The document discusses support vector machines (SVMs) for classification. It defines classifiers as using an object's characteristics to identify its class. SVMs create decision boundaries that maximize the margin between classes to perform classification. They can learn both linearly and non-linearly separable data using kernels to transform the data into a higher dimension where a linear separator can be found. The document outlines how SVMs solve a quadratic optimization problem to find the optimal separating hyperplane that maximizes the margin between classes.
This document discusses optimal weighted distributions and their applications to financial time series. It addresses the problems of minimizing distances between weighted sample distributions and target distributions for 1) a single population, 2) two populations, and 3) N populations. Explicit formulas are derived for the optimal weights in each case. An iterative algorithm is presented to compute the target distribution as a fixed point. Applications to Google stock data are shown. Possible developments include extending the analysis to more populations, relaxing assumptions, and considering alternative distances.
This document discusses category theory and algebraic semantics for programming languages. It defines categories, functors, natural transformations, Ω-algebras, (Ω,E)-algebras, and monads. Ω-algebras allow modeling algebraic structures like groups. (Ω,E)-algebras satisfy equations E and have a free algebra. Monads correspond to algebraic theories and define Kleisli categories of algebras. Algebraic semantics uses axioms and rules to prove equalities in (Ω,E)-algebras.
This document summarizes a presentation on matrix computations in machine learning. It discusses how matrix computations are used in traditional machine learning algorithms like regression, PCA, and spectral graph partitioning. It also covers specialized machine learning algorithms that involve numerical optimization and matrix computations, such as Lasso, non-negative matrix factorization, sparse PCA, matrix completion, and co-clustering. Finally, it discusses how machine learning can be applied to improve graph clustering algorithms.
A tutorial on the Frobenious Theorem, one of the most important results in differential geometry, with emphasis in its use in nonlinear control theory. All results are accompanied by proofs, but for a more thorough and detailed presentation refer to the book of A. Isidori.
In this talk, we give an overview of results on numerical integration in Hermite spaces. These spaces contain functions defined on $\mathbb{R}^d$, and can be characterized by the decay of their Hermite coefficients. We consider the case of exponentially as well as polynomially decaying Hermite coefficients. For numerical integration, we either use Gauss-Hermite quadrature rules or algorithms based on quasi-Monte Carlo rules. We present upper and lower error bounds for these algorithms, and discuss their dependence on the dimension $d$. Furthermore, we comment on open problems for future research.
The generation of Gaussian random fields over a physical domain is a challenging problem in computational mathematics, especially when the correlation length is short and the field is rough. The traditional approach is to make use of a truncated Karhunen-Loeve (KL) expansion, but the generation of even a single realisation of the field may then be effectively beyond reach (especially for 3-dimensional domains) if the need is to obtain an expected L2 error of say 5%, because of the potentially very slow convergence of the KL expansion. In this talk, based on joint work with Ivan Graham, Frances Kuo, Dirk Nuyens, and Rob Scheichl, a completely different approach is used, in which the field is initially generated at a regular grid on a 2- or 3-dimensional rectangle that contains the physical domain, and then possibly interpolated to obtain the field at other points. In that case there is no need for any truncation. Rather the main problem becomes the factorisation of a large dense matrix. For this we use circulant embedding and FFT ideas. Quasi-Monte Carlo integration is then used to evaluate the expected value of some functional of the finite-element solution of an elliptic PDE with a random field as input.
This document discusses optimal weighted distributions and their applications to financial time series. It addresses the problems of minimizing distances between weighted sample distributions and target distributions for 1) a single population, 2) two populations, and 3) N populations. Explicit formulas are derived for the optimal weights in each case. An iterative algorithm is presented to compute the target distribution as a fixed point. Applications to Google stock data are shown. Possible developments include extending the analysis to more populations, relaxing assumptions, and considering alternative distances.
This document discusses category theory and algebraic semantics for programming languages. It defines categories, functors, natural transformations, Ω-algebras, (Ω,E)-algebras, and monads. Ω-algebras allow modeling algebraic structures like groups. (Ω,E)-algebras satisfy equations E and have a free algebra. Monads correspond to algebraic theories and define Kleisli categories of algebras. Algebraic semantics uses axioms and rules to prove equalities in (Ω,E)-algebras.
This document summarizes a presentation on matrix computations in machine learning. It discusses how matrix computations are used in traditional machine learning algorithms like regression, PCA, and spectral graph partitioning. It also covers specialized machine learning algorithms that involve numerical optimization and matrix computations, such as Lasso, non-negative matrix factorization, sparse PCA, matrix completion, and co-clustering. Finally, it discusses how machine learning can be applied to improve graph clustering algorithms.
A tutorial on the Frobenious Theorem, one of the most important results in differential geometry, with emphasis in its use in nonlinear control theory. All results are accompanied by proofs, but for a more thorough and detailed presentation refer to the book of A. Isidori.
In this talk, we give an overview of results on numerical integration in Hermite spaces. These spaces contain functions defined on $\mathbb{R}^d$, and can be characterized by the decay of their Hermite coefficients. We consider the case of exponentially as well as polynomially decaying Hermite coefficients. For numerical integration, we either use Gauss-Hermite quadrature rules or algorithms based on quasi-Monte Carlo rules. We present upper and lower error bounds for these algorithms, and discuss their dependence on the dimension $d$. Furthermore, we comment on open problems for future research.
The generation of Gaussian random fields over a physical domain is a challenging problem in computational mathematics, especially when the correlation length is short and the field is rough. The traditional approach is to make use of a truncated Karhunen-Loeve (KL) expansion, but the generation of even a single realisation of the field may then be effectively beyond reach (especially for 3-dimensional domains) if the need is to obtain an expected L2 error of say 5%, because of the potentially very slow convergence of the KL expansion. In this talk, based on joint work with Ivan Graham, Frances Kuo, Dirk Nuyens, and Rob Scheichl, a completely different approach is used, in which the field is initially generated at a regular grid on a 2- or 3-dimensional rectangle that contains the physical domain, and then possibly interpolated to obtain the field at other points. In that case there is no need for any truncation. Rather the main problem becomes the factorisation of a large dense matrix. For this we use circulant embedding and FFT ideas. Quasi-Monte Carlo integration is then used to evaluate the expected value of some functional of the finite-element solution of an elliptic PDE with a random field as input.
This document provides an overview of linear models for classification. It discusses discriminant functions including linear discriminant analysis and the perceptron algorithm. It also covers probabilistic generative models that model class-conditional densities and priors to estimate posterior probabilities. Probabilistic discriminative models like logistic regression directly model posterior probabilities using maximum likelihood. Iterative reweighted least squares is used to optimize logistic regression since there is no closed-form solution.
