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added the paper on bandwidth tuning

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vasiloglou
2007-07-31 14:25:51 +00:00
parent e800d3272c
commit 408c73bf19
28 changed files with 190926 additions and 480 deletions
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fastlib/u/nvasil/docs/proposal/fig1.1a.eps
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fastlib/u/nvasil/docs/proposal/fig1.1b.eps
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fastlib/u/nvasil/docs/proposal/fig1.1c.eps
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Appendix A.
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Appendix B.
***************** References ******************
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8211 multiletter control sequences out of 60000
28619 words of font info for 89 fonts, out of 1000000 for 2000
14 hyphenation exceptions out of 8191
34i,8n,36p,341b,521s stack positions out of 5000i,500n,10000p,200000b,32768s
Output written on proposal.dvi (34 pages, 135100 bytes).
Output written on proposal.dvi (32 pages, 133300 bytes).
fastlib/u/nvasil/docs/proposal/proposal.lot
+6 -6
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@@ -1,6 +1,6 @@
\contentsline {table}{\numberline {Table\nobreakspace 4.1}{\ignorespaces Pseudo-code for the dual-tree all nearest neighbor algorithm}}{20}{table.52}
\contentsline {table}{\numberline {Table\nobreakspace 4.2}{\ignorespaces CPU time (in hours) for evaluating all 20 nearest neighbors on TIMIT database}}{23}{table.63}
\contentsline {table}{\numberline {Table\nobreakspace 4.3}{\ignorespaces Nearest k-neighbor classfier for TIMIT database}}{25}{table.64}
\contentsline {table}{\numberline {Table\nobreakspace 4.4}{\ignorespaces Nearest k-neighbor classfier for NTIMIT database}}{25}{table.65}
\contentsline {table}{\numberline {Table\nobreakspace 4.5}{\ignorespaces Nearest k-neighbor classfier for Leave One Speaker out Cross Validation in TIMIT database}}{26}{table.66}
\contentsline {table}{\numberline {Table\nobreakspace 4.6}{\ignorespaces Nearest k-neighbor classfier for Leave One Speaker Out Cross validation in NTIMIT database}}{26}{table.67}
\contentsline {table}{\numberline {Table\nobreakspace 3.1}{\ignorespaces Pseudo-code for the dual-tree all nearest neighbor algorithm}}{18}{table.41}
\contentsline {table}{\numberline {Table\nobreakspace 3.2}{\ignorespaces CPU time (in hours) for evaluating all 20 nearest neighbors on TIMIT database}}{22}{table.52}
\contentsline {table}{\numberline {Table\nobreakspace 3.3}{\ignorespaces Nearest k-neighbor classfier for TIMIT database}}{23}{table.53}
\contentsline {table}{\numberline {Table\nobreakspace 3.4}{\ignorespaces Nearest k-neighbor classfier for NTIMIT database}}{23}{table.54}
\contentsline {table}{\numberline {Table\nobreakspace 3.5}{\ignorespaces Nearest k-neighbor classfier for Leave One Speaker out Cross Validation in TIMIT database}}{24}{table.55}
\contentsline {table}{\numberline {Table\nobreakspace 3.6}{\ignorespaces Nearest k-neighbor classfier for Leave One Speaker Out Cross validation in NTIMIT database}}{24}{table.56}
fastlib/u/nvasil/docs/proposal/proposal.out
+26 -27
View File
@@ -6,30 +6,29 @@
\BOOKMARK [1][-]{section.11}{Section 2 \204 Origin and history of the problem}{}
\BOOKMARK [1][-]{section.12}{Section 3 \204 Preliminary Research}{}
\BOOKMARK [2][-]{subsection.13}{The Nearest Neighbor Problem}{section.12}
\BOOKMARK [2][-]{subsection.25}{Kernel methods}{section.12}
\BOOKMARK [1][-]{section.37}{Section 4 \204 Proposed Research}{}
\BOOKMARK [2][-]{subsection.38}{Large scale trees, for out of core memory}{section.37}
\BOOKMARK [2][-]{subsection.39}{Memory layout of Kd-trees}{section.37}
\BOOKMARK [2][-]{subsection.40}{System architecture}{section.37}
\BOOKMARK [2][-]{subsection.41}{Memory Manager Architecture}{section.37}
\BOOKMARK [2][-]{subsection.42}{Memory Mapped Files}{section.37}
\BOOKMARK [2][-]{subsection.43}{User defined cache, with TPIE}{section.37}
\BOOKMARK [2][-]{subsection.44}{Customizing Kernels with fast gradient descent}{section.37}
\BOOKMARK [2][-]{subsection.45}{Scalable Support Vector Machines}{section.37}
\BOOKMARK [2][-]{subsection.46}{Computing Gaussian Kernel Matrix with trees}{section.37}
\BOOKMARK [2][-]{subsection.47}{Intrinsic dimension of speech}{section.37}
\BOOKMARK [2][-]{subsection.48}{Speech recognition with nearest neighbor search}{section.37}
\BOOKMARK [2][-]{subsection.49}{Introduction}{section.37}
\BOOKMARK [2][-]{subsection.50}{Speech features}{section.37}
\BOOKMARK [2][-]{subsection.51}{Fast N-body methods for feature comparison}{section.37}
\BOOKMARK [2][-]{subsection.55}{Kernel PCA on a manifold}{section.37}
\BOOKMARK [2][-]{subsection.56}{Experimental Results}{section.37}
\BOOKMARK [2][-]{subsection.57}{All nearest neighbor performance}{section.37}
\BOOKMARK [2][-]{subsection.58}{Comparison of NRAF and MFCC}{section.37}
\BOOKMARK [2][-]{subsection.60}{Discussion}{section.37}
\BOOKMARK [2][-]{subsection.68}{Acoustic Environment adaptation}{section.37}
\BOOKMARK [1][-]{section.69}{Section 5 \204 Work remaining to be done}{}
\BOOKMARK [1][-]{section.70}{Section 6 \204 Facilities and Equipment Needed}{}
\BOOKMARK [1][-]{section.71}{Appendix A \204 This is my first appendix}{}
\BOOKMARK [1][-]{section.72}{Appendix B \204 This is my second appendix}{}
\BOOKMARK [1][-]{section*.73}{References}{}
\BOOKMARK [2][-]{subsection.20}{Large scale trees, for out of core memory}{section.12}
\BOOKMARK [2][-]{subsection.21}{Memory layout of Kd-trees}{section.12}
\BOOKMARK [2][-]{subsection.22}{System architecture}{section.12}
\BOOKMARK [2][-]{subsection.23}{Memory Manager Architecture}{section.12}
\BOOKMARK [2][-]{subsection.24}{Memory Mapped Files}{section.12}
\BOOKMARK [2][-]{subsection.25}{User defined cache, with TPIE}{section.12}
\BOOKMARK [2][-]{subsection.26}{Kernel methods}{section.12}
\BOOKMARK [2][-]{subsection.36}{Forming manifolds from data for clustering through non-parametric density estimation}{section.12}
\BOOKMARK [2][-]{subsection.37}{Speech recognition with nearest neighbor search}{section.12}
\BOOKMARK [2][-]{subsection.38}{Introduction}{section.12}
\BOOKMARK [2][-]{subsection.39}{Speech features}{section.12}
\BOOKMARK [2][-]{subsection.40}{Fast N-body methods for feature comparison}{section.12}
\BOOKMARK [2][-]{subsection.44}{Kernel PCA on a manifold}{section.12}
\BOOKMARK [2][-]{subsection.45}{Experimental Results}{section.12}
\BOOKMARK [2][-]{subsection.46}{All nearest neighbor performance}{section.12}
\BOOKMARK [2][-]{subsection.47}{Comparison of NRAF and MFCC}{section.12}
\BOOKMARK [2][-]{subsection.49}{Discussion}{section.12}
\BOOKMARK [1][-]{section.57}{Section 4 \204 Proposed Research}{}
\BOOKMARK [2][-]{subsection.58}{Customizing Kernels with fast gradient descent}{section.57}
\BOOKMARK [2][-]{subsection.64}{Computing Gaussian Kernel Matrix with trees}{section.57}
\BOOKMARK [2][-]{subsection.65}{Intrinsic dimension of speech}{section.57}
\BOOKMARK [2][-]{subsection.66}{Speech recognition with nearest neighbor search}{section.57}
\BOOKMARK [2][-]{subsection.67}{Acoustic Environment adaptation}{section.57}
\BOOKMARK [1][-]{section.68}{Section 5 \204 Work remaining to be done}{}
\BOOKMARK [1][-]{section.69}{Appendix A \204 This is my first appendix}{}
\BOOKMARK [1][-]{section*.70}{References}{}
fastlib/u/nvasil/docs/proposal/proposal.tex
+529 -151
View File
@@ -358,143 +358,7 @@ In general the dual tree algorithm as we call it since the query and
the reference set lie on trees (they might share the same tree if
the query and the reference set are the same), gives linear
complexity over the number of data. This is empirical complexity and
\subsubsection{Fast kernel summation with trees}
Kernel summation is on of the most common tasks in kernel methods.
