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then there exits 0 = 0 < 1 < · · · m = 1 and a sequence of pruned trees R1, ..,Rm
(with |R1| > |R2| ... > |Rm| = 1), such that 8i 2 {1, ..,m},
T ( ) = Ri, 8 2 [ i−1, i) . (3.22)
This results characterizes the full range of solutions for (3.21) as a function of the rela-tive
weight between the fidelity and penalization terms (or the optimal achievable cost-fidelity
region [8]). The problem of finding { 0, , .. m−1} and the associated solutions
{R1, ..,Rm} was coined as the “family pruning problem” under general size based penal-ties
[61]. It is not difficult to see that Rj is an admissible solution of the minimum cost
tree pruning (3.18) for kj = |Rj |, i.e. Rj = Tkj , and consequently we can consider
that {R1, ..,Rm}
Tk : k = 1, .., 2K−o
. Interestingly, if the cost tree functional is
additive, the following result can be stated.
THEOREM 3.3 If (|T|) = |T|, then the solution of the family pruning problem
admits an embedded structure, i.e., Rm Rm−1 · · · R1.
Indirectly, the proof of this theorem can be obtained from the fact that this set of
solutions characterizes an operational rate-distortion region associated with two mono-tone
affine tree functionals: one being the MI, our fidelity criterion, and the other, the
cost of the tree given by |T|, which is additive15. We derived an alternative algebraic
proof for this result based on Breiman et al.’s derivations (Chapter 10.2) [5], presented
in Appendix 3.8.7.
By Theorem 3.3, we have that the family pruning problem admits a nested solution.
Consequently, a simpler algorithm can be used for finding {R1, ..,Rm}. The algorithm is
15In the general case of monotone affine functionals, Chou et al. (Lemma 1) [8] showed that the solution
for an operational rate distortion problem allows a nested sub-tree structure.
57
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 70 |
| Contributing entity | University of Southern California |
| Repository email | cisadmin@lib.usc.edu |
| Full text | then there exits 0 = 0 < 1 < · · · m = 1 and a sequence of pruned trees R1, ..,Rm (with |R1| > |R2| ... > |Rm| = 1), such that 8i 2 {1, ..,m}, T ( ) = Ri, 8 2 [ i−1, i) . (3.22) This results characterizes the full range of solutions for (3.21) as a function of the rela-tive weight between the fidelity and penalization terms (or the optimal achievable cost-fidelity region [8]). The problem of finding { 0, , .. m−1} and the associated solutions {R1, ..,Rm} was coined as the “family pruning problem” under general size based penal-ties [61]. It is not difficult to see that Rj is an admissible solution of the minimum cost tree pruning (3.18) for kj = |Rj |, i.e. Rj = Tkj , and consequently we can consider that {R1, ..,Rm} Tk : k = 1, .., 2K−o . Interestingly, if the cost tree functional is additive, the following result can be stated. THEOREM 3.3 If (|T|) = |T|, then the solution of the family pruning problem admits an embedded structure, i.e., Rm Rm−1 · · · R1. Indirectly, the proof of this theorem can be obtained from the fact that this set of solutions characterizes an operational rate-distortion region associated with two mono-tone affine tree functionals: one being the MI, our fidelity criterion, and the other, the cost of the tree given by |T|, which is additive15. We derived an alternative algebraic proof for this result based on Breiman et al.’s derivations (Chapter 10.2) [5], presented in Appendix 3.8.7. By Theorem 3.3, we have that the family pruning problem admits a nested solution. Consequently, a simpler algorithm can be used for finding {R1, ..,Rm}. The algorithm is 15In the general case of monotone affine functionals, Chou et al. (Lemma 1) [8] showed that the solution for an operational rate distortion problem allows a nested sub-tree structure. 57 |
