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Gaussian process classification for segmenting and annotating sequences


Abstract and Figures

Many real-world classification tasks involve the prediction of multiple, inter-dependent class labels. A prototypical case of this sort deals with prediction of a sequence of labels for a sequence of observations. Such problems arise naturally in the context of annotating and segmenting observation sequences. This paper generalizes Gaussian Process classification to predict multiple labels by taking dependencies between neighboring labels into account. Our approach is motivated by the desire to retain rigorous probabilistic semantics, while overcoming limitations of parametric methods like Conditional Random Fields, which exhibit conceptual and computational difficulties in high-dimensional input spaces. Experiments on named entity recognition and pitch accent prediction tasks demonstrate the competitiveness of our approach.
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Gaussian Process Classification for
Segmenting and Annotating Sequences
Yasemin Altun
Department of Computer Science, Brown University, Providence, RI 02912 USA
Thomas Hofmann
Department of Computer Science, Brown University, Providence, RI 02912 USA
Max Planck Institute for Biological Cybernetics, 72076 T¨ubingen, Germany
Alexander J. Smola
Machine Learning Group, RSISE , Australian National University, Canberra, ACT 0200, Australia
Many real-world classification tasks involve
the prediction of multiple, inter-dependent
class labels. A prototypical case of this sort
deals with prediction of a sequence of la-
bels for a sequence of observations. Such
problems arise naturally in the context of
annotating and segmenting observation se-
quences. This paper generalizes Gaussian
Process classification to predict multiple la-
bels by taking dependencies between neigh-
boring labels into account. Our approach
is motivated by the desire to retain rigor-
ous probabilistic semantics, while overcom-
ing limitations of parametric methods like
Conditional Random Fields, which exhibit
conceptual and computational difficulties in
high-dimensional input spaces. Experiments
on named entity recognition and pitch ac-
cent prediction tasks demonstrate the com-
petitiveness of our approach.
1. Introduction
Multiclass classification refers to the problem of as-
signing class labels to instances where labels belong
to some finite set of elements. Often, however, the
instances to be labeled do not occur in isolation, but
rather in observation sequences. One is then interested
in predicting the joint label configuration, i.e. the se-
quence of labels corresponding to a sequence of ob-
Appearing in Proceedings of the 21 st International Confer-
ence on Machine Learning, Banff, Canada, 2004. Copyright
2004 by the first author.
servations, using models that take possible interde-
pendencies between label variables into account. This
scenario subsumes problems of sequence segmentation
and annotation, which are ubiquitous in areas such as
natural language processing, speech recognition, and
computational biology.
The most common approach to sequence labeling is
based on Hidden Markov Models (HMMs), which de-
fine a generative probabilistic model for labeled obser-
vation sequences. In recent years, the state-of-the-art
method for sequence learning is Conditional Random
Fields (CRFs) introduced by Lafferty et al. (Lafferty
et al., 2001). In most general terms, CRFs define a
conditional model over label sequences given an ob-
servation sequence in terms of an exponential family;
they are thus a natural generalization of logistic re-
gression to the problem of label sequence prediction.
Other related work on this subject includes Maximum
Entropy Markov models (McCallum et al., 2000) and
the Markovian model of (Punyakanok & Roth, 2000).
There have also been attempts to extend other dis-
criminative methods such as AdaBoost (Altun et al.,
2003a), perceptron learning (Collins, 2002), and Sup-
port Vector Machines (SVMs) (Altun et al., 2003b;
Taskar et al., 2004) to the label sequence learning prob-
lem. The latter have experimentally compared favor-
ably to other discriminative methods, including CRFs.
Moreover, they have the conceptual advantage of be-
ing compatible with implicit data representations via
kernel functions.
In this paper, we investigate the use of Gaussian
Process (GP) classification (Gibbs & MacKay, 2000;
Williams & Barber, 1998) for label sequences. The
main motivation for pursuing this direction is to com-
bine the best of both worlds from CRFs and SVMs.
