Compression and Predictive Distributions for Large Alphabets

107 Pages·2015·2.95 MB·English

Abstract
Compression and Predictive Distributions for
Large Alphabets
Xiao Yang
2015
Data generated from large alphabet exist almost everywhere in our life, for exam­
ple, texts, images and videos. Traditional universal compression algorithms mostly
involve small alphabets and assume implicitly an asymptotic condition under which
the extra bits induced in the compression process vanishes as an infinite number of
data come. In this thesis, we put the focus on compression and prediction for large
alphabets with the alphabet size comparable or larger than the sample size.
Wc first consider sequences of random variables independent and identically gen­
erated from a large alphabet. In particular, the size of the sample is allowed to be
variable. A product distribution based on Poisson sampling and tilting is proposed
as the coding distribution, which highly simplifies the implementation and analysis
through independence. Moreover, we characterize the behavior of the coding dis­
tribution through a condition on the tail sum of the ordered counts, and apply it
to sequences satisfying this condition. Further, we apply this method to envelope
classes. This coding distribution provides a convenient method to approximately
compute the Shtarkov’s normalized maximum likelihood (NML) distribution. And
the extra price paid for this convenience is small compared to the total cost. Fur­
thermore, we find this coding distribution can also be used to calculate the NML
distribution by a Monte Carlo method with no extra price. And this calculation
remains simple due to the independence of the coding distribution.
2
Further, we consider a more realistic class - the Markov class, and in particular,
tree sources. A context tree based algorithm is designed to describe the dependencies
among the contexts. It is a greedy algorithm which seeks for the greatest savings
in codelength when constructing the tree. Compression and prediction of individual
counts associated with the contexts uses the same coding distribution as in the i.i.d
case. Combining these two procedures, we demonstrate a compression algorithm
based on the tree model.
Results of simulation and real data experiments for both the i.i.d model and
Markov model have been included to illustrate the performance of the proposed
algorithm.
Compression and Predictive Distributions
for Large Alphabets
A Dissertation
Presented to the Faculty of the Graduate School
of
Yale University
in Candidacy for the Degree of
Doctor of Philosophy
by
Xiao Yang
Dissertation Director: Andrew R Barron
May 2015
UMI Number: 3663558
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Contents
1 Introduction 1
1.1 Universal compression .................................................................................. 2
1.2 Normalized maximum likelihood distribution........................................ 4
2 i.i.d model 5
2.1 Introduction................................................................................................ 6
2.2 The Poisson M odel.................................................................................... 20
2.3 Results.......................................................................................................... 22
2.3.1 Regret............................................................................................... 22
2.3.2 Subset of sequences with partitioned counts............................... 25
2.3.3 Envelope class................................................................................... 29
2.3.4 Regret with unknown total count................................................ 31
2.3.5 Conditional distributions induced by the tilted Stirling ratio
distribution........................................................................................ 34
2.3.6 Computational simplicity................................................................ 35
2.3.7 Computating Shtarkov’s NML distribution using Qa .................. 36
2.3.8 Prediction......................................................................................... 37
2.4 Application................................................................................................. 39
2.4.1 Simulation......................................................................................... 39
iii
2.4.2 Real d a ta ........................................................................................ 42
2.5 Discussion.................................................................................................... 42
3 Markov model 45
3.1 Introduction................................................................................................. 46
3.2 i.i.d class....................................................................................................... 54
3.3 Tree source ................................................................................................. 55
3.3.1 Coding cost..................................................................................... 55
3.3.2 Description cost............................................................................... 56
3.3.3 Using codelength to construct the tre e ...................................... 56
3.4 A real example........................................................................................... 57
3.5 Conclusion.................................................................................................... 57
3.6 Discussion.................................................................................................... 58
4 Summary and future work 60
Appendices 63
A Proof of Theorems 64
A.l Some facts.................................................................................................... 65
A.2 Proof of Theorem 3.2................................................................................. 74
A.3 Proof of Pythagorean Equality................................................................... 79
A.4 Redundancy................................................................................................. 81
A.5 Proof of Theorem 2.3.................................................................................. 84
B Supplementary materials 87
B.l Incompatibility of ................................................................................. 88
P n
B.2 Computation complexity........................................................................... 88
iv
B.3 Approximation of c
List of Figures
2.1 Relationship between a and Ca.................................................................. 14
2.2 Relationship between a* and ^ .................................................................. 16
2.3 Regret for case m ~ n................................................................................. 26
2.4 Relationship between a and ................................................................... 33
Va
2.5 Regret of using tilted Stirling ratio distribution for algebraically de­
creasing counts 40
2.6 Regret of using tilted Stirling ratio distribution for an algebraically
decreasing envelope class.............................................................................. 41
2.7 Regret of Qa^ for L from 1 to m................................................................ 43
3.1 An example context tree with A = {a, b, c, d} where • represents “oth­
ers” .................................................................................................................. 53
3.2 Context tree for Fortress Beseiged............................................................. 59
A.l tilted distribution and the T(l,1) density with a = 0.01....................... 68
A.2 Tilted distribution and the Gamma density. The relevant sum is only
to the left of T with a = 0.01..................................................................... 70
A.3 Tilted distribution and the Gamma density. The relevant sum is only
to the right of T with a = 0.01.................................................................. 71
vi
Acknowledgements
The past six years is a long journey, but I am very happy I made it. At the end
of this trip, I want to sincerely thank my advisor Prof.Andrew Barron. He guided
me onto this challenging and interesting path, and gave me endless patience and
support. I used to think he is a genius. Since it looks like he knows everything in
statistics and can always find the crux, but he doesn’t seem to research a lot (mostly
because he has other responsibilities like teaching and advising students). Later on,
I heard a story about him. It said when he was a graduate student, he read tons of
papers and sometimes slept in the office. This is not a unique story, but it told me
that the distance between “genius” and me is this determination and commitment.
Besides dedication to research, his generous and positive personality also influenced
me a lot. He never hesitates to share ideas or to compliment others. I still remember
once I had some small progress in my research, he was so happy that he gave me a
high five. I am extremely grateful to have him as my advisor.
I also wish to thank my committee Prof. Joseph Chang and Prof. Mokshay
Madiman. One thing I learn from Joe is no matter how complex one thing is, he can
always explain it in a simple and intuitive manner. Only people who truly capture
the essence and turn into their own understandings can do this. To me, it is a magic
ability. Mokshay also gave me strong support. He never said no when I requested to
meet him even when he was very busy. Because of their presence, I am able to reach
the destination.
Moreover, I would like to thank Prof. Peter Jones, Prof. Jun’ichi Takeuchi, Prof.
Wojciech Szpankowski, Prof. Narayana Santhanam and Prof. Teemu Roos. They