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6.1 General Relational Ranking Framework
6.1.1 Relational Ranking SVM
By considering the relationship in the ranking process, the ranking function can be
refined as follows, where the function h is named the relational ranking function [ 6 ],
h f( x ), R .
h( x )
=
(6.1)
Here f can still be understood as a scoring function operating on each single
document, and the matrix R describes the relationship between documents. There
are then several ways of parameterizing h( x ) . For example, one can let f contain
a parameter w , while the way of integrating f and R is pre-defined. One can also
regard f as a fixed function, and h has its own parameter. The most complex case
is that both f and h contain parameters. In [ 6 ], the first case is discussed, and it is
assumed that
h f(w, x ), R =
l 1 f(w, x ), z +
βl 2 (R, z ) ,
h( x )
=
arg min
z
(6.2)
where l 1 is an objective function to guarantee that the ranking results given by h
should not be too different from those given by f , and l 2 is another objective func-
tion to guarantee that the ranking results should be consistent with the relationship
requirement R as much as possible.
Then, as for topic distillation and pseudo relevance feedback, the relational rank-
ing function h is further realized and discussed. A nice thing is the existence of
the closed-form solution of the relational ranking function in these two cases. For
pseudo relevance feedback, one can eventually obtain,
h f(w, x ), R = I + β(D R) 1 f(w, x ), (6.3)
where I denotes an identity matrix, and D is a diagonal matrix with D i,i = j R i,j .
For topic distillation, one has
h f(w, x ), R = 2 I + β 2 D R R T 1 2 f(w, x ) βr ,
(6.4)
where r j = i R i,k j R k,j .
One may have noticed that in the relational ranking framework, there are two
different inputs: one is the features of the documents x , and the other is the relation-
ship matrix R . This seems quite different from the setting of conventional learning to
rank mentioned in previous chapters. However, if one notices that x = φ(q,d) and
R = ψ(d,d) , one will realize that the inputs are still documents and queries. The
only difference lies in that now we also care about the inter-dependency between
documents, while previously each document was dealt with in an independent man-
ner.
The relational ranking function can be substituted into any learning-to-rank al-
gorithm introduced in this topic for learning. In particular, the demonstration of
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