A Fast
Fastlk?ap:
Algorithm
Visualization
of Traditional
Christos
AT&T
for
Indexing,
and
Dept.
Laboratories
Murray
Hill,
Multimedia
King-Ip
Faloutsos”
Bell
NJ
Univ.
Abstract
N),
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promising
space,
using
domain
fine-tuned
[25].
spatial
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Thus,
of Maryland,
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type
pairs’
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objective
and
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query
extraction
functions
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expert
Given
obvious
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objects
of this
to map objects
into
(k is user-defined),
preserved.
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and
now
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that
We introduce
namely,
for
met hod.
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problem
proposed
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and
faster
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as opposed
[51];
data
for
algorithm
to solve
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show
than
database
[44],
tion
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size
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of insertion,
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timefeature-
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k-dimensional
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eg., the
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have
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work.
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to
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typically
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to
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gadish,
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Overcoming
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indeed
significantly
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excerpts,
extraction
although
yardstick
it allows
synthetic
other
where
to
available.
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k-dimensional
easy
substitutions
other).
voice
pat te~n recognition,
as
the
should
revealing
distance
and
warping
(MDS)
it
to
the objects
and
editing
words,
into
problem
of algorithms
always
An
[25],
object
the
functions.
English
deletions
as discussed
space,
typed
is the
(a)
for.
from
use
of
datasets.
k feature-extraction
displaying
is not
a retrieval
of traditional
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each
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it
provide
derive
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and
there
to
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ies are
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pot ential
paper.
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ret rieval,
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the topic
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collections
multimedia
to
space.
retrieving
which
and
experts
thus
the distance
is
suggested
domain
to assess the
work
for large
‘exotic’
However,
is exactly
efficient
on
this
tool
idea,
k-dimensional
to map
of
as
functions,
be
points.
space
well
excellent
the nearest-neighbor
[8]);
of two objects.
though,
algorithm
the
By
distances
Introduction
rely
feature
It is relatively
This
can
join
to answer
the
Park
of the data-set.
1
by a
use highly
etc.
similarity/distance
into
‘Query
query);
to a spatial
designing
information
the
Lin
Science
in k-d
provided
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(David)
and
points
functions,
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query,
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into
we can subsequently
including
to
in traditional
objects
access methods
of queries,
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searching
is to map
k feature-extraction
expert
types
idea for fast
databases
and
Datasets
of Computer
structure
multimedia
Data-Mining
points,
provided
D(*,
*).
case of features,
between
between
two
the
us
Notice
by using
feature
vectors
corresponding
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jects.
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Given
CDR-
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IRI-9205273),
with
and Thinking
Inc.
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users
query
are most
the
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a set of objects
would
like
object,
similar
(b)
space,
to
of
objects
in order
and
to
the
find
to find
to each other,
distribution
chosen
(a)
the
pairs
as well
into
check
distance
objects
of objects
to
that
as (c) to visualize
some
for
function
similar
appropriately
clusters
and
other
regularities.
Next,
Definition
163
we shall
1
use the following
The
k-dimensional
terminology:
point
Pi
that
corre-
a
spends
to
object
Oi.
the
Definition
Oi,
will
Some
will
be called
‘the
image’
search
of
is, P; ~ (z; ,1, z;, z, . . .,x~,~ )
2 The
‘images’
work
object
That
k-dimensional
be called
of the
target
next.
containing
the
space.
applications
are listed
space
it.
that
Some
motivated
distance
the
present
functions
are
also
and,
a collection
shapes
multimedia
shapes
[25]
to
images,
similar
the
shapes
feature
inertia
(voice,
that
are
clip,
Once
video
immediately
2-d
databases,
images
and
of
typically
(eg.,
the
we consider
warping
makes
it difficult
describe
a point
series,
prices,
in feature
sales
eg.
series
geological,
etc.,
of
In
‘jind
companies
such
whose
typical
stock
past days in which
showed
patterns
goal
patterns
we
that
used
errors)
similar
is to aid
the
to
have
the
Euclidean
as the
distance
solar
today’s
appeared
distance
function
find
ones
the
the
case of spelling,
tion
[26].
