Page 103 |
Save page Remove page | Previous | 103 of 196 | Next |
|
small (250x250 max)
medium (500x500 max)
Large (1000x1000 max)
Extra Large
large ( > 500x500)
Full Resolution
All (PDF)
|
This page
All
|
estimation, considering an adaptive partition scheme that approximates empirical statis-tical
equivalent intervals relative to the reference measure. Sufficient conditions on the
statistically equivalent partitions were stipulated to guarantee the strong consistency of
the estimate for the scalar case — where probability measures are defined on the real
line. This construction and consistency result was also extended to the case of Ergodic
sources, in this case for the estimation of relative entropy rate. Independently, Nguyen,
Wainwright and Jordan [48] proposed a nobel estimate based on a variational charac-terization
of the divergence [32, 20]. The proposed approach reduces to estimating the
likelihood ratio or Radon-Nicodym derivative of the probability measures involved by
the solution of a risk minimization problem. Under some approximation assumptions
and smoothness condition on the likelihood-ratio, strong consistency of the proposed
estimate was obtained. Remarkable, under those conditions an optimal minimax rate
for convergence for the likelihood-ratio and asymptotic rate of convergence for the pro-posed
plug-in divergence estimate were established.
In this paper, we extend the work of Wang et al. [78], by exploring in more gen-eral
terms the problem of histogram-based divergence estimation. We propose a gen-eral
histogram-based divergence estimate based on data dependent partitions. The main
result characterizes general sufficient conditions on the data-dependent partition scheme
that guarantee the divergence estimate to be universally strongly consistent. This con-sistency
result does not stipulate any major assumption other than requiring the two
probability measures to be absolutely continuous with respect to the Lebesgue mea-sure.
This direction was motivated by the seminal work of Lugosi and Nobel [46] and
Nobel [49], where general sufficient conditions were established — based on combi-natorial
notions of complexity for partition families and the extension of the Vapnik-
Chervonenkis inequality [71, 70] —- on data-dependent partition schemes to obtain
90
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 103 |
| Contributing entity | University of Southern California |
| Repository email | cisadmin@lib.usc.edu |
| Full text | estimation, considering an adaptive partition scheme that approximates empirical statis-tical equivalent intervals relative to the reference measure. Sufficient conditions on the statistically equivalent partitions were stipulated to guarantee the strong consistency of the estimate for the scalar case — where probability measures are defined on the real line. This construction and consistency result was also extended to the case of Ergodic sources, in this case for the estimation of relative entropy rate. Independently, Nguyen, Wainwright and Jordan [48] proposed a nobel estimate based on a variational charac-terization of the divergence [32, 20]. The proposed approach reduces to estimating the likelihood ratio or Radon-Nicodym derivative of the probability measures involved by the solution of a risk minimization problem. Under some approximation assumptions and smoothness condition on the likelihood-ratio, strong consistency of the proposed estimate was obtained. Remarkable, under those conditions an optimal minimax rate for convergence for the likelihood-ratio and asymptotic rate of convergence for the pro-posed plug-in divergence estimate were established. In this paper, we extend the work of Wang et al. [78], by exploring in more gen-eral terms the problem of histogram-based divergence estimation. We propose a gen-eral histogram-based divergence estimate based on data dependent partitions. The main result characterizes general sufficient conditions on the data-dependent partition scheme that guarantee the divergence estimate to be universally strongly consistent. This con-sistency result does not stipulate any major assumption other than requiring the two probability measures to be absolutely continuous with respect to the Lebesgue mea-sure. This direction was motivated by the seminal work of Lugosi and Nobel [46] and Nobel [49], where general sufficient conditions were established — based on combi-natorial notions of complexity for partition families and the extension of the Vapnik- Chervonenkis inequality [71, 70] —- on data-dependent partition schemes to obtain 90 |