This document introduces a dependent Dirichlet process (DDP) model that allows the cluster weights and locations to vary based on a covariate x. It defines a measure of dependence between data points based on x, and derives a Polya urn-style predictive rule. It then presents a novel DDP construction based on simulating gamma random variables, which allows for easy posterior computation. This model generalizes previous dependent DP work and can handle multidimensional covariates.
An implicit partial pivoting gauss elimination algorithm for linear system of...Alexander Decker
This document proposes a new method for solving fully fuzzy linear systems of equations (FFLS) using Gauss elimination with implicit partial pivoting. It begins by introducing concepts of fuzzy sets theory and arithmetic operations on fuzzy numbers. It then presents the FFLS problem and shows how it can be reduced to a crisp linear system. The key steps of the proposed Gauss elimination method with implicit partial pivoting are outlined. Finally, the method is illustrated on a numerical example, showing the steps to obtain solutions for the variables x, y and z.
The document provides an introduction to linear algebra concepts for machine learning. It defines vectors as ordered tuples of numbers that express magnitude and direction. Vector spaces are sets that contain all linear combinations of vectors. Linear independence and basis of vector spaces are discussed. Norms measure the magnitude of a vector, with examples given of the 1-norm and 2-norm. Inner products measure the correlation between vectors. Matrices can represent linear operators between vector spaces. Key linear algebra concepts such as trace, determinant, and matrix decompositions are outlined for machine learning applications.
QMC algorithms usually rely on a choice of “N” evenly distributed integration nodes in $[0,1)^d$. A common means to assess such an equidistributional property for a point set or sequence is the so-called discrepancy function, which compares the actual number of points to the expected number of points (assuming uniform distribution on $[0,1)^{d}$) that lie within an arbitrary axis parallel rectangle anchored at the origin. The dependence of the integration error using QMC rules on various norms of the discrepancy function is made precise within the well-known Koksma--Hlawka inequality and its variations. In many cases, such as $L^{p}$ spaces, $1<p<\infty$, the best growth rate in terms of the number of points “N” as well as corresponding explicit constructions are known. In the classical setting $p=\infty$ sharp results are absent for $d\geq3$ already and appear to be intriguingly hard to obtain. This talk shall serve as a survey on discrepancy theory with a special emphasis on the $L^{\infty}$ setting. Furthermore, it highlights the evolution of recent techniques and presents the latest results.
This document summarizes a talk given by Yoshihiro Mizoguchi on developing a Coq library for relational calculus. The talk introduces relational calculus and its applications. It describes implementing definitions and proofs about relations, Boolean algebras, relation algebras, and Dedekind categories in Coq. The library provides a formalization of basic notions in relational theory and can be used to formally verify properties of relations and prove theorems automatedly.
International Journal of Computational Engineering Research(IJCER)ijceronline
The document presents some fixed point theorems for expansion mappings in complete metric spaces. It begins with definitions of terms like metric spaces, complete metric spaces, Cauchy sequences, and expansion mappings. It then summarizes several existing fixed point theorems for expansion mappings established by other mathematicians. The main result proved in this document is Theorem 3.1, which establishes a new fixed point theorem for expansion mappings under certain conditions on the metric space and mapping. It shows that if the mapping satisfies the given inequality, then it has a fixed point. The proof of this theorem constructs a sequence to show that it converges to a fixed point.
The document discusses square matrices and determinants. It begins by noting that square matrices are the only matrices that can have inverses. It then presents an algorithm for calculating the inverse of a square matrix A by forming the partitioned matrix (A|I) and applying Gauss-Jordan reduction. The document also discusses determinants, defining them recursively as the sum of products of diagonal entries with signs depending on row/column position, for matrices larger than 1x1. Complexity increases exponentially with matrix size.
Lattice coverings are a simple case of covering problems. We will first expose methods for finding the covering density of a given lattice. If one considers a space of possible lattices, then one gets the theory of L-type. We will explain how this theory works out and how the over a given L-type the problem is a
semidefinite programming problem. Finally, we will explore the covering maxima of lattices, i.e. local behavior of the covering density function.
Object Detection with Discrmininatively Trained Part based Modelszukun
The document describes an object detection method using deformable part-based models that are discriminatively trained. The models consist of root filters and deformable part filters at multiple resolutions. Latent SVM training is used to learn the filters and deformation costs from weakly labeled images. The method achieved state-of-the-art results on the PASCAL object detection challenge, outperforming other methods in accuracy and speed.
This document summarizes research on using deformable models for object recognition. It discusses using deformable part models to detect objects by optimizing part locations. Efficient algorithms like dynamic programming and min-convolutions are used for matching. Non-rigid objects are modeled using triangulated polygons that can deform individual triangles. Hierarchical shape models capture shape variations. The document applies these techniques to the PASCAL visual object recognition challenge, achieving state-of-the-art results on 10 of 20 object categories through discriminatively trained, multiscale deformable part models.
This document provides an overview of continuous probability distributions including:
- The probability of an event occurring between two values a and b is defined by the shaded area under the probability density function curve between a and b.
- Key properties of continuous distributions include the mean, median, standard deviation, and cumulative distribution function.
- The normal distribution is discussed as an important continuous distribution with properties like symmetry and effectiveness in modeling real-world data.
- Methods for calculating normal probabilities using tables or Excel are presented with examples.
- An example problem involving a quality control scenario and modeling the number of defective items using a binomial distribution is provided.
CVPR2010: Advanced ITinCVPR in a Nutshell: part 5: Shape, Matching and Diverg...zukun
The document discusses using divergence measures like the Jensen-Shannon divergence to align multiple point sets represented as probability density functions. It motivates using the JS divergence by modeling point sets as mixtures of density functions, and shows how the likelihood ratio between models leads to the JS divergence. It then formulates the problem of group-wise point set registration as minimizing the JS divergence between density functions, combined with a regularization term. Experimental results on aligning multiple 3D hippocampus point sets are also presented.
1. Geodesic sampling and meshing techniques can be used to generate adaptive triangulations and meshes on Riemannian manifolds based on a metric tensor.
2. Anisotropic metrics can be defined to generate meshes adapted to features like edges in images or curvature on surfaces. Triangles will be elongated along strong features to better approximate functions.
3. Farthest point sampling can be used to generate well-spaced point distributions over manifolds according to a metric, which can then be triangulated using geodesic Delaunay refinement.
