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    Kernel Methods: Fundamentals and Applications
    Kernel Methods: Fundamentals and Applications
    Kernel Methods: Fundamentals and Applications
    Ebook107 pages54 minutes

    Kernel Methods: Fundamentals and Applications

    By Fouad Sabry

    ()

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    About this ebook

    What Is Kernel Methods


    In the field of machine learning, kernel machines are a class of methods for pattern analysis. The support-vector machine (also known as SVM) is the most well-known member of this group. Pattern analysis frequently makes use of specific kinds of algorithms known as kernel approaches. Utilizing linear classifiers in order to solve nonlinear issues is what these strategies entail. Finding and studying different sorts of general relations present in datasets is the overarching goal of pattern analysis. Kernel methods, on the other hand, require only a user-specified kernel, which can be thought of as a similarity function over all pairs of data points computed using inner products. This is in contrast to many algorithms that solve these tasks, which require the data in their raw representation to be explicitly transformed into feature vector representations via a user-specified feature map. According to the Representer theorem, although the feature map in kernel machines has an unlimited number of dimensions, all that is required as user input is a matrix with a finite number of dimensions. Without parallel processing, computation on kernel machines is painfully slow for data sets with more than a few thousand individual cases.


    How You Will Benefit


    (I) Insights, and validations about the following topics:


    Chapter 1: Kernel method


    Chapter 2: Support vector machine


    Chapter 3: Radial basis function


    Chapter 4: Positive-definite kernel


    Chapter 5: Sequential minimal optimization


    Chapter 6: Regularization perspectives on support vector machines


    Chapter 7: Representer theorem


    Chapter 8: Radial basis function kernel


    Chapter 9: Kernel perceptron


    Chapter 10: Regularized least squares


    (II) Answering the public top questions about kernel methods.


    (III) Real world examples for the usage of kernel methods in many fields.


    (IV) 17 appendices to explain, briefly, 266 emerging technologies in each industry to have 360-degree full understanding of kernel methods' technologies.


    Who This Book Is For


    Professionals, undergraduate and graduate students, enthusiasts, hobbyists, and those who want to go beyond basic knowledge or information for any kind of kernel methods.

    LanguageEnglish
    PublisherOne Billion Knowledgeable
    Release dateJun 23, 2023
    Kernel Methods: Fundamentals and Applications

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      Book preview

      Kernel Methods - Fouad Sabry

      Chapter 1: Kernel method

      The support-vector machine is the most well-known member of the class of pattern analysis techniques known as kernel machines in machine learning (SVM). Algorithms used for pattern analysis are called kernel methods. These techniques rely on linear classifiers to address nonlinear issues. Finding and analyzing common types of relationships (such as clusters, ranks, principal components, correlations, and classifications) in datasets is the main goal of pattern analysis. In contrast to kernel methods, which only need a user-specified kernel, or a similarity function over all pairs of data points computed using inner products, many algorithms that solve these tasks require that the data in raw representation be explicitly transformed into feature vector representations via a user-specified feature map. According to the Representer theorem, the infinitely dimensional feature map in kernel machines only needs a finite dimensional matrix from user input. Without parallel processing, kernel machines are sluggish to calculate for datasets larger than a few thousand samples.

      The use of kernel functions, which enables them to operate in a high-dimensional, implicit feature space without ever computing the coordinates of the data there, gives kernel methods their name. Instead, all pairs of data in the feature space are simply computed by computing the inner products between their respective images. The explicit computation of the coordinates is frequently computationally more expensive than this process. The kernel trick is the name of this strategy. There are now kernel functions for vectors, text, pictures, graphs, and sequence data.

      The kernel perceptron, support-vector machines (SVM), Gaussian processes, principal components analysis (PCA), canonical correlation analysis, ridge regression, spectral clustering, linear adaptive filters, and many others are examples of algorithms that can work with kernels.