There,
typing
given
string
[30]
databases,
and
a wrong
OCR
string,
we
(with
at-
etc.),
detect
any
we
clusters,
demographic
data
two
types
of
queries
requests
Specifically:
query-by-example
‘similarity
form:
are
(or,
equiva-
will
signify
a desirable
object
query’)
Given
search
a collection
of objects
within
a user- dejined
to
distance
e
object.
4 The term
join’)
wiIl
of objects,
distance
all pairs
signify
query
(or,
equivalently
of
the
form:
queries
find
the pairs
c from
of objects
other.
each
In
a
which
Again,
are
c is user-
the above
into
two
can
major
is
Access
[7] and
the
the
we
search
can
Methods
z-ordering
searching
by a mapping
Such
a mapping
for
time
for
employ
highly
(SAMS),
like
[37].
range
These
queries
queries.
The
fine-tuned
the
R*-trees
methods
provide
as well
as spatial
[8].
can
dataset,
[53].
general
linear
help
with
visualization,
Plotting
can
objects
reveal
such
as the
shape
clustering
as points
much
of
existence
Gaussian)
powerful
etc..
insights
discovering
rules.
as discussed
before,
in
the
and
k=2
(linear
These
data-
or
structure
of major
of the distribution
versus
provide
and
benefit
k-d space.
benefits:
that
Spatial
would
in some
accelerate
mensions
[1]
applications
points
3 diof
clusters,
versus
curvi-
observations
in formulating
the
the
can
hypotheses
time
as in
error
that
query
mining:
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two
object),
as follows.
in
applications.
patients
‘query-by-example’
or
query
wind
In
query’
(termed
Thus,
searching
to trans-
blood-pressure
descriptions,
following
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be
series.
Similarity
of
symptoms,
desirable:
of the
joins
similar
of
of insertions,
visualization
physician
among
queries
fast
similarly’,
past.
(sum
the
3 The term
reason
data,
would
pattern>
between
In
the
second.
age,
queries.
‘range
1. It
[11],
magnettc
in the
[4].
are needed
records
gender,
above
pairs’
provides
as stock
by examining
[2] and
to help
the
‘all
All
map
[53]
move
against
is typically
number
that
is a
alphabet
candidates
distance
to the
given
to be very
of objects
databases,
queries
there
diseases.
within
would
weather
prices
forecasting,
may
that
scientific
data,
where
best
to
approxi-
defined.
This
therefore,
such
like
collection
that
[50].
astrophysics
databases,
or ‘find
The
or
sensor
environmental,
etc.,
be
are properly
data,
[3],
like
Definition
space).
financial
numbers
time
(and
example,
‘spatial
some
sure
features
image
MRI
distance
differences
to find
each
with,
bones)
mining
For
from
teaching
make
Data
Definition
be
would
the
string
of
a four-letter
minimum
first
lently
retrieve
requiring
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structures
before
adequately
that
images,
aligned,
(eg.,
as for medical
ie.,
form
and
ECGS),
to
the
the
or substitutions
From
video
can
symptoms
Notice
two
anatomical
images
distance
strings
case
has to be matched
find
deletions
seem
function
Ability
as well
complicated,
the
3-d
from
string
to
closest
the
databases,
applications,
the
the
is
clips
or
(eg.,
these
and
users
method
all
a new
strings,
or correlations
audio
or video
objects
similar
purposes.
are
warping
with
we
is
with
score
old
would
moments
dis-similarity)
stored.
for diagnosis,
functions
Time
There
example,
music
and
are
cases with
research
it into
For
scores,
l-d
X-rays)
[5]
past
valuable
the
(eg.,
scans)
quickly
with
appropriately
proposed
where
of strings
tributes
applied.
Medical
brain
[35].
[33].
our
of
images
databases,
(or
determined,
find
attributes,
a target
in DNA
collection
editing
find
Search-by-content
music
matching
large
the
find
In
to
collection
between
(color
similarity
like
a
texture
etc.
to
the
in
etc.)
to retrieve,
similar
been
or
in multimedia
music),
want
databases:
would
to
distance
shape,
desirable
might
like
vectors
for
we
one;
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Euclidean
selected
highly
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colors,
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general,
of
similar
color
of
in
to
identical
mate
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described.
