引言

部分哈希技术使用的是缩写，具体的见论文中的表达；

非完整的笔记...

Introduction

背景知识

数据独立
- LSH，MinHash：这些基于随机投影或排列的哈希算法，编码后的哈希码的长度 r>1000 才能有良好的检索效果
数据非独立
- 非监督：
  - SH，ITQ，ISQH，DGH等：通过考虑数据潜在的数据结构，数据分布，拓扑信息来学习 coding functions【哈希函数簇】
- 监督：
  - Binary Reconstructive Embedding(BRE) [7],
  - Minimal Loss Hashing (MLH) [16],
  - Kernelbased Supervised Hashing (KSH) [13],
  - Hamming Distance Metric Learning (HDML) [17],
  - Ranking-based Supervised Hashing (RSH) [27],
  - Column Generation Hashing (CGH)[11],
  - Rank Preserving Hashing (RPH) [20]

：learn coding functions by incorporating supervised information【1. instance-level labels 2. pair-level labels 3. triplet-level ranks.】

要点

pointwise pairswise在训练哈希函数时，标签信息可以被保存在汉明空间中，但是没有达到最优的搜索任务，因为排序信息没有被充分利用；
rank supervised approaches (e.g., HDML [17], RSH [27], CGH[11], and RPH [20]) which employ triplet-level ranks are developed；
前面提到的这些rank supervised hashing approaches可以高效的处理大规模数据任务。但是他们很少达到精度很高的排序列表【根据生成的哈希码计算汉明距离】；

核心思想：train the disciplined coding functions by which the mistakes at the top of a Hamming-distance ranking list are penalized
more than those at the bottom.

Top Rank Supervised Binary Coding

准备工作

Since our introduced rank preserving objective is discrete in nature and its associated optimization problem is combinatorially difficult, we relax the original discrete objective to a continuous and differentiable surrogate, and accordingly derive a stochastic gradient descent method to optimize the surrogate objective.【因为前面的算法是离散的，导致优化非常麻烦。我们将离散映射转换为可微的替代品，并运用SGD方法来优化目标函数】

an online learning algorithm to optimize the surrogate objective efficiently

公式推导

总结：

I(·) is an indicator function which returns 1 if the condition in the parenthesis is satisfied and returns 0 otherwise.
function R(x_i, x_j) explicitly measures the number of the incorrectly ranked dissimilar samples xs’s which are closer to the query x_i than the similar sample x_j in terms of Hamming distance and therefore indicates the position of x_j in a Hamming distance ranking list with respect to the query x_i.
L(R) is a one to one monotonic increasing function with first order derivative L′(R) large than zero and second order derivative L′′(R) smaller than zero
the relative similarity of the triplets are measured with Hamming distance which is discrete and discontinuous【其他的基于rank的哈希函数都是离散的，这样很难优化，所以文中提出了log形式的排序函数】
our ranking loss is continuous differentiable while the previous ones are discrete and difficult to optimize.

Optimization

问题

coding functions 是离散的映射；
汉明范数也是在离散的空间中；

解决离散问题的松弛操作

将sgn(·)换成tanh函数；
Hamming norm is relaxed to ℓ1 norm which is convex【将汉明范数换成L1范数】
将 I(·) 换成sigmoid函数；

最后得到：

SGD

如果使用sub-gradient descent approach来优化上面得到的目标函数，可能不会收敛，因为全局梯度计算成本非常大，故采用随机梯度下降。

给定数据集的三元组相似信息的集合Ｄ；
随机选取一个点 $x_{i}$ 和它的相似点 $x_j$ ;
固定 $x_{i}$ 和 $x_j$ 。随机选取p个点 $x_s$ 【符合 $\left \{ x_i,x_j,x_s\right \}_{s=1}^p$ 】
假设所有负样本均匀分布， $\bar{R}(x_i,x_j)\approx \left \lfloor {\frac{\left | N \right |}{p}} \right \rfloor\cdot \sum_{s=1}^sG(T_{ij}-T_{is}))$