'='d(d)
An
Analysis
Based
on
Z.W.
Wang,
of Vector
Space
Computational
S.K.M.
Department
Geometry
Wong
and
of Computer
University Regina,
Models
Y.Y.
Yao
Science
of Regina Saskatchewan
Canada S4S 0A2
Abstract
Earlier the
This paper analyzes the properties, structures and limitations of vector-based models for information retrieval from the computational ometry point of view. the pseudo-cosine
ge-
It is shown that both
and the standard
vector
of a generalized linear model.
More impor-
tantly, both the necessary and sufficient con-
where
the
used
model.
alyzed
67 similarity
all
the
The
On
sures
solution
the
Introduction
Their
and
queries this
effective systems
hand,
to
into
either
al-
studies et al.,
Furnas
(1987)
of seven
similarity
mea-
algebraic
and
those
investigation
are Similar
and
an-
them
(Schneider
Jones
analysis
extensions (1979)
class
similar.
LIVE-project
other
complementary
revealed
among
many
similarity
gebraic
and
geometric
analyses
of
vector-based
these
information
successfully
fact,
the
measure
and
divided
each
point
measures.
major
empirical differences
Clearly,
both
the
al-
is suggested.
Vector-based used
in
of
con-
algebraic
Noreault
and
and
of the
similarity
and
or very
a geometric
understanding
1
Koll
members
made
an
variations
measures
equivalent
also
1986).
several
McGill,
classes.
from
of dif-
Raghavan
analysis
inner-product
to examine
of the
model
emphasized
evaluations
For example, a critical
of
view,
models
empirical
measures. presented space
studies.
for finding
and
vector
provided
tion region for acceptable ranking is analyzed
vector-based
ventional
were
rank the documents. The structure of the soluand an algorithm
(1986)
gebraically
pseudo-cosine, Dice, covariance and productmoment correlation measures can be used to
vectors
similarity
Wong
24
ditions have been identified, under which ranking functions such as the inner-product, cosine,
on
structure
ferent
was
space models can be viewed as special cases
studies
algebraic
for
(Salton, category
ranking (Salton,
the
retrieval
1971; Salton of retrieval strategy
models
representation
than
and models the
have
been
of documents McGill, provide standard
1983).
studies
In
suggested
the
use of different
not
identified
ticular
the
invaluable
for
models.
some
similarity
the
similarity
only
are
useful
guidelines
measures,
precise
conditions
measure
should
the
However,
but
under
they
on have
which
a par-
similarity
mea-
be used.
a more To justify
Boolean
sure,
1989).
Wong, Permission to copy without fee all or part of this matarial is granted provided that the copies are not made or distributed for direct commercial advantage, tha ACM copyright notice and tha title of the publication and its date appear, and notica is given that copying is by permission of the Association for Computing Machinary. To copy otherwise, or to republish, requires a fee and/or specific permission. 15th Ann Int’1 SIGIR ‘92/Denmark-6/92 01992 ACM 0-89791-524-0/92/0006/01 52...$1.50
tion
the choice
Wong,
Yao
1987;
of user
scription
Wong
identified. have
152
Bollmann
and
preference,
Yao,
(1988; 1990)
which
gives
of the user judgments
than
of relevance. referred
of an appropriate
and
In this
to as the It
a strong
approach,
perfect
was shown influence
and
two
introduced a more
that
these design
and the
accurate
the standard ranking
acceptable
on the
Bollmann
node-
notion
strategies, rankings,
ranking
are
strategies
of information
re-
trieval
systems.
basis
for
the
identified, ficient for
Their
in some
for
perfect
some
and
well
analyze
ranking
the
by adopting
This briefly the
paper
is organized the
acceptable
generalized two
linear
linear
models,
vector
space
models
strated
that
a minor
tions
may
lead
structures. sufficient tor
are analyzed
Section
conditions
for
the
for
lyze the structure vector
space
the
the
in detail.
of the solution
system
In
vector-based
sented ponent
other
term
each
a set
that
there
user
information
preference
of
exists
vectors
the
Rn.
Rnx
3,
query
write
m(q,
function
Rn-+R,
vector
needs
structure
the
According
subscript
defined d,d’
the
q emphasizes
>g is defined
q. However,
we will
c,f
1991).
that
such
a
as a real funca document
-+ m(q,
and
function
vec-
lowing
is said
d)
d).
becomes
emphasize
which
document
(2)
this
is referred
a func-
point,
we
to as a ranking
vectors. preference,
(Wong
to provide
condition
m(q,
To
to the user
two
and Yao,
ranking
1990).
a perfect
strate-
A ranking
ranking
if the
fol-
holds:
d>
d’ ~m~(d)
function
if a weaker
The
is said
> m~(d’). to provide
condition
(3)
an acceptable
rank-
holds:
it
ranking
In
other
the
antees
that
ahead
of the
time.
by a binary
fore,
by the The
the
ing
user
preferred.
in the
c D,
(qlqz
present
user
prefers
with drop
d’.
the fact respect
document
that
the subscript
are not
guarranked
argued
is more
with
On the
only
(1988)
retrieval
concerned
of query
and
and
d = (~ld~”
measure
that
appropriate
systems.
There-
acceptable
document “
is defined
ranlk-
“1~7~)”
vectors, suppose
q = a
lin-
as:
cl (1)
2=1 With
the preference
to a particular
ranking
and
study.
a pair “%)
et el.
of information
preferred
rank.
strateg,y
documents
Wong
we are primarily
the
less preferred,
to the same ranking
of acceptable
design
Consider
relation
belong
acceptable
that
of the
the less preferred
the paradigm for
ensures
ahead
documents
hand,
a query
is assumed
strategy
are ranked
indifferent
of imd.
perfect
documents
as a vector
governed
at a particular
for
degree
document
D,
to document
relation
d.
d) as mg(d)
for
ear similarity
The
system
measure.
q and
(qjd)
q is fixed,
variable
gies can be identified
q in Rn.
vectors
a preference
~
map
Yao,
assume
vector
accor-
d):
of one
each com-
Similarly,
as a vector
can be formally
d>qd’
in is to
and
m can be viewed
(a query
information
the relation
Bollmann we may
tion
is repre-
where
is represented space
document
>~ on D as follows:
onto
by a similarity
of two
(Roberts,
task
can be generated
we ana-
a document
indicating
vector
can also be represented Given
The
loss of generality,
measure
preference
>)
documents
mapping
If the
preference
necessary
ti in describing
document
in a n-dimensional
>)
the
(D, of an
the
preference. (D,
> d’)
for the standard
dz, . . . . d~),
number
of index
words,
models,
d = (dI,
di is a real
portance
and
objectives
(Wong,
=(d
Models
retrieval
by a vector
D system
rank
i.e.
d’ are indzflerent.
Without
A ranking
Retrieval
user
(R, >)
ing
Linear
to
real
numbers
vectors
main
is
the
of
model.
2
of the
system with
preference, d and
a relational
the relational
representa-
Finally,
region
One
form
m:
standard
of a solution
measure.
of document
A similarity
It is demon-
user the
existence
1976).
of strict
we say that
and
notion
the
in document
linear
>
tion
In Section
different
set
2, we
as linearity
and
4, we present
generalized
models
In Section
pseudo-cosine
The
vector
is introduced.
to dramatically
In
to is to
linear
Then,
difference
and
for computational
such
strategy.
the
inner-
respect
> d),
relation
dance
1985).
as follows.
measures
as the
In the absence
V(d’
retrieval
of
objective
of these
concepts
ranking
the
existence
with
Another
Shames,
basic
measure
covariance,
developed
and
review
such
Dice,
properties
(Preparata
the
measures,
the techniques
geometry
for
functions
strategy.
geometric
and
suf-
is to formulate
conditions
correlation
the acceptable
paper
pseudo-cosine,
product-moment
and
of a linear
arises.
they
ranking.
of this
similarity
cosine,
necessary
existence
sufficient
a theoretical
In particular,
the
or acceptable
known
product,
cases,
the
One of the objectives necessary
established
measures.
special
conditions
either
work
use of linear
respect
functions
query
linear.
q if no confusion
153
such
to
the
above
definition,
cosine
and
as the
We believe
that
this
Dice
definition
many
similarity
measures is too
are not
restrictive.
We
now
introduce
measures, tions
a more
which
of linear
will
general
enable
definition
of linear
us to broaden
the
where
applica-
a is real
thus
models.
no
mapped longer
1.
Rn x Rn.
Let
If there
rn(q,
exist
d)
be a similarity
two
measure
on
Nq:Rn+Rn,
(6)
q+q=~g(q)
such
as an
[ad]
and
d = Nd(d)
(7)
that
then
d) = Ng(q)
we say m(q, can
vector,
be
the
new
measure.
Nd
and
shown
in d
the
they
S“.
The of
function the
We
same
are may
now view
is a representative
of
space.
measure
pseudo-cosine
similarity
measure
can
be
ex-
as:
In this
case, the
normalization
functions
are:
query
(14)
of the
many
are
similarity
indeed
measures
linear
under
where
this
Iql
becomes
of linearity.