Image
a dictionary
Conceptually
to highlight
the
correc-
known:
should
General
164
We
the
the general
shall
refer
to
fact
that
only
Problem
(’distance’
it
problem
as the
the
distance
case)
is defined
‘ distance
case’,
function
is
Given
N objects
and distance
them
(eg.,
N
simply
the distance
an
N
x
information
distance
function
D(*,
has
about
matrix,
or
*) between
two
been
ences,
solve
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objects)
Find
N
points
such
that
in a k-dimensional
the
distances
are maintained
been
as well
We
expect
ity.
In
the
Euclidean
tions.
that
the
symmetric
‘target’
it
distance
like
the
the
space,
because
Alternative
function
obeys
(k-d)
distance,
Lp metrics,
distance
and
V()
is
triangular
we
is invariant
metrics
L1 (’city-block’
use
under
could
features
case
from
projection,
is
the
when
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of the
or ‘Manhattan’
‘features’
have
but
we
objects,
usually
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because
curse’).
the
We
already
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features
shall
are
refer
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from
There
to
Given
N
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such
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vectors
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distances
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dimensionality
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spaces
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to use the
In
the
fulfill
Lp
where
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but
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(L2
ideal
and
for
should
large
gives
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suffer
from,
crepancies
(low
To
should
or O(N
because
distances,
‘stress’
should
a new
The
provide
the
leading
- see (Eq.
object
cost
to
a
random).
dis-
smal.
k-d
The
is vital
outline
(MDS),
of this
a brief
related
SVD
etc)
for
O(1)
and
our
method.
In
results
on real
list
conclusions.
to
methods.
section
and
its
O(log
is as follows.
pointers
access
or
to
N).
its
In
Here
distances
the
point
older
present
its
to
attempts
cuss the
to
solve
In
section
literature
on
section
in
function:
datasets.
background
(K-
information
(MDS)
First
The
above
have
termines
5 we
[28]
different
we
MDS
dis-
including
that
165
space
algorithm
heuristic,
or
point,
to
even
at
and
k-d
method
of
moves
‘steepest
k-d
points.
as a ‘spring’
algorithm
points
the
distances.
of
the
distance
the
k-d
item
between
estimated
then,
is
computes
– 1 points
each pair-wise
and
works
each
the
of the
a guess
improvement
assigns
positions
version
of
scaling
[51],
MDS
Several
proposed
tries
to re-
to minimize
the
the
to the
for
that
the
metric
basic
[48],
MDS
where
MDS,
semantic
algorithm:
de-
and
Kruskal
the
distance
Young
which
[55]
incor-
corresponding
of the
has been used in numerous,
are
exten-
automatically
messures,
perception
mul-
distances
and
qualitatively;
difference
distance
following:
above
k; Shepard
specified
individual
observers’
called
generalizations
non-metric
are
is
because
a method
value
multiple
the
with
discrepancy
the
the
points;
a good
the
in k-d
springs.
been
items
N
the
[29] proposed
proposed
function
‘stress’
every
other
employs
as numbers.
Kruskal
about
method
two
distance
The
the
some
and
positions
of the
sions
experimental
the
it treats
‘stress’
given
present
In section
problem.
Scaling
object
(Euclidean)
starts
examines
minimize
tidimensional
clustering
3 we
some
2
Scaling
methods
4 we give
the
to a point
between
no further
originally
using
it
MDS
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porates
Multidimensional
items,
desirable
stress
distances
version,
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update
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arrange
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until
simplest
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to
between
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is the
MDS
dissimilarities
between
Survey
we
each object
and
that
it,
(eg.,
the
describes
2
‘distance’
of N
(c) the
y measure
&
goal,
Then,
Intuitively,
This
reduction
In
synthetic
the
(eg.,
average.
improves
descent’
image.
