Word Vector Compositionality based Relevance Feedback using Kernel Density Estimation

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Word Vector Compositionality based Relevance
Feedback using Kernel Density Estimation
Dwaipayan Roy, Debasis Ganguly,
Mandar Mitra, Gareth J.F. Jones
Indian Statistical Institue Dublin City University
Kolkata, India Dublin, Ireland
D. Roy (ISI, Kolkata) KDE-RLM 1 / 48

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Christopher Manning - SIGIR 2016 Keynote
Source: Natural Language Inference, Reading Comprehension and Deep Learning - Chris Manning - SIGIR 2016

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Outline
1 Motivation
2 Kernel Density Estimation
3 Embedding based Relevance Feedback
4 Evaluation
5 Conclusion

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Outline
1 Motivation
4 Evaluation
5 Conclusion

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Relevance based Language Model (RLM)1
Assumes that both query and relevant documents are sampled from a
latent relevance model R.
The task is to find (estimate) the density function for R.
1
Lavrenko and Croft - SIGIR 2001

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RM32
RM3
A mixture model of RLM and query likelihood model.
P′
(w|R) = µP(w|R) + (1 − µ)P(w|Q)
2
UMass at TREC 2004: Novelty and HARD - Jaleel et al.

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Relevance Model
Relevance Model
Captures co-occurence information of terms with query in top docs.

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Word Embedding
Figure source: T-SNE Output

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Word Embedding
Represents every word as a low dimensional vector in an abstract
space.
Figure source: T-SNE Output

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Word Embedding
space.
Similarity between the word vectors can reflect the semantic similarity
between the constituent terms.
Figure source: “A Simple Introduction to Word Embeddings” - B. Mitra

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Word Embedding
space.
Effect of conceptual composition by simple addition of the embedded
vectors.
u
v
w1
w2

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Word Embedding
space.
Effect of conceptual composition by simple addition of the embedded
vectors.
u
v
w1
w2
German + Airlines ∼ Lufthansa

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Word Embedding
Word Embedding
Captures the semantic relationship between terms.

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Motivation of this work
Word Vector Compositionality based Relevance Feedback
using Kernel Density Estimation

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Motivation of this work
Word Vector Compositionality based Relevance Feedback
using Kernel Density Estimation
Relevance feedback:
Captures co-occurence information of terms with query in top docs.
Word embedding:

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Outline
1 Motivation
4 Evaluation
5 Conclusion

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Kernel Density Estimation (KDE)
76543210-1
5
4
3
2
1
0
76543210-1
5
4
3
2
1
0
76543210-1
5
4
3
2
1
0
Estimate a distribution that generates the given data (data points).
Place kernel function (e.g. Gaussian) centered around each observed
data point.
Combine the Gaussians to get a function peaked at the observed data
point.
f(x) =
1
nh
n∑
i=1
K(
x − xi
h
)

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f(x) =
1
nh
n∑
i=1
K(
x − xi
h
)
h - bandwidth (smoothing parameter)
n - number of data points
xi - ith data point
K() - the kernel

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KDE with Gaussian Kernel
f(x) =
1
nh
n∑
i=1
K(
x − xi
h
)

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f(x) =
1
nh
n∑
i=1
K(
x − xi
h
)
f(x) =
1
nh
n∑
i=1
1
√
2πσ
e− 1
2σ2 (
x−xi
h
)
2

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f(x) =
1
nh
n∑
i=1
K(
x − xi
h
)
f(x) =
1
nh
n∑
i=1
1
√
2πσ
e− 1
2σ2 (
x−xi
h
)
2
h and σ: tunable parameters

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Relevance Model
Relevance Model
Estimates the density function that generates the relevance model R.

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Kernel Density Estimation
Given a set of n data samples, estimates the probability density function
that generates the observed data.

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KDE in Relevance Feedback Scenario
f(x) =
1
nh
n∑
i=1
K(
x − xi
h
)
The Points
Observed data points (xi): Q = {q1, . . . , qn}
Points at which probability (density) is to be estimated (x):
w → Candidate expansion term

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KDE in Relevance Feedback Scenario
f(x) =
1
nh
n∑
i=1
K(
x − xi
h
)
The Points
f(w) =
1
nh
n∑
i=1
K(
w − qi
h
)

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Relevance Feedback with KDE
f(w2)
f(q2)
f(w1)
q1 q2 q3
w1 w2
One dimensional projection of embedded vectors

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Kernel Density Estimate (Unweighted)
f(w) =
1
nh
n∑
i=1
K(
w − qi
h
)
f(w2)
f(q2)
f(w1)
q1 q2 q3
w1 w2
n - number of samples
h - bandwidth
K(·) - kernel function

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Kernel Density Estimate (Weighted)
f(w, α) =
1
nh
n∑
i=1
αiK(
w − qi
h
)
f(w2)
f(q2)
f(w1)
q1 q2 q3
w1 w2
n - number of samples
h - bandwidth
K(·) - kernel function
αi - weight of local
Kernel function around
ith data point