This document provides an introduction to systems of linear equations and matrix operations. It defines key concepts such as matrices, matrix addition and multiplication, and transitions between different bases. It presents an example of multiplying two matrices using NumPy. The document outlines how systems of linear equations can be represented using matrices and discusses solving systems using techniques like Gauss-Jordan elimination and elementary row operations. It also introduces the concepts of homogeneous and inhomogeneous systems.
slides for "Supervised Model Learning with Feature Grouping based on a Discre...Kensuke Mitsuzawa
The document discusses a supervised machine learning method that performs feature grouping to reduce model complexity. It does this by:
1) Defining a set of possible discrete values (e.g. -4 to 4) that feature weights can take on.
2) Incorporating a discrete constraint during model training to force weights to values in the predefined set, grouping similar features.
3) Using dual decomposition to solve the optimization problem, as directly optimizing with the discrete constraint is NP-hard. It decomposes the problem by introducing auxiliary variables and adding equality constraints.
Image sciences, image processing, image restoration, photo manipulation. Image and videos representation. Digital versus analog imagery. Quantization and sampling. Sources and models of noises in digital CCD imagery: photon, thermal and readout noises. Sources and models of blurs. Convolutions and point spread functions. Overview of other standard models, problems and tasks: salt-and-pepper and impulse noises, half toning, inpainting, super-resolution, compressed sensing, high dynamic range imagery, demosaicing. Short introduction to other types of imagery: SAR, Sonar, ultrasound, CT and MRI. Linear and ill-posed restoration problems.
The document discusses support vector machines (SVMs) for classification. It begins by defining classifiers and the difference between classification and clustering. It then introduces SVMs, explaining that they find optimal decision boundaries that separate classes through mapping data points into higher dimensional space. The document outlines linear and non-linear SVMs, describing how non-linear SVMs can find more complex separating structures through kernels. It also discusses supervised learning with SVMs and how to solve the optimization problem to train linear SVMs, including soft-margin classification to handle non-separable data.
Instant intimacy or fake personas? Meaningful, trusted connections or surface ties? How much of ourselves are we really willing to share with our social networks?
Examining the purchasing experience in a multi-channel environment; maintaining contact with today’s consumer and meet the expectations of tomorrow’s consumer.
This document provides an overview of linear models for classification. It discusses discriminant functions including linear discriminant analysis and the perceptron algorithm. It also covers probabilistic generative models that model class-conditional densities and priors to estimate posterior probabilities. Probabilistic discriminative models like logistic regression directly model posterior probabilities using maximum likelihood. Iterative reweighted least squares is used to optimize logistic regression since there is no closed-form solution.
This document introduces a dependent Dirichlet process (DDP) model that allows the cluster weights and locations to vary based on a covariate x. It defines a measure of dependence between data points based on x, and derives a Polya urn-style predictive rule. It then presents a novel DDP construction based on simulating gamma random variables, which allows for easy posterior computation. This model generalizes previous dependent DP work and can handle multidimensional covariates.
An implicit partial pivoting gauss elimination algorithm for linear system of...Alexander Decker
This document proposes a new method for solving fully fuzzy linear systems of equations (FFLS) using Gauss elimination with implicit partial pivoting. It begins by introducing concepts of fuzzy sets theory and arithmetic operations on fuzzy numbers. It then presents the FFLS problem and shows how it can be reduced to a crisp linear system. The key steps of the proposed Gauss elimination method with implicit partial pivoting are outlined. Finally, the method is illustrated on a numerical example, showing the steps to obtain solutions for the variables x, y and z.
The document provides an introduction to linear algebra concepts for machine learning. It defines vectors as ordered tuples of numbers that express magnitude and direction. Vector spaces are sets that contain all linear combinations of vectors. Linear independence and basis of vector spaces are discussed. Norms measure the magnitude of a vector, with examples given of the 1-norm and 2-norm. Inner products measure the correlation between vectors. Matrices can represent linear operators between vector spaces. Key linear algebra concepts such as trace, determinant, and matrix decompositions are outlined for machine learning applications.
QMC algorithms usually rely on a choice of “N” evenly distributed integration nodes in $[0,1)^d$. A common means to assess such an equidistributional property for a point set or sequence is the so-called discrepancy function, which compares the actual number of points to the expected number of points (assuming uniform distribution on $[0,1)^{d}$) that lie within an arbitrary axis parallel rectangle anchored at the origin. The dependence of the integration error using QMC rules on various norms of the discrepancy function is made precise within the well-known Koksma--Hlawka inequality and its variations. In many cases, such as $L^{p}$ spaces, $1<p<\infty$, the best growth rate in terms of the number of points “N” as well as corresponding explicit constructions are known. In the classical setting $p=\infty$ sharp results are absent for $d\geq3$ already and appear to be intriguingly hard to obtain. This talk shall serve as a survey on discrepancy theory with a special emphasis on the $L^{\infty}$ setting. Furthermore, it highlights the evolution of recent techniques and presents the latest results.
This document summarizes a talk given by Yoshihiro Mizoguchi on developing a Coq library for relational calculus. The talk introduces relational calculus and its applications. It describes implementing definitions and proofs about relations, Boolean algebras, relation algebras, and Dedekind categories in Coq. The library provides a formalization of basic notions in relational theory and can be used to formally verify properties of relations and prove theorems automatedly.
International Journal of Computational Engineering Research(IJCER)ijceronline
The document presents some fixed point theorems for expansion mappings in complete metric spaces. It begins with definitions of terms like metric spaces, complete metric spaces, Cauchy sequences, and expansion mappings. It then summarizes several existing fixed point theorems for expansion mappings established by other mathematicians. The main result proved in this document is Theorem 3.1, which establishes a new fixed point theorem for expansion mappings under certain conditions on the metric space and mapping. It shows that if the mapping satisfies the given inequality, then it has a fixed point. The proof of this theorem constructs a sequence to show that it converges to a fixed point.
The document discusses square matrices and determinants. It begins by noting that square matrices are the only matrices that can have inverses. It then presents an algorithm for calculating the inverse of a square matrix A by forming the partitioned matrix (A|I) and applying Gauss-Jordan reduction. The document also discusses determinants, defining them recursively as the sum of products of diagonal entries with signs depending on row/column position, for matrices larger than 1x1. Complexity increases exponentially with matrix size.
Lattice coverings are a simple case of covering problems. We will first expose methods for finding the covering density of a given lattice. If one considers a space of possible lattices, then one gets the theory of L-type. We will explain how this theory works out and how the over a given L-type the problem is a
semidefinite programming problem. Finally, we will explore the covering maxima of lattices, i.e. local behavior of the covering density function.
Object Detection with Discrmininatively Trained Part based Modelszukun
The document describes an object detection method using deformable part-based models that are discriminatively trained. The models consist of root filters and deformable part filters at multiple resolutions. Latent SVM training is used to learn the filters and deformation costs from weakly labeled images. The method achieved state-of-the-art results on the PASCAL object detection challenge, outperforming other methods in accuracy and speed.