of Multi-Dimensional
dimensionality
spatial
object)
map
to
‘queries-by-example’,
paper
survey
and
the
be
map
Pi
error
on the
point
actual
algorithm
and
as follows:
will
be
Oi
achieve
log N),
1)).
fast
a query
should
requirement
L,
(eg.,
algorithm
we present
a very
data
them.
method
a set
and
to minimize
‘images’
Technically,
3. It
will
dissimilarity
relative
iteratively
databases.
preserve
basic
(a)
(dis)similarities
of
among
Following
expects
space,
their
of
metric).
mapping
O(N)
or higher,
the
discover
set
k.
object
possible.
to compute:
0(N2)
prohibitive
2. It
method
is the
we
in either
As before,
requirements:
be fast
but
next.
the algorithm
roughly
1, It should
to
a
as well
vectors
metric.
distance
problems,
following
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be any
Euclidean
above
the
between
could
of
each,
space,
Oi
the
two
the
used
information
variations,
described
pair-wise
between
Again,
access meth-
case)
in a k-dimensional
the
has
case).
(MDS)
is
structure
(dis)similarity
are several
[29] ) is
(b) their
as possible.
the
the
a k dimensional
Problem
closely
that
algorithms.
(MDS)
(spatial)
case setting,
case:
Specialized
the
of spatial
to
present
(’features’
Scaling
scaling
underlying
items
extracted
want
survey
we
and
to clustering
sci-
[55])
(SVD)
reduction
Multi-Dimensional
see
special
transform
a brief
social
Then,
Decomposition
as pointers
Multidimensional
tance).
A
problem.
(K-L)
we provide
as well
2.1
the
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be any
case
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(eg.,
physics
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inequal-
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fields
research,
used for dimensionality
ods,
negative,
diverse
market
Singular
Finally,
as possible.
in several
Karhunen-Lo&we
related
space,
used
psychology,
to
dat a’s difference.
diverse
applications,
structure
analysis
of
words;
perceived
operating
on 60 different
perception
of
‘trusting’);
what
shifts)
However,
the
our
‘warm’
texture
analysis
applications,
and
of
of
spins
from
Value
Appendix
as
well
the
K-L
Our
the
related
[49, 39, 19]
implementation
[54]
is
‘mosaic’
/pub/SRC/
transform
on
(SVD)
as on
approx-
is closely
Mathematical
edu
and
projections
operation
matrix.
in
A,
order,
its
Decomposition
cs. umd.
However,
drawbacks:
with
The
transform
//olympos.
two
eigenvalue
vector
object-feature
K-L
in
suffers
Singular
the
the
[40].
MDS
data
k eigenvectors.
to the
(determining
in decreasing
each
first
pattern
type
science
them
imates
people’s
spectra
different
sorts
[41],
and
(like
gamma-ray
political
[55];
for
traits
together
(nuclear
relationships);
ideological
relationships
personality
recognizing
their
trait
goes
physics
recognition,
and
personality
kl
of
available
(URL:
f tp:
two
draw-
m).
suffers
from
backs:
It
●
requires
In
the
Its
) time,
it
for
‘query-by-example’
be
mapped
algorithm
is
search/add
a point
this
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a new
at best.
query
would
Thus,
query
k-d
the
number
●
it can not
●
even
the
complexity
latter
situation
trieval
and
MDS
spond
to
to
lary
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would
of answering
as sequential
this
above
two
present
MDS
and
drawbacks
paper.
Despite
as a yardstick,
‘stress’
are the
the
against
of our
a
motivation
behind
problems,
such
the
‘features’
we measure
extensively
in statistical
algebra,
The
(’K-L’)
transform
the
where
and
sense
the
its
Figure
it
the speed
to map
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and matrix
points
a
method
1 shows
a set
of
2 directions
suggests:
If we are allowed
to project
(x’
K-L
mean
between
on is the
2-d
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normalization
to the
technical
MAT
Wustl
of Information
group
logarithmic
news-
wuarchive.
function
document
from
on the Internet
electronically
tance
to unit
are
first
with
and
above
groups
the
method
also
ware
The
Comparison
algorithms
recipes.
(October
Sale
basketball
of the Bible
the
REC:
East
4.1
In
ports.