More specifically, we would like to preserve the main
strength of CRFs, which we see in its rigorous prob-
abilistic semantics. There are two important advan-
tages of a probabilistic model. First, it is very intuitive
to incorporate prior knowledge within a probabilistic
framework. Second, in addition to predicting the best
labels, one can compute posterior label probabilities
and thus derive confidence scores for predictions. This
is a valuable property in particular for applications
requiring a cascaded architecture of classifiers. Con-
fidence scores can be propagated to subsequent pro-
cessing stages or used to abstain on certain predic-
tions. The other design goal is the ability to use kernel
functions in order to construct and learn in Reproduc-
ing Kernel Hilbert Spaces (RKHS), thereby overcom-
ing the limitations of (finite-dimensional) parametric
statistical models.
A second, independent objective of our work is to
gain clarification with respect to two aspects on which
CRFs and the SVM-based methods differ, the first as-
pect being the loss function (logistic loss vs. hinge
loss), and the second aspect being the mechanism
used for constructing the hypothesis space (parametric
vs. RKHS).
GPs are non-parametric tools to perform Bayesian in-
ference, which – like SVMs – make use of the kernel
trick to work in high (possibly infinite) dimensional
spaces. Like other discriminative methods, GPs pre-
dict single variables and do not take into account any
dependency structure in case of multiple label predic-
tions. Our goal is to generalize GPs to predict label
sequences. While computationally demanding, recent
progress on sparse approximation methods for GPs,
e.g. (Csat’o & Opper, 2002; Smola & Bartlett, 2000;
Seeger et al., 2003), suggest that scalable GP label se-
quence learning may be an achievable goal. Exploiting
the compositionality of the kernel function, we derive a
gradient-based optimization method for GP sequence
classification. Moreover, we present a column genera-
tion algorithm that performs a sparse approximation
of the solution.
The rest of the paper is organized as follows: In Sec-
tion 2, we introduce Gaussian Process classification.
Then, we present our formulation of Gaussian Pro-
cess sequence classification (GPSC) in Section 3 and
describe the proposed optimization algorithms in Sec-
tion 4. Finally, we report some experimental results
using real-world data for named entity classification
and pitch accent prediction in Section 5.
2. Gaussian Process Classification
In supervised classification, we are given a training set
of nlabeled instances or observations (xi, yi) with yi
{1,...,m}, drawn i.i.d. from an unknown, but fixed,
joint probability distribution p(x, y). We denote the
training observations and labels by X= (x1,...,xn)
and y= (y1,...,yn), respectively.
GP classification constructs a two-stage model for the
conditional probability distribution p(y|x) by intro-
ducing an intermediate, unobserved stochastic process
u(u(x, y)) where u(x, y) can be considered a com-
patibility measure of an observation xand a label y.
Given an instantiation of the stochastic process, we
assume that the conditional probability p(y|x,u) only
depends on the values of uat the input xvia a multi-
nomial response model, i.e.
p(y|x,u) = p(y|u(x,·)) = exp(u(x, y))
y0=1 exp(u(x, y0)) (1)
It is furthermore assumed that the stochastic process u
is a zero mean Gaussian process with covariance func-
tion C, typically a kernel function. An additional as-
sumption typically made in multiclass GP classifica-
tion is that the processes u(·, y) and u(·, y 0) are uncor-
related for y6=y0(Williams & Barber, 1998).
For notational convenience, we will identify uwith the
relevant restriction of uto the training patterns Xand
represent it as a n×mmatrix. For simplicity we will
(in slight abuse of notation) also think of uas a vec-
tor with multi-index (i, y). Moreover we will denote
by Kthe kernel matrix with entries1K(i,y),(j,y0)=
C((xi, y),(xj, y0)). Notice that under the above as-
sumptions Khas a block diagonal structure with
blocks K(y) = (Kij (y)), Kij (y)Cy(xi,xj), where
Cyis a class-specific covariance function.