The general formula
\begin{equation}
\label{kernel_sum}
G(x)=\sum_{q=0}^{N} w_i\exp(\frac{(x-x_q)^2}{\sigma^2})
\end{equation}
appears in many cases, with the most interesting case the
eigenvalue/eigenvector computation. The kernel summation is a very
computationally intensive. For Gaussian kernels there are
approximations that can speed up computations orders of magnitude.
We will briefly discuss them here. Besides computation saving we can
save memory as well. The gaussian kernel has infinite support so the
kernel matrix $K, K(x,y)=\exp(\frac{(x-y)^2}{\sigma^2}$ is not
sparse. Numerical iterative methods for eigenvector computation are
based on the general iteration $x_n=x_{n-1}+\lambda Kx_{n-1}$, where
the multiplication of each row of $K$ with $x_{n-1}$ is a kernel
summation problem. So we don't really need to store the kernel
matrix, we just have to compute the kernel sum for every row using
$x_{n-1}$ as a the wight vector.
The gaussian can be approximated with the help of Hermite
polynomials. The Hermite polynomials $H_n(t)$ are defined by the
Rodrigues formula:
\begin{equation}
H_n(t)=(-1)^n\exp(t^2)D^n\exp(-t^2), t\in \Re
\end{equation}
where $D=d/dt$.
The gaussian can now be expressed as
\begin{equation}
\exp^(-(t-s)^2)=\sum_{n=0}^{\inf}\frac{s^n}{n!}h_n(t)
\end{equation}
where $h_n(t)$ is defined by
\begin{equation}
h_n(t)=\exp(-t^2)H_n(t).
\end{equation}
\subsection{Kernel methods}
The kernel or Gram matrix is an informational representation for a
set of data. There are a different interpretations for the kernel
matrix. The most mathematical interpretation is the dot product
matrix. Every element of the kernel matrix is the dot product
between the two data points. It is a positive semidefinite matrix
and it can be considered as the adjacency matrix of a graph where
the points are nodes. This graph can also be considered as an
approximation of the surface that data lie on. Positive
semidefinitness is the only property required for the gram matrix.
In most of the methods that we discuss in this section, we prefer
the interpretation of the dot product matrix, in the general Hilbert
sense. The dot products in most of the cases are non-linear.
\subsubsection{Kernel principal component analysis}
Kernel Principal Component Analysis (KPCA), is an extension of
principal component analysis where the data are mapped into a
different feature space through the kernel. In Principal Component
Analysis the goal is to find the principal components for the
covariance matrix $C$ for a given dataset $S={x_j, x_j\in \Re^d, j=1
\dots M}$, where $C=\frac{1}{M}\sum_{j=1}^{M}x_j x_j^T$. If we
define a mapping
\begin{equation}
\Phi : x \rightarrow X , x \in \Re^d, X\in \Re^f
\end{equation}
usually $f>>d$. So usually the nonlinear mapping, maps the data in a
higher dimensional space sometimes in an infinite dimensional space.
At this point it should be clarified that the mapping increases the
extrinsic dimension of the data not the intrinsic. In kernel PCA we
want to find the principal components of the covariance matrix
\begin{equation}
\tilde{C} = \frac{1}{M}\sum_{j=1}^{M}\phi(x_j)\phi(x_j)^T
\end{equation}
It turns out that the mapping $\Phi$ doesn't have to be known
explicitly, all we need is the kernel that represents the dot
product.
\begin{equation}
k(x, y) = \phi(x)\phi(y)^T
\end{equation}
For some kernels there is an analytic factorization while for others
there is not. Not every function can be a valid kernel (dot
product). There are conditions that must be fulfilled. Basically for
every dataset they must give a valid kernel matrix as described in
previous sections.
Kernel PCA has many cousins such as Laplacian eigenmaps, Diffusion
maps Isomap, Local Linear embedding
\subsubsection{Customized Kernels}
As we described in the previous section there are infinite functions
that can act as valid kernels. The most popular kernels are:
\begin{enumerate}
\item The gaussian kernel $k(x, y)=e^{-\frac{||x-y||^2}{\sigma^2)}}$
\item The polynomial kernel $k(x, y)=(xy)^n$
\item The epanechnikov kernel $k(x, y)=1-||x-y||^2/\sigma^2, ||x-y||<\sigma, else 0$
\item The k-nearest neighbor kernel
\end{enumerate}
Although these are valid kernels for any kernel method including
Kernel PCA they are not appropriate for every task. In some cases we
want to map data in higher dimensional spaces preserving distances
(isometry) while in other cases we want to make small distances
smaller and send high distances to infinity (clustering). The
gaussian kernel for example tends to behave better in clustering
since it fades out very quickly. On the other hand it fails to
unfold the data manifold so that the intrinsic dimension can be
estimated. Manifold unfolding is a very important procedure since it
can reveal the true dimension of the dataset.
In [?] the authors introduced the idea of building the kernel matrix
from scratch without using any fixed kernel. As we have already
mentioned the kernel matrix has to be a positive semidefinite
matrix. Initially we have to define a k neighborhood for every
point. So at first we have to connect points to their neighbors and
build a initialize the non zero values of the matrix to 1. Then we
maximize the trace of the kernel matrix under the constraint that
the the distances in the kernel matrix are preserved. This leads to
a semidefinite programming problem.
\subsubsection{Stochastic optimization}
Stochastic optimization or on line learning. On line svd is one key
aspect.
\subsubsection{Support Vector Machines}
\section{Proposed Research}
\label{proposed}
The goal of this thesis is to make kernel methods scalable.
Efficient computation of the kernel matrix for large amounts of
data. Special emphasis is given in high dimensional data $d>30$. One
of the goals is to use Kernel PCA for unfolding the speech manifold.
In other words estimate the intrinsic dimensionality of speech by
using big databases like timit, wall street journal and broadcast
news. Implicitly dimensionality reduction will lead to new feature
generation we will explain it in the following sections. The
ultimate goal is to do speech recognition by nearest neighbor
search.
\subsection{Large scale trees, for out of core memory}
Kd-trees have been proven to be be quite successful in
multidimensional indexing and turn out outperform several other
@@ -672,20 +536,461 @@ to address efficiently this. Pages are write protected in memory and
if a write occurs a signal is raised and the page is marked as
modified.
\subsection{Customizing Kernels with fast
gradient descent}
As we described in section ? it is possible to customize the kernel
for a given dataset, through semidefinite programming. Although this
method is very promising and gives very good results it is not
scalable. Semidefinite programming is solved with the intrinsic
point method which is of polynomial complexity.
\subsection{Scalable Support Vector Machines}
\subsection{Computing Gaussian Kernel Matrix with trees}
\subsection{Intrinsic dimension of speech}
\subsection{Kernel methods}
The kernel or Gram matrix is an informational representation for a
set of data. There are a different interpretations for the kernel
matrix. The most mathematical interpretation is the dot product
matrix. Every element of the kernel matrix is the dot product
between the two data points. It is a positive semidefinite matrix
and it can be considered as the adjacency matrix of a graph where
the points are nodes. This graph can also be considered as an
approximation of the surface that data lie on. Positive
semidefinitness is the only property required for the gram matrix.