In
measure measure
a document
of similarity
d is defined
between
and
Idl
~~=1
~idi
this
the
are
all
vector
11 norms.
d
the
=
the entire
vectors
d/Id
vector
a plane
by:
Rn
{d
I XV=l
pointing
Thus,
equation
(13)
= ~ . d.
model,
unit
measure, q and
is
(8)
as a normalization
that
1987)
Furnas,
inner-product
query
class;
quotient
having
because
R*
is, one can
(15)
inner-product
The
Rn
vector
space
That
vectors.
definition
The
= q “ ~>
as a normalization
function
It can be easily (Jones
is a hnear
regarded
and
document
d)
Nd(d)
equivalence
vector S“.
in
lengths,
same
pseudo-cosine
The
entire
sphere
vectors
the
Sn is the
pressed rn(q,
N~
d ~
unit
different by
[ad]
The
and Nd:Rn+Rn,
but
represented
functions:
The
the
distinguish
direction Definition
number.
onto
di
in the
=
1.
same
ad
are
mapped
1. In
contrast
to
space
Rn is mapped
That
is,
any
direction
but
onto
the
two
cosine
onto
the
vectors
in
having
different
n
inner(q,
d) = ~
qidi
= q ~d
lengths
(9)
?=1 Obviously, tity
the
normalizations
functions,
hence
the
The
given
=
q,
d =
is a linear
den-
case are the Nd(d)
=
d
and
in
the
The
measure
information
which
was
retrieval
first
system
used
(Salton,
1971)
(van
is defined
a plane.
Rijsbergen,
1979;
Salton
and
vector
q =
i.e.,
Iql =
by:
normalization
functions
\\ d II are 12 or
above
be written
normalizations]
as ~~=1
with
the following
q
=
N~(q)
d
=
N~(d)
query
the 11 norm,
normalizations: (17)
= 2q,
(11) equation
(12)
the
1. By
qi =
the
are:
Nd(d)”&=A’
II q I\ and
case in which
(lo)
_—
‘=
a special
. . . , ~~) is normalized
(W,92)
is
‘=Nq(q)=&-ll:ll’ can
measure
1983)
Consider
~~=1
Cos(q]’) =mm
applying
on such
measure.
x:=,qid
where
vector
measure
Dice
McGill,
by:
The
Dice
as the same
measure
cosine
SMART
Ng(q)
inner-product
cosine
The
q =
The in this
are regarded
Euclidean the
qi & = q
norms.
cosine
d, which
By
(16)
can be written
fore,
in this
special
The
measures
case, the
of
The
is therefore
related
linear.
~~~~ = Q.d.
measure
There-
is linear.
and
correlation
covariance (Jones
Dice
covariance
product-moment
measure
as ~~=1
and
and
correlation
Furnas,
measures
1987).
are closely
Let
n
With ad
are
this
representation
mapped
onto
the
of documents, unit
vector
T=:g%,
all the vectors d
=
d/
II d
II,
‘2=1
154
d-$di.
(19) t=l
The
formula
for
the
co~(q,
covariance
d)
measure
is:
where
5(w–T)(4 – Z).
=
2X is the
power
~-’(b)
(20)
set of X,
and
= {a I (a E X)
A(?(a)
= b)};
of a under
f.
(29)
i=l Let
1 = (1, 1, . . . . 1).
are defined
the
~~=1
@i&=
The
the
normalization
f-1 (~)is called
functions
by:
Thus,
fined
Here,
As already
q=
Ng(q)
= q–ql,
(21)
d=
N~(d)
=d–~1.
(22)
covariance
measure
d ~~, and
measure
of
(20)
hence
can
be expressed
it is a linear
product-moment
measure
mg,
as
measure.
correlation
is de-
by: D’=l(%
d) = /x=,(9i
Similarly,
- m(~t -Q)-27”
- ~)
z,=,(~i
-~)
Definition
3.
Any
- 1)2 ‘
The
a linear
’25)
Space
into
section,
which
Models
that
slightly
lead to dramatically 3.1
Contour
Before haps
useful
first to our
Definition Consider
models,
normalizations
user preference
we
2.
notion
to review
of contour
structures.
the
concepts
sets, about
a function
f
from
X
it is perfunctions
X The
is called range
the off
domain
of ~ and
nonempty
let
(26)
~(a). codomatn
The
tnverse
mappmg, ~-1
: f(X)
A (CIC .Y)},
plane
notion
of .f is defined
+2X,
b–
f-l(b),
similarity normal
of
a measure
nz~
R},
of U and curve.
a contour
The
contour
curves.
then
contour
of j.
generates
(27)
different.
b
not
155
preferred
smaller
this
in the
r values.
curve
of the unit
sphere
contour
to the
the
and
generates
a per-
contour
contour
in
contour
one
can
set only
smaller
in the
contour
the mea-
one can still
same
set
sets with
If a similarity
sets with
documents
the
(Jones
documents
ranking,
however,
analyzing
same
in the to
in the
case,
for
in the
r values.
an acceptable
preferred
larger
(28)
documents
documents In
=
A contour
preference
measure
documents
are
sets with
that
user
and
r values
; U = S’
sphere.
sets is useful
of the
If a similarity
ranking,
larger
d/&~
intersection
of contour
1987).
=
q . d = r.
structure
documents
f‘1,
I r=
a contour
Nd(d)
the
sure
= b)
this
=
planes:
1} is a unit
by r E Rn is the
Furnas,
sets.
is:
= {b ~ Y \ (f(a)
mf(d)
q is the
sets of such
of all contour
=
and
clude ~(X)
is called
S“
fect
1“ the
of r under
intersection
defined
The
to Y: a +
space
of R“.
The
example,
I ~~
are indifferent,
f:x-Y,
vec-
vector
q, suppose
q. d = r, where
a partition 4.
geometric .Y, Y are two
set
the
of Rn.
vector
= {rn;l(r)
of U consists
For
discussion. Suppose
maps
may
{d the
-+
m~ is called
IVd for the dcjcument
of parallel
U n rn; l (r),
map
sets
introducing
pertinent
different
different
define
Definition linear
same
q. d,
Standard
two typical
into
the
mg : R“
of r under
(7),
the contour
of a family
space
m;l(r).
query
rn~l(R)
and
by studying
demonstrate
Thus,
of
value.
function,
U = Nd(Rn)
contour
values
in
similarity
preimage
functic,n
the
document
a ranking
function
the
on
documents
same
by equation
is the plane
plane.
set, In this
Then
q, a similarity
as a ranking
entire that
the
The
a subspace
consist
transformed
= r.
measure.
Pseudo-Cosine
Vector
d.
the
have
a normalized
measure
can be immediately
is therefore
onto
q
the such
normalization
Given
query
Based
set of r, written
as defined
R“
‘=‘d(d) “a’ (23)
contour
vectors.
Consider
Let rn~ (d)
(24)
Nq(q)= VU=,(%
3
class
tors q–ql
q=
which
equivalence
’23)
if we let
equation
classes
R.
morn(q,
document . . can partltlon
we
for a fixed
d) can be considered
of
equivalence
the
prezmage
mentioned,
m(q,
~~ (d)
the
con-
are insay that
r values
are
sets with
3.2
The
In
this
pseudo-cosine
model,
resented
by
q . d.
IVd(d)
that
1989;
the
Wong,
case of the
Let
document
=
the ranking
Note
Yao,
a normalized
d =
Accordingly,
model
d/ldl,
function
expected
and
pseudo-cosine
Idl
=
is defined
utility
Bollmann
vector
where
model
Yao,
d,.
by rnI(d)
=
difference
vectors
are on the
and
is a special
map
The
of
subspace
Solution
U
let
U is defined
U = Nd(ft”)
by:
Clearly,
U is a plane query
Since
ranking
the
is also
= {d
I ~di
U n m~l
(r)
= 1}.
can
be shown
preferred the
document
the
least
by
Therefore, line,
that
b
=
d – &
of d
the
I ~~=1
subspace
U, but
they
b, = O}.
and
in this
Q . d,
the the
there
subset
1(r)
Any
curve
map
document
vector
> on a document
I (d, d’~
q is a solution of linear
D)
vector
set D,
vectors: A(d>
d’)}.
(32)
if it is a solution
of the
inequalities:
solution
of U
is referred
most
is a generator
tor. or
b E BP(D).
of the
ranking. to
as the
any closed
solution
b>O,
vector
an acceptable
in practice,
of U, the
d–d’
q.
is no most
However,
relation
the set of difference
= {b=
1(r).
m;
contour
contour
model
in U.
set D is a finite
denote
3,
is m;
that
vector
preferred
Definition
to Definition
lines.
document
{b
a preference
A query
set of r E R“
is defined
Rn.
of parallel
are outside
plane,
BP (D)
system
According
function in
is a straight
is a family It
in Rn.
q, the contour
a plane
vector
vector
BP(D)
i=l
for a fixed
difference
model. Given
Contour
the
is rep~~=1
(Wong
1991)
b denote
above
The
For example,
of all
region.
convex
q(1), q(2) ,
vectors
inequalities
set
solutzon
(open)
(33) provides
solution It
vectors
is easy
combination
to
see
of k distinct
. . . q(~) is also a solution
let q = ~~=
~ q[’~.
Obviously,
vet.
for
any
b E BP(D),
(see
k
q.b=~q(z).b>O.