This document summarizes research on using deformable models for object recognition. It discusses using deformable part models to detect objects by optimizing part locations. Efficient algorithms like dynamic programming and min-convolutions are used for matching. Non-rigid objects are modeled using triangulated polygons that can deform individual triangles. Hierarchical shape models capture shape variations. The document applies these techniques to the PASCAL visual object recognition challenge, achieving state-of-the-art results on 10 of 20 object categories through discriminatively trained, multiscale deformable part models.
This document provides an overview of continuous probability distributions including:
- The probability of an event occurring between two values a and b is defined by the shaded area under the probability density function curve between a and b.
- Key properties of continuous distributions include the mean, median, standard deviation, and cumulative distribution function.
- The normal distribution is discussed as an important continuous distribution with properties like symmetry and effectiveness in modeling real-world data.
- Methods for calculating normal probabilities using tables or Excel are presented with examples.
- An example problem involving a quality control scenario and modeling the number of defective items using a binomial distribution is provided.
CVPR2010: Advanced ITinCVPR in a Nutshell: part 5: Shape, Matching and Diverg...zukun
The document discusses using divergence measures like the Jensen-Shannon divergence to align multiple point sets represented as probability density functions. It motivates using the JS divergence by modeling point sets as mixtures of density functions, and shows how the likelihood ratio between models leads to the JS divergence. It then formulates the problem of group-wise point set registration as minimizing the JS divergence between density functions, combined with a regularization term. Experimental results on aligning multiple 3D hippocampus point sets are also presented.
1. Geodesic sampling and meshing techniques can be used to generate adaptive triangulations and meshes on Riemannian manifolds based on a metric tensor.
2. Anisotropic metrics can be defined to generate meshes adapted to features like edges in images or curvature on surfaces. Triangles will be elongated along strong features to better approximate functions.
3. Farthest point sampling can be used to generate well-spaced point distributions over manifolds according to a metric, which can then be triangulated using geodesic Delaunay refinement.
This document provides an introduction to systems of linear equations and matrix operations. It defines key concepts such as matrices, matrix addition and multiplication, and transitions between different bases. It presents an example of multiplying two matrices using NumPy. The document outlines how systems of linear equations can be represented using matrices and discusses solving systems using techniques like Gauss-Jordan elimination and elementary row operations. It also introduces the concepts of homogeneous and inhomogeneous systems.
slides for "Supervised Model Learning with Feature Grouping based on a Discre...Kensuke Mitsuzawa
The document discusses a supervised machine learning method that performs feature grouping to reduce model complexity. It does this by:
1) Defining a set of possible discrete values (e.g. -4 to 4) that feature weights can take on.
2) Incorporating a discrete constraint during model training to force weights to values in the predefined set, grouping similar features.
3) Using dual decomposition to solve the optimization problem, as directly optimizing with the discrete constraint is NP-hard. It decomposes the problem by introducing auxiliary variables and adding equality constraints.
Image sciences, image processing, image restoration, photo manipulation. Image and videos representation. Digital versus analog imagery. Quantization and sampling. Sources and models of noises in digital CCD imagery: photon, thermal and readout noises. Sources and models of blurs. Convolutions and point spread functions. Overview of other standard models, problems and tasks: salt-and-pepper and impulse noises, half toning, inpainting, super-resolution, compressed sensing, high dynamic range imagery, demosaicing. Short introduction to other types of imagery: SAR, Sonar, ultrasound, CT and MRI. Linear and ill-posed restoration problems.
The document discusses support vector machines (SVMs) for classification. It begins by defining classifiers and the difference between classification and clustering. It then introduces SVMs, explaining that they find optimal decision boundaries that separate classes through mapping data points into higher dimensional space. The document outlines linear and non-linear SVMs, describing how non-linear SVMs can find more complex separating structures through kernels. It also discusses supervised learning with SVMs and how to solve the optimization problem to train linear SVMs, including soft-margin classification to handle non-separable data.
Instant intimacy or fake personas? Meaningful, trusted connections or surface ties? How much of ourselves are we really willing to share with our social networks?
Examining the purchasing experience in a multi-channel environment; maintaining contact with today’s consumer and meet the expectations of tomorrow’s consumer.
Paul Hudson evaluates the significance of the pace of change we’ve witnessed in the last 10 years and predicts the likely pace of change over the next 10 years. What can we expect in the future and what does this means for your business?
This document compares FPGAs and microcontrollers. FPGAs can provide much higher performance per watt than microcontrollers due to their ability to perform thousands of calculations per clock cycle. However, microcontrollers are better suited for floating point calculations and have an advantage for tasks that require dynamic parallelism. FPGAs are well-suited for problems that can be parallelized, while microcontrollers may be preferable for unpredictable tasks. Both device types have pros and cons related to factors like power usage, programming difficulty, cost, and interfaces. The selection depends on the specific application requirements and developer resources.
Our guest speaker, Nicola Millard – Customer Experience Futurologist at BT, looks at how traditional business models are being challenged by both customers and employees as new technologies and new ways of collaboration emerge.
The document proposes establishing an International Nuclear Fuel Association (INFCA) that would lease areas to control sensitive nuclear fuel cycle facilities like enrichment plants. Key points:
1) INFCA would establish "Internationally-Secured Leased Areas" to conduct enrichment and later reprocessing, ensuring nonproliferation compliance.
2) States would need to join INFCA and strengthen IAEA safeguards to receive fuel cycle services.
3) INFCA would certify legitimate producers and track critical component end-uses, without replacing IAEA functions or access.
This document discusses the characteristics of the "M-Age" generation, born since 1997, who have grown up immersed in mobile technology. They comprise 24% of the UK population by 2018. As "digital natives", they are self-directed learners who prefer learning by doing and figuring things out through social interaction. They are highly aware of both the benefits and risks of technology like the internet and understand the need to verify information. Companies will need to understand their skills and relationship to an increasingly mobile-centric internet in order to effectively engage with this demographic.
Photoshop CS3 allows users to create simple animations. Key steps include creating a new 125x125 pixel file, adding 3 frames of 3 seconds each, using the Animation window to set the duration to 0 seconds, duplicating selected frames to copy them, and playing the animation to view the duplicated frames. The animation can be saved using File > Save for Web & Devices and shortcut ALT CTRL SHIFT S. ImageReady is also included with Photoshop CS3.
This document contains a personal history form for an applicant to the United Nations. It requests biographical information such as name, date of birth, citizenship, education history, employment history, languages spoken, and references. It also asks whether the applicant has any limitations that would impact their ability to perform the job duties or travel internationally. The applicant must certify that their statements are true and complete.
