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4.3 Data-Dependent Partition for Divergence Estima-tion:
Problem Statement
Let P and Q be probability measures in (Rd, B(Rd)) absolutely continuous with respect
to the Lebesgue measure, such that D(P||Q) < 1. Let = { 1, 2, · · · } be a partition
scheme for Rd, and let us consider X1, ..,Xn and Y1, .., Ym i.i.d. realizations of random
variables with values in Rd and distributions P and Q, respectively. Then the proposed
candidate for the empirical divergence is given by
ˆD
n,m(P||Q)
X
A2 m(Y1,..,Ym)
Pn(A) · log
Pn(A)
Qm(A)
, (4.9)
where Pn and Qm denote the empirical distributions induced by X1, ..,Xn and Y1, .., Ym
respectively by (4.6), defined in the sub-sigma field ( m(Y1, .., Ym)) B(Rd). Note
that this data-dependent partition is a function only of the i.i.d. realizations associ-ated
with the reference measure Q. Loosely speaking the choice of using only partial
information can be justified by the assumption that P Q. The next section will
formally support this choice by introducing the notion of universally sufficient data-dependent
partition and showing that the proposed construction satisfies this desired
asymptotic property. In addition, we impose in the condition that Qm(A) > 0,
8A 2 m(Y1, .., Ym), that ensures that ˆD
n,m(P||Q) < 1.
ˆD
n,m(P||Q) is a measurable function of X1, ..,Xn and Y1, .., Ym, and consequently
we are interested in studying the strong — almost surely with respect to the joint distri-bution
of {Xn, n 2 N} and {Ym,m 2 N}—universal consistency of ˆD
n,m(P||Q) as m
and n tend to infinity and as a function of the aforementioned notions of combinatorial
complexity for .
96
Object Description
| Title | On optimal signal representation for statistical learning and pattern recognition |
| Author | Silva, Jorge |
| Author email | jorgesil@usc.edu; josilva@ing.uchile.cl |
| Degree | Doctor of Philosophy |
| Document type | Dissertation |
| Degree program | Electrical Engineering |
| School | Viterbi School of Engineering |
| Date defended/completed | 2008-06-23 |
| Date submitted | 2008 |
| Restricted until | Unrestricted |
| Date published | 2008-10-21 |
| Advisor (committee chair) | Narayanan, Shrikanth S. |
| Advisor (committee member) |
Kuo, C.-C. Jay Ordóñez, Fernando I. |
| Abstract | This work presents contributions on two important aspects of the role of signal representation in statistical learning problems, in the context of deriving new methodologies and representations for speech recognition and the estimation of information theoretic quantities.; The first topic focuses on the problem of optimal filter bank selection using Wavelet Packets (WPs) for speech recognition applications. We propose new results to show an estimation-approximation error tradeoff across sequence of embedded representations. These results were used to formulate the minimum probability of error signal representation (MPE-SR) problem as a complexity regularization criterion. Restricting this criterion to the filter bank selection, algorithmic solutions are provided by exploring the dyadic tree-structure of WPs. These solutions are stipulated in terms of a set of conditional independent assumptions for the acoustic observation process, in particular, a Markov tree property across the indexed structure of WPs. In the technical side, this work presents contributions on the extension of minimum cost tree pruning algorithms and their properties to affine tree functionals. For the experimental validation, a phone classification task ratifies the goodness of Wavelet Packets as an analysis scheme for non-stationary time-series processes, and the effectiveness of the MPE-SR to provide cost effective discriminative filter bank solution for pattern recognition.; The second topic addresses the problem of data-dependent partitions for the estimation of mutual information and Kullback-Leibler divergence (KLD). This work proposes general histogram-based estimates considering non-product data-driven partition schemes. The main contribution is the stipulation of sufficient conditions to make these histogram-based constructions strongly consistent for both problems. The sufficient conditions consider combinatorial complexity indicator for partition families and the use of large deviation type of inequalities (Vapnik-Chervonenkis inequalities). On the application side, two emblematic data-dependent constructions are derived from this result, one based on statistically equivalent blocks and the other, on a tree-structured vector quantization scheme. A range of design values was stipulated to guarantee strongly consistency estimates for both framework. Furthermore, experimental results under controlled settings demonstrate the superiority of these data-driven techniques in terms of a bias-variance analysis when compared to conventional product histogram-based and kernel plug-in estimates. |
| Keyword | signal representation in statistical learning; Bayes decision theory; basis selection; tree-structured bases and Wavelet packet (WP); complexity regularization; minimum cost tree pruning; family pruning problem; mutual information estimation; divergence estimation; data-dependent partitions; statistical learning theory; concentration inequalities; tree-structured vector quantization. |
| Language | English |
| Part of collection | University of Southern California dissertations and theses |
| Publisher (of the original version) | University of Southern California |
| Place of publication (of the original version) | Los Angeles, California |
| Publisher (of the digital version) | University of Southern California. Libraries |
| Provenance | Electronically uploaded by the author |
| Type | texts |
| Legacy record ID | usctheses-m1684 |
| Contributing entity | University of Southern California |
| Rights | Silva, Jorge |
| Repository name | Libraries, University of Southern California |
| Repository address | Los Angeles, California |
| Repository email | cisadmin@lib.usc.edu |
| Filename | etd-Silva-2450 |
| Archival file | uscthesesreloadpub_Volume32/etd-Silva-2450.pdf |
Description
| Title | Page 109 |
| Contributing entity | University of Southern California |
| Repository email | cisadmin@lib.usc.edu |
| Full text | 4.3 Data-Dependent Partition for Divergence Estima-tion: Problem Statement Let P and Q be probability measures in (Rd, B(Rd)) absolutely continuous with respect to the Lebesgue measure, such that D(P||Q) < 1. Let = { 1, 2, · · · } be a partition scheme for Rd, and let us consider X1, ..,Xn and Y1, .., Ym i.i.d. realizations of random variables with values in Rd and distributions P and Q, respectively. Then the proposed candidate for the empirical divergence is given by ˆD n,m(P||Q) X A2 m(Y1,..,Ym) Pn(A) · log Pn(A) Qm(A) , (4.9) where Pn and Qm denote the empirical distributions induced by X1, ..,Xn and Y1, .., Ym respectively by (4.6), defined in the sub-sigma field ( m(Y1, .., Ym)) B(Rd). Note that this data-dependent partition is a function only of the i.i.d. realizations associ-ated with the reference measure Q. Loosely speaking the choice of using only partial information can be justified by the assumption that P Q. The next section will formally support this choice by introducing the notion of universally sufficient data-dependent partition and showing that the proposed construction satisfies this desired asymptotic property. In addition, we impose in the condition that Qm(A) > 0, 8A 2 m(Y1, .., Ym), that ensures that ˆD n,m(P||Q) < 1. ˆD n,m(P||Q) is a measurable function of X1, ..,Xn and Y1, .., Ym, and consequently we are interested in studying the strong — almost surely with respect to the joint distri-bution of {Xn, n 2 N} and {Ym,m 2 N}—universal consistency of ˆD n,m(P||Q) as m and n tend to infinity and as a function of the aforementioned notions of combinatorial complexity for . 96 |