In most of the methods that we discuss in this section, we prefer
the interpretation of the dot product matrix, in the general Hilbert
sense. The dot products in most of the cases are non-linear.
\subsubsection{Kernel principal component analysis}
Kernel Principal Component Analysis (KPCA), is an extension of
principal component analysis where the data are mapped into a
different feature space through the kernel. In Principal Component
Analysis the goal is to find the principal components for the
covariance matrix $C$ for a given dataset $S={x_j, x_j\in \Re^d, j=1
\dots M}$, where $C=\frac{1}{M}\sum_{j=1}^{M}x_j x_j^T$. If we
define a mapping
\begin{equation}
\Phi : x \rightarrow X , x \in \Re^d, X\in \Re^f
\end{equation}
usually $f>>d$. So usually the nonlinear mapping, maps the data in a
higher dimensional space sometimes in an infinite dimensional space.
At this point it should be clarified that the mapping increases the
extrinsic dimension of the data not the intrinsic. In kernel PCA we
want to find the principal components of the covariance matrix
\begin{equation}
\tilde{C} = \frac{1}{M}\sum_{j=1}^{M}\phi(x_j)\phi(x_j)^T
\end{equation}
It turns out that the mapping $\Phi$ doesn't have to be known
explicitly, all we need is the kernel that represents the dot
product.
\begin{equation}
k(x, y) = \phi(x)\phi(y)^T
\end{equation}
For some kernels there is an analytic factorization while for others
there is not. Not every function can be a valid kernel (dot
product). There are conditions that must be fulfilled. Basically for
every dataset they must give a valid kernel matrix as described in
previous sections.
Kernel PCA has many cousins such as Laplacian eigenmaps, Diffusion
maps Isomap, Local Linear embedding
\subsubsection{Customized Kernels}
As we described in the previous section there are infinite functions
that can act as valid kernels. The most popular kernels are:
\begin{enumerate}
\item The gaussian kernel $k(x, y)=e^{-\frac{||x-y||^2}{\sigma^2)}}$
\item The polynomial kernel $k(x, y)=(xy)^n$
\item The epanechnikov kernel $k(x, y)=1-||x-y||^2/\sigma^2, ||x-y||<\sigma, else 0$
\item The k-nearest neighbor kernel
\end{enumerate}
Although these are valid kernels for any kernel method including
Kernel PCA they are not appropriate for every task. In some cases we
want to map data in higher dimensional spaces preserving distances
(isometry) while in other cases we want to make small distances
smaller and send high distances to infinity (clustering). The
gaussian kernel for example tends to behave better in clustering
since it fades out very quickly. On the other hand it fails to
unfold the data manifold so that the intrinsic dimension can be
estimated. Manifold unfolding is a very important procedure since it
can reveal the true dimension of the dataset.
In [?] the authors introduced the idea of building the kernel matrix
from scratch without using any fixed kernel. As we have already
mentioned the kernel matrix has to be a positive semidefinite
matrix. Initially we have to define a k neighborhood for every
point. So at first we have to connect points to their neighbors and
build a initialize the non zero values of the matrix to 1. Then we
maximize the trace of the kernel matrix under the constraint that
the the distances in the kernel matrix are preserved. This leads to
a semidefinite programming problem.
\subsection{Parameter estimation for manifold learning, through density estimation}
Manifold learning turns out to be a very useful tool for many
applications of machine learning, such as classification.
Unfortunately the existing algorithms use ad hoc selection of the
parameters that define the geometry of the manifold. The parameter
choice affects significantly the performance of manifold learning
algorithms. Recent theoretical work has proven the equivalence
between the Mercer kernel learning methods and the kernel in kernel
density estimation. Based on this fact the problem of kernel
parameter estimation for manifold learning is addressed based on the
nonparametric statistical theory estimation. An automatic way of
determining the local bandwidths that define the geometry is
introduced. The results show that the automatic bandwidth selection
leads to improved clustering performance and reduces the
computational load versus ad hoc selection.
\subsection{INTRODUCTION}
Manifold learning has been active recently through the work of
\cite{Lafon} and \cite{Belkin}. Both works present algorithms for
computing the Laplace-Beltrami operator on a submanifold $\Gamma$
embedded in $\Re^d$. Lafon has also introduced a diffusion process
on the manifold based on a kernel $k(x,y)$ that defines the local
geometry on the points that belong to the manifold. The choice of
the kernel affects essentially the results of the diffusion process,
since it affects the notion of the neighborhood around the points
which is critical for the creation of the proximity graph of the
manifold. In their work Belkin and Lafon don't deal with the problem
of training the local bandwidth of the kernel, assuming that the
manifold has been sufficiently sampled. Jenssenl \cite{jenssen17lpd}
have proven that there is theoretical equivalence between the
Mercer kernel and the kernel in kernel density estimation. Based on
that fact we performed several experiments using the optimal
bandwidth for kernel density estimation. In this paper the Adaptive
Kernel Based Density Estimation Algorithm (AKDEA) \cite{Silverman}
was chosen. The proposed algorithm is a partial solution because it
doesn't deal with the final objective function directly, which in
some applications of manifold learning it could be clustering or
dimensionality reduction.
The connection between manifold
learning and Kernel Density Estimation (KDE) has also been addressed
in \cite{girolami2002osd} where the connection between manifold
learning and density estimation was also considered by using
orthogonal series density estimation rather than kernel density
estimation
In section 2 the diffusion operator is reviewed based on the work
of Lafon \cite{Lafon}. Section 3 outlines the adaptive kernel
estimation algorithm. The last section presents examples of the
diffusion operator when the kernel bandwidth is trained.
\subsection{Geometric diffusion on a manifold} Let $(\Gamma,\mu)$ be
a measure space, where $\Gamma$ is a finite set of $d$-dimensional
points and $\mu$ is a counting measure that represents the
distribution of the points on the data set. In other words $\Gamma$
is a submanifold of $\Re^{d}$. Assume that the geometry of $\Gamma$
is defined by a kernel $k(x,y)$. The kernel $k(x,y)$ measures the
degree of similarity between two points $x,y$. The kernel satisfies
the following conditions:
\begin{itemize}
\item $k$ is symmetric: $k(x,y)=k(y,x)$,
\item $k$ is positivity-preserving: for all $x,y$ in
$\Gamma, k(x,y)\geq 0$
\item $k$ is positive semi-definite: for all bounded
functions $f$ defined on $\Gamma$,
\[
\int_{\Gamma} \int_{\Gamma} k(x,y)f(x)f(y)d\mu(x)d\mu(y)\geq 0\]
\end{itemize}
In this paper it is assumed that $\Gamma$ is a subset of the
Euclidean space $\Re^{d}$. So for $x,y\in\Gamma$ the kernel
(similarity measure) is a function of the Euclidean distance
$\parallel x-y\parallel:$
\[
k(x,y)=\eta\left(\frac{\parallel x-y\parallel}{h}\right)
\]
In order to study the geometry of the submanifold $\Gamma$, an
oriented graph $G$ is formed. Every node corresponds to a data
point. Let $b(x,y)$ be the associated adjacency matrix, where
$b(x,y)=1$ if $x$ is in the neighborhood of $y$ and $b(x,y)=0$ if
$x$ is not in the neighborhood of $y$. The kernel $k$ defines the
notion of neighborhood between the points.
Let $u^2(x)=\int_{\Gamma}k(x,y)d\mu(y)$, then the normalized kernel
$a(x,y)=\frac{k(x,y)}{u(x)u(y)}$ is stochastic since:
\begin{equation}\label{kernel}
\int_{\Gamma}a(x,y)d\mu(y)=1.
