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A review of dimension reduction techniques
, 1997
"... The problem of dimension reduction is introduced as a way to overcome the curse of the dimensionality when dealing with vector data in high-dimensional spaces and as a modelling tool for such data. It is defined as the search for a low-dimensional manifold that embeds the high-dimensional data. A cl ..."
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Cited by 42 (4 self)
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The problem of dimension reduction is introduced as a way to overcome the curse of the dimensionality when dealing with vector data in high-dimensional spaces and as a modelling tool for such data. It is defined as the search for a low-dimensional manifold that embeds the high-dimensional data. A classification of dimension reduction problems is proposed. A survey of several techniques for dimension reduction is given, including principal component analysis, projection pursuit and projection pursuit regression, principal curves and methods based on topologically continuous maps, such as Kohonen’s maps or the generalised topographic mapping. Neural network implementations for several of these techniques are also reviewed, such as the projection pursuit learning network and the BCM neuron with an objective function. Several appendices complement the mathematical treatment of the main text.
Continuous latent variable models for dimensionality reduction and sequential data reconstruction
, 2001
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Efficient Algorithms for Function Approximation with Piecewise Linear Sigmoidal Networks
, 1998
"... This paper presents a computationally efficient algorithm for function approximation with piecewise linear sigmoidal nodes. A one hidden layer network is constructed one node at a time using the well-known method of fitting the residual. The task of fitting an individual node is accomplished using ..."
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Cited by 21 (1 self)
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This paper presents a computationally efficient algorithm for function approximation with piecewise linear sigmoidal nodes. A one hidden layer network is constructed one node at a time using the well-known method of fitting the residual. The task of fitting an individual node is accomplished using a new algorithm that searches for the best fit by solving a sequence of Quadratic Programming problems. This approach offers significant advantages over derivative-based search algorithms (e.g. backpropagation and its extensions). Unique characteristics of this algorithm include: finite step convergence, a simple stopping criterion, solutions that are independent of initial conditions, good scaling properties and a robust numerical implementation. Empirical results are included to illustrate these characteristics.
Beyond maximum likelihood and density estimation: A sample-based criterion for unsupervised learning of complex models
"... The goal of many unsupervised learning procedures is to bring two probability distributions into alignment. Generative models such as Gaussian mixtures and Boltzmann machines can be cast in this light, as can recoding models such as ICA and projection pursuit. ..."
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Cited by 3 (0 self)
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The goal of many unsupervised learning procedures is to bring two probability distributions into alignment. Generative models such as Gaussian mixtures and Boltzmann machines can be cast in this light, as can recoding models such as ICA and projection pursuit.
A Hybrid Learning System for Image Deblurring
"... In this paper we propose a 3-stage hybrid learning system with unsupervised learning to cluster data in the rst stage, supervised learning in the middle stage to determine network parameters and nally a decision making stage using voting mechanism. We take this opportunity to study the role of vario ..."
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Cited by 2 (1 self)
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In this paper we propose a 3-stage hybrid learning system with unsupervised learning to cluster data in the rst stage, supervised learning in the middle stage to determine network parameters and nally a decision making stage using voting mechanism. We take this opportunity to study the role of various supervised learning systems that constitute the middle stage. Speci cally, we focus on one-hidden layer neural network with sigmoidal activation function, radial basis function network with Gaussian activation function and projection pursuit learning network with Hermite polynomial as the activation function. These learning systems rank in increasing order of complexity. We train and test each system with identical data sets. Experimental results show that learning ability of a system is controlled by the shape of the activation function when other parameters remain xed. We observe that clustering in the input space leads to better system performance. Experimental results provide compelling evidences in favor of use of the hybrid learning system and the committee machines with gating network.
unknown title
, 2001
"... Continuous latent variable models for dimensionality reduction and sequential data reconstruction by ..."
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Continuous latent variable models for dimensionality reduction and sequential data reconstruction by
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"... The continuous latent variable modelling formalism This chapter gives the theoretical basis for continuous latent variable models. Section 2.1 defines intuitively the concept of latent variable models and gives a brief historical introduction to them. Section 2.2 uses a simple example, inspired by t ..."