5).
i=l Operations
on
To analyze following
the
It
characteristics
of U, we introduce
the
operations.
Definition d(~)
U
5.
in Rn,
Given
the
k distinct
vectors
d[l~,
is
1)
model.
First,
l.d–l.
dtz~, . . ..
tour k
C=
{
i.=1
the
convex d(k).
convex
set
generated
Every
element
combination
by
the
d(z),
any
plane
the
= {d
{ is called convex d(k)
I (ai
> O) A(~a,
i=l the
Note
to be
. . . . d(~).
= 1)
2=1
interior
of C’, and
combination
of
the
every
d E C“
generators,
the
vector
the
pseudo-cosine
1 b
1
=
= 1. (d–~)
is, lisnot with
the
con-
we can
Moreover,
the following
+bll(a>O)A(b
=
asolution
U coincides
I 1. d = 1}.
1 to generate
Q={q”=aq that
for allb
Therefore,
The
set: d– – ~aid(i)
in
b E BP(D),
subspace
m~l(l)
vector
that role
That
c BP(D),
q* . b >0
q* in Q produces
&=
note
a special
set: ER)}.
generators,
d E C is said
of d(l),
for
Second,
use the
(30)
i=l
is called dad,., a closed
1
d=~aid(iJl(ai~O)A(~ai=l)
to
plays
~=l–l=O.
vector.
set: k
interesting
(1,1,...,
1
(31)
can
be used
one
can
non-negative
the same
to show
always
q“.b
= (aq+bl).b
@ q . b >0, ranking
that
construct
components
namely,
from
= aq.b. every
as q. This any
another
solution
solution
( Wong
and
vector
property vector
vector
Yao,
q, with
1991).
is an open 3.3
d[l J, d~z), . . ..
In
The the
standard
standard
vector vector
space
space
model
model,
the
Euclidean
norm: If k distinct the
subspace
convex U.
This
nation
vectors
d(l),
U = {d
combination means is closed
that
I ~$=1 of these
d(z),
. . . . d(k)
di =
1},any
generators
the operation
in the
subspace
of the
are taken closed also
from
belongs
convex
r-Z-(34)
(open) to is used
combi-
d=
U.
156
to
N~(d)
normalize = d/
the
II d Il.
document
vectors.
That
is,
Contour
map
The
of
subspace
U
In
U is defined
contrast,
in
the
by:
A query
vector
of the system
S“
and
in this
curve
model
latttudes
that
function
function and
is a circle; unit
the
on
contour
a minimum. the
Clearly,
the weighted
curve
vectors
ql,
is a family
U = S“.
Operations
on
must
mg (d)
must
therefore
least
the subspace
set
Since
In other
and
sphere
of
continuous
have
is a real
words,
have
there
preferred
solution
In this
model,
vector;
however,
ranking
a maxi-
U is not
the convex
exist
the
document
necessarily
reason,
the
combination
a closed
most
vectors
following
of vectors
operation
operations
in
Definition
6.
Let
numbers.
J(l),d(z),...,
The d[~)
U defined
R+
the
U.
from
For
is an operation
sum
@ from
presenting
utational
geometry 7. side
(R+
x U)k
otherwise.
to
of Rn,
d(’)11
as ado
difference,
b(–d’).
simply
referred
When
a =
to as the
as ad (B bd’.
written
adeb~,
b =
1, d@
spherical
sum
It
is important e
Solution
that
in the
the
d’
(de
d’)
is
(difference).
spherical
spherzcal B.(D)
difference = {b=
e
vectors ded’
is a spherical
tained
by
the
subspace
length
of d – d’ is zero,
model,
a >0.
for the generalnotions
Shames,
is the
portion
A polyhedral set of closed
in comp1985).
of Rn lying
set
in Rn
is the
half-spaces.
operations
convex
set.
We
polyhedral
set
will as a
or a polytope).
an arbitrary
huli
polyhedron
polyhedral
n-dimensional
Given
subset
of vectors
smallest,
convex
set
set of vectors
in Rn
is a
of L is the
L
L.
convex polytope;
convex
hull
of
of a finite conversely,
a finite
set
a convex
polytope
vectors
(McMullen
of
is the and
1971).
A convex
a set of
hull
convex
form
we define
set; a convex
(a n-polytope
8. the
The
is defined
set U U {O}.
to the pseudo-cosine
same
The
vector
Similar
Since
to note
are closed
some and
set is a convex
a bounded
Shephard,
@ and
the
the
theorem
define
of a (bounded)
n-polytope
cent aining
spherical
aq with
(Preparata
of a finite
Definition
~:=, ala(’)
weighted
provide
Similarity
A half-space
A polyhedral
to
convex
be written
we first
of a plane.
intersection
refer
can
~).
be a query
of k vectors
by:
k = 2, @j=la,d(z)
can
for
the existence
measure,
Definition
o
For
operation
q could
vectors
Linear
ized linear
is an instance
11~:=,
the
That
this
set of non-negative
spherzcal
vector
solution
vector.
Measure
are introduced.
denote
weighted
under
vector
Theorem
Generalized
in
on one
real
is closed
those
(36)
a solution
non-zero
no other
Existence
4
a maximum
U
model,
if it iIs a solution
sum of k distinct
is also
region
as q, except
zero.
continuous
Before
In this
vector
spherical
any
11 norm
b E BC(D).
q2, . . . . qk
is, the
the
is always
inequalities:
b>O,
unit
map
vector
q is a solution
a contour
set and any real
a compact
m~ (a)
on Sn,
of the Thus,
sphere.
a minimum.
preferred
1(r).
Sn is a compact
defined and
intersection
Sn n m;
on the
Note
mum
is the
a plane:
model,
difference
of linear q.
A contour
pseudo-cosine
lb I = Id – d’ I of any
the
polytope
boundary
vex
set
face
of smaller
denotes
dirnensional,
is defined of the
dimension
Each
than
a k-dimensional its
by a set of faces
polytope.
face.
its If
(n – 1 )-faces
are called
A supporting
plane
face
which
is a con-
polytope; a polytope
a liYis n-
facets.
as follows:
I (d, d’ E D) difference, U
the
=
Sn.
if and
only
A(d set In
Bc( D)
9.
set V is a plane
such
that
W’ of a polyhedral
W n V # 0 and
V lies entirely
on one side of W.
is con-
particular,
if d = d’,
Definition
(35)
> d’)}.
the Given
i.e.,
measure
I[d-d’ll=oed=d’.
a preference
relation
m(q,
objective
d),
our
on > for the existence
157
of a query
> on D
and
a similarity
is to seek the conditions vector
q which
provides
an acceptable
ranking.
conditions
the
for
d > d’ ~
That
is, we are looking
existence
rn(q,
of a query
d) > m(q,
for
q such
d’),
the
nj F3.
that:
Since
FjO .
d, d’ c D.
Let
and
(37)
that
the generalized
linear
m(q> d) = ~g(q) Thus,
finding
a solution
the following
system q.
measure
. IV~(d)
query
is defined
by:
properly
= Q . d.
q is equivalent
to
P+
P+.
That plane
The
solution
of the
to the
Theorem
inequalities
1.
There
Let
exists
ceptable
>
ranking,
D)}.
linear
measure
for
the
can be
Since
B(D)
of a
B(D),
q
existence
q which
only
if for
all
relation
on
D.
provides
an
ac-
K
>
0 vectors
c B(D),
5
The
In
adaptive
query
# O,
a~ > 0.
set
Proofi q satisfying
the
condiS*,
and
d>q.
is equivalent
every
b(i)
d’,
to the
c B(D).
d,d’
statement:
If ~~1
a,bi we
that
#
for
all
ai
>
O,
may
assume
O @ C,
>
0 for
denote
● B(D),
O, b(’)
{b(l),
b(2),
Since
{b(l),b(z),
C
. . . . b(~)},
Let
which defined
P+.
C
that
is, for
half-spaces, normal
vector
of
Therefore, d,d’
for
every
b
..,,
>0.
Hence,
~~1
b(l~,
b(z),...,
0.
sib(i)
is the
# O
b(K)
Without ~~=1
c
lose
at
convex
of
set
is a finite
of
the the
basic
d and
the
sphertcal
D.
In
on the
geometric
from
vector
this
a given
space
model
ideas. set of document relation
on D.
d’ CT)}
difference H(b)
vectors
denote
the
for
a given
open
half-
d’ in the unit distance in
radians,
+1
q II II
sphere
Sn.
between then
d
This and
O
Cos(q,
d’).
C’ is a poly -
ilFj
denote
lies.
plane
the set of facets
8Fj,
Let
the Fj
supporting
denote
containing
the C.
of C.
For
plane
Cos(q,
Thus,
C
d) >
Cos(q,
d’) e
Q’(q,
d)
d’)A(d,
Let
D
vectors,
b c B.(T); b is the pole of the half. d’) denote the angle between two vec-
. cr(d,
Let
on a sam-
set
obtained
the standard
D.
containing
a solution
preference
discussion
=dGd’l(d
~
methods,
user
vectors
is a finite
T
Vectors
document
a brief
+ a preference
tors
general-
generated
set,
learning
from
solution
the D
is measured B(D),
1. This
=
and Al is the cardinality
. . . . b(M)}
, .fN}
=
~ D,
Also, {tl,~z,...
is
say,
P-,
of Solution
We choose
set
sphere.
tope.
j=l,2,
bEP.
the set of spherical
sample
i=l
that
where
~
of the set.
sphere
Suppose
~~1 ity,
=
K
O=qO=q(~aib(i))=~a;q.b(’) i=l
(+)
q.b=O
BC(T)={b
q . b[i~
a,b(i)
K
is a contradiction.