\end{equation}
The kernel $a(x,y)$ can be considered as a transition matrix of a
Markov process on the submanifold $\Gamma$. So any admissible kernel
can be associated with a random walk on $\Gamma$. The operator
\[
Af(x)=\int_{\Gamma}a(x,y)f(y)d\mu(y)
\]
is called a diffusion operator. The eigenvalue analysis of the graph
with the adjacency matrix $a(x,y)$ is the basis for embedding the
data to a lower dimensional space.
\[
a(x,y)=\sum_{j=0}^{M} \lambda_{j}\phi_{j}(x)\phi_{j}(y)
\]
where $M$ is the number of data points, $x,y$ are nodes of the graph
and $\phi_j$ is the $j_{th}$ eigenvector of $a$ that corresponds to
the $\lambda_j$ eigenvalue.
\section{Non parametric density estimation}
The problem of density estimation of finite data set is very
significant for most of the well known machine learning algorithms.
Lafon has proven that the Laplace Beltrami operator on a manifold
can be computed without the knowledge of the density. Although the
density is not necessary for the computation of the operator it can
be useful in the estimation of the parameters of the kernel
[\ref{kernel}].
The density estimate $f$ of a finite set of data $X={x_i\ni
\Re^n,i=1,\dots,M}$ can be expressed with the following equation
\cite{Silverman}:
\begin{equation}\label{powerdensity}
f(x)=\frac{1}{M}\sum_{i=1}^{M}\frac{1}{h^d}k\left(\frac{\parallel x-x_i
\parallel}{h}\right)
\end{equation}
The shape of the density function varies significantly with
different values of $h$, which is the global bandwidth. Higher
values of $h$ lead to smoothed versions of the density. An
automatic way to determine the optimum value of $h$ is to maximize
the Leave One Out Cross Validation (LOOCV) criterion. In other words
the density is formed according to (\ref{powerdensity}) by leaving
one point out and then the probability of this point is evaluated
according to the density. The LOOCV can be mathematically expressed:
\begin{equation}
M(h)=\sum_{i=1}^{M}\log\sum_{i\neq j}f_j(x_i)
\end{equation}
where $f_j$ is the density computed according to
(\ref{powerdensity}) by leaving $x_j$ out.
Another restriction that can be imposed to the estimated density is
the minimization of the power of the second derivative of the
density $\int f''(x)^2$. This restriction tends to give smoother
densities and avoids the formation of isolated islands of points.
With the standard optimization techniques $h$ can be chosen.
The value of $h$ defines the neighborhood around a point through the
kernel $k$. For dense areas $h$ has to be small, while in sparse
areas $h$ must be high since the points are far away from each
other. This leads to the conclusion that $h$ should be different for
every point. The density can be written in the form:
\[
f(x)=\frac{1}{M}\sum_{i=1}^{M}\frac{1}{(\lambda_i h)^d}k(\frac{\parallel x-x_i
\parallel}{\lambda_i h})
\]
The $\lambda_i$ are called local bandwidths. The local bandwidths
can be estimated from the following algorithm:
\begin{itemize}
\item Find a pilot estimate $\tilde{f}(x)$ that satisfies
$\tilde{f(x_i)>0, i=1,\dots,M}$
\item The local bandwidth $\lambda_i$ is given by:
\[
\lambda_i=(\tilde{f}(x_i)/g)^{\frac{1}{d}}
\]
where g is the geometric mean of the $\tilde{f}(x_i)$:
\[
g=\sqrt[M]{\prod_{i=1}^{M}\tilde{f}(x_i)}
\]
\item The adaptive kernel estimate $f(x)$ is defined by:
\[
f(x)=\frac{1}{M}\sum_{i=1}^{M}\frac{1}{(\lambda_i
h)^d} k\left( \frac{\parallel x-x_i \parallel}{\lambda_i h} \right)
\]
\end{itemize}
The pilot estimate can be any ad hoc density estimate. A common
estimate is the \emph{kth nearest neighbor estimate} defined by:
\[
\tilde{f}(x)=\frac{1}{Md_k(x)}\sum_{i=1}^{M}k\left(\frac{
\parallel x-x_i \parallel}{d_k(x)}\right)
\]
where $d_k(x)$ is the distance between $x$ and the $kth$ nearest
neighbor. It turns out that the choice of $k$ doesn't affect the
values of $\lambda_i$. The adaptive kernel estimate can be improved
if we iterate the above procedure by replacing in every iteration
the pilot density estimate with the adaptive kernel estimate
computed in the last step.
\section{Eigenvalue analysis of the diffusion graph}
In the previous section we saw that the non-parametric estimation
can define the local bandwidth for every point. This information
can be used in the construction of the adjacency matrix $G$. The
local bandwidths as derived from the previous section will usually
give a graph that has connected components, or equivalently the
manifold is connected. A safe test to prove that is to look into
the eigenvalues also known as the spectrum of the graph. According
to \cite{Cvetkovic} the multiplicity of the first eigenvalue
reflects the number of disconnected components in the graph. There
are cases where the local bandwidths might not give a connected
manifold, then it is necessary to increase $h$ until the
multiplicity of the first eigenvalue becomes 1.
\subsection{Kernel choice}
There are many classes of kernels that can be used for the diffusion
operator on the manifold. The most popular is the Gaussian kernel:
\[
k(x)=\frac{1}{(2\pi)^{\frac{-d}{2}}}e^{(\frac{-1}{2}xx^T)}
\]
The problem with the gaussian kernel is that it is of infinite
support and it leads to adjacency matrices that are not sparse. It
would be preferable to use a kernel of finite support. The most
popular one is the Epanechnikov kernel \cite{Silverman}:
\begin{equation}
k(x)=\left\{
\begin{array}{ll}
\frac{1}{2}c_{d}^{-1}(d+2)(1-x^Tx) & x^Tx<1 \\
0 & x^Tx\geq 1 \\
\end{array}
\right.
\end{equation}
\subsection{Examples}
In this section two examples illustrated in the two dimensional
space are presented and a third one on real speech data. The first
example fig.~\ref{fig1} contains two clusters of data. Each cluster
contains 300 points uniformly distributed in a unity area square.
The Average Nearest Neighbor Distance (ANND) as a choice for the
bandwidth for the Epanechnikov kernel turns out to be inadequate
since it leads to a disconnected graph with 4 disjoint components.
This is probably because the two classes are sparsely sampled. The
AKDE based on maximization of LOOCV gives on the average local
bandwidths that are an order of magnitude higher than the ANND, and
lead to a smoother density. As it is shown in fig.~\ref{fig1}b the
density is not flat as it was expected to be and this is because of
the fact that the number of points that sample the manifold is
small. In fig.~\ref{fig1}c the first non trivial eigenvector is
depicted discriminating very sharply the two classes. In
fig.~\ref{fig1.1} the same data set is analyzed with a higher global
bandwidth. The results are very poor since there is high intra-class
deviation and the centroids of the two classes are closer.
The second example fig.~\ref{fig2} illustrates two classes of data
that lie on two halves of a circle, slightly separated. The clusters
are highly the sampled around the angle of $90^0$ degrees. This
fact makes very difficult to chose a universal value for the local
bandwidths since the notion of the locality is significantly
different in different areas of the manifold. In fig.~\ref{fig2}b
the density of the data points is shown, reflecting the high
concentration of points around the angle of $90^0$. In this case the
AKDE trains the local bandwidths $\lambda_i$ and the global $h$, so
that the LOOCV is maximized. The resulting values give a manifold
graph that is disconnected in two classes. A slight increase in the
global bandwidth makes the graph connected. The first non-trivial
eigenvector of the diffusion operator identifies very clearly the
two classes as shown in fig.~\ref{fig2}c. If the global bandwidth is
increased by an order of magnitude, then still the first non-trivial
eigenvector identifies the two classes, but the classes are not
compact and their centroids are closer fig.\ref{fig3}c. Another
disadvantage of keeping $h$ large is that the adjacency matrix
$a(x,y)$ fig.~\ref{fig5}b of the manifold graph becomes dense. This
increases tremendously the memory requirement and the computation of
the eigenvectors.