Products and services are purchased at a very local level while increasingly delivered globally. How do we understand and prepare for service delivery that is both universal and locally sensitive?
Este documento presenta la información de un colegio técnico en Ecuador, incluyendo su dirección, teléfono y escudo. También lista a tres estudiantes de primer año de bachillerato y su profesor, e identifica el año lectivo 2010-2011. Explica los pasos para elaborar un proyecto en Power Point sobre el colegio agregando imágenes e información en diapositivas y grabando la narración.
Vandalism is defined as the willful or malicious destruction, injury, disfigurement, or defacement of public or private property without the owner's consent. Common acts of vandalism include graffiti, breaking or throwing objects, damaging property, setting fires, and tampering with equipment. Locations that are often targets of vandalism include schools, cemeteries, buildings, houses, libraries, parks, trains, cars and other transportation. A survey was conducted to understand perceptions of vandalism and over half of respondents considered graffiti to be an act of vandalism. Most thought the laws around vandalism should be tougher and knew someone personally who had committed an act of vandalism.
This document contains a personal history form for an application to the United Nations. It requests basic biographical information such as name, date of birth, citizenship, education history, employment record, languages, and availability to travel. The applicant is male, born in 1986 in Pakistan, holds Pakistani citizenship, and has a bachelor's degree in electrical engineering. He is currently employed as an assistant and is seeking work in telecommunications and IT.
This document provides an introduction and overview of support vector machines (SVMs) for text classification. It discusses how SVMs find an optimal separating hyperplane between classes that maximizes the margin between the classes. It explains how SVMs can handle non-linear classification through the use of kernels to map data into higher-dimensional feature spaces. The document also discusses evaluation metrics like precision, recall, and accuracy for text classification and the differences between micro-averaging and macro-averaging of these metrics.
This document summarizes support vector machines (SVMs). It explains that SVMs find the optimal separating hyperplane that maximizes the margin between two classes of data points. The hyperplane is determined by support vectors, which are the data points closest to the hyperplane. SVMs can be solved as a quadratic programming problem. The document also discusses how kernels can map data into higher dimensional spaces to make non-separable problems separable by SVMs.
This document provides an overview of linear classifiers and support vector machines (SVMs) for text classification. It explains that SVMs find the optimal separating hyperplane between classes by maximizing the margin between the hyperplane and the closest data points of each class. The document discusses how SVMs can be extended to non-linear classification through kernel methods and feature spaces. It also provides details on solving the SVM optimization problem and using SVMs for classification.
This document provides an introduction to support vector machines (SVMs) for text classification. It discusses how SVMs find an optimal separating hyperplane that maximizes the margin between classes. SVMs can handle non-linear classification through the use of kernels, which map data into a higher-dimensional feature space. The document outlines the mathematical formulations of linear and soft-margin SVMs, explains how the kernel trick allows evaluating inner products implicitly in that feature space, and summarizes how SVMs are used for classification tasks.
This document discusses support vector machines (SVMs) for classification tasks. It describes how SVMs find the optimal separating hyperplane with the maximum margin between classes in the training data. This is formulated as a quadratic optimization problem that can be solved using algorithms that construct a dual problem. Non-linear SVMs are also discussed, using the "kernel trick" to implicitly map data into higher-dimensional feature spaces. Common kernel functions and the theoretical justification for maximum margin classifiers are provided.
support vector machine algorithm in machine learningSamGuy7
The objective of the support vector machine algorithm is to find a hyperplane in an N-dimensional space(N — the number of features) that distinctly classifies the
SVMs are known for their effectiveness in high-dimensional spaces and their ability to handle complex data patterns. data points
This document discusses support vector machines (SVMs) for classification tasks. It describes how SVMs find the optimal separating hyperplane with the maximum margin between classes in the training data. This is formulated as a quadratic optimization problem that can be solved using algorithms that construct a dual problem. Non-linear SVMs are also discussed, using the "kernel trick" to implicitly map data to higher dimensions where a linear separator can be found.
This document provides an introduction to support vector machines (SVMs). It discusses how SVMs can be used for binary classification, regression, and multi-class problems. SVMs find the optimal separating hyperplane that maximizes the margin between classes. Soft margins allow for misclassified points by introducing slack variables. Kernels are discussed for mapping data into higher dimensional feature spaces to perform linear separation. The document outlines the formulation of SVMs for classification and regression and discusses model selection and different kernel functions.
SVM is a supervised learning method that finds a hyperplane with maximum margin to separate classes. It uses kernels to map data to higher dimensions to allow for nonlinear separation. The objective is to minimize training error and model complexity by maximizing the margin between classes. SVMs solve a convex optimization problem that finds support vectors and determines the separating hyperplane using kernels, slack variables, and a cost parameter C to balance margin and errors. Parameter selection, like the kernel and its parameters, affects performance and is typically done through grid search and cross-validation.
This document provides a tutorial on support vector machines (SVM) for binary classification. It outlines the key concepts of SVM including linear separable and non-separable cases, soft margin classification, solving the SVM optimization problem, kernel methods for non-linear classification, commonly used kernel functions, and relationships between SVM and other methods like logistic regression. Example code for using SVM from the scikit-learn Python package is also provided.
Kernel based models for geo- and environmental sciences- Alexei Pozdnoukhov –...Beniamino Murgante
This document discusses using kernel methods, specifically support vector machines (SVMs), for environmental and geoscience applications. It provides an overview of SVMs, including how they find the optimal separating hyperplane with the maximum margin to perform classification and regression. It discusses how SVMs can handle nonlinear decision boundaries using the kernel trick. The document gives examples of applying SVMs to problems like porosity mapping, temperature inversion mapping, and landslide susceptibility modeling. It demonstrates how SVMs can extract patterns from high-dimensional environmental data and produce predictive spatial models.
This document provides an introduction to support vector machines (SVM). It discusses the history and key concepts of SVM, including how SVM finds the optimal separating hyperplane with maximum margin between classes to perform linear classification. It also describes how SVM can learn nonlinear decision boundaries using kernel tricks to implicitly map inputs to high-dimensional feature spaces. The document gives examples of commonly used kernel functions and outlines the steps to perform classification with SVM.
These notes are a basic introduction to SVM, assuming almost no prior exposure. They contain some derivations, details, and explanations that not many SVM tutorials usually delve into. Thus, they're meant to augment primary course material (textbook or lecture notes) on SVMs and to help digest the course material.