GP+.
bE
we present
and
e D,
ai > 0, then:
This
O~b
u
a subset
Suppose
tion:
which
q be the
(inductive)
to illustrate is a vector
d+d’~q.
open
Let
Structure
structure
there
b>
g Fjo
T,
sample
Suppose
of the
half-
13FJ0,Fjo
to
that
is determined
section,
(39)
i=l
(+)
Rn into two open
P is parallel
q.b0,
5
ple sib(i)
O @
origin,
Hence,
(38)
a preference
vector
if and
btKl
the
P +.
is, FJO C P such
that
through
(38). be
a query
bib,...,
for
conditions
Since in one
q.
[ (d>d’)A(d,d’6 theorem
in terms
divides
P–.
a F’O such
passing
to solving
b E B(D),
={b=d–~
existence
stated
t3Fjo; P
and
where B(D)
plane
of inequalities:
b>O,
exists
the
contained
the
there
denote
parallel
spaces Recall
O @ C,
P
if q provides
an acceptable
ranking,
we obtain:
For
Given
d > d’,
a(q,
d’)
let
holds.
0, then
for
b =
Let
any
de
P(T)
given
d’.
If q E II(b),
a(q,
= rlb~~c(~)~(b).
d)
If P(T)
q.
d)
Vb E Be(T)
should
finding to
< cx(q, d’)
That
is, every
vector
we call P(T) sample
P(T)
vector.
the solution
every
implies there
6
Hence,
region
the same
solution
To produces sample
set;
solutions.
set.
region
the
the
true
It can be easily
set.
cussions
vec-
itly
P(D)
document
solution
cases
region
subset
of
required
for
for
properly tion mation
Yao,
chosen
about
user
help
P(T)
n
~ D,
under
o P(T’),
the
region
the methods
Wong,
following
(polygon) given
Algorithm
preference
and
which
region
valuable
structure. a lmore
Ya,o
of a
in Preparata
and
measures structure
Such
The
infor-
the
method
(Wong
to construct
the
ometry
are useful
practical
this
similar-
and
suggested
an
the investigations
structures measures.
techniques
aspects
is that these
We also analyzed
vectors
complements
of similarity that
an
it.
study
suggest
or the
produce
point
adopting
the
~asthe inner-
would
retrieval.
solution
lead
identified,
covari ante,
important for
that
is that
been
such
Dice,
stan-
structure.
paper
have
measure,
basis
algebraic/geometric
ations
and
the
may
preference of this
correlation
for finding
particular,
representations
in information
present
speciid
indicates
pseudo-cosine,
of the
In
analysis
function
The
are
This
user
dis-
We have explic-
measure.
conditions
the
and
Our
and
ranking
ranking.
algorithm
1991). algorithm
a linear
structures models.
pseudo-cosine
contributions
establishes
the
informa-
effective
feedback
of a sample
and
Our
developed
for
the
cm
empirical
evalu-
preliminary
results
for computational
study
of information
of the
theoretical
retrieval
systems.
geand
set T by adopting Shames
References
(1985).
Bollmann,
k=o; B.(T)
and
of computationid
measures
linear
different
P. and
Wong,
ear information 2. While
algorithm
Wang
models.
in document
cosine,
ity
point
retrieval
the
and sufficient
acceptable
FindPolygon
l.v=u;
by
properties,
models.
product-moment
n P(T’),
solution
contains
sets for relevance
Wan
We suggest
set
the
space
One of the main
P(T).
that
us to design
training
1990;
solution
sample
the
may
to select
indicate
in
an efficient
similarity
in detail
change
necessary
all T, T’
the
many
to a dramatically
T~D
properties
if one
vantage
generalized
vector
a minor
To a mzntmul
information
of the
dard
the
on the linear
that
we examined
set To defines
no proper
= P(T)
=
that
was suggested
analyzed
centered
shown
result (iii)
is interested
versa.
vector,
of vector-based
product,
T’)
vice
all ve r-
1991).
from
we
limitations
region.
that
P(Tu
et al.,
paper,
infinite
we call
T C T’ =+
(ii)
out
task
we know of these
Conclusion
In this
of the
Euclidean
entire
the
and
all the solution
verified
(i)
These
this
Wong
a solution
If a sample
solutions,
To contains
determining
is an
then
as D,
same
In
exists If the
set,
solution
set.
set
there
as a sample true
open
open
whenever
are infinite
is the
an
nonempty
that
set D is used P(D)
is
and
one solution
perform
geometry
definition,
topology,
tor,
a solution
= fI&Bc(@(b)
vector,
polygon,
combination
d’.
set T.
By
This
P(T)
q in
solution
convex
be pointed
only
(1990,
the
open
is a solution
It q G P(T)
found
Any
# @ Do
S.K. M.
retrieval
ACM
SIGIR
ment
tn Information
(1987).
models.
Conference
Adaptive
Proceedings
on Research
Retrteval,
linof the
and Develop-
157-163.
Begin b(k)
=b
V =
Vnlf(b(
B.(T)
Jones,
E B.(T);
= B.(T)
evance:
k)); – b(k);
k=k+l
159
Furnas,
a geometric of the
Science,
38,
G.W.
(1987).
analysis
Amertcan
Pictures
of similarity Soczety
for
of rel-
measure. Information
420-446.
M. J., Koll,
evaluation
v.
and
Journal
McGill,
End; 3. output
W.P.
M.,
of factors
and
Noreault,
affecting
T.
(1979).
document
ranking
An by
information
reirieval
tion
Studies,
York
13210.
McMullen,
Syracuse
University,
Shephard,
G.C.
the
Bound
and
bridge:
School
Wong,
of Informa-
Syracuse,
S.K.M.
and
distribution
New
Upper
Cambridge
(1971).
University
Wong,
Convex
Conjecture,
and
in
for
Yao.
linear
American
Press.
Y.Y.
Processing
S.K.M.
lation
Cam-
Yao.
model
formatton
P. and
Polyiope
systems.
(1989). information
and Y.Y.
retrieval
Socaety
for
A
probability
retrieval.
ln-
25,
39-53.
Management, (1990).
Query
models.
Journal
Information
Sciencej
formuof the 41, 334-
341. Preparata,
F.P.
tational
and
Sharnos,
M.I.
Geometry-an
(1985).
conlp’u-
Introduclaon,
New
Wong,
York:
S.K.M.
inference
Springer-Verlag.
tion Raghavan,
V.V.
analysis
and
Wong,
of vector
S.K.M.
space
(1986).
model
trieval.
Journal
of the American
mation
Science,
37,
Roberts,
F.S.
York: Salton, —
G.
Salton,
in
Automatic NJ:
McGill,
Modern
for
Infor-
New
Tezt
M.J.
Retrteval
System
Document
Publishing
Process-
Processing,
Read-
Company. Introduction
(1983).
Information
Retrieval,
New
York:
McGraw-Hill. Schneider, E.,
H.-J.,
Bollmann,
Reiner,
U.,
iungsbewertung @IVE).
and
P., Jochum, Weissmann,
Projelctbericht,
F.,
V.
von mformatzon
Konrad,
(1986).
retrteval
Technische
Leis-
verfahren Universitat,
Berlin. van
Rijsbergen,
C.J.
London: Wan,
S. J., and
Wong,
To appear and
S. K. M.,
Information theory. 19, Wong,
(1991).
S.K.M
learning
Analysis Wong,
Retrieval,
Butterworth.
pervised tem.
Information
(1979).
algorithm in IEEE
Machtne
Bollmann, retrieval
International
A
for linear
partially
su-
separable
sys-
Transactions
on Pattern
Intelligence. P., and based
Yao,
Y.Y.
(1991).
on axiomatic
Journal
of General
decision Systems,
101-117. S. K. M.,
Linear ings
Yao,
structure of the llth
Conference mation
Y. Y.,
Annual
on Research
Retrieval,
and
Bollmann,
in information
P. (1988).
retrieval.
Interl~attonal
model
Systems,
ACM
and Development
S. K. M.,
(1991).
Prentice-Hall.
Automatic
Wesley
G. and
to
Wong,
re-
Theory,
SMART
The
Cliffs,
(1989).
MA:
Soczety
Measurement
Ed.).
Englewood
ing,
information
Press.
Experiments
ing, Salton,
(1971,
in
Yao.
Y.Y.
(1991).
for information 16,
A probabilistic retrieval.
Informa-
301-321.
A critical
279-287.
(1976).
Academic
G.
and
ProceedSIGIR zn Infor-
219-232.
160
Yao.
Y. Y.,
Evaluation
Journal
of the
Science,
42,
pp.
Salton,
G. and
of an adaptive
American 723-730.
Society
Buckley, linear
for
C.
model.
