_inference_.mutable_cpu_data());
      Dtype soft_maximum = log(caffe_cpu_dot(num_negatives, summer_vec_.cpu_data(), loss_aug_inference_.mutable_cpu_data())) + max_elem;

      // hinge the soft_maximum - S_ij (positive pair similarity)
      Dtype this_loss = std::max(soft_maximum + dist_pos, Dtype(0.0));

      // squared hinge
      loss += this_loss * this_loss; 
      num_constraints += Dtype(1.0);

这里主要进行的计算是：

positive i 与 negetive k 间的 ${ \alpha- D_{i, k} }$</p>
positive j 与 negetive k 间的 ${ \alpha- D_{j, k} }$</p>
采用 Log-Sum-Exp 的计算技巧</p>
<p>soft_maximum(sm): ${ log(\sum_{(i, k) \in \mathcal{N}} exp(\alpha- D_{i, k}) + \sum_{(j, l) \in \mathcal{N}} exp(\alpha- D_{j, l})) }$</p>
<p>this_loss: ${ \hat{J} = max(0, sm + D_{i, j}) }$</p>

<p>loss: ${ \hat{J} = max(0, sm + D_{i, j})^2 }$ 最终的loss是：

loss = loss / num_constraints / Dtype(2.0);

${ J = \frac{1}{2 \mathcal{P}} \sum_{(i,j) \in \mathcal{P}} max(0, sm + D_{i, j})^2 }$</p></li>
</ul>

<p>反向传播计算：

      // 3. 梯度计算
      Dtype sum_exp = caffe_cpu_dot(num_negatives, summer_vec_.cpu_data(), loss_aug_inference_.mutable_cpu_data());

      // update from positive distance dJ_dD_{ij}; update x_i, x_j
      Dtype scaler(0.0);

      scaler = Dtype(2.0)*this_loss / dist_pos;
      // update x_i
      caffe_axpy(K_, scaler  Dtype(1.0), blob_pos_diff_.cpu_data(), bout + iK_);
      // update x_j
      caffe_axpy(K_, scaler  Dtype(-1.0), blob_pos_diff_.cpu_data(), bout + jK_);

      // update from negative distance dJ_dD_{ik}; update x_i, x_k
      neg_idx = 0;
      Dtype dJ_dDik(0.0);
      for (int k=0; k<N_; k++){
        if (!label_mati){
          caffe_sub(K_, bin + iK_, bin + kK_, blob_neg_diff_.mutable_cpu_data());

          dJ_dDik = Dtype(2.0)this_loss  Dtype(-1.0)* loss_aug_inference_.cpu_data()[neg_idx] / sum_exp;
          neg_idx++;

          scaler = dJ_dDik / sqrt(dot_.cpu_data()[i*N_ + k]);

          // update x_i
          caffe_axpy(K_, scaler  Dtype(1.0), blob_neg_diff_.cpu_data(), bout + iK_);
          // update x_k
          caffe_axpy(K_, scaler  Dtype(-1.0), blob_neg_diff_.cpu_data(), bout + kK_);
        }
      }

      // update from negative distance dJ_dD_{jk}; update x_j, x_k
      Dtype dJ_dDjk(0.0);
      for (int k=0; k<N_; k++){
        if (!label_matj){
          caffe_sub(K_, bin + jK_, bin + kK_, blob_neg_diff_.mutable_cpu_data());

          dJ_dDjk = Dtype(2.0)this_loss  Dtype(-1.0)*loss_aug_inference_.cpu_data()[neg_idx] / sum_exp;
          neg_idx++;

          scaler = dJ_dDjk / sqrt(dot_.cpu_data()[j*N_ + k]);

          // update x_j
          caffe_axpy(K_, scaler  Dtype(1.0), blob_neg_diff_.cpu_data(), bout + jK_);
          // update x_k
          caffe_axpy(K_, scaler  Dtype(-1.0), blob_neg_diff_.cpu_data(), bout + kK_);
          }
        }
      } // close this postive pair

dist_pos: ${ D_{i,j} }$

第 7 行的 scaler: ${ \frac{\partial J}{\partial D_{i,j}} }$

Loss ${ J }$ 关于 ${ x_i }$ 的梯度计算：

Loss ${ J }$ 关于 ${ x_j }$ 的梯度计算：

Loss ${ J }$ 关于 ${ D_{i,k} }$ 的梯度计算：

${ \frac{\partial J}{\partial D_{i,k}} = J_{i,j} \cdot \frac{-exp(\alpha- D_{i, k}) }{exp (J_{i,j}- D_{i,j}) } }$

反向传播计算流程：

<h2>6. Related</h2>

[1] - 论文阅读 - Deep Metric Learning via Lifted Structured Feature Embedding

CaffeLoss - Lifted Struct Similarity Softmax Layer

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