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The continuous latent variable modelling formalism This chapter gives the theoretical basis for continuous latent variable models. Section 2.1 defines intuitively the concept of latent variable models and gives a brief historical introduction to them. Section 2.2 uses a simple example, inspired by the mechanics of a mobile point, to justify and explain latent variables. Section 2.3 gives a more rigorous definition, which we will use throughout this thesis. Section 2.6 describes the most important specific continuous latent variable models and section 2.7 defines mixtures of continuous latent variable models. The chapter discusses other important topics, including parameter estimation, identifiability, interpretability and marginalisation in high dimensions. Section 2.9 on dimensionality reduction will be the basis for part II of the thesis. Section 2.10 very briefly mentions some applications of continuous latent variable models for dimensionality reduction. Section 2.11 shows a worked example of a simple continuous latent variable model. Section 2.12 give some complementary mathematical results, in particular the derivation of a diagonal noise GTM model and of its EM algorithm. 2.1 Introduction and historical overview of latent variable models Latent variable models are probabilistic models that try to explain a (relatively) high-dimensional process in
unknown title
"... The continuous latent variable modelling formalism This chapter gives the theoretical basis for continuous latent variable models. Section 2.1 defines intuitively the concept of latent variable models and gives a brief historical introduction to them. Section 2.2 uses a simple example, inspired by t ..."
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The continuous latent variable modelling formalism This chapter gives the theoretical basis for continuous latent variable models. Section 2.1 defines intuitively the concept of latent variable models and gives a brief historical introduction to them. Section 2.2 uses a simple example, inspired by the mechanics of a mobile point, to justify and explain latent variables. Section 2.3 gives a more rigorous definition, which we will use throughout this thesis. Section 2.6 describes the most important specific continuous latent variable models and section 2.7 defines mixtures of continuous latent variable models. The chapter discusses other important topics, including parameter estimation, identifiability, interpretability and marginalisation in high dimensions. Section 2.9 on dimensionality reduction will be the basis for part II of the thesis. Section 2.10 very briefly mentions some applications of continuous latent variable models for dimensionality reduction. Section 2.11 shows a worked example of a simple continuous latent variable model. Section 2.12 give some complementary mathematical results, in particular the derivation of a diagonal noise GTM model and of its EM algorithm. 2.1 Introduction and historical overview of latent variable models Latent variable models are probabilistic models that try to explain a (relatively) high-dimensional process in
Chapter 4 Dimensionality reduction
"... This chapter introduces and defines the problem of dimensionality reduction, discusses the topics of the curse of the dimensionality and the intrinsic dimensionality and then surveys non-probabilistic methods for dimensionality reduction, that is, methods that do not define a probabilistic model for ..."
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This chapter introduces and defines the problem of dimensionality reduction, discusses the topics of the curse of the dimensionality and the intrinsic dimensionality and then surveys non-probabilistic methods for dimensionality reduction, that is, methods that do not define a probabilistic model for the data. These include linear methods (PCA, projection pursuit), nonlinear autoassociators, kernel methods, local dimensionality reduction, principal curves, vector quantisation methods (elastic net, self-organising map) and multidimensional scaling methods. One of these methods (the elastic net) does define a probabilistic model but not a continuous dimensionality reduction mapping. If one is interested in stochastically modelling the dimensionality reduction mapping then the natural choice are latent variable models, discussed in chapter 2. We close the chapter with a summary and with some thoughts on dimensionality reduction with discrete variables. Consider an application in which a system processes data in the form of a collection of real-valued vectors: speech signals, images, etc. Suppose that the system is only effective if the dimension of each individual vector—the number of components of the vector—is not too high, where high depends on the particular application. The problem of dimensionality reduction appears when the data are in fact of a higher dimension
unknown title
, 2001
"... Continuous latent variable models for dimensionality reduction and sequential data reconstruction by ..."
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Continuous latent variable models for dimensionality reduction and sequential data reconstruction by
