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4.10.1 Reducing the problem to a bounded measurable space
Note that the partition scheme is monotone transformation invariant [21], in the
sense that for any function F : Rd ! Rd that can be expressed by F(x) =
(f1(x(1)), · · · , fd(x(d))) for some collection of strictly increasing real functions
{fi(·) : i = 1, .., d}, any partition rule m = (Tm) of satisfies that 8x 2 Rd,
8ym
1 2 Rd·m, m(x|y1, ..ym) = m(F(x)|F(y1), ..F (ym)).
We can consider fi(·) to be the distribution function of the marginal probability
Q restricted to events on the i-coordinate, 8i 2 {1, .., d}. Without loss of generality
we can restrict to the case when {fi(·) : i = 1, .., d} are strictly increasing functions.
Consequently, the induced distributions of the transform space, denoted by ¯Qand ¯ P
respectively, have support on [0, 1]d and satisfies that [32]
D(P||Q) = D( ¯ P||¯Q
), (4.82)
because F(·) is one-to-one continuous mapping from Rd to [0, 1]d (more precisely
F−1(A) : A 2 B([0, 1]d)
= B(Rd)). Moreover, by construction of the divergence
estimator, Section 4.3, if we apply in the transform domain, i.e. we estimate the
empirical distributions using the transform i.i.d. realizations F(X1), .., F(Xm) and
F(Y1), .., F(Ym) — denoted by ¯ Pn and ¯Q
m on ( (F(Y1), .., F(Ym))), and estimate
the divergence ˆD
m,n( ¯ Pn, ¯Q
m) by (4.9), it is simple to check that
ˆD
m,n(Pn,Qm) = ˆD
m,n( ¯ Pn, ¯Q
m). (4.83)
In other words, the empirical divergence based on our TSP is invariant under mono-tone
coordinate transformations. Then from (4.82) and (4.83), we can reduce the prob-lem
to checking the asymptotic sufficient nature of and consequently the shrinking
132
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 145 |
| Contributing entity | University of Southern California |
| Repository email | cisadmin@lib.usc.edu |
| Full text | 4.10.1 Reducing the problem to a bounded measurable space Note that the partition scheme is monotone transformation invariant [21], in the sense that for any function F : Rd ! Rd that can be expressed by F(x) = (f1(x(1)), · · · , fd(x(d))) for some collection of strictly increasing real functions {fi(·) : i = 1, .., d}, any partition rule m = (Tm) of satisfies that 8x 2 Rd, 8ym 1 2 Rd·m, m(x|y1, ..ym) = m(F(x)|F(y1), ..F (ym)). We can consider fi(·) to be the distribution function of the marginal probability Q restricted to events on the i-coordinate, 8i 2 {1, .., d}. Without loss of generality we can restrict to the case when {fi(·) : i = 1, .., d} are strictly increasing functions. Consequently, the induced distributions of the transform space, denoted by ¯Qand ¯ P respectively, have support on [0, 1]d and satisfies that [32] D(P||Q) = D( ¯ P||¯Q ), (4.82) because F(·) is one-to-one continuous mapping from Rd to [0, 1]d (more precisely F−1(A) : A 2 B([0, 1]d) = B(Rd)). Moreover, by construction of the divergence estimator, Section 4.3, if we apply in the transform domain, i.e. we estimate the empirical distributions using the transform i.i.d. realizations F(X1), .., F(Xm) and F(Y1), .., F(Ym) — denoted by ¯ Pn and ¯Q m on ( (F(Y1), .., F(Ym))), and estimate the divergence ˆD m,n( ¯ Pn, ¯Q m) by (4.9), it is simple to check that ˆD m,n(Pn,Qm) = ˆD m,n( ¯ Pn, ¯Q m). (4.83) In other words, the empirical divergence based on our TSP is invariant under mono-tone coordinate transformations. Then from (4.82) and (4.83), we can reduce the prob-lem to checking the asymptotic sufficient nature of and consequently the shrinking 132 |