In this third example the effect of the local bandwidth on the
clustering problem of three phoneme classes is presented. The points
correspond to the Fast Fourier Transform of 20 msec speech frames
sampled at 8KHz, of the TIMIT database. 200 points from the three
phoneme classes /aa/,/ih/,/sh/ where chosen. In fig.~\ref{fig4}a the
data are plotted in 2 dimensions after the dimensionality reduction
process. It is clear that the diffusion process separates the
classes very well. In fig.~\ref{fig4}b where the bandwidth is an
order of magnitude larger than the optimal, the diffusion process
still separates the classes, but the clusters are not very compact.
When the local bandwidth is an order of magnitude smaller than the
optimal, the results are very poor. The corresponding graph
fig.~\ref{fig5}c shows that the points are isolated and not
connected.
\section{Discussion}
In this paper the problem of constructing the diffusion graph of a
manifold defined by data points in $\Re^d$ was addressed. The
parameters of the kernel that define the geometry over the manifold
were automatically trained through the estimation of the adaptive
kernel based density algorithm. Experiments on synthetic and real
data showed that if the global bandwidth is smaller than the one
that maximizes the LOOCV, then the graph that represents the
manifold is disconnected into many spurious clusters. On the other
hand if the bandwidth is higher, then the diffusion operator will
still identify the clusters, but their distance is smaller and they
are not as compact compared to the optimal bandwidth. Moreover high
bandwidth lead to dense graphs and as a consequence more
computations.
The approach followed is still heuristic, although it has an
underlying theoretical justification on the work described at
\cite{jenssen17lpd}. It is the first time that experimental results
are presented that show the effect of optimal kernel density
estimation on the manifold learning.
In terms of algorithmic
complexity, most of the operations performed are $O(n^2)$ with the
naive approach. Both kernel density estimation and graph formation
fall under the general case of N-body problem computations. The
speed can be significantly improved by using KD-trees as described
in \cite{gray2001nbp} and reduce the complexity up to $O(n)$.
\begin{figure}[h!]
\centerline{\includegraphics[height=4.5cm]{fig1.1b.eps}(a)}
\centerline{\includegraphics[height=4.5cm]{fig1.1c.eps}(b)}
\caption{The same data with fig.~\ref{fig1}a analyzed with a
larger global bandwidth. (a) The estimated density, (b) The first
non-trivial eigenvector }
\label{fig1.1}
\end{figure}
\bibliographystyle{IEEEbib}
\bibliography{main}
\begin{figure}[tb]
\centerline{\includegraphics[height=4.4cm]{fig1a.eps}(a)}
\centerline{\includegraphics[height=4.4cm]{fig1b.eps}(b)}
\centerline{\includegraphics[height=4.4cm]{fig1c.eps}(c)}
\caption{(a) Two clusters of uniformly distributed 2-D points,
(b) The estimated density, (c) The first non-trivial eigenvector }
\label{fig1}
\end{figure}
\begin{figure}[tb]
\centerline{\includegraphics[height=6.0cm]{fig2a.eps}(a)}
\centerline{\includegraphics[height=6.0cm]{fig2b.eps}(b)}
\centerline{\includegraphics[height=6.0cm]{fig2c.eps}(c)}
\caption{(a) Two clusters of non-uniformly distributed 2-D points,
(b)The estimated density, (c) The first non-trivial eigenvector
\vspace{1.2cm}}
\label{fig2}
\end{figure}
\begin{figure}[tb]
\centerline{\includegraphics[height=6cm]{fig3a.eps}(a)}
\centerline{\includegraphics[height=6cm]{fig3b.eps}(b)}
\centerline{\includegraphics[height=6cm]{fig3c.eps}(c)}
\caption{The same data set with fig.~\ref{fig2} processed with a
bandwidth one order of magnitude larger. Although the first
eigenvector still separates the two classes, the classes are closer
and not compact (a) Two clusters of non-uniformly distributes 2-D
points, (b)The estimated density, (c) The first non trivial
eigenvector }
\label{fig3}
\end{figure}
\begin{figure}[tb]
\centerline{\includegraphics[height=6cm]{fig4a.eps}(a)}
\centerline{\includegraphics[height=6cm]{fig4c.eps}(b)}
\centerline{\includegraphics[height=6cm]{fig4e.eps}(c)}
\caption{(a) The 3 phoneme classes after the dimensionality reduction
with the optimal bandwidth, plotted in two dimensions, (b) The 3
phoneme classes with a larger bandwidth,(c) The 3 phoneme classes
with a smaller bandwidth}
\label{fig4}
\end{figure}
\begin{figure}[tb]
\centerline{\includegraphics[height=6cm]{fig4b.eps}(a)}
\centerline{\includegraphics[height=6cm]{fig4d.eps}(b)}
\centerline{\includegraphics[height=6cm]{fig4f.eps}(c)}
\caption{The diffusion graph for three different local bandwidths
(a)Optimal, (b)An order of magnitude larger, (c)An order of
magnitude smaller}
\label{fig5}
\end{figure}
\subsection{Speech recognition with nearest neighbor search}
In this paper we present a computationally efficient means of
@@ -1043,18 +1348,91 @@ MFCC features}{8cm}
} \label{kneighborLOOCVNTIMIT}
\end{table}
\section{Proposed Research}
\label{proposed}
The goal of this thesis is to make kernel methods scalable.
Efficient computation of the kernel matrix for large amounts of
data. Special emphasis is given in high dimensional data $d>30$. One
of the goals is to use Kernel PCA for unfolding the speech manifold.
In other words estimate the intrinsic dimensionality of speech by
using big databases like timit, wall street journal and broadcast
news. Implicitly dimensionality reduction will lead to new feature
generation we will explain it in the following sections. The
ultimate goal is to do speech recognition by nearest neighbor
search.
\subsection{Customizing Kernels with fast
gradient descent}
As we described in section ? it is possible to customize the kernel
for a given dataset, through semidefinite programming. Although this
method is very promising and gives very good results it is not
scalable. Semidefinite programming is solved with the intrinsic
point method which is of polynomial complexity.
\subsubsection{Fast kernel summation with trees}
Kernel summation is on of the most common tasks in kernel methods.
The general formula
\begin{equation}
\label{kernel_sum}
G(x)=\sum_{q=0}^{N} w_i\exp(\frac{(x-x_q)^2}{\sigma^2})
\end{equation}
appears in many cases, with the most interesting case the
eigenvalue/eigenvector computation. The kernel summation is a very
computationally intensive. For Gaussian kernels there are
approximations that can speed up computations orders of magnitude.
We will briefly discuss them here. Besides computation saving we can
save memory as well. The gaussian kernel has infinite support so the
kernel matrix $K, K(x,y)=\exp(\frac{(x-y)^2}{\sigma^2}$ is not
sparse. Numerical iterative methods for eigenvector computation are
based on the general iteration $x_n=x_{n-1}+\lambda Kx_{n-1}$, where
the multiplication of each row of $K$ with $x_{n-1}$ is a kernel
summation problem. So we don't really need to store the kernel
matrix, we just have to compute the kernel sum for every row using
$x_{n-1}$ as a the wight vector.
The gaussian can be approximated with the help of Hermite
polynomials. The Hermite polynomials $H_n(t)$ are defined by the
Rodrigues formula:
\begin{equation}
H_n(t)=(-1)^n\exp(t^2)D^n\exp(-t^2), t\in \Re
\end{equation}
where $D=d/dt$.
The gaussian can now be expressed as
\begin{equation}
\exp^(-(t-s)^2)=\sum_{n=0}^{\inf}\frac{s^n}{n!}h_n(t)
\end{equation}
where $h_n(t)$ is defined by
\begin{equation}
h_n(t)=\exp(-t^2)H_n(t).
\end{equation}
\subsection{Computing Gaussian Kernel Matrix with trees}
Direct computation of the kernel(Gaussian) kernel leads to a
non-sparse matrix. There is a way though to compute the
inverse/eigenvalues without explicitly computing the kernel matrix.
Several optimizations can be done for high dimensions
\subsection{Intrinsic dimension of speech}
\subsection{Speech recognition with nearest neighbor search}
Implement a merge reduce scheme to create different descriptions of
speech. Use nearest neighbor algorithms to replace gaussian mixtures
in hidden markov models.
