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Remark 5.4 It is well known that if a collection of measurable events C has a finite VC
dimension, let say V > 0, then 8n > V [22, 21],
Sn(C) (n + 1)V , (5.24)
and consequently 8 2 (0, 1), lim!1
1
n log Sn(C) = 0.
Then for the finite VC dimensional case the growth function is dominated by a polyno-mial
behavior. Considering the condition c.1 of Theorem 5.2, it is interesting to extend
this idea to a sequence of measurable events. The following proposition guarantees c.1
of Theorem 5.2 when a collections of measurable events have finite VC-dimensions.
Proposition 5.2 Let {Cn : n 2 N} be a collection of measurable events with finite VC
dimension sequence (Vn)n2N. If (Vn)n2N is o
n o
log(n+1)
for some o 2 (0, 1), then4
lim
n!1
1
n o
log Sn(Cn) = 0. (5.25)
Proof: From the fact that (Vn)n2N is o
n o
log(n+1)
, there exists N, such that 8n >
N, n > Vn. Then from the definition of VC dimension [21], 8n > N, Sn(Cn)
(n + 1)Vn. Then lim supn
1
n o log Sn(Cn) limn
Vn log(n+1)
n o = 0 by the hypothesis.
Then a valid question to ask is how this general result translates into specific design
conditions when working with some specific data-dependent constructions. In other
words, is it possible to find strongly consistent mutual information estimates based on
this result? This will be the focus for the rest of the paper, where we show how these gen-eral
conditions provide specific design setting in the implementation of two widely used
and emblematic partition schemes: tree-structured partitions and statistically equivalent
blocks.
4A nonnegative sequence (an)n2N is o(f(n)), for some non-negative increasing function f(·) : N !
R if, limn!1
an
f(n) = 0.
152
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 165 |
| Contributing entity | University of Southern California |
| Repository email | cisadmin@lib.usc.edu |
| Full text | Remark 5.4 It is well known that if a collection of measurable events C has a finite VC dimension, let say V > 0, then 8n > V [22, 21], Sn(C) (n + 1)V , (5.24) and consequently 8 2 (0, 1), lim!1 1 n log Sn(C) = 0. Then for the finite VC dimensional case the growth function is dominated by a polyno-mial behavior. Considering the condition c.1 of Theorem 5.2, it is interesting to extend this idea to a sequence of measurable events. The following proposition guarantees c.1 of Theorem 5.2 when a collections of measurable events have finite VC-dimensions. Proposition 5.2 Let {Cn : n 2 N} be a collection of measurable events with finite VC dimension sequence (Vn)n2N. If (Vn)n2N is o n o log(n+1) for some o 2 (0, 1), then4 lim n!1 1 n o log Sn(Cn) = 0. (5.25) Proof: From the fact that (Vn)n2N is o n o log(n+1) , there exists N, such that 8n > N, n > Vn. Then from the definition of VC dimension [21], 8n > N, Sn(Cn) (n + 1)Vn. Then lim supn 1 n o log Sn(Cn) limn Vn log(n+1) n o = 0 by the hypothesis. Then a valid question to ask is how this general result translates into specific design conditions when working with some specific data-dependent constructions. In other words, is it possible to find strongly consistent mutual information estimates based on this result? This will be the focus for the rest of the paper, where we show how these gen-eral conditions provide specific design setting in the implementation of two widely used and emblematic partition schemes: tree-structured partitions and statistically equivalent blocks. 4A nonnegative sequence (an)n2N is o(f(n)), for some non-negative increasing function f(·) : N ! R if, limn!1 an f(n) = 0. 152 |