Information
Analysis
Based
on
Z.W.
Wang,
of Vector
Space
Computational
S.K.M.
Department
Geometry
Wong
and
of Computer
University Regina,
Models
Y.Y.
Yao
Science
of Regina Saskatchewan
Canada S4S 0A2
Abstract
Earlier the
This paper analyzes the properties, structures and limitations of vector-based models for information retrieval from the computational ometry point of view. the pseudo-cosine
ge-
It is shown that both
and the standard
vector
of a generalized linear model.
More impor-
tantly, both the necessary and sufficient con-
where
the
used
model.
alyzed
67 similarity
all
the
The
On
sures
solution
the
Introduction
Their
and
queries this
effective systems
hand,
to
into
either
al-
studies et al.,
Furnas
(1987)
of seven
similarity
mea-
algebraic
and
those
investigation
are Similar
and
an-
them
(Schneider
Jones
analysis
extensions (1979)
class
similar.
LIVE-project
other
complementary
revealed
among
many
similarity
gebraic
and
geometric
analyses
of
vector-based
these
information
successfully
fact,
the
measure
and
divided
each
point
measures.
major
empirical differences
Clearly,
both
the
al-
is suggested.
Vector-based used
in
of
con-
algebraic
Noreault
and
and
of the
similarity
and
or very
a geometric
understanding
1
Koll
members
made
an
variations
measures
equivalent
also
1986).
several
McGill,
classes.
from
of dif-
Raghavan
analysis
inner-product
to examine
of the
model
emphasized
evaluations
For example, a critical
of
view,
models
empirical
measures. presented space
studies.
for finding
and
vector
provided
tion region for acceptable ranking is analyzed
vector-based
ventional
were
rank the documents. The structure of the soluand an algorithm
(1986)
gebraically
pseudo-cosine, Dice, covariance and productmoment correlation measures can be used to
vectors
similarity
Wong
24
ditions have been identified, under which ranking functions such as the inner-product, cosine,
on
structure
ferent
was
space models can be viewed as special cases
studies
algebraic
for
(Salton, category
ranking (Salton,
the
retrieval
1971; Salton of retrieval strategy
models
representation
than
and models the
have
been
of documents McGill, provide standard
1983).
studies
In
suggested
the
use of different
not
identified
ticular
the
invaluable
for
models.
some
similarity
the
similarity
only
are
useful
guidelines
measures,
precise
conditions
measure
should
the
However,
but
under
they
on have
which
a par-
similarity
mea-
be used.
a more To justify
Boolean
sure,
1989).
Wong, Permission to copy without fee all or part of this matarial is granted provided that the copies are not made or distributed for direct commercial advantage, tha ACM copyright notice and tha title of the publication and its date appear, and notica is given that copying is by permission of the Association for Computing Machinary. To copy otherwise, or to republish, requires a fee and/or specific permission. 15th Ann Int’1 SIGIR ‘92/Denmark-6/92 01992 ACM 0-89791-524-0/92/0006/01 52...$1.50
tion
the choice
Wong,
Yao
1987;
of user
scription
Wong
identified. have
152
Bollmann
and
preference,
Yao,
(1988; 1990)
which
gives
of the user judgments
than
of relevance. referred
of an appropriate
and
In this
to as the It
a strong
approach,
perfect
was shown influence
and
two
introduced a more
that
these design
and the
accurate
the standard ranking
acceptable
on the
Bollmann
node-
notion
strategies, rankings,
ranking
are
strategies
of information
re-
trieval
systems.
basis
for
the
identified, ficient for
Their
in some
for
perfect
some
and
well
analyze
ranking
the
by adopting
This briefly the
paper
is organized the
acceptable
generalized two
linear
linear
models,
vector
space
models
strated
that
a minor
tions
may
lead
structures. sufficient tor
are analyzed
Section
conditions
for
the
for
lyze the structure vector
space
the
the
in detail.
of the solution
system
In
vector-based
sented ponent
other
term
each
a set
that
there
user
information
preference
of
exists
vectors
the
Rn.
Rnx
3,
query
write
m(q,
function
Rn-+R,
vector
needs
structure
the
According
subscript
defined d,d’
the
q emphasizes
>g is defined
q. However,
we will
c,f
1991).
that
such
a
as a real funca document
-+ m(q,
and
function
vec-
lowing
is said
d)
d).
becomes
emphasize
which
document
(2)
this
is referred
a func-
point,
we
to as a ranking
vectors. preference,
(Wong
to provide
condition
m(q,
To
to the user
two
and Yao,
ranking
1990).
a perfect
strate-
A ranking
ranking
if the
fol-
holds:
d>
d’ ~m~(d)
function
if a weaker
The
is said
> m~(d’). to provide
condition
(3)
an acceptable
rank-
holds:
it
ranking
In
other
the
antees
that
ahead
of the
time.
by a binary
fore,
by the The
the
ing
user
preferred.
in the
c D,
(qlqz
present
user
prefers
with drop
d’.
the fact respect
document
that
the subscript
are not
guarranked
argued
is more
with
On the
only
(1988)
retrieval
concerned
of query
and
and
d = (~ld~”
measure
that
appropriate
systems.
There-
acceptable
document “
is defined
ranlk-
“1~7~)”
vectors, suppose
q = a
lin-
as:
cl (1)
2=1 With
the preference
to a particular
ranking
and
study.
a pair “%)
et el.
of information
preferred
rank.
strateg,y
documents
Wong
we are primarily
the
less preferred,
to the same ranking
of acceptable
design
Consider
relation
belong
acceptable
that
of the
the less preferred
the paradigm for
ensures
ahead
documents
hand,
a query
is assumed
strategy
are ranked
indifferent
of imd.
perfect
documents
as a vector
governed
at a particular
for
degree
document
D,
to document
relation
d.
d) as mg(d)
for
ear similarity
The
system
measure.
q and
(qjd)
q is fixed,
variable
gies can be identified
q in Rn.
vectors
a preference
~
map
Yao,
assume
vector
accor-
d):
of one
each com-
Similarly,
as a vector
can be formally
d>qd’
in is to
and
m can be viewed
(a query
information
the relation
Bollmann we may
tion
is repre-
where
is represented space
document
>~ on D as follows:
onto
by a similarity
of two
(Roberts,
task
can be generated
we ana-
a document
indicating
vector
can also be represented Given
The
loss of generality,
measure
preference
>)
documents
mapping
If the
preference
necessary
ti in describing
document
in a n-dimensional
>)
the
(D, of an
the
preference. (D,
> d’)
for the standard
dz, . . . . d~),
number
of index
words,
models,
d = (dI,
di is a real
portance
and
objectives
(Wong,
=(d
Models
retrieval
by a vector
D system
rank
i.e.
d’ are indzflerent.
Without
A ranking
Retrieval
user
(R, >)
ing
Linear
to
real
numbers
vectors
main
is
the
of
model.
2
of the
system with
preference, d and
a relational
the relational
representa-
Finally,
region
One
form
m:
standard
of a solution
measure.
of document
A similarity
It is demon-
user the
existence
1976).
of strict
we say that
and
notion
the
in document
linear
>
tion
In Section
different
set
2, we
as linearity
and
4, we present
generalized
models
In Section
pseudo-cosine
The
vector
is introduced.
to dramatically
In
to is to
linear
Then,
difference
and
for computational
such
strategy.
the
inner-
respect
> d),
relation
dance
1985).
as follows.
measures
as the
In the absence
V(d’
retrieval
of
objective
of these
concepts
ranking
the
existence
with
Another
Shames,
basic
measure
covariance,
developed
and
review
such
Dice,
properties
(Preparata
the
measures,
the techniques
geometry
for
functions
strategy.
geometric
and
suf-
is to formulate
conditions
correlation
the acceptable
paper
pseudo-cosine,
product-moment
and
of a linear
arises.
they
ranking.
of this
similarity
cosine,
necessary
existence
sufficient
a theoretical
In particular,
the
or acceptable
known
product,
cases,
the
One of the objectives necessary
established
measures.
special
conditions
either
work
use of linear
respect
functions
query
linear.
q if no confusion
153
such
to
the
above
definition,
cosine
and
as the
We believe
that
this
Dice
definition
many
similarity
measures is too
are not
restrictive.
We
now
introduce
measures, tions
a more
which
of linear
will
general
enable
definition
of linear
us to broaden
the
where
applica-
a is real
thus
models.
no
mapped longer
1.
Rn x Rn.
Let
If there
rn(q,
exist
d)
be a similarity
two
measure
on
Nq:Rn+Rn,
(6)
q+q=~g(q)
such
as an
[ad]
and
d = Nd(d)
(7)
that
then
d) = Ng(q)
we say m(q, can
vector,
be
the
new
measure.
Nd
and
shown
in d
the
they
S“.
The of
function the
We
same
are may
now view
is a representative
of
space.
measure
pseudo-cosine
similarity
measure
can
be
ex-
as:
In this
case, the
normalization
functions
are:
query
(14)
of the
many
are
similarity
indeed
measures
linear
under
where
this
Iql
becomes
of linearity.