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Outline
1 Motivation
4 Evaluation
5 Conclusion

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One dimensional KDE
f(w2)
f(q2)
f(w1)
q1 q2 q3
w1 w2
f(w, α) =
1
nh
n∑
i=1
αi
1
√
2πσ
e−
(w−qi)2
2σ2h2
αi : weight of the ith query term

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One dimensional KDE
f(w, α) =
1
nh
n∑
i=1
αi
1
√
2πσ
e−
(w−qi)2
2σ2h2

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One dimensional KDE
f(w, α) =
1
nh
n∑
i=1
αi
1
√
2πσ
e−
(w−qi)2
2σ2h2
(w − qi)2
= (w − qi)T
(w − qi)

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One dimensional KDE
f(w, α) =
1
nh
n∑
i=1
αi
1
√
2πσ
e−
(w−qi)2
2σ2h2
(w − qi)2
= (w − qi)T
(w − qi)
αi = P(w|M)P(qi|M)
M : set of terms of the union of M top ranked documents

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One dimensional KDE
f(w, α) =
n∑
i=1
P(w|M)P(qi|M)
1
σ
√
2π
exp(−
(w − qi)T(w − qi)
2σ2h2
)
Captures co-occurence information of terms with query in top
docs.

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Two dimensional KDE
In 1D-KDE, set of all top ranked documents considered as a single
document model (M).
We now generalize the 1D-KDE to two dimensions so as to weight
the contributions obtained from different documents in the ranked list
separately.

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Two dimensional KDE
q1 q2 q3 Terms
Documents
D1
D2
2
1
D1
D3
The x-axis corresponds to the query term vectors similar to one
dimensional KDE.
The y-axis represents the normalized term frequency of query terms in
respective feedback documents.

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Two dimensional KDE
The Points: One Dimensional KDE

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Two dimensional KDE
The Points: Two Dimensional KDE
Observed data points (xij) = (qi, Dj): Encapsulates word vector for
i-th query word qi and its normalized term frequency in feedback
document Dj (P(qi|Dj)).
Points at which probability (density) is to be estimated (x)
= (w, Dj): Encapsulates word vector for candidate expansion term w
and its normalized term frequency in feedback document Dj
(P(w|Dj)).

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Two dimensional KDE
f(x, α) =
n∑
i=1
M∑
j=1
P(w|Dj)P(qi|Dj)
2πσ2
exp(
(w − qi)2 + (P(w|Dj)−P(qi|Dj))2
−2σ2h2
)

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Two dimensional KDE
f(x, α) =
n∑
i=1
M∑
j=1
P(w|Dj)P(qi|Dj)
2πσ2
exp(
(w − qi)2 + (P(w|Dj)−P(qi|Dj))2
−2σ2h2
)
w in document Dj, denoted as x, will get a high value of the density
function if:
(i) w is semanticlly close to query terms qi;
(ii) w frequently co-occurs with query terms in each top ranked
document Dj.

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Term Composition in KDE
Extending the set of pivot points:
q1 q2q1+q2
w
(a) Without composition
q1 q2q1+q2
w
(b) With composition

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Importance of Composition
ISI Inter Services
Intelligence
Infrared
Spatial In-
terferometer
Indian
Statistical
Institute
Institute for
Scientific
Interchange
Islamic
State of Iraq
Integral
Satcom
Initiative
Ice Skating
Institute
Islamic
School
of Irving

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Importance of Composition
ISI
Kolkata
Inter Services
Intelligence
Infrared
Spatial In-
terferometer
Indian
Statistical
Institute
Institute for
Scientific
Interchange
Islamic
State of Iraq
Integral
Satcom
Initiative
Ice Skating
Institute
Islamic
School
of Irving

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Summary
Given:
Corpora : Set of documents.
Query Q : q1, . . . , qn.
Vector embeddings of each of the terms.
First level retrieval performed using LM.
ET = All terms of top ranked M documents considered as potential
expansion terms.
Calculate KDE(w) for each w ∈ ET.
Terms with high estimated value taken as expansion terms.
Linearly interpolate derived density function with underlying query
model.

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Outline
1 Motivation
4 Evaluation
5 Conclusion

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Evaluation Dataset
Document Document #Docs Vocab Query Query Set Dev Test
Collection Type Size Fields Set Set
TREC
News 528,155 242,036 Title
TREC 6 ad-hoc ✓
TREC 7 ad-hoc ✓
Disks 4, 5 TREC 8 ad-hoc ✓
TREC Robust ✓
WT10G Web pages 1,692,096 1,659,231 Title
TREC 9 Web ✓
TREC 10 Web ✓
Table: Dataset Overview

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Baseline Methods
Standard LM with Jelinek Mercer Smoothing (weak baseline).
K-NN based Query Expansion using Word Embeddings.
Relevance Feedback baseline method: RM3.

