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are computed accordingly. Hence, we can obtain the empirical MIs using the formula
below17,
ˆIQ1,2×3
N (X1,X2,X3; Y ) =
1
N
XN
i=1
log
ˆ PN(Ai
1,2 × Ai
3 × {y})
ˆ PN(Ai
1,2 × Ai
3) · ˆ PN({y})
(3.24)
ˆIQ3
N (X3; Y ) =
1
N
XN
i=1
log
ˆ PN(Ai
3 × {y})
ˆ PN(Ai
3) · ˆ PN({y})
(3.25)
and consequently the empirical CMI by ˆ N(Q1,2×3) = IˆQ1,2×3
N (X1,X2,X3; Y ) −
ˆIQ3
N (X3; Y ).
Considering the mentioned product sufficient partition sequence for the CMI and
sufficient number of samples points, from the weak law of large numbers [73, 4] it is
simple to show that ˆ N(Q1,2×3) can be arbitrarily close to in probability, which is
a desired weak consistency result. However, in practice we need to deal with the non-asymptotic
case of having a finite amount of training data. In this context, the problem
of finding a good estimation for across a sequence of nested partitions needs to
consider an approximation and estimation error tradeoff, as with any other statistical
learning problem [21]. To address this issue, we follow the data-dependent partition
framework proposed by Darbellay et al. [18].
3.5.2 Darbellay-Vajda Data-Dependent Partition
The Darbellay-Vajda algorithm partitions the observation space, by iterating a splitting
rule that generates a sequence of tree-indexed nested partitions [18]. To illustrate the
idea, let X and Y be continuous finite alphabet random variables and let us consider
the problem of estimating I(X; Y ). In addition, let {(xi, yi) : i = 1, ..,N} denote the
training data and ˆ PN be the empirical probability with distribution function denoted by
17The sub-script indices on the probabilities are omitted to simplify notation in (3.24) and (3.25).
61
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 74 |
| Contributing entity | University of Southern California |
| Repository email | cisadmin@lib.usc.edu |
| Full text | are computed accordingly. Hence, we can obtain the empirical MIs using the formula below17, ˆIQ1,2×3 N (X1,X2,X3; Y ) = 1 N XN i=1 log ˆ PN(Ai 1,2 × Ai 3 × {y}) ˆ PN(Ai 1,2 × Ai 3) · ˆ PN({y}) (3.24) ˆIQ3 N (X3; Y ) = 1 N XN i=1 log ˆ PN(Ai 3 × {y}) ˆ PN(Ai 3) · ˆ PN({y}) (3.25) and consequently the empirical CMI by ˆ N(Q1,2×3) = IˆQ1,2×3 N (X1,X2,X3; Y ) − ˆIQ3 N (X3; Y ). Considering the mentioned product sufficient partition sequence for the CMI and sufficient number of samples points, from the weak law of large numbers [73, 4] it is simple to show that ˆ N(Q1,2×3) can be arbitrarily close to in probability, which is a desired weak consistency result. However, in practice we need to deal with the non-asymptotic case of having a finite amount of training data. In this context, the problem of finding a good estimation for across a sequence of nested partitions needs to consider an approximation and estimation error tradeoff, as with any other statistical learning problem [21]. To address this issue, we follow the data-dependent partition framework proposed by Darbellay et al. [18]. 3.5.2 Darbellay-Vajda Data-Dependent Partition The Darbellay-Vajda algorithm partitions the observation space, by iterating a splitting rule that generates a sequence of tree-indexed nested partitions [18]. To illustrate the idea, let X and Y be continuous finite alphabet random variables and let us consider the problem of estimating I(X; Y ). In addition, let {(xi, yi) : i = 1, ..,N} denote the training data and ˆ PN be the empirical probability with distribution function denoted by 17The sub-script indices on the probabilities are omitted to simplify notation in (3.24) and (3.25). 61 |