\subsection{Acoustic Environment adaptation}
\section{Work remaining to be done}
\label{remains}
\section{Facilities and Equipment Needed}
\label{facitlies}
\end{Body}
\begin{EndMatter}
\appendix{This is my first appendix}
\appendix{This is my second appendix}
%% Below are some optional environments that can be used here
\bibliographystyle{ieeetr}
\references % Generates the bibliography page
fastlib/u/nvasil/docs/proposal/proposal.tex.bak
+191 -150
View File
@@ -156,7 +156,37 @@ literature dimensionality is used without the prefix intrinsic or
extrinsic, causing confusion. The extrinsic dimensionality is just
the number of variables needed to describe a point, i.e. the
extrinsic dimensionality of $x\in \Re^d$ is just $d$. Each variable
of the d-dimensional vector will be called dimension $1\dots d$
of the d-dimensional vector will be called dimension $1\dots d$. In
reality some of the dimensions might be linearly or non-linearly
dependent on other dimensions. So less than $d$ dimensions are
sufficient to describe a point. We can define the intrinsic
dimensionality of a dataset as the minimum number of "independent"
variables needed to describe a point in space. Still the term is not
very clear, for the scope of this thesis we mean that there is no
algorithm that can predict a dimension from the others. Principal
Component Analysis for example finds the most important dimensions
on a dataset that are linearly independent. In practice because of
noise all dimensions appear to be strictly independent. Modern
kernel methods can capture non-lineal dependencies between
dimensions. Kernel PCA is the father of non-linear dimensionality
reduction methods.
Although Kernel PCA seems to be a very powerful tool it suffers from
polynomial complexity. It requires $O(N^2)$ kernel computations,
which make it unsuitable for large scale problems. In this thesis we
will show how we are going to use multidimensional trees to reduce
the complexity, along with other approximations to bring it down to
linear.
The driving force for investigating dimensionality reduction was the
estimation of the true dimension of speech. Many representations
have been recommended for speech MFCC, NRAFF, LPC, etc. The true
dimension is yet unknown. Meanwhile there is theoretical evidence
that speech signals are embedded in high dimensional manifolds. In
this thesis we will attempt to estimate the true dimensionality of
speech based on several representations. We will see in the future
sections how this can help in developing new directions in speech
recognition.
@@ -328,143 +358,7 @@ In general the dual tree algorithm as we call it since the query and
the reference set lie on trees (they might share the same tree if
the query and the reference set are the same), gives linear
complexity over the number of data. This is empirical complexity and
\subsubsection{Fast kernel summation with trees}
Kernel summation is on of the most common tasks in kernel methods.
The general formula
\begin{equation}
\label{kernel_sum}
G(x)=\sum_{q=0}^{N} w_i\exp(\frac{(x-x_q)^2}{\sigma^2})
\end{equation}
appears in many cases, with the most interesting case the
eigenvalue/eigenvector computation. The kernel summation is a very
computationally intensive. For Gaussian kernels there are
approximations that can speed up computations orders of magnitude.
We will briefly discuss them here. Besides computation saving we can
save memory as well. The gaussian kernel has infinite support so the
kernel matrix $K, K(x,y)=\exp(\frac{(x-y)^2}{\sigma^2}$ is not
sparse. Numerical iterative methods for eigenvector computation are
based on the general iteration $x_n=x_{n-1}+\lambda Kx_{n-1}$, where
the multiplication of each row of $K$ with $x_{n-1}$ is a kernel
summation problem. So we don't really need to store the kernel
matrix, we just have to compute the kernel sum for every row using
$x_{n-1}$ as a the wight vector.
The gaussian can be approximated with the help of Hermite
polynomials. The Hermite polynomials $H_n(t)$ are defined by the
Rodrigues formula:
\begin{equation}
H_n(t)=(-1)^n\exp(t^2)D^n\exp(-t^2), t\in \Re
\end{equation}
where $D=d/dt$.
The gaussian can now be expressed as
\begin{equation}
\exp^(-(t-s)^2)=\sum_{n=0}^{\inf}\frac{s^n}{n!}h_n(t)
\end{equation}
where $h_n(t)$ is defined by
\begin{equation}
h_n(t)=\exp(-t^2)H_n(t).
\end{equation}
\subsection{Kernel methods}
The kernel or Gram matrix is an informational representation for a
set of data. There are a different interpretations for the kernel
matrix. The most mathematical interpretation is the dot product
matrix. Every element of the kernel matrix is the dot product
between the two data points. It is a positive semidefinite matrix
and it can be considered as the adjacency matrix of a graph where
the points are nodes. This graph can also be considered as an
approximation of the surface that data lie on. Positive
semidefinitness is the only property required for the gram matrix.
In most of the methods that we discuss in this section, we prefer
the interpretation of the dot product matrix, in the general Hilbert
sense. The dot products in most of the cases are non-linear.
\subsubsection{Kernel principal component analysis}
Kernel Principal Component Analysis (KPCA), is an extension of
principal component analysis where the data are mapped into a
different feature space through the kernel. In Principal Component
Analysis the goal is to find the principal components for the
covariance matrix $C$ for a given dataset $S={x_j, x_j\in \Re^d, j=1
\dots M}$, where $C=\frac{1}{M}\sum_{j=1}^{M}x_j x_j^T$. If we
define a mapping
\begin{equation}
\Phi : x \rightarrow X , x \in \Re^d, X\in \Re^f
\end{equation}
usually $f>>d$. So usually the nonlinear mapping, maps the data in a
higher dimensional space sometimes in an infinite dimensional space.
At this point it should be clarified that the mapping increases the
extrinsic dimension of the data not the intrinsic. In kernel PCA we
want to find the principal components of the covariance matrix
\begin{equation}
\tilde{C} = \frac{1}{M}\sum_{j=1}^{M}\phi(x_j)\phi(x_j)^T
\end{equation}
It turns out that the mapping $\Phi$ doesn't have to be known
explicitly, all we need is the kernel that represents the dot
product.
\begin{equation}
k(x, y) = \phi(x)\phi(y)^T
\end{equation}
For some kernels there is an analytic factorization while for others
there is not. Not every function can be a valid kernel (dot
product). There are conditions that must be fulfilled. Basically for
every dataset they must give a valid kernel matrix as described in
previous sections.
Kernel PCA has many cousins such as Laplacian eigenmaps, Diffusion
maps Isomap, Local Linear embedding
\subsubsection{Customized Kernels}
As we described in the previous section there are infinite functions
that can act as valid kernels. The most popular kernels are:
\begin{enumerate}
\item The gaussian kernel $k(x, y)=e^{-\frac{||x-y||^2}{\sigma^2)}}$
\item The polynomial kernel $k(x, y)=(xy)^n$
\item The epanechnikov kernel $k(x, y)=1-||x-y||^2/\sigma^2, ||x-y||<\sigma, else 0$
\item The k-nearest neighbor kernel
\end{enumerate}
Although these are valid kernels for any kernel method including
Kernel PCA they are not appropriate for every task. In some cases we
want to map data in higher dimensional spaces preserving distances
(isometry) while in other cases we want to make small distances
smaller and send high distances to infinity (clustering). The
gaussian kernel for example tends to behave better in clustering
since it fades out very quickly. On the other hand it fails to
unfold the data manifold so that the intrinsic dimension can be
estimated. Manifold unfolding is a very important procedure since it
can reveal the true dimension of the dataset.
In [?] the authors introduced the idea of building the kernel matrix
from scratch without using any fixed kernel. As we have already
mentioned the kernel matrix has to be a positive semidefinite
matrix. Initially we have to define a k neighborhood for every
point. So at first we have to connect points to their neighbors and
build a initialize the non zero values of the matrix to 1. Then we
maximize the trace of the kernel matrix under the constraint that
the the distances in the kernel matrix are preserved. This leads to
a semidefinite programming problem.
\subsubsection{Stochastic optimization}
Stochastic optimization or on line learning. On line svd is one key
aspect.