In
measure measure
a document
of similarity
d is defined
between
and
Idl
~~=1
~idi
this
the
are
all
vector
11 norms.
d
the
=
the entire
vectors
d/Id
vector
a plane
by:
Rn
{d
I XV=l
pointing
Thus,
equation
(13)
= ~ . d.
model,
unit
measure, q and
is
(8)
as a normalization
that
1987)
Furnas,
inner-product
query
class;
quotient
having
because
R*
is, one can
(15)
inner-product
The
Rn
vector
space
That
vectors.
definition
The
= q “ ~>
as a normalization
function
It can be easily (Jones
is a hnear
regarded
and
document
d)
Nd(d)
equivalence
vector S“.
in
lengths,
same
pseudo-cosine
The
entire
sphere
vectors
the
Sn is the
pressed rn(q,
N~
d ~
unit
different by
[ad]
The
and Nd:Rn+Rn,
but
represented
functions:
The
the
distinguish
direction Definition
number.
onto
di
in the
=
1.
same
ad
are
mapped
1. In
contrast
to
space
Rn is mapped
That
is,
any
direction
but
onto
the
two
cosine
onto
the
vectors
in
having
different
n
inner(q,
d) = ~
qidi
= q ~d
lengths
(9)
?=1 Obviously, tity
the
normalizations
functions,
hence
the
The
given
=
q,
d =
is a linear
den-
case are the Nd(d)
=
d
and
in
the
The
measure
information
which
was
retrieval
first
system
used
(Salton,
1971)
(van
is defined
a plane.
Rijsbergen,
1979;
Salton
and
vector
q =
i.e.,
Iql =
by:
normalization
functions
\\ d II are 12 or
above
be written
normalizations]
as ~~=1
with
the following
q
=
N~(q)
d
=
N~(d)
query
the 11 norm,
normalizations: (17)
= 2q,
(11) equation
(12)
the
1. By
qi =
the
are:
Nd(d)”&=A’
II q I\ and
case in which
(lo)
_—
‘=
a special
. . . , ~~) is normalized
(W,92)
is
‘=Nq(q)=&-ll:ll’ can
measure
1983)
Consider
~~=1
Cos(q]’) =mm
applying
on such
measure.
x:=,qid
where
vector
measure
Dice
McGill,
by:
The
Dice
as the same
measure
cosine
SMART
Ng(q)
inner-product
cosine
The
q =
The in this
are regarded
Euclidean the
qi & = q
norms.
cosine
d, which
By
(16)
can be written
fore,
in this
special
The
measures
case, the
of
The
is therefore
related
linear.
~~~~ = Q.d.
measure
There-
is linear.
and
correlation
covariance (Jones
Dice
covariance
product-moment
measure
as ~~=1
and
and
correlation
Furnas,
measures
1987).
are closely
Let
n
With ad
are
this
representation
mapped
onto
the
of documents, unit
vector
T=:g%,
all the vectors d
=
d/
II d
II,
‘2=1
154
d-$di.
(19) t=l
The
formula
for
the
co~(q,
covariance
d)
measure
is:
where
5(w–T)(4 – Z).
=
2X is the
power
~-’(b)
(20)
set of X,
and
= {a I (a E X)
A(?(a)
= b)};
of a under
f.
(29)
i=l Let
1 = (1, 1, . . . . 1).
are defined
the
~~=1
@i&=
The
the
normalization
f-1 (~)is called
functions
by:
Thus,
fined
Here,
As already
q=
Ng(q)
= q–ql,
(21)
d=
N~(d)
=d–~1.
(22)
covariance
measure
d ~~, and
measure
of
(20)
hence
can
be expressed
it is a linear
product-moment
measure
mg,
as
measure.
correlation
is de-
by: D’=l(%
d) = /x=,(9i
Similarly,
- m(~t -Q)-27”
- ~)
z,=,(~i
-~)
Definition
3.
Any
- 1)2 ‘
The
a linear
’25)
Space
into
section,
which
Models
that
slightly
lead to dramatically 3.1
Contour
Before haps
useful
first to our
Definition Consider
models,
normalizations
user preference
we
2.
notion
to review
of contour
structures.
the
concepts
sets, about
a function
f
from
X
it is perfunctions
X The
is called range
the off
domain
of ~ and
nonempty
let
(26)
~(a). codomatn
The
tnverse
mappmg, ~-1
: f(X)
A (CIC .Y)},
plane
notion
of .f is defined
+2X,
b–
f-l(b),
similarity normal
of
a measure
nz~
R},
of U and curve.
a contour
The
contour
curves.
then
contour
of j.
generates
(27)
different.
b
not
155
preferred
smaller
this
in the
r values.
curve
of the unit
sphere
contour
to the
the
and
generates
a per-
contour
contour
in
contour
one
can
set only
smaller
in the
contour
the mea-
one can still
same
set
sets with
If a similarity
sets with
documents
the
(Jones
documents
ranking,
however,
analyzing
same
in the to
in the
case,
for
in the
r values.
an acceptable
preferred
larger
(28)
documents
documents In
=
A contour
preference
measure
documents
are
sets with
that
user
and
r values
; U = S’
sphere.
sets is useful
of the
If a similarity
ranking,
larger
d/&~
intersection
of contour
1987).
=
q . d = r.
structure
documents
f‘1,
I r=
a contour
Nd(d)
the
sure
= b)
this
=
planes:
1} is a unit
by r E Rn is the
Furnas,
sets.
is:
= {b ~ Y \ (f(a)
mf(d)
q is the
sets of such
of all contour
=
and
clude ~(X)
is called
S“
fect
1“ the
of r under
intersection
defined
The
to Y: a +
space
of R“.
The
example,
I ~~
are indifferent,
f:x-Y,
vec-
vector
q, suppose
q. d = r, where
a partition 4.
geometric .Y, Y are two
set
the
of Rn.
vector
= {rn;l(r)
of U consists
For
discussion. Suppose
maps
may
{d the
-+
m~ is called
IVd for the dcjcument
of parallel
U n rn; l (r),
map
sets
introducing
pertinent
different
different
define
Definition linear
same
q. d,
Standard
two typical
into
the
mg : R“
of r under
(7),
the contour
of a family
space
m;l(r).
query
rn~l(R)
and
by studying
demonstrate
Thus,
of
value.
function,
U = Nd(Rn)
contour
values
in
similarity
preimage
functic,n
the
document
a ranking
function
the
on
documents
same
by equation
is the plane
plane.
set, In this
Then
q, a similarity
as a ranking
entire that
the
The
a subspace
consist
transformed
= r.
measure.
Pseudo-Cosine
Vector
d.
the
have
a normalized
measure
can be immediately
is therefore
onto
q
the such
normalization
Given
query
Based
set of r, written
as defined
R“
‘=‘d(d) “a’ (23)
contour
vectors.
Consider
Let rn~ (d)
(24)
Nq(q)= VU=,(%
3
class
tors q–ql
q=
which
equivalence
’23)
if we let
equation
classes
R.
morn(q,
document . . can partltlon
we
for a fixed
d) can be considered
of
equivalence
the
prezmage
mentioned,
m(q,
~~ (d)
the
con-
are insay that
r values
are
sets with
3.2
The
In
this
pseudo-cosine
model,
resented
by
q . d.
IVd(d)
that
1989;
the
Wong,
case of the
Let
document
=
the ranking
Note
Yao,
a normalized
d =
Accordingly,
model
d/ldl,
function
expected
and
pseudo-cosine
Idl
=
is defined
utility
Bollmann
vector
where
model
Yao,
d,.
by rnI(d)
=
difference
vectors
are on the
and
is a special
map
The
of
subspace
Solution
U
let
U is defined
U = Nd(ft”)
by:
Clearly,
U is a plane query
Since
ranking
the
is also
= {d
I ~di
U n m~l
(r)
= 1}.
can
be shown
preferred the
document
the
least
by
Therefore, line,
that
b
=
d – &
of d
the
I ~~=1
subspace
U, but
they
b, = O}.
and
in this
Q . d,
the the
there
subset
1(r)
Any
curve
map
document
vector
> on a document
I (d, d’~
q is a solution of linear
D)
vector
set D,
vectors: A(d>
d’)}.
(32)
if it is a solution
of the
inequalities:
solution
of U
is referred
most
is a generator
tor. or
b E BP(D).
of the
ranking. to
as the
any closed
solution
b>O,
vector
an acceptable
in practice,
of U, the
d–d’
q.
is no most
However,
relation
the set of difference
= {b=
1(r).
m;
contour
contour
model
in U.
set D is a finite
denote
3,
is m;
that
vector
preferred
Definition
to Definition
lines.
document
{b
a preference
A query
set of r E R“
is defined
Rn.
of parallel
are outside
plane,
BP (D)
system
According
function in
is a straight
is a family It
in Rn.
q, the contour
a plane
vector
vector
BP(D)
i=l
for a fixed
difference
model. Given
Contour
the
is rep~~=1
(Wong
1991)
b denote
above
The
For example,
of all
region.
convex
q(1), q(2) ,
vectors
inequalities
set
solutzon
(open)
(33) provides
solution It
vectors
is easy
combination
to
see
of k distinct
. . . q(~) is also a solution
let q = ~~=
~ q[’~.
Obviously,
vet.
for
any
b E BP(D),
(see
k
q.b=~q(z).b>O.
