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3.8.8 Asymptotically Sufficient Results for the Product CMI Con-struction
Here we provide theoretical support for the product partition presented in Section 3.5.1,
for the CMI estimation problem. In particular based on the product construction, we
prove a desired asymptotic sufficient result and show a weak consistency argument for
our histogram based CMI estimate.
It is well known that quantization reduces the magnitude of information quantities
[32, 18], which is also the case for our quantized CMI construction, i.e., (Q1,2×3)
= I(X1,X2; Y |X3). Then it is interesting to study the approximation properties of
the proposed product CMI construction. In other words, find if this suggested construc-tion
can achieve by systematically increasing the resolution of a sequence of product
quantizers— a notion of asymptotic sufficient partitions for the CMI estimation.
Let us start with some basic definitions and results.
Definition 3.1 (from [18]) Let X be a continuous random variable taking values in
(Rk, B(Rk)) with probability measure PX. Let {Qn : n 2 N} be a nested sequence of
partitions for (Rk, B(Rk))23. The sequence is asymptotic sufficient for X, if 8A 2
B(Rk), 8 > 0, there exists n( ) 2 N and A¯ 2 (Qn( )), such that
PX(A4A¯) < , (3.55)
where (Qn) B(Rk) denotes the sigma field induced by Qn.24
23{Qn : n 2 N} is nested if for any n 2 N, Qn+1 is a refinement of Qn., i.e., every atom in Qn is
disjoint union of atoms in Qn+1.
24A sigma field induced by a set Q is the smallest sigma field that contains Q [34]. For the case of
finite and countable partitions is the collection of set that can be constructed as the union of cells in Q.
80
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 93 |
| Contributing entity | University of Southern California |
| Repository email | cisadmin@lib.usc.edu |
| Full text | 3.8.8 Asymptotically Sufficient Results for the Product CMI Con-struction Here we provide theoretical support for the product partition presented in Section 3.5.1, for the CMI estimation problem. In particular based on the product construction, we prove a desired asymptotic sufficient result and show a weak consistency argument for our histogram based CMI estimate. It is well known that quantization reduces the magnitude of information quantities [32, 18], which is also the case for our quantized CMI construction, i.e., (Q1,2×3) = I(X1,X2; Y |X3). Then it is interesting to study the approximation properties of the proposed product CMI construction. In other words, find if this suggested construc-tion can achieve by systematically increasing the resolution of a sequence of product quantizers— a notion of asymptotic sufficient partitions for the CMI estimation. Let us start with some basic definitions and results. Definition 3.1 (from [18]) Let X be a continuous random variable taking values in (Rk, B(Rk)) with probability measure PX. Let {Qn : n 2 N} be a nested sequence of partitions for (Rk, B(Rk))23. The sequence is asymptotic sufficient for X, if 8A 2 B(Rk), 8 > 0, there exists n( ) 2 N and A¯ 2 (Qn( )), such that PX(A4A¯) < , (3.55) where (Qn) B(Rk) denotes the sigma field induced by Qn.24 23{Qn : n 2 N} is nested if for any n 2 N, Qn+1 is a refinement of Qn., i.e., every atom in Qn is disjoint union of atoms in Qn+1. 24A sigma field induced by a set Q is the smallest sigma field that contains Q [34]. For the case of finite and countable partitions is the collection of set that can be constructed as the union of cells in Q. 80 |