MAT:
0,1,
5 documents):
versus
want
the
to
Figure
again
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over MDS,
study
in
the
the
of
subset
the
logarithmic
1 ics . uci . edu: // ftp/pub/machine-learn
achieves
performance
k
60-point
7 shows
FastMap
even for small
dimensionality
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2 to 6.
k,
conclusion
savings
time
scales.
the
and
for
datasets.
of
target
each
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each
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k
method
that
ing-databases/wine
the
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time
of
while
our
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for
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8:
for
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the
Response
FastMap
Io?J(N)‘b
WINE
database
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size
with
N
for
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to
savings
in time,
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its
first
k for
the
WINE
FastMap
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each
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gives
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variable
‘ideal’
it
time,
closer
(0,0).
quality),
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for
a method
much
but
give
lower
the
FastMap
of magnitude
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‘stress’,
origin
closer
present
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then
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can
zero
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value
produce
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of the
graphs,
of
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that,
fz,
(=
a mapping
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170
ask
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stand
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for
k=3
for
the
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detect
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first
two
the
for
plots,
by
the
of
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6
the
same
vs $2,
and
(c)
dimensions
4 clusters;
can
clusters
Figure
showing
‘target’
In
clusters
two
k=3,
forming
‘ FastMap-attributes’.
roughly
the
for
of ~1 vs.
that
using
be
are
disjoint
9(c)
all
the
completely
in
at
confirms
the
6 clusters
are
space.
it uses a fictitious
ability
synthetic
points,
indicated
three
scatter-plots.
observation,
the
the
scatter-plot
scatter-plot
with
3-d
with
ones.
cluster).
are
gives
because
in the
and
The
that
and
on
we
mapping
per
cluster
we can
Although
tion
point
results
(N=120
plot
one of the
previous
show
.fZ and
resulting
9(a)
scatter-plot
least
is to
visualization
stated,
real
dataset
gives
Notice
next
goal
results
the
20 points
3-d
~1 and
lustrates
better
‘stress’
(b)
the
of
Data
the
Figure
while
significant
quality.
for
the
.fI,
the
with
of a given
disjoint
time.
‘ideal’
gives
with
gives
our
present
that
we
separated,
reduce
Figure
independent
in
to the
MDS
of each
k. In these
zero
of
stress.
as a function
scales.
goes to the
for
we see that
stress
algorithm
of time.
points
is that,
achieves
‘ FastMap-attributes’.
9
clusters,
and
k,
it gives
can each
method,
is in general
Alternatively,
the
k,
and
properties
otherwise
GAUSSIAN5D
letter.
dimensionality
dimensionality
should
(solid)
‘price/performance’
in logarithmic
FastMap
an order
estimate
amount
each
dimensions
- MDS
to
same
the
is, how
was the
method
is.
is
of
of experiments
is useful
Synthetic
Figure
axes logarithmic.
as we saw,
in a given
‘stress’
response
Both
is to find
that
‘stress’,
The
longer,
question
algorithm,
the
For
takes
number
(N=60)
line).
experiment
method.
clearly
subset
(dashed
final
vs.
varying
(solid)
logarithmic.
loss in output
we
Recall
4.2.1
the
group
of experiments
Unless
First
time
axes
FastMap
algorithm
Here
datasets
56
of this
without
group
dimensions.
Response
with
- MDS
Map
proposed
datasets.
7:
Both
linearity,
clustering.
Figure
stress
N=60
Clustering/visualization
this
the
4
vs.
with
line).
conclusion
thanks
Fast
3
Iw(k)
time
subset
(dashed
4.2
2
lC.W
the
Both
logarithmic.
:-
Iwl
Ikw%)
I
*--,:.,-:.:
dataset,
of FastMap
this
to help
example
with
il-
visualiza-
clustering.
next
Figure
10(a)
shows
the
Notice
that
experiment
plots
result
the
the
of
involves
original
FastMap
projections
the
SPIRAL
dataset
in
for
(Figure
k=2
10(b))
dataset.
