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Figure 4.1: A: Example of Gessaman’s statistically equivalent partition for a two dimen-sional
bounded space. B: Example of a tree-structured data dependent partition and its
tree-indexed structure. Each internal node has a label indicating the spatial coordinate
used to split its associated rectangular set.
this sequence tends to zero as m tends to infinity — considering that by construction
(lm) (m). The second term on the RHS of (4.34) behaves asymptotically like −m1−l ·
log(1 − 1/lm) which is upper bounded by the sequence m1−l
lm
· 1
1−1/lm
— using that
log(x) x − 1, 8x > 0. This upper bound tends to zero because (lm) (m0.5+l/2) and
l > 1/3. Consequently from (4.34), limm!1m−l log(
m(Am)) = 0.
Finally concerning condition d), Lugosi et al. [46] (Theorem 4) proved that it is
sufficient to show that limm!1
lm
m = 0, which is the case considering that l < 1.
Remark 4.3 Under the condition presented in Theorem 4.5 we have that lm ! 1
and lm/m ! 1 as m tends to infinity, which are the sufficient conditions presented
by Lugosi and Nobel [46] for the lm-based histogram based density estimation to be
strongly consistent in the L1 sense. The fact that stronger sufficient conditions are
needed to get strong consistency for the divergence estimation, can be explained by
the fact that the divergence is not continuous function of the densities with respect to the
L1 norm [22].
108
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 121 |
| Contributing entity | University of Southern California |
| Repository email | cisadmin@lib.usc.edu |
| Full text | Figure 4.1: A: Example of Gessaman’s statistically equivalent partition for a two dimen-sional bounded space. B: Example of a tree-structured data dependent partition and its tree-indexed structure. Each internal node has a label indicating the spatial coordinate used to split its associated rectangular set. this sequence tends to zero as m tends to infinity — considering that by construction (lm) (m). The second term on the RHS of (4.34) behaves asymptotically like −m1−l · log(1 − 1/lm) which is upper bounded by the sequence m1−l lm · 1 1−1/lm — using that log(x) x − 1, 8x > 0. This upper bound tends to zero because (lm) (m0.5+l/2) and l > 1/3. Consequently from (4.34), limm!1m−l log( m(Am)) = 0. Finally concerning condition d), Lugosi et al. [46] (Theorem 4) proved that it is sufficient to show that limm!1 lm m = 0, which is the case considering that l < 1. Remark 4.3 Under the condition presented in Theorem 4.5 we have that lm ! 1 and lm/m ! 1 as m tends to infinity, which are the sufficient conditions presented by Lugosi and Nobel [46] for the lm-based histogram based density estimation to be strongly consistent in the L1 sense. The fact that stronger sufficient conditions are needed to get strong consistency for the divergence estimation, can be explained by the fact that the divergence is not continuous function of the densities with respect to the L1 norm [22]. 108 |