      The majority of kernel techniques are statistically sound and are based on eigenproblems or convex optimization. Typically, statistical learning theory is used to study their statistical characteristics (for example, using Rademacher complexity).

      Kernel methods can be compared to instance-based learners because they don't learn a set of parameters that are fixed and correspond to the features of their inputs, they instead remember the i -th training example (\mathbf {x} _{i},y_{i}) and learn for it a corresponding weight w_{i} .

      Prediction for inputs without labels, i.e, individuals who are not on the practice set, is treated by the application of a similarity function k , known as a kernel, between the unlabeled input \mathbf {x'} and each of the training inputs \mathbf {x} _{i} .

      For instance, A weighted sum of similarities is often calculated by a kernelized binary classifier.

      {\hat {y}}=\operatorname {sgn} \sum _{i=1}^{n}w_{i}y_{i}k(\mathbf {x} _{i},\mathbf {x'} ) , where

      {\hat {y}}\in \{-1,+1\} is the kernelized binary classifier's predicted label for the unlabeled input \mathbf {x'} whose hidden true label y is of interest; k\colon {\mathcal {X}}\times {\mathcal {X}}\to \mathbb {R} is the kernel function that measures similarity between any pair of inputs \mathbf {x} ,\mathbf {x'} \in {\mathcal {X}} ; the sum ranges over the n labeled examples \{(\mathbf {x} _{i},y_{i})\}_{i=1}^{n} in the classifier's training set, with y_{i}\in \{-1,+1\} ; the w_{i}\in \mathbb {R} are the weights for the training examples, according to what the learning algorithm decides; the sign function \operatorname {sgn} determines whether the predicted classification {\hat {y}} comes out positive or negative.

      With the development of the kernel perceptron in the 1960s, kernel classifiers were first reported. They gained significant notoriety when the support-vector machine (SVM) gained popularity in the 1990s and proved to be competitive with neural networks on tasks like handwriting recognition.

      The kernel method avoids the explicit mapping required to train linear learning algorithms to recognize a decision boundary or nonlinear function.

      For all \mathbf {x} and \mathbf {x'} in the input space {\mathcal {X}} , certain functions k(\mathbf {x} ,\mathbf {x'} ) can be expressed as an inner product in another space {\mathcal {V}} .

      The function k\colon {\mathcal {X}}\times {\mathcal {X}}\to \mathbb {R} is often referred to as a kernel or a kernel function.

      In mathematics, the term kernel refers to a weighting function for a weighted sum or integral.

      Certain problems in machine learning have more structure than an arbitrary weighting function k .

      The computation is made much simpler if the kernel can be written in the form of a feature map \varphi \colon {\mathcal {X}}\to {\mathcal {V}} which satisfies

      k(\mathbf {x} ,\mathbf {x'} )=\langle \varphi (\mathbf {x} ),\varphi (\mathbf {x'} )\rangle _{\mathcal {V}}.

      The key restriction is that \langle \cdot ,\cdot \rangle _{\mathcal {V}} must be a proper inner product.

      As opposed to that,, an explicit representation for \varphi is not necessary, as long as {\mathcal {V}} is an inner product space.

      The alternative follows from Mercer's theorem: an implicitly defined function \varphi exists whenever the space {\mathcal {X}} can be equipped with a suitable measure ensuring the function k satisfies Mercer's condition.

      A generalization of the linear algebraic finding that assigns an inner product to any positive-definite matrix is what Mercer's theorem is akin to.

      In fact, We may simplify the situation for Mercer by using this example.

      If we choose as our measure the counting measure \mu (T)=|T| for all T\subset X , which counts the number of points inside the set T , then Mercer's theorem's integral becomes a summation.

      \sum _{i=1}^{n}\sum _{j=1}^{n}k(\mathbf {x} _{i},\mathbf {x} _{j})c_{i}c_{j}\geq 0.

      If this summation holds for all finite sequences of points (\mathbf {x} _{1},\dotsc ,\mathbf {x} _{n}) in {\mathcal {X}} and all

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