3-d
and
(b)
dimensions.
give
much
f3
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17.5
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17.5
15
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15
D
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5
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Figure
9: FustMap
information
to form
2.5
5
7.5
type
GAUSSIAN5D
the original
curve,
10
12.5
15
fl
17.5
(c)
(b)
on the
about
a l-d
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&
fl
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some
‘:@@q’%
E
PF
&
‘-”p:
z,:
~
,51
with
dataset
dat aset:
(a)
the points
no obvious
clusters,
~z vs ~1 (b)
and
~3 vs jl
(c) the
3-d
scatter-plot
(~1, .f2, ~3)
seems
and
with
.,
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of oscillation.
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(a)
Figure
Figure
and
10: (a)
(b)
the
3-d points
result
on a spiral
for
of FastMap,
(SPIRAL
12: The
space
dataset)
box
k=2
(a) The
Real
Next,
in
Data
present
we
11.
Figure
The
GAU~SIAN5D
using
gives
the
two
scatter-plot
previous
two
first,
the
first
the
The
second
three
clusters
separate
gives
for
third
class,
one
Figure
9 for
a 2~d
~1,
~s and
points
the
scatter-plot
(c)
~z,
labeled
even
picture
to
Notice
separate
with
more
‘?’).
The
and separates
of the
The
3-d
our
Figure
in
its
last
12.
dataset,
The
entirety
to illustrate
documents
separated
figure
and
that
shows
(b)
after
FastMap
of each
well,
DOCS,
class.
in only
k=3
the
the
results
almost
3-d
zooming
manages
Notice
that
are shown
scatter-plot,
into
the
to cluster
the
objects
space,
in
(a)
center,
well
the
7 classes
are
to
171
dashed
our
for
each
queries,
‘all
queries
[42]
optimized
spatial
access
(R-trees
[20],
is useful
visualization
The
can
the
R*-trees
for
has
the
or spatial
etc.),
[7]
etc.).
data-mining,
algorithm
of this
that-
two
searching
or
joins
because
methods
contribution
a linear
appropriate
following
accelerate
are
for
‘range’
[9, 8], nearest
several,
highly
readily
available
Secondly,
such
cluster
of a high-dimensionality
main
the
function,
(’query-by-example’
queries
neighbor
mapping
algorithm,
infer
points
it
of queries
pairs’
expert
functions.
a distance
will
into
Firstly,
types
in non-
a domain
extraction
provide
algorithm
as possible.
object.
objects
several
[25],
into
distances
searching
cFastMap’
only
objects
the
as well
feature
proposed
need
which
features
provide
the
that
for similarity
databases
expert
FastMap,
dimensions!
in k=3-d
of the
to map
so
are preserved
approach
applications.
help
scatter-plot
the clusters
to
domain
Mapping
jl-~s
to
completely.
For
the
expected
from
that
one
information
better.
between
was
of the
FastMap
the contents
algorithm
traditional/multimedia
combines
members
a fast
k-dimensional
In an earlier
(b)
after
(b)
detail,
proposed
in
Thanks
denote
respectively.
some
clusters
the whole
‘?’)
manages
(the
dataset,
a 3-d scatter-plot.
provides
the
~1 and
‘Cl’,
scatter-plot
scatter-plot
gives
WINE
‘ FastMap-coordinates’
(’+’,
and
the
as in
is
(a)
into
symbols
~1-~z
layout
dataset:
the
for
results
dataset,
Conclusions
We have
the
DOCS
big picture
in more
5
4.2.2
(b)
(a)
;f
analysis
a
and
dataset.
paper
f~lfills
is the
all
design
the
d&ign
of
f3
+
1.5[
@
+
1.5
1.25
1
@
0 75
@
0.5
0.25
I
I
0.25
0.5
0 75
1.25
1
0.25
1.’75’1
1.5
0.75
0.5
(a)
1
1.25
1.5
1.75f]
(b)
Figure
11:
on WINE
FastMap
dataset
(a)
(c)
k=2
with
jz
vs .fl,
Finally,
goals:
we
output
1. it solves the general
eg.,
the
Value
than
the
database
time,
it
able
to
map
a new,
point
in O(k)
distance
the
only
(while,
synthetic
separate
solve
the
low
size,
and
therefore
Scaling
leads
to
fast
arbitrary
(MDS)
and
indexing,
object
calculations,
●
work
into
a k-d
regardless
of the
●
The
algorithm
uses theorems
(such
as the
cosine
each
object
on an appropriate
recursive
calls.