5).
i=l Operations
on
To analyze following
the
It
characteristics
of U, we introduce
the
operations.
Definition d(~)
U
5.
in Rn,
Given
the
k distinct
vectors
d[l~,
is
1)
model.
First,
l.d–l.
dtz~, . . ..
tour k
C=
{
i.=1
the
convex d(k).
convex
set
generated
Every
element
combination
by
the
d(z),
any
plane
the
= {d
{ is called convex d(k)
I (ai
> O) A(~a,
i=l the
Note
to be
. . . . d(~).
= 1)
2=1
interior
of C’, and
combination
of
the
every
d E C“
generators,
the
vector
the
pseudo-cosine
1 b
1
=
= 1. (d–~)
is, lisnot with
the
con-
we can
Moreover,
the following
+bll(a>O)A(b
=
asolution
U coincides
I 1. d = 1}.
1 to generate
Q={q”=aq that
for allb
Therefore,
The
set: d– – ~aid(i)
in
b E BP(D),
subspace
m~l(l)
vector
that role
That
c BP(D),
q* . b >0
q* in Q produces
&=
note
a special
set: ER)}.
generators,
d E C is said
of d(l),
for
Second,
use the
(30)
i=l
is called dad,., a closed
1
d=~aid(iJl(ai~O)A(~ai=l)
to
plays
~=l–l=O.
vector.
set: k
interesting
(1,1,...,
1
(31)
can
be used
one
can
non-negative
the same
to show
always
q“.b
= (aq+bl).b
@ q . b >0, ranking
that
construct
components
namely,
from
= aq.b. every
as q. This any
another
solution
solution
( Wong
and
vector
property vector
vector
Yao,
q, with
1991).
is an open 3.3
d[l J, d~z), . . ..
In
The the
standard
standard
vector vector
space
space
model
model,
the
Euclidean
norm: If k distinct the
subspace
convex U.
This
nation
vectors
d(l),
U = {d
combination means is closed
that
I ~$=1 of these
d(z),
. . . . d(k)
di =
1},any
generators
the operation
in the
subspace
of the
are taken closed also
from
belongs
convex
r-Z-(34)
(open) to is used
combi-
d=
U.
156
to
N~(d)
normalize = d/
the
II d Il.
document
vectors.
That
is,
Contour
map
The
of
subspace
U
In
U is defined
contrast,
in
the
by:
A query
vector
of the system
S“
and
in this
curve
model
latttudes
that
function
function and
is a circle; unit
the
on
contour
a minimum. the
Clearly,
the weighted
curve
vectors
ql,
is a family
U = S“.
Operations
on
must
mg (d)
must
therefore
least
the subspace
set
Since
In other
and
sphere
of
continuous
have
is a real
words,
have
there
preferred
solution
In this
model,
vector;
however,
ranking
a maxi-
U is not
the convex
exist
the
document
necessarily
reason,
the
combination
a closed
most
vectors
following
of vectors
operation
operations
in
Definition
6.
Let
numbers.
J(l),d(z),...,
The d[~)
U defined
R+
the
U.
from
For
is an operation
sum
@ from
presenting
utational
geometry 7. side
(R+
x U)k
otherwise.
to
of Rn,
d(’)11
as ado
difference,
b(–d’).
simply
referred
When
a =
to as the
as ad (B bd’.
written
adeb~,
b =
1, d@
spherical
sum
It
is important e
Solution
that
in the
the
d’
(de
d’)
is
(difference).
spherical
spherzcal B.(D)
difference = {b=
e
vectors ded’
is a spherical
tained
by
the
subspace
length
of d – d’ is zero,
model,
a >0.
for the generalnotions
Shames,
is the
portion
A polyhedral set of closed
in comp1985).
of Rn lying
set
in Rn
is the
half-spaces.
operations
convex
set.
We
polyhedral
set
will as a
or a polytope).
an arbitrary
huli
polyhedron
polyhedral
n-dimensional
Given
subset
of vectors
smallest,
convex
set
set of vectors
in Rn
is a
of L is the
L
L.
convex polytope;
convex
hull
of
of a finite conversely,
a finite
set
a convex
polytope
vectors
(McMullen
of
is the and
1971).
A convex
a set of
hull
convex
form
we define
set; a convex
(a n-polytope
8. the
The
is defined
set U U {O}.
to the pseudo-cosine
same
The
vector
Similar
Since
to note
are closed
some and
set is a convex
a bounded
Shephard,
@ and
the
the
theorem
define
of a (bounded)
n-polytope
cent aining
spherical
aq with
(Preparata
of a finite
Definition
~:=, ala(’)
weighted
provide
Similarity
A half-space
A polyhedral
to
convex
be written
we first
of a plane.
intersection
refer
can
~).
be a query
of k vectors
by:
k = 2, @j=la,d(z)
can
for
the existence
measure,
Definition
o
For
operation
q could
vectors
Linear
ized linear
is an instance
11~:=,
the
That
this
set of non-negative
spherzcal
vector
solution
vector.
Measure
are introduced.
denote
weighted
under
vector
Theorem
Generalized
in
on one
real
is closed
those
(36)
a solution
non-zero
no other
Existence
4
a maximum
U
model,
if it iIs a solution
sum of k distinct
is also
region
as q, except
zero.
continuous
Before
In this
vector
spherical
any
11 norm
b E BC(D).
q2, . . . . qk
is, the
the
is always
inequalities:
b>O,
unit
map
vector
q is a solution
a contour
set and any real
a compact
m~ (a)
on Sn,
of the Thus,
sphere.
a minimum.
preferred
1(r).
Sn is a compact
defined and
intersection
Sn n m;
on the
Note
mum
is the
a plane:
model,
difference
of linear q.
A contour
pseudo-cosine
lb I = Id – d’ I of any
the
polytope
boundary
vex
set
face
of smaller
denotes
dirnensional,
is defined of the
dimension
Each
than
a k-dimensional its
by a set of faces
polytope.
face.
its If
(n – 1 )-faces
are called
A supporting
plane
face
which
is a con-
polytope; a polytope
a liYis n-
facets.
as follows:
I (d, d’ E D) difference, U
the
=
Sn.
if and
only
A(d set In
Bc( D)
9.
set V is a plane
such
that
W’ of a polyhedral
W n V # 0 and
V lies entirely
on one side of W.
is con-
particular,
if d = d’,
Definition
(35)
> d’)}.
the Given
i.e.,
measure
I[d-d’ll=oed=d’.
a preference
relation
m(q,
objective
d),
our
on > for the existence
157
of a query
> on D
and
a similarity
is to seek the conditions vector
q which
provides
an acceptable
ranking.
conditions
the
for
d > d’ ~
That
is, we are looking
existence
rn(q,
of a query
d) > m(q,
for
q such
d’),
the
nj F3.
that:
Since
FjO .
d, d’ c D.
Let
and
(37)
that
the generalized
linear
m(q> d) = ~g(q) Thus,
finding
a solution
the following
system q.
measure
. IV~(d)
query
is defined
by:
properly
= Q . d.
q is equivalent
to
P+
P+.
That plane
The
solution
of the
to the
Theorem
inequalities
1.
There
Let
exists
ceptable
>
ranking,
D)}.
linear
measure
for
the
can be
Since
B(D)
of a
B(D),
q
existence
q which
only
if for
all
relation
on
D.
provides
an
ac-
K
>
0 vectors
c B(D),
5
The
In
adaptive
query
# O,
a~ > 0.
set
Proofi q satisfying
the
condiS*,
and
d>q.
is equivalent
every
b(i)
d’,
to the
c B(D).
d,d’
statement:
If ~~1
a,bi we
that
#
for
all
ai
>
O,
may
assume
O @ C,
>
0 for
denote
● B(D),
O, b(’)
{b(l),
b(2),
Since
{b(l),b(z),
C
. . . . b(~)},
Let
which defined
P+.
C
that
is, for
half-spaces, normal
vector
of
Therefore, d,d’
for
every
b
..,,
>0.
Hence,
~~1
b(l~,
b(z),...,
0.
sib(i)
is the
# O
b(K)
Without ~~=1
c
lose
at
convex
of
set
is a finite
of
the the
basic
d and
the
sphertcal
D.
In
on the
geometric
from
vector
this
a given
space
model
ideas. set of document relation
on D.
d’ CT)}
difference H(b)
vectors
denote
the
for
a given
open
half-
d’ in the unit distance in
radians,
+1
q II II
sphere
Sn.
between then
d
This and
O
Cos(q,
d’).
C’ is a poly -
ilFj
denote
lies.
plane
the set of facets
8Fj,
Let
the Fj
supporting
denote
containing
the C.
of C.
For
plane
Cos(q,
Thus,
C
d) >
Cos(q,
d’) e
Q’(q,
d)
d’)A(d,
Let
D
vectors,
b c B.(T); b is the pole of the half. d’) denote the angle between two vec-
. cr(d,
Let
on a sam-
set
obtained
the standard
D.
containing
a solution
preference
discussion
=dGd’l(d
~
methods,
user
vectors
is a finite
T
Vectors
document
a brief
+ a preference
tors
general-
generated
set,
learning
from
solution
the D
is measured B(D),
1. This
=
and Al is the cardinality
. . . . b(M)}
, .fN}
=
~ D,
Also, {tl,~z,...
is
say,
P-,
of Solution
We choose
set
sphere.
tope.
j=l,2,
bEP.
the set of spherical
sample
i=l
that
where
~
of the set.
sphere
Suppose
~~1 ity,
=
K
O=qO=q(~aib(i))=~a;q.b(’) i=l
(+)
q.b=O
BC(T)={b
q . b[i~
a,b(i)
K
is a contradiction.