Following a Bayesian approach, the prediction of a la-
bel for a new observation xis obtained by computing
the posterior probability distribution over labels and
selecting the label that has the highest probability:
p(y|X,y,x) = Zp(y|u(x,·)) p(u|X,y)du(2)
Thus, one needs to integrate out all n·mlatent
variables of u. Since this is in general intractable,
it is common to perform a saddle-point approxima-
tion of the integral around the optimal point esti-
1Here and below, we will make extensive use of multi-
indices. We will put parentheses around a comma-
separated list of indices to denote a multi-index and use
two comma-separated multi-indices to refer to matrix ele-
mate, which is the maximum a posterior (MAP) es-
timate: p(y|X,y,x)p(y|umap(x,·)) where umap =
argmaxulog p(u|X,y). Exploiting the conditional in-
dependence assumptions, the posterior of ucan – up
to a multiplicative constant – be written as
p(yi|u(xi,·)) (3)
Combining the GP prior over uand the conditional
model in (1) yields the more specific expression
log p(u|X,y) =
i=1 "u(xi, yi)log X
exp(u(xi, y))#
2uTK1u+ const. (4)
The Representer Theorem (Kimeldorf & Wahba, 1971)
guarantees that the maximizer of (4) is of the form
umap(xi, y) =
with suitably chosen coefficients α. In the block diago-
nal case, K(i,y),(j,y0)= 0 for y6=y0and this reduces to
the simpler form umap(xi, y) = Pn
j=1 α(j,y)Cy(xi,xj).
Using the representation in (5), we can rewrite the
optimization problem as an objective Rparameter-
ized by α. Let e(i,y)be the (i, y)-th unit vector, then
αTKe(i,y)=Pj,y0α(j,y0)K(i,y ),(j,y0)and the negative
of Eq. (4) can be written as follows:
R(α|X,y) = αTKα
log p(yi|xi, α) (6)
log X
A comparison between (6) and a similar multiclass
SVM formulation (Crammer & Singer, 2001; Weston
& Watkins, 1999) clarifies the connection between GP
classification and SVMs. Their difference lies primar-
ily in the utilized loss functions: logistic loss vs. hinge
loss. Because the hinge loss truncates values smaller
than ²to 0, it enforces sparseness in terms of the α
parameters. This is not the case for logistic regression
as well as other choices of loss functions.2
For non-linear link functions like the one induced by
Eq. (1), umap cannot be found analytically and one
2Several studies focused on finding sparse solutions of
Eq. (6) or optimization problems similar to Eq. (6) (Ben-
nett et al., 2002; Girosi, 1997; Smola & Sch¨olkopf, 2000).
has to resort to approximate solutions. Various ap-
proximation schemes have been studied to that ex-
tent: Laplace approximation (Williams & Barber,
1998; Williams & Seeger, 2000), variational methods
(Jaakkola & Jordan, 1996), mean field approximations
(Opper & Winther, 2000), and expectation propaga-
tion (Minka, 2001; Seeger et al., 2003). Performing
these methods usually involves the computation of the
Hessian matrix as well as the inversion of K, a nm×nm
matrix, which is not tractable for large data sets (of
size n) and/or large label sets (of size m). Several tech-
niques have been proposed to approximate Ksuch that
the inversion of the approximating matrix is tractable
(cf. (Sch¨olkopf & Smola, 2002) for references on such
methods). One can also try to solve (6) using greedy
optimization methods as proposed in (Bennett et al.,
3. GP Sequence Classification (GPSC)
3.1. Sequence Labeling and GPC
In sequence classification, our goal is to learn a dis-
criminant function for sequences, i.e. a mapping from
observation sequences X= (x1,x2,...,xt,...,xT) to
label sequences y= (y1, y2,...,yt,...,yT). There ex-
ists a label ytΣ = {1,...,r}for every observation
xtin the sequence. Thus, we have Tmulticlass classifi-
cation problems. Because of the sequence structure of
the labels ( i.e. every label ytdepends on its neighbor-
ing labels ), one needs to solve these Tclassification
problems jointly. Then, the problem can be consid-
ered as a multiclass classification where for an obser-
vation sequence of length l, the possible label set Yis
of size m=rl.3We call Σ label set of observations
or micro-label set, and Ythe set of label sequences of
observation sequences or macro-label set.
We assume that a training set of nlabeled sequences
Z≡ {(Xi,yi)|i= 1,...,n}is available. Using the
notation introduced in the context of GP classification,
we define p(yi|u(Xi)) as in (1), treating every macro
label as a separate label in GP multiclass classification
and using the whole sequence Xias the input.
3.2. Kernels for Labeled Sequences
The fundamental design decision is then the engineer-
ing of the kernel function kthat determines the kernel
matrix K. Notice that the use of a block diagonal ker-
nel matrix is not an option in the current setting, since
it would prohibit generalizing across label sequences
that differ in as little as a single micro-label.
