Manifold learning theory and applications pdf

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manifold learning theory and applications pdf

Manifold learning theory and applications (eBook, ) []

Manifold alignment is a class of machine learning algorithms that produce projections between sets of data, given that the original data sets lie on a common manifold. The concept was first introduced as such by Ham, Lee, and Saul in , [1] adding a manifold constraint to the general problem of correlating sets of high-dimensional vectors. Manifold alignment assumes that disparate data sets produced by similar generating processes will share a similar underlying manifold representation. By learning projections from each original space to the shared manifold, correspondences are recovered and knowledge from one domain can be transferred to another. Most manifold alignment techniques consider only two data sets, but the concept extends to arbitrarily many initial data sets. The projection functions are denoted:. This is usually encoded as the heat kernel of the adjacency matrix of a k -nearest neighbor graph.
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3. Introduction to Statistical Learning Theory

Manifold Learning

Manifold learning, a groundbreaking technique designed to tackle these issues of dimensionality reduction, finds widespread application in machine learning, neural networks, pattern recognition, image processing, and computer vision. Reviews Review Policy. Published on. Flowing text, Google-generated PDF. Learn More. Flag as inappropriate. It syncs automatically with your account and allows you to read online or offline wherever you are.

Manifold learning methods are one of the most exciting developments in machine learning in recent years. The central idea underlying these methods is that although natural data is typically represented in very high-dimensional spaces, the process generating the data is often thought to have relatively few degrees of freedom. A natural mathematical characterization of this intuition is to model the data as lying on or near a low-dimensional manifold. Recently, manifold learning has also been applied in utilizing both labeled and unlabeled data for classification, that is, semi-supervised learning. For example, once the manifold is estimated, then the Laplace—Beltrami operator may be used to provide a basis for maps intrinsically defined on this manifold and then the appropriate classifier map is estimated on the basis of the labeled examples.

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Filling a void in the literature, Manifold Learning Theory and Applications incorporates state-of-the-art techniques in manifold learning with a solid theoretical and.
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Manifold learning theory has seen a surge of interest in the modeling of large and extensive datasets in medical imaging since they capture the essence of data in a way that fundamentally outperforms linear methodologies, the purpose of which is to essentially describe things that are flat. This problematic is particularly relevant with medical imaging data, where linear techniques are frequently unsuitable for capturing variations in anatomical structures. In many cases, there is enough structure in the data CT, MRI, ultrasound so a lower dimensional object can describe the degrees of freedom, such as in a manifold structure. Still, complex, multivariate distributions tend to demonstrate highly variable structural topologies that are impossible to capture with a single manifold learning algorithm. This chapter will present recent techniques developed in manifold theory for medical imaging analysis, to allow for statistical organ shape modeling, image segmentation and registration from the concept of navigation of manifolds, classification, as well as disease prediction models based on discriminant manifolds. We will present the theoretical basis of these works, with illustrative results on their applications from various organs and pathologies, including neurodegenerative diseases and spinal deformities.


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