GP+.
bE
we present
and
e D,
ai > 0, then:
This
O~b
u
a subset
Suppose
tion:
which
q be the
(inductive)
to illustrate is a vector
d+d’~q.
open
Let
Structure
structure
there
b>
g Fjo
T,
sample
Suppose
of the
half-
13FJ0,Fjo
to
that
is determined
section,
(39)
i=l
(+)
Rn into two open
P is parallel
q.b0,
5
ple sib(i)
O @
origin,
Hence,
(38)
a preference
vector
if and
btKl
the
P +.
is, FJO C P such
that
through
(38). be
a query
bib,...,
for
conditions
Since in one
q.
[ (d>d’)A(d,d’6 theorem
in terms
divides
P–.
a F’O such
passing
to solving
b E B(D),
={b=d–~
existence
stated
t3Fjo; P
and
where B(D)
plane
of inequalities:
b>O,
exists
the
contained
the
there
denote
parallel
spaces Recall
O @ C,
P
if q provides
an acceptable
ranking,
we obtain:
For
Given
d > d’,
a(q,
d’)
let
holds.
0, then
for
b =
Let
any
de
P(T)
given
d’.
If q E II(b),
a(q,
= rlb~~c(~)~(b).
d)
If P(T)
q.
d)
Vb E Be(T)
should
finding to
< cx(q, d’)
That
is, every
vector
we call P(T) sample
P(T)
vector.
the solution
every
implies there
6
Hence,
region
the same
solution
To produces sample
set;
solutions.
set.
region
the
the
true
It can be easily
set.
cussions
vec-
itly
P(D)
document
solution
cases
region
subset
of
required
for
for
properly tion mation
Yao,
chosen
about
user
help
P(T)
n
~ D,
under
o P(T’),
the
region
the methods
Wong,
following
(polygon) given
Algorithm
preference
and
which
region
valuable
structure. a lmore
Ya,o
of a
in Preparata
and
measures structure
Such
The
infor-
the
method
(Wong
to construct
the
ometry
are useful
practical
this
similar-
and
suggested
an
the investigations
structures measures.
techniques
aspects
is that these
We also analyzed
vectors
complements
of similarity that
an
it.
study
suggest
or the
produce
point
adopting
the
~asthe inner-
would
retrieval.
solution
lead
identified,
covari ante,
important for
that
is that
been
such
Dice,
stan-
structure.
paper
have
measure,
basis
algebraic/geometric
ations
and
the
may
preference of this
correlation
for finding
particular,
representations
in information
present
speciid
indicates
pseudo-cosine,
of the
In
analysis
function
The
are
This
user
dis-
We have explic-
measure.
conditions
the
and
Our
and
ranking
ranking.
algorithm
1991). algorithm
a linear
structures models.
pseudo-cosine
contributions
establishes
the
informa-
effective
feedback
of a sample
and
Our
developed
for
the
cm
empirical
evalu-
preliminary
results
for computational
study
of information
of the
theoretical
retrieval
systems.
geand
set T by adopting Shames
References
(1985).
Bollmann,
k=o; B.(T)
and
of computationid
measures
linear
different
P. and
Wong,
ear information 2. While
algorithm
Wang
models.
in document
cosine,
ity
point
retrieval
the
and sufficient
acceptable
FindPolygon
l.v=u;
by
properties,
models.
product-moment
n P(T’),
solution
contains
sets for relevance
Wan
We suggest
set
the
space
One of the main
P(T).
that
us to design
training
1990;
solution
sample
the
may
to select
indicate
in
an efficient
similarity
in detail
change
necessary
all T, T’
the
many
to a dramatically
T~D
properties
if one
vantage
generalized
vector
a minor
To a mzntmul
information
of the
dard
the
on the linear
that
we examined
set To defines
no proper
= P(T)
=
that
was suggested
analyzed
centered
shown
result (iii)
is interested
versa.
vector,
of vector-based
product,
T’)
vice
all ve r-
1991).
from
we
limitations
region.
that
P(Tu
et al.,
paper,
infinite
we call
T C T’ =+
(ii)
out
task
we know of these
Conclusion
In this
of the
Euclidean
entire
the
and
all the solution
verified
(i)
These
this
Wong
a solution
If a sample
solutions,
To contains
determining
is an
then
as D,
same
In
exists If the
set,
solution
set.
set
there
as a sample true
open
open
whenever
are infinite
is the
an
nonempty
that
set D is used P(D)
is
and
one solution
perform
geometry
definition,
topology,
tor,
a solution
= fI&Bc(@(b)
vector,
polygon,
combination
d’.
set T.
By
This
P(T)
q in
solution
convex
be pointed
only
(1990,
the
open
is a solution
It q G P(T)
found
Any
# @ Do
S.K. M.
retrieval
ACM
SIGIR
ment
tn Information
(1987).
models.
Conference
Adaptive
Proceedings
on Research
Retrteval,
linof the
and Develop-
157-163.
Begin b(k)
=b
V =
Vnlf(b(
B.(T)
Jones,
E B.(T);
= B.(T)
evance:
k)); – b(k);
k=k+l
159
Furnas,
a geometric of the
Science,
38,
G.W.
(1987).
analysis
Amertcan
Pictures
of similarity Soczety
for
of rel-
measure. Information
420-446.
M. J., Koll,
evaluation
v.
and
Journal
McGill,
End; 3. output
W.P.
M.,
of factors
and
Noreault,
affecting
T.
(1979).
document
ranking
An by
information
reirieval
tion
Studies,
York
13210.
McMullen,
Syracuse
University,
Shephard,
G.C.
the
Bound
and
bridge:
School
Wong,
of Informa-
Syracuse,
S.K.M.
and
distribution
New
Upper
Cambridge
(1971).
University
Wong,
Convex
Conjecture,
and
in
for
Yao.
linear
American
Press.
Y.Y.
Processing
S.K.M.
lation
Cam-
Yao.
model
formatton
P. and
Polyiope
systems.
(1989). information
and Y.Y.
retrieval
Socaety
for
A
probability
retrieval.
ln-
25,
39-53.
Management, (1990).
Query
models.
Journal
Information
Sciencej
formuof the 41, 334-
341. Preparata,
F.P.
tational
and
Sharnos,
M.I.
Geometry-an
(1985).
conlp’u-
Introduclaon,
New
Wong,
York:
S.K.M.
inference
Springer-Verlag.
tion Raghavan,
V.V.
analysis
and
Wong,
of vector
S.K.M.
space
(1986).
model
trieval.
Journal
of the American
mation
Science,
37,
Roberts,
F.S.
York: Salton, —
G.
Salton,
in
Automatic NJ:
McGill,
Modern
for
Infor-
New
Tezt
M.J.
Retrteval
System
Document
Publishing
Process-
Processing,
Read-
Company. Introduction
(1983).
Information
Retrieval,
New
York:
McGraw-Hill. Schneider, E.,
H.-J.,
Bollmann,
Reiner,
U.,
iungsbewertung @IVE).
and
P., Jochum, Weissmann,
Projelctbericht,
F.,
V.
von mformatzon
Konrad,
(1986).
retrteval
Technische
Leis-
verfahren Universitat,
Berlin. van
Rijsbergen,
C.J.
London: Wan,
S. J., and
Wong,
To appear and
S. K. M.,
Information theory. 19, Wong,
(1991).
S.K.M
learning
Analysis Wong,
Retrieval,
Butterworth.
pervised tem.
Information
(1979).
algorithm in IEEE
Machtne
Bollmann, retrieval
International
A
for linear
partially
su-
separable
sys-
Transactions
on Pattern
Intelligence. P., and based
Yao,
Y.Y.
(1991).
on axiomatic
Journal
of General
decision Systems,
101-117. S. K. M.,
Linear ings
Yao,
structure of the llth
Conference mation
Y. Y.,
Annual
on Research
Retrieval,
and
Bollmann,
in information
P. (1988).
retrieval.
Interl~attonal
model
Systems,
ACM
and Development
S. K. M.,
(1991).
Prentice-Hall.
Automatic
Wesley
G. and
to
Wong,
re-
Theory,
SMART
The
Cliffs,
(1989).
MA:
Soczety
Measurement
Ed.).
Englewood
ing,
information
Press.
Experiments
ing, Salton,
(1971,
in
Yao.
Y.Y.
(1991).
for information 16,
A probabilistic retrieval.
Informa-
301-321.
A critical
279-287.
(1976).
Academic
G.
and
ProceedSIGIR zn Infor-
219-232.
160
Yao.
Y. Y.,
Evaluation
Journal
of the
Science,
42,
pp.
Salton,
G. and
of an adaptive
American 723-730.
Society
Buckley, linear
for
C.
model.
Information