\subsubsection{Support Vector Machines}
\section{Proposed Research}
\label{proposed}
The goal of this thesis is to make kernel methods scalable.
Efficient computation of the kernel matrix for large amounts of
data. Special emphasis is given in high dimensional data $d>30$. One
of the goals is to use Kernel PCA for unfolding the speech manifold.
In other words estimate the intrinsic dimensionality of speech by
using big databases like timit, wall street journal and broadcast
news. Implicitly dimensionality reduction will lead to new feature
generation we will explain it in the following sections. The
ultimate goal is to do speech recognition by nearest neighbor
search.
\subsection{Large scale trees, for out of core memory}
Kd-trees have been proven to be be quite successful in
multidimensional indexing and turn out outperform several other
@@ -642,19 +536,93 @@ to address efficiently this. Pages are write protected in memory and
if a write occurs a signal is raised and the page is marked as
modified.
\subsection{Customizing Kernels with fast
gradient descent}
As we described in section ? it is possible to customize the kernel
for a given dataset, through semidefinite programming. Although this
method is very promising and gives very good results it is not
scalable. Semidefinite programming is solved with the intrinsic
point method which is of polynomial complexity.
\subsection{Scalable Support Vector Machines}
\subsection{Computing Gaussian Kernel Matrix with trees}
\subsection{Intrinsic dimension of speech}
\subsection{Kernel methods}
The kernel or Gram matrix is an informational representation for a
set of data. There are a different interpretations for the kernel
matrix. The most mathematical interpretation is the dot product
matrix. Every element of the kernel matrix is the dot product
between the two data points. It is a positive semidefinite matrix
and it can be considered as the adjacency matrix of a graph where
the points are nodes. This graph can also be considered as an
approximation of the surface that data lie on. Positive
semidefinitness is the only property required for the gram matrix.
In most of the methods that we discuss in this section, we prefer
the interpretation of the dot product matrix, in the general Hilbert
sense. The dot products in most of the cases are non-linear.
\subsubsection{Kernel principal component analysis}
Kernel Principal Component Analysis (KPCA), is an extension of
principal component analysis where the data are mapped into a
different feature space through the kernel. In Principal Component
Analysis the goal is to find the principal components for the
covariance matrix $C$ for a given dataset $S={x_j, x_j\in \Re^d, j=1
\dots M}$, where $C=\frac{1}{M}\sum_{j=1}^{M}x_j x_j^T$. If we
define a mapping
\begin{equation}
\Phi : x \rightarrow X , x \in \Re^d, X\in \Re^f
\end{equation}
usually $f>>d$. So usually the nonlinear mapping, maps the data in a
higher dimensional space sometimes in an infinite dimensional space.
At this point it should be clarified that the mapping increases the
extrinsic dimension of the data not the intrinsic. In kernel PCA we
want to find the principal components of the covariance matrix
\begin{equation}
\tilde{C} = \frac{1}{M}\sum_{j=1}^{M}\phi(x_j)\phi(x_j)^T
\end{equation}
It turns out that the mapping $\Phi$ doesn't have to be known
explicitly, all we need is the kernel that represents the dot
product.
\begin{equation}
k(x, y) = \phi(x)\phi(y)^T
\end{equation}
For some kernels there is an analytic factorization while for others
there is not. Not every function can be a valid kernel (dot
product). There are conditions that must be fulfilled. Basically for
every dataset they must give a valid kernel matrix as described in
previous sections.
Kernel PCA has many cousins such as Laplacian eigenmaps, Diffusion
maps Isomap, Local Linear embedding
\subsubsection{Customized Kernels}
As we described in the previous section there are infinite functions
that can act as valid kernels. The most popular kernels are:
\begin{enumerate}
\item The gaussian kernel $k(x, y)=e^{-\frac{||x-y||^2}{\sigma^2)}}$
\item The polynomial kernel $k(x, y)=(xy)^n$
\item The epanechnikov kernel $k(x, y)=1-||x-y||^2/\sigma^2, ||x-y||<\sigma, else 0$
\item The k-nearest neighbor kernel
\end{enumerate}
Although these are valid kernels for any kernel method including
Kernel PCA they are not appropriate for every task. In some cases we
want to map data in higher dimensional spaces preserving distances
(isometry) while in other cases we want to make small distances
smaller and send high distances to infinity (clustering). The
gaussian kernel for example tends to behave better in clustering
since it fades out very quickly. On the other hand it fails to
unfold the data manifold so that the intrinsic dimension can be
estimated. Manifold unfolding is a very important procedure since it
can reveal the true dimension of the dataset.
In [?] the authors introduced the idea of building the kernel matrix
from scratch without using any fixed kernel. As we have already
mentioned the kernel matrix has to be a positive semidefinite
matrix. Initially we have to define a k neighborhood for every
point. So at first we have to connect points to their neighbors and
build a initialize the non zero values of the matrix to 1. Then we
maximize the trace of the kernel matrix under the constraint that
the the distances in the kernel matrix are preserved. This leads to
a semidefinite programming problem.
\subsection{Forming manifolds from data for clustering through
non-parametric density estimation}
\subsection{Speech recognition with nearest neighbor search}
@@ -1013,18 +981,91 @@ MFCC features}{8cm}
} \label{kneighborLOOCVNTIMIT}
\end{table}
\section{Proposed Research}
\label{proposed}
The goal of this thesis is to make kernel methods scalable.
Efficient computation of the kernel matrix for large amounts of
data. Special emphasis is given in high dimensional data $d>30$. One
of the goals is to use Kernel PCA for unfolding the speech manifold.
In other words estimate the intrinsic dimensionality of speech by
using big databases like timit, wall street journal and broadcast
news. Implicitly dimensionality reduction will lead to new feature
generation we will explain it in the following sections. The
ultimate goal is to do speech recognition by nearest neighbor
search.
\subsection{Customizing Kernels with fast
gradient descent}
As we described in section ? it is possible to customize the kernel
for a given dataset, through semidefinite programming. Although this
method is very promising and gives very good results it is not
scalable. Semidefinite programming is solved with the intrinsic
point method which is of polynomial complexity.
\subsubsection{Fast kernel summation with trees}
Kernel summation is on of the most common tasks in kernel methods.
The general formula
\begin{equation}
\label{kernel_sum}
G(x)=\sum_{q=0}^{N} w_i\exp(\frac{(x-x_q)^2}{\sigma^2})
\end{equation}
appears in many cases, with the most interesting case the
eigenvalue/eigenvector computation. The kernel summation is a very
computationally intensive. For Gaussian kernels there are
approximations that can speed up computations orders of magnitude.
We will briefly discuss them here. Besides computation saving we can
save memory as well. The gaussian kernel has infinite support so the
kernel matrix $K, K(x,y)=\exp(\frac{(x-y)^2}{\sigma^2}$ is not
sparse. Numerical iterative methods for eigenvector computation are
based on the general iteration $x_n=x_{n-1}+\lambda Kx_{n-1}$, where
the multiplication of each row of $K$ with $x_{n-1}$ is a kernel
summation problem. So we don't really need to store the kernel
matrix, we just have to compute the kernel sum for every row using
$x_{n-1}$ as a the wight vector.
The gaussian can be approximated with the help of Hermite
polynomials. The Hermite polynomials $H_n(t)$ are defined by the
Rodrigues formula:
\begin{equation}
H_n(t)=(-1)^n\exp(t^2)D^n\exp(-t^2), t\in \Re
\end{equation}
where $D=d/dt$.
The gaussian can now be expressed as
\begin{equation}
\exp^(-(t-s)^2)=\sum_{n=0}^{\inf}\frac{s^n}{n!}h_n(t)
\end{equation}
where $h_n(t)$ is defined by
\begin{equation}
h_n(t)=\exp(-t^2)H_n(t).
\end{equation}
\subsection{Computing Gaussian Kernel Matrix with trees}
Direct computation of the kernel(Gaussian) kernel leads to a
non-sparse matrix. There is a way though to compute the
inverse/eigenvalues without explicitly computing the kernel matrix.