This document provides an overview of Linear Discriminant Analysis (LDA) for dimensionality reduction. LDA seeks to perform dimensionality reduction while preserving class discriminatory information as much as possible, unlike PCA which does not consider class labels. LDA finds a linear combination of features that separates classes best by maximizing the between-class variance while minimizing the within-class variance. This is achieved by solving the generalized eigenvalue problem involving the within-class and between-class scatter matrices. The document provides mathematical details and an example to illustrate LDA for a two-class problem.
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Support vector machines are widely used binary classifiers known for its ability to handle high dimensional data that classifies data by separating classes with a hyper-plane that maximizes the margin between them. The data points that are closest to hyper-plane are known as support vectors. Thus the selected decision boundary will be the one that minimizes the generalization error (by maximizing the margin between classes).
The document discusses support vector machines (SVMs). SVMs find the optimal separating hyperplane between classes that maximizes the margin between them. They can handle nonlinear data using kernels to map the data into higher dimensions where a linear separator may exist. Key aspects include defining the maximum margin hyperplane, using regularization and slack variables to deal with misclassified examples, and kernels which implicitly map data into other feature spaces without explicitly computing the transformations. The regularization and gamma parameters affect model complexity, with regularization controlling overfitting and gamma influencing the similarity between points.
Support Vector Machine topic of machine learning.pptxCodingChamp1
Support Vector Machines (SVM) find the optimal separating hyperplane that maximizes the margin between two classes of data points. The hyperplane is chosen such that it maximizes the distance from itself to the nearest data points of each class. When data is not linearly separable, the kernel trick can be used to project the data into a higher dimensional space where it may be linearly separable. Common kernel functions include linear, polynomial, radial basis function (RBF), and sigmoid kernels. Soft margin SVMs introduce slack variables to allow some misclassification and better handle non-separable data. The C parameter controls the tradeoff between margin maximization and misclassification.
UiPath Test Automation using UiPath Test Suite series, part 5DianaGray10
Welcome to UiPath Test Automation using UiPath Test Suite series part 5. In this session, we will cover CI/CD with devops.
Topics covered:
CI/CD with in UiPath
End-to-end overview of CI/CD pipeline with Azure devops
Speaker:
Lyndsey Byblow, Test Suite Sales Engineer @ UiPath, Inc.
HCL Notes und Domino Lizenzkostenreduzierung in der Welt von DLAUpanagenda
Webinar Recording: https://www.panagenda.com/webinars/hcl-notes-und-domino-lizenzkostenreduzierung-in-der-welt-von-dlau/
DLAU und die Lizenzen nach dem CCB- und CCX-Modell sind für viele in der HCL-Community seit letztem Jahr ein heißes Thema. Als Notes- oder Domino-Kunde haben Sie vielleicht mit unerwartet hohen Benutzerzahlen und Lizenzgebühren zu kämpfen. Sie fragen sich vielleicht, wie diese neue Art der Lizenzierung funktioniert und welchen Nutzen sie Ihnen bringt. Vor allem wollen Sie sicherlich Ihr Budget einhalten und Kosten sparen, wo immer möglich. Das verstehen wir und wir möchten Ihnen dabei helfen!
Wir erklären Ihnen, wie Sie häufige Konfigurationsprobleme lösen können, die dazu führen können, dass mehr Benutzer gezählt werden als nötig, und wie Sie überflüssige oder ungenutzte Konten identifizieren und entfernen können, um Geld zu sparen. Es gibt auch einige Ansätze, die zu unnötigen Ausgaben führen können, z. B. wenn ein Personendokument anstelle eines Mail-Ins für geteilte Mailboxen verwendet wird. Wir zeigen Ihnen solche Fälle und deren Lösungen. Und natürlich erklären wir Ihnen das neue Lizenzmodell.
Nehmen Sie an diesem Webinar teil, bei dem HCL-Ambassador Marc Thomas und Gastredner Franz Walder Ihnen diese neue Welt näherbringen. Es vermittelt Ihnen die Tools und das Know-how, um den Überblick zu bewahren. Sie werden in der Lage sein, Ihre Kosten durch eine optimierte Domino-Konfiguration zu reduzieren und auch in Zukunft gering zu halten.
Diese Themen werden behandelt
- Reduzierung der Lizenzkosten durch Auffinden und Beheben von Fehlkonfigurationen und überflüssigen Konten
- Wie funktionieren CCB- und CCX-Lizenzen wirklich?
- Verstehen des DLAU-Tools und wie man es am besten nutzt
- Tipps für häufige Problembereiche, wie z. B. Team-Postfächer, Funktions-/Testbenutzer usw.
- Praxisbeispiele und Best Practices zum sofortigen Umsetzen
TrustArc Webinar - 2024 Global Privacy SurveyTrustArc
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See how organizational priorities and strategic approaches to data security and privacy are evolving around the globe.
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- Key themes to consider in developing and maintaining your privacy program
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* Practical use cases across various industries
* Step-by-step implementation guide
* Live demos with code snippets
* Enhancing LLM capabilities with vector search
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Perfect for developers, AI enthusiasts, and tech leaders. Learn how to leverage MongoDB Atlas to deliver highly relevant, context-aware search results, transforming your data retrieval process. Stay ahead in tech innovation and maximize the potential of your applications.
#MongoDB #VectorSearch #AI #SemanticSearch #TechInnovation #DataScience #LLM #MachineLearning #SearchTechnology
In the rapidly evolving landscape of technologies, XML continues to play a vital role in structuring, storing, and transporting data across diverse systems. The recent advancements in artificial intelligence (AI) present new methodologies for enhancing XML development workflows, introducing efficiency, automation, and intelligent capabilities. This presentation will outline the scope and perspective of utilizing AI in XML development. The potential benefits and the possible pitfalls will be highlighted, providing a balanced view of the subject.
We will explore the capabilities of AI in understanding XML markup languages and autonomously creating structured XML content. Additionally, we will examine the capacity of AI to enrich plain text with appropriate XML markup. Practical examples and methodological guidelines will be provided to elucidate how AI can be effectively prompted to interpret and generate accurate XML markup.
Further emphasis will be placed on the role of AI in developing XSLT, or schemas such as XSD and Schematron. We will address the techniques and strategies adopted to create prompts for generating code, explaining code, or refactoring the code, and the results achieved.
The discussion will extend to how AI can be used to transform XML content. In particular, the focus will be on the use of AI XPath extension functions in XSLT, Schematron, Schematron Quick Fixes, or for XML content refactoring.
The presentation aims to deliver a comprehensive overview of AI usage in XML development, providing attendees with the necessary knowledge to make informed decisions. Whether you’re at the early stages of adopting AI or considering integrating it in advanced XML development, this presentation will cover all levels of expertise.
By highlighting the potential advantages and challenges of integrating AI with XML development tools and languages, the presentation seeks to inspire thoughtful conversation around the future of XML development. We’ll not only delve into the technical aspects of AI-powered XML development but also discuss practical implications and possible future directions.
