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|Title: ||Statistical learning algorithms for regression and regularized spectral clustering|
|Other Titles: ||Tong ji xue xi zhong hui gui he zheng ze hua pu ju lei suan fa de yan jiu|
|Authors: ||Lü, Shaogao ( 呂紹高)|
|Department: ||Department of Mathematics|
|Degree: ||Doctor of Philosophy|
|Issue Date: ||2011|
|Publisher: ||City University of Hong Kong|
|Subjects: ||Machine learning -- Statistical methods.|
|Notes: ||CityU Call Number: Q325.5 .L8 2011|
vi, 79 leaves 30 cm.
Thesis (Ph.D.)--City University of Hong Kong, 2011.
Includes bibliographical references (leaves -79)
|Abstract: ||In this thesis we investigate several algorithms in statistical learning theory, and our
contributions consist of the following three parts.
First we focus on the least square regularized regression learning algorithm in a
setting of unbounded sampling. Our task is to establish learning rates by means of
integral operators. By imposing a moment hypothesis on the unbounded sampling
outputs and a function space condition associated with the marginal distribution, we
derive learning rates which are consistent with those in the bounded sampling setting.
Then we consider the spectral clustering algorithms by learning with a regularization
scheme in a sample data hypothesis space and l1-regularizer. The data dependent
space spanned by means of the kernel function provides great flexibility for learning.
The main difficulty in studying spectral clustering in our setting is that the hypothesis
space not only depends on a sample, but also depends on some constrained conditions.
The technical difficultly is solved by a local polynomial reproduction formula and a
construction method. The consistency of spectral clustering algorithms is stated in
terms of properties of the data space, the underlying measure, the kernel as well as the
regularity of a target function.
Finally, we take a learning theory viewpoint to study a family of learning schemes
for regression related to positive linear operators in approximation theory. Such a
learning scheme is generated from a random sample by a kernel function parameterized
by a scaling parameter. The essential difference between this algorithm and the classical approximation schemes is the randomness of the sampling points, which breaks
the condition for good distribution of sampling points often required in approximation
theory. We investigate efficiency of the learning algorithm in a regression setting and
present learning rates stated in terms of the smoothness of the regression function, sizes
of variances, and distances of kernel centers from regular grids. The error analysis is
conducted by estimating the sample error and approximation error. Two examples with
kernel functions related to continuous Bernstein bases and Jackson kernels are studied
in detail and concrete learning rates are obtained.|
|Online Catalog Link: ||http://lib.cityu.edu.hk/record=b4086788|
|Appears in Collections:||MA - Doctor of Philosophy |
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