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8k 2
1, .., |Tfull| = 2Ko
. Let us define Tv as the largest branch of Tfull rooted at
v 2 Tfull and Tk
v Tv as the solution of the following more general optimal tree
pruning problem,
Tk
v = arg max
T Tv
|T|=k
T (v), 8k 2 {1, .., |Tv|} . (3.19)
The next result presents a DP solution for this problem using the additive properties of
our fidelity indicator.
THEOREM 3.2 Let us consider an arbitrary internal node v 2 I(Tfull) and
denote its left and right children by l(v) and r(v) respectively13. Assuming that
we know the solution of (3.19) for the child nodes l(v) and r(v), i.e., we know
n
Tk1
l(v),Tk2
r(v) : k1 = 1, ..,
Tl(v)
k2 = 1, ..,
Tr(v)
o
, the solution of (3.19) for the par-ent
node is given by Tk
v =
h
v, T
ˆk1
l(v) , T
ˆk2
r(v)
i
, where14
(ˆk1, ˆk2) = arg max
(k1,k2)2{1,..,|Tl(v)|}×{1,..,|Tr(v)|}
k1+k2=k
T
k1
l(v)
(l(v)) + T
k2
r(v)
(r(v))
, (3.20)
8k 2 {1, .., |Tv|}. In particular, when v is equal to the root of Tfull the solution for the
optimal pruning problem in (3.18), is given by Tk =
h
vroot,T
ˆk1
l(vroot),T
ˆk2
r(vroot)
i
. The
proof is presented in Appendix 3.8.6.
This DP solution is a direct consequence of solving (3.19) for the parent node
as a function of the solutions of the same problem for its direct descendants. In
13Based on our previous notation any internal node v 2 I(Tfull) can be written as (l, j) where l(v) =
(l + 1, 2j) and r(v) = (l + 1, 2j + 1)
14Using Scott’s nomenclature [61], the notation [v, T1, T2] represents a binary tree T with root v,
Tl(v) = T1 and Tr(v) = T2.
55
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 68 |
| Contributing entity | University of Southern California |
| Repository email | cisadmin@lib.usc.edu |
| Full text | 8k 2 1, .., |Tfull| = 2Ko . Let us define Tv as the largest branch of Tfull rooted at v 2 Tfull and Tk v Tv as the solution of the following more general optimal tree pruning problem, Tk v = arg max T Tv |T|=k T (v), 8k 2 {1, .., |Tv|} . (3.19) The next result presents a DP solution for this problem using the additive properties of our fidelity indicator. THEOREM 3.2 Let us consider an arbitrary internal node v 2 I(Tfull) and denote its left and right children by l(v) and r(v) respectively13. Assuming that we know the solution of (3.19) for the child nodes l(v) and r(v), i.e., we know n Tk1 l(v),Tk2 r(v) : k1 = 1, .., Tl(v) k2 = 1, .., Tr(v) o , the solution of (3.19) for the par-ent node is given by Tk v = h v, T ˆk1 l(v) , T ˆk2 r(v) i , where14 (ˆk1, ˆk2) = arg max (k1,k2)2{1,..,|Tl(v)|}×{1,..,|Tr(v)|} k1+k2=k T k1 l(v) (l(v)) + T k2 r(v) (r(v)) , (3.20) 8k 2 {1, .., |Tv|}. In particular, when v is equal to the root of Tfull the solution for the optimal pruning problem in (3.18), is given by Tk = h vroot,T ˆk1 l(vroot),T ˆk2 r(vroot) i . The proof is presented in Appendix 3.8.6. This DP solution is a direct consequence of solving (3.19) for the parent node as a function of the solutions of the same problem for its direct descendants. In 13Based on our previous notation any internal node v 2 I(Tfull) can be written as (l, j) where l(v) = (l + 1, 2j) and r(v) = (l + 1, 2j + 1) 14Using Scott’s nomenclature [61], the notation [v, T1, T2] represents a binary tree T with root v, Tl(v) = T1 and Tr(v) = T2. 55 |
