Ranking SVM
In machine learning, a ranking SVM is a variant of the support vector machine algorithm, which is used to solve certain ranking problems (via learning to rank). The ranking SVM algorithm was published by Thorsten Joachims in 2002. 
The original purpose of the algorithm was to improve the performance of an internet search engine. However, it was found that ranking SVM also can be used to solve other problems such as Rank SIFT.

Description
The ranking SVM algorithm is a learning retrieval function that employs pairwise ranking methods to adaptively sort results based on how 'relevant' they are for a specific query. The ranking SVM function uses a mapping function to describe the match between a search query and the features of each of the possible results. This mapping function projects each data pair (such as a search query and clicked web-page, for example) onto a feature space. These features are combined with the corresponding click-through data (which can act as a proxy for how relevant a page is for a specific query) and can then be used as the training data for the ranking SVM algorithm.

Generally, ranking SVM includes three steps in the training period: 
 It maps the similarities between queries and the clicked pages onto a certain feature space.
 It calculates the distances between any two of the vectors obtained in step 1.
 It forms an optimization problem which is similar to a standard SVM classification and solves this problem with the regular SVM solver.

Background

Ranking method

Suppose  is a data set containing  elements .  is a ranking method applied to . Then 
the  in  can be represented as a  binary matrix. If the rank of  is higher than the rank of , i.e. , the corresponding position of this matrix is set to value of "1". Otherwise the element in that position will be set as the value "0".

Kendall's tauA. Mood, F. Graybill, and D. Boes. Introduction to the Theory of Statistics. McGraw-Hill, 3rd edition, 1974

Kendall's Tau also refers to Kendall tau rank correlation coefficient, which is commonly used to compare two ranking methods for the same data set.

Suppose  and  are two ranking method applied to data set , the Kendall's Tau between  and  can be represented as follows:

 

where  is the number of concordant pairs  and  is the number of discordant pairs (inversions). A pair  and  is concordant if both  and  agree in how they order  and  . It is discordant if they disagree.

Information retrieval qualityY. Yao. "Measuring retrieval effectiveness based on user preference of documents." Journal of the American Society for Information Science, 46(2): 133–145, 1995.

Information retrieval quality is usually evaluated by the following three measurements:
 Precision
 Recall
 Average precision

For a specific query to a database, let  be the set of relevant information elements in the database and  be the set of the retrieved information elements. Then the above three measurements can be represented as follows:

 

where  is the  of .

Let  and  be the expected and proposed ranking methods of a database respectively, the lower bound of Average Precision of method  can be represented as follows:

 

where  is the number of different elements in the upper triangular parts of matrices of  and  and  is the number of relevant elements in the data set.

SVM classifier
Suppose  is the element of a training data set, where  is the feature vector and  is the label (which classifies the category of ). A typical SVM classifier for such data set can be defined as the solution of the following optimization problem.

 

The solution of the above optimization problem can be represented as a linear combination of the feature vectors s.

 

where  is the coefficients to be determined.

Ranking SVM algorithm

Loss function
Let  be the Kendall's tau between expected ranking method  and proposed method , it can be proved that maximizing  helps to minimize the lower bound of the Average Precision of .
 Expected loss function
The negative  can be selected as the loss function to minimize the lower bound of average precision of 

 

where  is the statistical distribution of  to certain query .

 Empirical loss function
Since the expected loss function is not applicable, the following empirical loss function is selected for the training data in practice.

Collecting training data
 i.i.d. queries are applied to a database and each query corresponds to a ranking method. The training data set has  elements. Each element contains a query and the corresponding ranking method.

Feature space

A mapping function  is required to map each query and the element of database to a feature space. Then each point in the feature space is labelled with certain rank by ranking method.

Optimization problem
The points generated by the training data are in the feature space, which also carry the rank information (the labels). These labeled points can be used to find the boundary (classifier) that specifies the order of them. In the linear case, such boundary (classifier) is a vector.

Suppose  and  are two elements in the database and denote  if the rank of  is higher than  in certain ranking method . Let vector  be the linear classifier candidate in the feature space. Then the ranking problem can be translated to the following SVM classification problem. Note that one ranking method corresponds to one query.

 

The above optimization problem is identical to the classical SVM classification problem, which is the reason why this algorithm is called Ranking-SVM.

Retrieval function
The optimal vector  obtained by the training sample is

 

So the retrieval function could be formed based on such optimal classifier. 
For new query , the retrieval function first projects all elements of the database to the feature space. Then it orders these feature points by the values of their inner products with the optimal vector. And the rank of each feature point is the rank of the corresponding element of database for the query .

Application of ranking SVM
Ranking SVM can be applied to rank the pages according to the query. The algorithm can be trained using click-through data, where consists of the following three parts: 
 Query.
 Present ranking of search results
 Search results clicked on by user
The combination of 2 and 3 cannot provide full training data order which is needed to apply the full SVM algorithm. Instead, it provides a part of the ranking information of the training data. So the algorithm can be slightly revised as follows.

 

The method  does not provide ranking information of the whole dataset, it's a subset of the full ranking method. So the condition of optimization problem becomes more relax compared with the original Ranking-SVM.

References

Support vector machines