3For notational convenience we will assume that all
training sequences are of the same length l.
We define the kernel function for labeled sequences
with respect to the feature representation. Inspired
by HMMs, we use two types of features: Features that
capture the dependency of the micro-labels on the at-
tributes of the observations Φ(xs) and features that
capture the inter-dependency of micro-labels. As in
other discriminative methods, Φ(xs) can include over-
lapping attributes of xsas well as attributes of obser-
vations xtwhere t6=s. Using stationarity, the inner
product between the feature vectors of two observation
sequences can be stated as: k=k1+k2, where
[[ys= ¯yt]]hΦ(xs),Φ(¯
[[ys= ¯ytys+1 = ¯yt+1]] (7b)
k1couples observations in both sequences that are
classified with the same micro-labels at respective po-
sitions. k2simply counts the number of consecutive
label pairs both label sequences have in common (ir-
respective of the inputs). One can generalize (7) in
various ways, e.g. by using higher order terms between
micro-labels in both contributions, without posing ma-
jor conceptual challenges.
kis a linear kernel function for labeled sequences. This
can be generalized to non-linear kernel functions for
labeled sequences by replacing hΦ(xs),Φ(¯
xt)iwith a
standard kernel function defined over input patterns.
We can naively follow the same line of argumentation
as in the GPC case of Section 2, evoke the Representer
Theorem and ultimately arrive at the objective in (6).
Since we need it for subsequent derivations, we will
restate the objective here
R(α|Z) = αTKα
log X
exp ¡αTKe(i,y)¢(8)
Notice that in the third term, the sum ranges over
the macro-label set, Y, which grows exponentially in
the sequence length. Therefore, this view suffers from
the large cardinality of Y. In order to re-establish
tractability of this formulation, we use a trick simi-
lar to the one deployed in (Taskar et al., 2004) and
reparametrize the objective in terms of an equivalent
lower dimensional set of parameters. The crucial ob-
servation is that the definition of kin (7) is homo-
geneous (or stationary). Thus, the absolute positions
of patterns and labels in the sequence are irrelevant.
This observation can be exploited by re-arranging the
sums inside the kernel function with the outer sums,
i.e. the sums in the objective function.
3.3. Exploiting Kernel Structure
In order to carry out this reparameterization more for-
mally we proceed in two steps. The first step consists
of finding an appropriate low-dimensional summary of
α. In particular, we are looking for a parameteriza-
tion that does not scale with m=rl. The second step
consists of re-writing the objective function in terms
of these new parameters.
As we will prove subsequently, the following linear map
Λextracts the information in αthat is relevant for
solving (8):
γΛα, Λ∈ {0,1}n·l·r2×n·m(9)
λ(j,t,σ,τ),(i,y)δij [[yt=σyt+1 =τ]] (10)
Notice that each variable λ(j,t,σ,τ),(i,y )encodes
whether the input sequence is the j-th training se-
quence and whether the label sequence ycontains
micro-labels σand τat position tand t+ 1, respec-
tively. Hence, γ(j,t,σ,τ )is simply the sum of all α(j,y)
over label sequences ythat contain the στ-motif at
position t.
We define two reductions derived from γvia further
linear dimension reduction,
γ(1) Pγ, with P(i,s,σ ),(j,t,τ,ρ)=δij δst δστ ,(11a)
γ(2) Qγ, with Q(i,σ,ζ ),(j,t,τ,ρ)=δij δστ δζ ρ .(11b)
Intuitively, γ(2)
(i,σ,τ)is the sum of all α(i,y)over ev-
ery position in the sequence ythat contains στ -motif.
i,s,σ, on the other hand, is the sum of all α(i,y)that
has σmicro-label at position sin macro-label y.
We can now show how to represent the kernel matrix
using the previously defined matrices Λ,P,Qand the
gram matrix Gwith G(i,s),(j,t)=g(xs
Proposition 1. With the definitions from above:
where H= diag(G,...,G).
Proof. By elementary comparison of coefficients.