GraphSummit Singapore | The Art of the Possible with Graph - Q2 2024Neo4j
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UiPath Test Automation using UiPath Test Suite series, part 6DianaGray10
Welcome to UiPath Test Automation using UiPath Test Suite series part 6. In this session, we will cover Test Automation with generative AI and Open AI.
UiPath Test Automation with generative AI and Open AI webinar offers an in-depth exploration of leveraging cutting-edge technologies for test automation within the UiPath platform. Attendees will delve into the integration of generative AI, a test automation solution, with Open AI advanced natural language processing capabilities.
Throughout the session, participants will discover how this synergy empowers testers to automate repetitive tasks, enhance testing accuracy, and expedite the software testing life cycle. Topics covered include the seamless integration process, practical use cases, and the benefits of harnessing AI-driven automation for UiPath testing initiatives. By attending this webinar, testers, and automation professionals can gain valuable insights into harnessing the power of AI to optimize their test automation workflows within the UiPath ecosystem, ultimately driving efficiency and quality in software development processes.
What will you get from this session?
1. Insights into integrating generative AI.
2. Understanding how this integration enhances test automation within the UiPath platform
3. Practical demonstrations
4. Exploration of real-world use cases illustrating the benefits of AI-driven test automation for UiPath
Topics covered:
What is generative AI
Test Automation with generative AI and Open AI.
UiPath integration with generative AI
Speaker:
Deepak Rai, Automation Practice Lead, Boundaryless Group and UiPath MVP
Building Production Ready Search Pipelines with Spark and MilvusZilliz
Spark is the widely used ETL tool for processing, indexing and ingesting data to serving stack for search. Milvus is the production-ready open-source vector database. In this talk we will show how to use Spark to process unstructured data to extract vector representations, and push the vectors to Milvus vector database for search serving.
Cosa hanno in comune un mattoncino Lego e la backdoor XZ?Speck&Tech
ABSTRACT: A prima vista, un mattoncino Lego e la backdoor XZ potrebbero avere in comune il fatto di essere entrambi blocchi di costruzione, o dipendenze di progetti creativi e software. La realtà è che un mattoncino Lego e il caso della backdoor XZ hanno molto di più di tutto ciò in comune.
Partecipate alla presentazione per immergervi in una storia di interoperabilità, standard e formati aperti, per poi discutere del ruolo importante che i contributori hanno in una comunità open source sostenibile.
BIO: Sostenitrice del software libero e dei formati standard e aperti. È stata un membro attivo dei progetti Fedora e openSUSE e ha co-fondato l'Associazione LibreItalia dove è stata coinvolta in diversi eventi, migrazioni e formazione relativi a LibreOffice. In precedenza ha lavorato a migrazioni e corsi di formazione su LibreOffice per diverse amministrazioni pubbliche e privati. Da gennaio 2020 lavora in SUSE come Software Release Engineer per Uyuni e SUSE Manager e quando non segue la sua passione per i computer e per Geeko coltiva la sua curiosità per l'astronomia (da cui deriva il suo nickname deneb_alpha).
Communications Mining Series - Zero to Hero - Session 1DianaGray10
This session provides introduction to UiPath Communication Mining, importance and platform overview. You will acquire a good understand of the phases in Communication Mining as we go over the platform with you. Topics covered:
• Communication Mining Overview
• Why is it important?
• How can it help today’s business and the benefits
• Phases in Communication Mining
• Demo on Platform overview
• Q/A
2. CONTENTS
Classifiers
Difference b/w Classification and Clustering
What is SVM
Supervised learning
Linear SVM
NON Linear SVM
Features and Application
3. C LASSIFIERS
The the goal of Classifiers is to use an object's
characteristics to identify which class (or group) it
belongs to.
Have labels for some points
Supervised learning
Genes
Proteins
Feature Y
Feature X
4. D IFFERENCE B / W C LASSIFICATION
AND C LUSTERING
In general, in classification you have a set of
predefined classes and want to know which class
a new object belongs to.
Clustering tries to group a set of object.
In the context of machine learning, classification
is supervised learning and clustering
is unsupervised learning.
5. W HAT I S SVM?
Support Vector Machines are based on the
concept of decision planes that define decision
boundaries.
A decision plane is one that separates between a
set of objects having different class
memberships.
6. the objects belong either to class GREEN or RED.
The separating line defines a boundary on the right side of
which all objects are GREEN and to the left of which all
objects are RED. Any new object (white circle) falling to
the right is labeled, i.e., classified, as GREEN (or classified
as RED should it fall to the left of the separating line).
This is a classic example of a linear classifier
7. Most classification tasks are not as simple, as we have
seen in previous example
More complex structures are needed in order to make an
optimal separation
Full separation of the GREEN and RED objects would
require a curve (which is more complex than a line).
8.
9. In fig. we can see the original objects (left side of the
schematic) mapped, i.e., rearranged, using a set of
mathematical functions, known as kernels.
The process of rearranging the objects is known as
mapping (transformation). Note that in this new setting,
the mapped objects (right side of the schematic) is linearly
separable and, thus, instead of constructing the complex
curve (left schematic), all we have to do is to find an
optimal line that can separate the GREEN and the RED
objects.
10. Support Vector Machine (SVM) is primarily a classier
method that performs classification tasks by
constructing hyperplanes in a multidimensional space
that separates cases of different class labels.
SVM supports both regression and classification tasks
and can handle multiple continuous and categorical
variables. For categorical variables a dummy variable is
created with case values as either 0 or 1. Thus, a
categorical dependent variable consisting of three
levels, say (A, B, C), is represented by a set of three
dummy variables:
A: {1 0 0}, B: {0 1 0}, C: {0 0 1}
12. S UPERVISED L EARNING
Training set: a number of expression profiles with known
labels which represent the true population.
Difference to clustering: there you don't know the labels,you
have to find a structure on your own.
Learning/Training:find a decision rule which explains the
training set well.
This is the easy part, because we know the labels of the
training set!
Generalisation ability: how does the decision rule learned
from the training set generalize to new specimen?
Goal: find a decision rule with high generalisation ability.
13. L INEAR S EPARATORS
Binary classification can be viewed as the
task of separating classes in feature space:
wTx + b = 0
wTx + b > 0
wTx + b < 0
f(x) = sign(wTx + b)
14. L INEAR SEPARATION OF THE
TRAINING SET
A separating hyperplane is
defined by
- the normal vector w and
- the offset b:
hyperplane = {x |<w,x>+ b = 0}
<.,.> is called inner product,
scalar product or dot product.
Training: Choose w and b from
the labelled examples in the
training set.