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Parameters
KDE specific parameters
h: bandwidth
σ: standard deviation of Gaussian Kernel

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Parameters
KDE specific parameters
h: bandwidth (Set to 1)
σ: standard deviation of Gaussian Kernel (Trained on dev. set)

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Parameter Tuning
Effect of varying σ:
(a) TREC 6 (b) TREC 9
Figure: Effect of varying σ (h set to 1) for KDE feedback models.

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Parameter Settings
Parameter Settings
Bandwidth h = 1
Standard Deviation σ = 0.6
LM Mixing Weight λ = 0.4
Query Model Mixing Weight µ = 0.6

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Evaluation: TREC 678, Robust
Dataset Method Parameters Metrics
#fdbk-docs #exp-terms MAP Recall P@5
TREC 6
LM n/a n/a 0.2355 0.5100 0.4040
k-NN n/a 40 0.2406 0.5214 0.4000
RM3 20 70 0.2634 0.5368 0.4360
1d KDE 10 80 0.2818†
0.5757†
0.4680†
2d KDE 10 80 0.2850†
0.5724†
0.4600†
TREC 7
LM n/a n/a 0.1787 0.4893 0.4040
k-NN n/a 40 0.1806 0.4833 0.4080
RM3 20 70 0.2151 0.5432 0.4160
1d KDE 10 80 0.2351†
0.6001†
0.4425†
2d KDE 10 80 0.2380 0.6108†‡
0.4400
TREC 8
LM n/a n/a 0.2466 0.5936 0.4560
k-NN n/a 40 0.2535 0.6178 0.4320
RM3 20 70 0.2701 0.6410 0.4760
1d KDE 10 80 0.2746 0.6749†
0.4888
2d KDE 10 80 0.2957†‡
0.6887†
0.5120†‡
TREC Rb
LM n/a n/a 0.2699 0.7938 0.4464
k-NN n/a 40 0.2842 0.8075 0.4949
RM3 20 70 0.3304‡
0.8559 0.4949
1d KDE 10 80 0.3228 0.8725 0.4929
2d KDE 10 80 0.3456†‡
0.8772†‡
0.5152†‡
Table: † and ‡ denote significance with respect to RM3 and 1d KDE respectively.

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Evaluation: WT10G
Dataset Method Parameters Metrics
#fdbk-docs #exp-terms MAP Recall P@5
TREC 9
LM n/a n/a 0.1814 0.6175 0.2839
k-NN n/a 40 0.1817 0.6224 0.3080
RM3 20 70 0.1930 0.6755 0.3233
1d KDE 10 80 0.1984 0.6851 0.3360
2d KDE 10 80 0.2145†‡
0.6878 0.3562†‡
TREC 10
LM n/a n/a 0.1625 0.6640 0.3224
k-NN n/a 40 0.1629 0.7033 0.3429
RM3 20 70 0.1759 0.7386 0.3347
1d KDE 10 80 0.2192†
0.7499 0.4004†
2d KDE 10 80 0.2213†
0.7502 0.4204†
Table: † and ‡ denote significance with respect to RM3 and 1d KDE respectively.

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Outline
1 Motivation
4 Evaluation
5 Conclusion

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Contributions
Integration of semantic information with co-occurrence statistics.
Integration of Kernel Density Estimate in Relevance Feedback.
Incorporation of Word Compositionality in the model.

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Future Work

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Future Work
Future Work
Applications of larger units of embeddings (e.g.
sentences, paragraphs) for further improvement
of retrieval performance.

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Thank You!

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Travel Support

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1D-KDE Details
P(w|M) : tf(w,M)
|M|
P(qi|M) : tf(qi,M)
|M|
h: Bandwidth set to one (1)
σ: Trained on the development set (TREC 6 and TREC 9)
Return

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2D-KDE Details
(w − qi)2
= (w − qi)T(w − qi)
P(w|Dj) :
tf(w,Dj)
|Dj|
P(qi|Dj) :
tf(qi,Dj)
|Dj|
h: Bandwidth set to one (1)
σ: Trained on the development set (TREC 6 and TREC 9)
covariance matrix σ as a diagonal matrix with equal covariance for
both the dimensions.
Return

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Parameter Details
Intuition behind setting h to 1
Distance between two word vectors always lies between 0 and 1
For each pivot point (query term), the points with distance of at
most 0.5 (h/2) from each pivot point estimated to be most similar to
the query term.
Back

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Why Not Using Larger Collection
Aimed to investigate whether semantic relationships between words
can improve IR effectiveness.
Tested our approach on collections with relevance judgements
collected with significant pool depths, mostly giving a chance for
semantically similar relevant documents to be considered for
evaluation
TREC ad-hoc, WT10G datasets use a pool depth of 100 for relevance
assessments
Did not use some of the newer collections, e.g. TREC Terabyte track
(pool-depth 50) TREC web track (pool-depth 25)

Word Vector Compositionality based Relevance Feedback using Kernel Density Estimation

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