We now have r2parameters for every observation xsin
the training data (nlr2parameters) and we can rewrite
the objective function in terms of these variables:
R(γ|X,y) = γTK0γ
log X
exp ¡γTK0Λe(i,y)¢(12)
3.4. GPSC and Other Label Sequence
Learning Methods
We now briefly point out the relationship between our
approach and the previous discriminative methods of
sequence learning, in particular, CRFs, HM-SVMs and
CRF is a natural generalization of logistic regression to
label sequence learning. The probability distribution
over label sequences given an observation sequence is
given in Eq. (1), where u(X,y) = hθ, Ψ(X,y)iis a
linear discriminative function over some feature repre-
sentation Ψ parameterized with θ. The objective func-
tion of CRFs is the minimization of the negative condi-
tional likelihood of training data. To avoid overfitting,
it is common to multiply the conditional likelihood by
a Gaussian with zero mean and diagonal covariance
matrix K, resulting in an additive term in log scale.
log p(θ|X,y) =
log p(yi|Xi, θ) + θTKθ(13)
From a Bayesian point of view, CRFs assume a uni-
form prior p(u), if there is no regularization term.
When regularized, CRFs define a Gaussian distribu-
tion over a finite vector space θ. In GPSC, on the
other hand, the prior is defined as a Gaussian dis-
tribution over the function space of possibly infinite
dimension. Thus, GPSC generalizes CRFs by defin-
ing a more sophisticated prior on the discriminative
function u. This prior leads to the ability of using
kernel function in order to construct and learn over
Reproducing Kernel Hilbert Spaces. So, GPSC, a non-
parametric Bayesian inference tool for sequence label-
ing, can overcome the limitations of CRFs, parametric
(linear) statistical models. When the kernel that de-
fines the covariance matrix Kin GPSC is linear, uin
both models become equivalent.
The difference between SVM and GP approaches to
sequence learning is the utilized loss function over the
training data, i.e. hinge loss vs. log loss. GPSC ob-
jective function parameterized with α(Eq. (8)) corre-
sponds to HM-SVMs where the number of parameters
scale exponentially with the length of sequences. The
objective function parameterized with γ(Eq. (12)) cor-
responds to MMMs, where the number of parameters
scale only linearly.
4. GPSC Optimization Algorithm
4.1. A Dense Algorithm
Using optimization methods described in Section 2 re-
quires the computation of the Hessian matrix. In se-
quence labeling, this corresponds to computing the ex-
pections of micro-labels within different cliques, which
is not tractable to compute exactly for large training
sets. In order to minimize Rwith respect to γ, we pro-
pose a 1storder exact optimization method, which we
call Dense Gaussian Process Sequence Classification
It is well-known that the derivatives of the log partition
function with respect to γis simply the expectation of
sufficient statistics:
log X
exp ¡γTK0Λe(i,y)¢
=EY£Λe(i,Y )¤(14)
where EYdenotes an expectation with respect to the
conditional distribution of the label sequence ygiven
the observation sequence Xi. Then, the gradients of
Ris trivially given by:
K0EY£Λe(i,Y )¤
The remaining challenge is to come-up with an efficient
way to compute the expectations. First of all, let us
more explicitly examine these quantities:
EY[(Λe(i,Y ))(j,t,σ,τ)]=δij EY£[[Yt=σYt+1 =τ]]¤(16)
In order to compute the above expectations one can
once again exploit the structure of the kernel and is
left with the problem of computing probabilities for
every neighboring micro-label pair (σ, τ ) at positions
(t, t + 1) for all training sequences Xi. The latter can
be accomplished by performing the forward-backward
algorithm over the training data using the transition
probability matrix Tand the observation probability
matrices O(i), which are simply decompositions and
reshapings of K0:
¯γ(2) Rγ(2) ,with R(σ,ζ),(i,τ,ρ)=δσ τ δζρ (17a)
Tγ(2) ]r,r (17b)
O(i)= [γ(1)]n·l,r G(i,.),(.,.)(17c)
where [x]m,n denotes the reshaping operation of a vec-
tor xinto an mnmatrix, AI,J denotes the |I| ∗ |J|
sub-matrix of Aand (.) denotes the set of all possible
A single optimization step of DGPS is described in
Algorithm 1. The complexity of one optimization step
is O(t2) dominated by the forward-backward algorithm
Algorithm 1 One optimization step of Dense Gaus-
sian Process Sequence Classification (DGPS)
Require: Training data (Xi,yi)i=1:n; Proposed pa-
rameter values γc
1: Initialize γ(1)
c, γ(2)
c(Eq. (11)).