15. P REDICT THE LABEL OF A
NEW POINT
Prediction: On which side
of the hyper-plane does
the new point lie?
Points in the direction of
the normal vector are
classified as POSITIVE.
Points in the opposite
direction are classified as
NEGATIVE.
17. C LASSIFICATION M ARGIN
wT xi b
Distance from example xi to the separator is r
w
Examples closest to the hyperplane are support
vectors.
ρ
Margin ρ of the
r
separator is the
distance between
support vectors.
18. M AXIMUM M ARGIN C LASSIFICATION
Maximizing the margin is good according to
intuition and PAC theory.
Implies that only support vectors matter; other
training examples are ignorable.
19. L INEAR SVM
M ATHEMATICALLY
Let training set {(xi, yi)}i=1..n, xi Rd, yi {-1, 1} be separated
by a hyperplane with margin ρ. Then for each training
example (xi, yi):
wTxi + b ≤ - ρ/2 if yi = -1 yi(wTxi + b) ≥ ρ/2
wTxi + b ≥ ρ/2 if yi = 1
For every support vector xs the above inequality is an
equality. After rescaling w and b by ρ/2 in the equality,
we obtain that distance between each xs and the
hyperplane is T
y s ( w x s b) 1
r
w w
Then the margin can be expressed through (rescaled) w
and b as: 2r
2
w
20. L INEAR SVM S
M ATHEMATICALLY ( CONT.)
Then we can formulate the quadratic optimization
problem:
Find w and b such that
2
is maximized
w
and for all (xi, yi), i=1..n : yi(wTxi + b) ≥ 1
Which can be reformulated as:
Find w and b such that
Φ(w) = ||w||2=wTw is minimized
and for all (xi, yi), i=1..n : yi (wTxi + b) ≥ 1
21. S OLVING THE O PTIMIZATION
P ROBLEM
Find w and b such that
Φ(w) =wTw is minimized
and for all (xi, yi), i=1..n : yi (wTxi + b) ≥ 1
Need to optimize a quadratic function subject to linear
constraints.
Quadratic optimization problems are a well-known class of
mathematical programming problems for which several (non-
trivial) algorithms exist.
The solution involves constructing a dual problem where a
Lagrange multiplier αi is associated with every inequality
constraint in the primal (original) problem:
Find α1…αn such that
Q(α) =Σαi - ½ΣΣαiαjyiyjxiTxj is maximized and
(1) Σαiyi = 0
(2) αi ≥ 0 for all αi
22. T HE O PTIMIZATION P ROBLEM
S OLUTION
Given a solution α1…αn to the dual problem, solution to the
primal is:
w =Σαiyixi b = yk - Σαiyixi Txk for any αk > 0
Each non-zero αi indicates that corresponding xi is a support
vector.
Then the classifying function is (note that we don’t need w
explicitly): f(x) = ΣαiyixiTx + b
Notice that it relies on an inner product between the test point
x and the support vectors xi – we will return to this later.
Also keep in mind that solving the optimization problem
involved computing the inner products xiTxj between all training
points.
23. S OFT M ARGIN
C LASSIFICATION
What if the training set is not linearly separable?
Slack variables ξi can be added to allow
misclassification of difficult or noisy examples,
resulting margin called soft.
ξi
ξi
24. S OFT M ARGIN C LASSIFICATION
M ATHEMATICALLY
The old formulation:
Find w and b such that
Φ(w) =wTw is minimized
and for all (xi ,yi), i=1..n : yi (wTxi + b) ≥ 1
Modified formulation incorporates slack variables:
Find w and b such that
Φ(w) =wTw + CΣξi is minimized
and for all (xi ,yi), i=1..n : yi (wTxi + b) ≥ 1 – ξi, , ξi ≥ 0
Parameter C can be viewed as a way to control overfitting: it
“trades off” the relative importance of maximizing the
margin and fitting the training data.
25. S OFT M ARGIN
C LASSIFICATION – S OLUTION
Dual problem is identical to separable case (would not be
identical if the 2-norm penalty for slack variables CΣξi2 was
used in primal objective, we would need additional
Lagrange multipliers for slack variables):
Find α1…αN such that
Q(α) =Σαi - ½ΣΣαiαjyiyjxiTxj is maximized and
(1) Σαiyi = 0
(2) 0 ≤ αi ≤ C for all αi
Again, xi with non-zero αi will be support vectors.
Solution to the dual problem is: Again, we don’t need
to compute w explicitly
w =Σαiyixi
for classification:
b= yk(1- ξk) - ΣαiyixiTxk for any k s.t. αk>0
f(x) = ΣαiyixiTx + b
26. T HEORETICAL J USTIFICATION
FOR M AXIMUM M ARGINS
Vapnik has proved the following:
The class of optimal linear separators has VC dimension h
bounded from above as 2
D
h min 2
, m0 1
where ρ is the margin, D is the diameter of the smallest sphere
that can enclose all of the training examples, and m0 is the
dimensionality.
Intuitively, this implies that regardless of dimensionality m0 we
can minimize the VC dimension by maximizing the margin ρ.
Thus, complexity of the classifier is kept small regardless of
dimensionality.
27. L INEAR SVM S : O VERVIEW
The classifier is a separating hyperplane.
Most “important” training points are support vectors;
they define the hyperplane.
Quadratic optimization algorithms can identify which
training points xi are support vectors with non-zero
Lagrangian multipliers αi.
Both in the dual formulation of the problem and in the
solution training points appear only inside inner
products:
Find α1…αN such that f(x) = ΣαiyixiTx + b
Q(α) =Σαi - ½ΣΣαiαjyiyjxiTxj is maximized and
(1) Σαiyi = 0
(2) 0 ≤ αi ≤ C for all αi
28. N ON - LINEAR SVM S
Datasets that are linearly separable with some noise
work out great:
0 x
But what are we going to do if the dataset is just too
hard?
0 x
How about… mapping data to a higher-dimensional
space: x2
0 x
29. N ON - LINEAR SVM S :
F EATURE SPACES
General idea: the original feature space can always be
mapped to some higher-dimensional feature space
where the training set is separable:
Φ: x → φ(x)
30. P ROPERTIES OF SVM
Flexibility in choosing a similarity function
Sparseness of solution when dealing with
large data sets
- only support vectors are used to specify the
separating hyperplane
Ability to handle large feature spaces
- complexity does not depend on the
dimensionality of the feature space
Guaranteed to converge to a single global
solution
31. SVM A PPLICATIONS
SVM has been used successfully in many real-world
problems
- text (and hypertext) categorization
- image classification
- bioinformatics (Protein classification,
Cancer classification)
- hand-written character recognition