Several optimizations can be done for high dimensions
\subsection{Intrinsic dimension of speech}
\subsection{Speech recognition with nearest neighbor search}
Implement a merge reduce scheme to create different descriptions of
speech. Use nearest neighbor algorithms to replace gaussian mixtures
in hidden markov models.
\subsection{Acoustic Environment adaptation}
\section{Work remaining to be done}
\label{remains}
\section{Facilities and Equipment Needed}
\label{facitlies}
\end{Body}
\begin{EndMatter}
\appendix{This is my first appendix}
\appendix{This is my second appendix}
%% Below are some optional environments that can be used here
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fastlib/u/nvasil/docs/proposal/proposal.toc
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\contentsline {subsubsection}{\numberline {3.1.3}Ball trees}{5}{subsubsection.16}
\contentsline {subsubsection}{\numberline {3.1.4}Nearest Neighbor Algorithm}{6}{subsubsection.18}
\contentsline {subsubsection}{\numberline {3.1.5}All nearest Neighbors}{7}{subsubsection.19}
\contentsline {subsubsection}{\numberline {3.1.6}Fast kernel summation with trees}{8}{subsubsection.20}
\contentsline {subsection}{\numberline {3.2}Kernel methods}{8}{subsection.25}
\contentsline {subsubsection}{\numberline {3.2.1}Kernel principal component analysis}{9}{subsubsection.26}
\contentsline {subsubsection}{\numberline {3.2.2}Customized Kernels}{10}{subsubsection.30}
\contentsline {subsubsection}{\numberline {3.2.3}Stochastic optimization}{11}{subsubsection.35}
\contentsline {subsubsection}{\numberline {3.2.4}Support Vector Machines}{11}{subsubsection.36}
\contentsline {subsection}{\numberline {3.2}Large scale trees, for out of core memory}{8}{subsection.20}
\contentsline {subsection}{\numberline {3.3}Memory layout of Kd-trees}{9}{subsection.21}
\contentsline {subsection}{\numberline {3.4}System architecture}{10}{subsection.22}
\contentsline {subsection}{\numberline {3.5}Memory Manager Architecture}{10}{subsection.23}
\contentsline {subsection}{\numberline {3.6}Memory Mapped Files}{11}{subsection.24}
\contentsline {subsection}{\numberline {3.7}User defined cache, with TPIE}{11}{subsection.25}
\contentsline {subsection}{\numberline {3.8}Kernel methods}{12}{subsection.26}
\contentsline {subsubsection}{\numberline {3.8.1}Kernel principal component analysis}{13}{subsubsection.27}
\contentsline {subsubsection}{\numberline {3.8.2}Customized Kernels}{14}{subsubsection.31}
\contentsline {subsection}{\numberline {3.9}Forming manifolds from data for clustering through non-parametric density estimation}{15}{subsection.36}
\contentsline {subsection}{\numberline {3.10}Speech recognition with nearest neighbor search}{15}{subsection.37}
\contentsline {subsection}{\numberline {3.11}Introduction}{15}{subsection.38}
\contentsline {subsection}{\numberline {3.12}Speech features}{16}{subsection.39}
\contentsline {subsection}{\numberline {3.13}Fast N-body methods for feature comparison}{17}{subsection.40}
\contentsline {subsection}{\numberline {3.14}Kernel PCA on a manifold}{18}{subsection.44}
\contentsline {subsection}{\numberline {3.15}Experimental Results}{19}{subsection.45}
\contentsline {subsection}{\numberline {3.16}All nearest neighbor performance}{20}{subsection.46}
\contentsline {subsection}{\numberline {3.17}Comparison of NRAF and MFCC}{20}{subsection.47}
\contentsline {subsection}{\numberline {3.18}Discussion}{21}{subsection.49}
\setcounter {tocdepth}{2}
\contentsline {section}{\numberline {\MakeUppercase {Section} 4}\MakeUppercase {Proposed Research}}{11}{section.37}
\contentsline {subsection}{\numberline {4.1}Large scale trees, for out of core memory}{11}{subsection.38}
\contentsline {subsection}{\numberline {4.2}Memory layout of Kd-trees}{13}{subsection.39}
\contentsline {subsection}{\numberline {4.3}System architecture}{13}{subsection.40}
\contentsline {subsection}{\numberline {4.4}Memory Manager Architecture}{14}{subsection.41}
\contentsline {subsection}{\numberline {4.5}Memory Mapped Files}{14}{subsection.42}
\contentsline {subsection}{\numberline {4.6}User defined cache, with TPIE}{15}{subsection.43}
\contentsline {subsection}{\numberline {4.7}Customizing Kernels with fast gradient descent}{16}{subsection.44}
\contentsline {subsection}{\numberline {4.8}Scalable Support Vector Machines}{16}{subsection.45}
\contentsline {subsection}{\numberline {4.9}Computing Gaussian Kernel Matrix with trees}{16}{subsection.46}
\contentsline {subsection}{\numberline {4.10}Intrinsic dimension of speech}{16}{subsection.47}
\contentsline {subsection}{\numberline {4.11}Speech recognition with nearest neighbor search}{16}{subsection.48}
\contentsline {subsection}{\numberline {4.12}Introduction}{17}{subsection.49}
\contentsline {subsection}{\numberline {4.13}Speech features}{18}{subsection.50}
\contentsline {subsection}{\numberline {4.14}Fast N-body methods for feature comparison}{18}{subsection.51}
\contentsline {subsection}{\numberline {4.15}Kernel PCA on a manifold}{20}{subsection.55}
\contentsline {subsection}{\numberline {4.16}Experimental Results}{21}{subsection.56}
\contentsline {subsection}{\numberline {4.17}All nearest neighbor performance}{21}{subsection.57}
\contentsline {subsection}{\numberline {4.18}Comparison of NRAF and MFCC}{22}{subsection.58}
\contentsline {subsection}{\numberline {4.19}Discussion}{23}{subsection.60}
\contentsline {subsection}{\numberline {4.20}Acoustic Environment adaptation}{25}{subsection.68}
\contentsline {section}{\numberline {\MakeUppercase {Section} 4}\MakeUppercase {Proposed Research}}{22}{section.57}
\contentsline {subsection}{\numberline {4.1}Customizing Kernels with fast gradient descent}{23}{subsection.58}
\contentsline {subsubsection}{\numberline {4.1.1}Fast kernel summation with trees}{24}{subsubsection.59}
\contentsline {subsection}{\numberline {4.2}Computing Gaussian Kernel Matrix with trees}{25}{subsection.64}
\contentsline {subsection}{\numberline {4.3}Intrinsic dimension of speech}{25}{subsection.65}
\contentsline {subsection}{\numberline {4.4}Speech recognition with nearest neighbor search}{25}{subsection.66}
\contentsline {subsection}{\numberline {4.5}Acoustic Environment adaptation}{25}{subsection.67}
\setcounter {tocdepth}{2}
\contentsline {section}{\numberline {\MakeUppercase {Section} 5}\MakeUppercase {Work remaining to be done}}{25}{section.69}
\contentsline {section}{\numberline {\MakeUppercase {Section} 5}\MakeUppercase {Work remaining to be done}}{25}{section.68}
\setcounter {tocdepth}{2}
\contentsline {section}{\numberline {\MakeUppercase {Section} 6}\MakeUppercase {Facilities and Equipment Needed}}{25}{section.70}
\setcounter {tocdepth}{2}
\contentsline {section}{\numberline {\MakeUppercase {Appendix} A}\MakeUppercase {This is my first appendix}}{27}{section.71}
\setcounter {tocdepth}{2}
\contentsline {section}{\numberline {\MakeUppercase {Appendix} B}\MakeUppercase {This is my second appendix}}{28}{section.72}
\contentsline {section}{\numberline {\MakeUppercase {Appendix} A}\MakeUppercase {This is my first appendix}}{26}{section.69}
\setcounter {tocdepth}{3}
\contentsline {section}{\MakeUppercase {References}}{29}{section*.73}
\contentsline {section}{\MakeUppercase {References}}{27}{section*.70}
\setcounter {tocdepth}{2}