2: Compute Twrt γ(2)
c(Eq. (17a), Eq. (17b)).
3: for i= 1,...,n do
4: Compute O(i)wrt γ(1)
c(Eq. (17c)).
5: Compute p(yi|Xi, γc) and
EY£[[Yt=σYt+1 =τ]]¤for all t, σ, τ via
forward-backward algorithm using O(i)and T
6: end for
7: Compute γR(Eq. (15)).
over all instances where t=nlr2. We propose to use
a quasi-Newton method for the optimization process.
Then, the overall complexity is given by O(ηt2) where
η < t2. The memory requirement is given by the size
of γ,O(t).
During inference, one can find the most likely label
sequence for an observation sequence Xby performing
Viterbi decoding using the transition and observation
probability matrices described above.
4.2. A Sparse Algorithm
While the above method is attractive for small data
sets, the computation or the storage of K0poses a
serious problem when the data set is large. Also, clas-
sification of a new observation involves evaluating the
covariance function at nl data points, which is more
than acceptable for many applications. Hence, as in
the case of standard Gaussian Process Classification
discussed in Section 2, one has to find a method for
sparse solutions in terms of the γparameters to speed
up the training and prediction stages.
We propose a sparse greedy method, Sparse Gaussian
Process Sequence Classification (SGPS), that is simi-
lar to the method presented by (Bennett et al., 2002).
SGPS starts with an empty matrix ˆ
K. At each iter-
ation, SGPS selects a training instance Xiand com-
putes the gradients of the parameters associated with
Xi,γ(i,.), to select the steepest descent direction(s)
of Rover this subspace. Then ˆ
Kis augmented with
these columns and SGPS performs optimization of the
current problem using a Quasi-Newton method. This
process is repeated until the gradients vanish (i.e. they
are smaller than a threshold value η) or a maximum
number of γcoordinates, p, are selected (i.e. some
sparseness level is achieved). Since the bottleneck of
this method is the computation of the expectations,
EY£[[Yt=σYt+1 =τ]]¤, we pick the steepest ddirec-
tions, once the expectations are computed.
One has two options to compute the optimal γat every
iteration: by updating all of the γparameters selected
until now, or alternatively, by updating only the pa-
rameters selected in the last iteration. We prefer the
latter because of its less expensive iterations. This ap-
proach is in the spirit of a boosting algorithm or the
cyclic coordinate optimization method.
Algorithm 2 Sparse Gaussian Process Sequence Clas-
sification (SGPS) algorithm.
Require: Training data (Xi,yi)i=1:n; Maximum
number of coordinates to be selected, p,p < nlr2;
Threshold value ηfor gradients
1: K[]
2: for i= 1,...,n do
3: Compute γ(i,.)R(Equation 15).
4: sSteepest ddirections of γ(i,.)R
5: ˆ
6: Optimize Rwrt s.
7: Return if γ< η or pcoordinates selected.
8: end for
SGPS is described in Algorithm 2. Its complexity is
O(p2t) where pis the maximum number of coordinates
5. Experiments
5.1. Pitch Accent Prediction
Pitch Accent Prediction is the task of identifying more
prominent words in a sentence. The micro-label set is
of size 2, accented and not-accented. We used phonet-
ically hand-transcribed Switchboard corpus consisting
of 1824 sentences (13K words) (Greenberg et al., 1996).
We extracted probabilistic, acoustic and textual infor-
mation from the current, previous and next words for
every position in the training data. We used 1storder
Markov features to capture the dependencies between
neighboring labels.
We compared the performance of CRFs and HM-
SVMs with the GPSC dense and sparse methods ac-
cording to their test accuracy in 5-fold cross valida-
tion. CRFs were regularized and optimized using lim-
ited memory BFGS, a limited memory Quasi-Newton
optimization method. When performing experiments
on DGPS, we used polynomial kernels with different
degrees (denoted with DGPSXin Figure 1a where
X∈ {1,2,3}is the degree of the polynomial kernel).