With
sured
the
‘stress’
by
FastMap
the
from
law),
it
respect
we
datasets:
The
‘stress’
levels
as MDS,
projects
●
at each of the
to quality
function),
k
of output
(mea-
experimented
with
result
is that
for
should
for
given
second
contribution
tools
matrix
from
algebra,
Scalzng
form
and
method
(or
though
(MDS)
these
database
research,
tion
function.
study
of its
is a good
SVD
tion
the
the
We
with
would
choice
and
for
the
the
general
algorithm.
to handle
for
K-L
problem
and
MDS
has been
into
Being
transform
provide
case
(although
of the
‘distance’
used
in
quadratic
on N
easily,
MDS
datasets.
The
optimal
solu-
unable
of
interactive
the
the
the
dis-
given
data
algorithm
for
thank
Labs
Dr.
for
mining
for
and
document
them;
and
Howard
for
Patrick
maintaining
Learning
Joseph
providing
algorithms
on
Aha
to
Bell
MDS
A
using
the
for
from
M.
the
Elman
and Doug
Kruskal
source
answering
Murphy
UC-Irvine
Databases
B.
the
code
several
and
David
Repository
and
Domain
Oard
for help
of
Theories;
with
SVD
of
visualiza-
k-d points
of small
dataset,
databases,
determine
algorithms.
al-
arsenal
the
Prof.
Al-
proposed
the
‘queries-by-example’
visualization
‘features’
to
like
AT&T
Machine
trans-
indexing
objects
W.
SVD).
as the
benefits
to multimedia
automatically
and
application
questions
and
Multi-Dimensional
added
datasets.
to map
for
intro-
Karhunen-Lo3ve
as fast
to help
it
sciences
Decomposition,
be
iterative
unable
or
the
space
Acknowledgments
of the
is that
social
the
could
applications
a quadratic,
and
tools
of non-traditional
diverse
and
Value
as general
gorithm,
paper
target
to
with
it achieves
a fraction
recognition,
specifically,
Singular
not
of the
pattern
even
retrieval.
time.
A
the
tance
from
duces
the
and
geom-
quickly
direction
on real
same
traditional
and
and
real
managed
clusters,
k of the
of the algorithm
clustering
etry
‘ FastMap’
existing
dimensionality
FastMap
features
size N.
on
includes:
Application
where
being
speed
algorithm
or 3 dimensions).
Future
much
the
of the
k=3
the
proposed
There,
or most
for
(c)
demonstrated
of our
datasets.
all
values
(k=2
j’s vs $1 and
have
quality
Singular
case))
Multi-Dimensional
same
database
can
(’features’
on the
case)
and
(SVD)
version
2. it is linear
faster
(’distance’
(K-L)
Decomposition
specialized
3. at
problem
Karhunen-Loeve
(b)
Karhunen-Lo&e
This
is the
code
for
Transform
the
K-L
transform
in
Mathemat-
ical [54]
(*
given
$n$
it
to handle
and
case).
a matrix
vectors
creates
their
attributes
172
mat_
of
$m$
a matrix
f lrst
(le.
$k$
, the
with
attributes,
with
$n$
most
‘ important’
vectors
K-L
expansions
of
these
$n$
vectors)
[10]
*)
KLexpansion[
mat_,
k_:2]
. Transpose[
mat
KL[mat,
k]
];
[11]
(*
given
a matrix
dimensions,
$h$
singular
of
KLC
with
the
mat_
{n,m,
K-L
, k_:2
1:=
Module[
$k$
3,
vec
translate
newmat=
{val,
vectors,
so
Table[
vecl=
[13]
n IIN;
the
mat[[ill
-
mean
avgvec,
and
{i,l,nll;
burg,
1;
[14]
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Christos
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