203 research outputs found
Positive Definite Kernels in Machine Learning
This survey is an introduction to positive definite kernels and the set of
methods they have inspired in the machine learning literature, namely kernel
methods. We first discuss some properties of positive definite kernels as well
as reproducing kernel Hibert spaces, the natural extension of the set of
functions associated with a kernel defined
on a space . We discuss at length the construction of kernel
functions that take advantage of well-known statistical models. We provide an
overview of numerous data-analysis methods which take advantage of reproducing
kernel Hilbert spaces and discuss the idea of combining several kernels to
improve the performance on certain tasks. We also provide a short cookbook of
different kernels which are particularly useful for certain data-types such as
images, graphs or speech segments.Comment: draft. corrected a typo in figure
Learning Dynamic Systems for Intention Recognition in Human-Robot-Cooperation
This thesis is concerned with intention recognition for a humanoid robot and investigates how the challenges of uncertain and incomplete observations, a high degree of detail of the used models, and real-time inference may be addressed by modeling the human rationale as hybrid, dynamic Bayesian networks and performing inference with these models. The key focus lies on the automatic identification of the employed nonlinear stochastic dependencies and the situation-specific inference
Kernel Methods and their derivatives: Concept and perspectives for the Earth system sciences
Kernel methods are powerful machine learning techniques which implement
generic non-linear functions to solve complex tasks in a simple way. They Have
a solid mathematical background and exhibit excellent performance in practice.
However, kernel machines are still considered black-box models as the feature
mapping is not directly accessible and difficult to interpret.The aim of this
work is to show that it is indeed possible to interpret the functions learned
by various kernel methods is intuitive despite their complexity. Specifically,
we show that derivatives of these functions have a simple mathematical
formulation, are easy to compute, and can be applied to many different
problems. We note that model function derivatives in kernel machines is
proportional to the kernel function derivative. We provide the explicit
analytic form of the first and second derivatives of the most common kernel
functions with regard to the inputs as well as generic formulas to compute
higher order derivatives. We use them to analyze the most used supervised and
unsupervised kernel learning methods: Gaussian Processes for regression,
Support Vector Machines for classification, Kernel Entropy Component Analysis
for density estimation, and the Hilbert-Schmidt Independence Criterion for
estimating the dependency between random variables. For all cases we expressed
the derivative of the learned function as a linear combination of the kernel
function derivative. Moreover we provide intuitive explanations through
illustrative toy examples and show how to improve the interpretation of real
applications in the context of spatiotemporal Earth system data cubes. This
work reflects on the observation that function derivatives may play a crucial
role in kernel methods analysis and understanding.Comment: 21 pages, 10 figures, PLOS One Journa
A Statistical Perspective of the Empirical Mode Decomposition
This research focuses on non-stationary basis decompositions methods in time-frequency analysis. Classical methodologies in this field such as Fourier Analysis and Wavelet Transforms rely on strong assumptions of the underlying moment generating process, which, may not be valid in real data scenarios or modern applications of machine learning. The literature on non-stationary methods is still in its infancy, and the research contained in this thesis aims to address challenges arising in this area. Among several alternatives, this work is based on the method known as the Empirical Mode Decomposition (EMD). The EMD is a non-parametric time-series decomposition technique that produces a set of time-series functions denoted as Intrinsic Mode Functions (IMFs), which carry specific statistical properties. The main focus is providing a general and flexible family of basis extraction methods with minimal requirements compared to those within the Fourier or Wavelet techniques. This is highly important for two main reasons: first, more universal applications can be taken into account; secondly, the EMD has very little a priori knowledge of the process required to apply it, and as such, it can have greater generalisation properties in statistical applications across a wide array of applications and data types. The contributions of this work deal with several aspects of the decomposition. The first set regards the construction of an IMF from several perspectives: (1) achieving a semi-parametric representation of each basis; (2) extracting such semi-parametric functional forms in a computationally efficient and statistically robust framework. The EMD belongs to the class of path-based decompositions and, therefore, they are often not treated as a stochastic representation. (3) A major contribution involves the embedding of the deterministic pathwise decomposition framework into a formal stochastic process setting. One of the assumptions proper of the EMD construction is the requirement for a continuous function to apply the decomposition. In general, this may not be the case within many applications. (4) Various multi-kernel Gaussian Process formulations of the EMD will be proposed through the introduced stochastic embedding. Particularly, two different models will be proposed: one modelling the temporal mode of oscillations of the EMD and the other one capturing instantaneous frequencies location in specific frequency regions or bandwidths. (5) The construction of the second stochastic embedding will be achieved with an optimisation method called the cross-entropy method. Two formulations will be provided and explored in this regard. Application on speech time-series are explored to study such methodological extensions given that they are non-stationary
Kernel Methods and Measures for Classification with Transparency, Interpretability and Accuracy in Health Care
Support vector machines are a popular method in machine learning. They learn from data about a subject, for example, lung tumors in a set of patients, to classify new data, such as, a new patient’s tumor. The new tumor is classified as either cancerous or benign, depending on how similar it is to the tumors of other patients in those two classes—where similarity is judged by a kernel.
The adoption and use of support vector machines in health care, however, is inhibited by a perceived and actual lack of rationale, understanding and transparency for how they work and how to interpret information and results from them. For example, a user must select the kernel, or similarity function, to be used, and there are many kernels to choose from but little to no useful guidance on choosing one.
The primary goal of this thesis is to create accurate, transparent and interpretable kernels with rationale to select them for classification in health care using SVM—and to do so within a theoretical framework that advances rationale, understanding and transparency for kernel/model selection with atomic data types. The kernels and framework necessarily co-exist.
The secondary goal of this thesis is to quantitatively measure model interpretability for kernel/model selection and identify the types of interpretable information which are available from different models for interpretation.
Testing my framework and transparent kernels with empirical data I achieve classification accuracy that is better than or equivalent to the Gaussian RBF kernels. I also validate some of the model interpretability measures I propose
Convergence of sparse variational inference in gaussian processes regression
Gaussian processes are distributions over functions that are versatile and mathematically convenient priors in Bayesian modelling. However, their use is often impeded for data with large numbers of observations, N, due to the cubic (in N) cost of matrix operations used in exact inference. Many solutions have been proposed that rely on M << N inducing variables to form an approximation at a cost of O(NM^2). While the computational cost appears linear in N, the true complexity depends on how M must scale with N to ensure a certain quality of the approximation. In this work, we investigate upper and lower bounds on how M needs to grow with N to ensure high quality approximations. We show that we can make the KL-divergence between the approximate model and the exact posterior arbitrarily small for a Gaussian-noise regression model with M<<N. Specifically, for the popular squared exponential kernel and D-dimensional Gaussian distributed covariates, M=O((log N)^D) suffice and a method with an overall computational cost of O(N(log N)^{2D}(\log\log N)^2) can be used to perform inference
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