We used third order polynomial kernel in HM-SVMs
(denoted with SVM3 in Figure 1). As expected, CRFs
Test Accuracy
0 1 2 3 4 5 6 7 8
Sparseness %
0.5 0.6 0.7 0.8 0.9 1
Figure 1. Pitch Accent Prediction task results a) Test accuracy of Pitch Accent Prediction task over a window of size
3 using 5-fold cross validation. b) Test accuracy of Pitch Accent Prediction w.r.t. the sparseness of the solution. c)
Precision-Recall curves for different threshold probabilities to abstain.
and DGPS1 performed very similar. When 2ndor-
der features were incorporated implicitly using second
degree polynomial kernel (DGPS2), the performance
increased dramatically. Extracting 2ndorder features
explicitly results in a 12 million dimensional feature
space, where CRFs slow down dramatically. We ob-
served that 3rdorder features do not provide signifi-
cant improvement over DGPS2. HM-SVM3 performs
slightly worse than DGPS2.
To investigate how the sparsity of SGPS affects its per-
formance, we report the test accuracy with respect to
the sparseness of SGPS solution in Figure 1b. Sparse-
ness is measured by the percentage of the parameters
selected by SGPS. The straight line is the performance
of DGPS using second degree polynomial kernel. Us-
ing 1% of the parameters, SGPS achieves 75% accu-
racy (1.48% less than the accuracy of DGPS). When
7.8% of the parameters are selected, the accuracy is
76.18% which is not significantly different than the
performance of DGPS (76.48%). We observed that
these parameters were related to 6.2% of the obser-
vations along with 1.13 label pairs on average. Thus,
during inference one needs to evaluate the kernel func-
tion only at 6% of the observations which reduces the
inference time dramatically.
In order to experimentally verify how useful the pre-
dictive probabilities are as confidence scores, we forced
DGPS to abstain from predicting a label when the
probability of a micro-label is lower than a threshold
value. In Figure 1c, we plot precision-recall values
for different thresholds. We observed that the error
rate for DGPS decreased 8.54%, abstaining on 14.93%
of the test data. The improvement on the error rate
shows the validity of the probabilities generated by
5.2. Named Entity Recognition
Named Entity Recognition (NER), a subtask of In-
formation Extraction, is finding phrases containing
names in a sentences. The micro-label set consists
of the beginning and continuation of person, location,
organization and miscellaneous names and non-name.
We used a Spanish newswire corpus, which was pro-
vided for the Special Session of CoNLL 2002 on NER,
to randomly select 1000 sentences (21K words). We
used the word and its spelling properties of the cur-
rent, previous and next observations.
Error 4.58 4.39 4.48 4.92 4.56
Table 1. Test error of NER over a window of size 3 using
5-fold cross validation.
The experimental setup was similar to pitch accent
prediction task. We compared the performance of
CRFs with and without the regularizer term (CRF-
R, CRF) with the GPSC dense and sparse methods.
Qualitatively, the behavior of the different optimiza-
tion methods is comparable to the pitch accent predic-
tion task. The results are summarized in Table 1. Sec-
ond degree polynomial DGPS outperformed the other
methods. We set the sparseness parameter of SGPS to
25%, i.e. p= 0.25nlr2, where r= 9 and nl = 21K on
average. SGPS with 25% sparseness achieves an accu-
racy that is only 0.1% below DGPS. We observed that
19% of the observations are selected along with 1.32
label pairs on average, which means that one needs to
compute only one fifth of the gram matrix.
We also tried a sparse algorithm that does not exploit
the kernel structure and optimizes Equation 8 to ob-
tain sparse solutions in terms of observation sequences
Xand label sequence y, as opposed to SPGS, where
the sparse solution is in terms of observations and label
pairs. This method achieved 92.7% of accuracy, hence,
was clearly outperformed by all the other methods.
6. Conclusion and Future Work
We presented GPSC, a generalization of Gaussian Pro-
cess classification to label sequence learning problem.
This method combines the advantages of the rigor-
ous probabilistic semantics of CRFs and overcomes
the curse of dimensionality problem using kernels in
order to construct and learn over RKHS. The experi-
ments on named entity recognition show the compet-
itiveness and the experiments on pitch accent predic-
tion show the superiority of our approach in terms of
the achieved error rate. We also experimentally veri-
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