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Mixtures of Probabilistic Principal Component Analysers
, 1998
"... Principal component analysis (PCA) is one of the most popular techniques for processing, compressing and visualising data, although its effectiveness is limited by its global linearity. While nonlinear variants of PCA have been proposed, an alternative paradigm is to capture data complexity by a com ..."
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Cited by 398 (6 self)
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Principal component analysis (PCA) is one of the most popular techniques for processing, compressing and visualising data, although its effectiveness is limited by its global linearity. While nonlinear variants of PCA have been proposed, an alternative paradigm is to capture data complexity by a combination of local linear PCA projections. However, conventional PCA does not correspond to a probability density, and so there is no unique way to combine PCA models. Previous attempts to formulate mixture models for PCA have therefore to some extent been ad hoc. In this paper, PCA is formulated within a maximumlikelihood framework, based on a specific form of Gaussian latent variable model. This leads to a welldefined mixture model for probabilistic principal component analysers, whose parameters can be determined using an EM algorithm. We discuss the advantages of this model in the context of clustering, density modelling and local dimensionality reduction, and we demonstrate its applicat...
A New Approach to Dimensionality Reduction: Theory and Algorithms
 SIAM Journal of Applied Mathematics
, 2000
"... Abstract. This paper applies Whitney’s embedding theorem to the data reduction problem and introduces a new approach motivated in part by the (constructive) proof of the theorem. The notion of a good projection is introduced which involves picking projections of the highdimensional system that are ..."
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Cited by 27 (2 self)
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Abstract. This paper applies Whitney’s embedding theorem to the data reduction problem and introduces a new approach motivated in part by the (constructive) proof of the theorem. The notion of a good projection is introduced which involves picking projections of the highdimensional system that are optimized such that they are easy to invert. The basic theory of the approach is outlined and algorithms for finding the projections are presented and applied to several test cases. A method for constructing the inverse projection is detailed and its properties, including a new measure of complexity, are discussed. Finally, wellknown methods of data reduction are compared with our approach within the context of Whitney’s theorem. Key words. dimensionality reduction, radial basis functions, Whitney’s embedding theorem, secant basis
Large Dimensionality Reduction Using Secantbased Projection Methods: The Induced Dynamics
, 2005
"... previous papers we have developed an approach to the data reduction problem which is based on a wellknown, constructive proof of Whitney’s embedding theorem [Broomhead, D. S. and Kirby, M., SIAM Journal of Applied Mathematics ..."
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Cited by 5 (0 self)
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previous papers we have developed an approach to the data reduction problem which is based on a wellknown, constructive proof of Whitney’s embedding theorem [Broomhead, D. S. and Kirby, M., SIAM Journal of Applied Mathematics
Visualization of Scientific Video Data Using KL Decomposition
"... Fast methods are developed for visualizing and classifying certain types of scientific video data. These techniques, which are based on KL decomposition, find a best coordinate system for a data set. When the data set represents a temporally ordered collection of images, the best coordinate system l ..."
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Cited by 4 (1 self)
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Fast methods are developed for visualizing and classifying certain types of scientific video data. These techniques, which are based on KL decomposition, find a best coordinate system for a data set. When the data set represents a temporally ordered collection of images, the best coordinate system leads to approximations that are separable in time and space. Practical methods for computing this best coordinate system are discussed, and physically significant visualizations for experimental video data are developed. The visualization techniques are applied to two experimental systems  one from combustion and the other from neurobiology to show how relevant information can be quickly extracted from video data. These techniques can be integrated into the video acquisition process to provide realtime feedback to the experimentalist during the operation of an experiment. KeywordsScientific visualization, realtime visualization, video analysis. I. INTRODUCTION The development of f...
DIRECT SEARCH METHODS OVER LIPSCHITZ MANIFOLDS ∗
, 2007
"... We extend direct search methods to optimization problems that include equality constraints given by Lipschitz functions. The equality constraints are assumed to implicitly define a Lipschitz manifold. Numerically implementing the inverse (implicit) function theorem allows us to define a new problem ..."
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Cited by 1 (1 self)
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We extend direct search methods to optimization problems that include equality constraints given by Lipschitz functions. The equality constraints are assumed to implicitly define a Lipschitz manifold. Numerically implementing the inverse (implicit) function theorem allows us to define a new problem on the tangent spaces of the manifold. We can then use a direct search method on the tangent spaces to solve the new optimization problem without any equality constraints. Solving this related problem implicitly solves the original optimization problem. Our main example utilizes the LTMADS algorithm for the direct search method. However, other direct search methods can be employed. Convergence results trivially carry over to our new procedure under mild assumptions.
Estimation of Topological Dimension
"... We present two extensions of the algorithm by Broomhead et al [2] which is based on the idea that singular values that scale linearly with the radius of the data ball can be exploited to develop algorithms for computing topological dimension and for detecting whether data models based on manifolds a ..."
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We present two extensions of the algorithm by Broomhead et al [2] which is based on the idea that singular values that scale linearly with the radius of the data ball can be exploited to develop algorithms for computing topological dimension and for detecting whether data models based on manifolds are appropriate. We present a geometric scaling property and dimensionality criterion that permit the automated application of the algorithm as well as a significant reduction in computational expense. For irregularly distributed data this approach can provide a detailed analysis of the structure of the data including an estimated dimension distribution function. We present our approach on several data sets.
massachusetts institute of technology, cambridge, ma 02139 usa — www.csail.mit.eduMultiscale Geometric Methods for Data Sets I: Multiscale SVD, Noise and Curvature
, 2012
"... Large data sets are often modeled as being noisy samples from probability distributions µ in R D, with D large. It has been noticed that oftentimes the support M of these probability distributions seems to be wellapproximated by lowdimensional sets, perhaps even by manifolds. We shall consider set ..."
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Large data sets are often modeled as being noisy samples from probability distributions µ in R D, with D large. It has been noticed that oftentimes the support M of these probability distributions seems to be wellapproximated by lowdimensional sets, perhaps even by manifolds. We shall consider sets that are locally well approximated by kdimensional planes, with k ≪ D, with kdimensional manifolds isometrically embedded in R D being a special case. Samples from µ are furthermore corrupted by Ddimensional noise. Certain tools from multiscale geometric measure theory and harmonic analysis seem wellsuited to be adapted to the study of samples from such probability distributions, in order to yield quantitative geometric information about them. In this paper we introduce and study multiscale covariance matrices, i.e. covariances corresponding to the distribution restricted to a ball of radius r, with a fixed center and varying r, and under rather general geometric assumptions we study how their empirical, noisy counterparts behave. We prove that in the range of scales where these covariance matrices are most informative, the empirical, noisy covariances are close to their expected, noiseless counterparts. In fact, this is true as soon as the number of samples in the balls where the covariance matrices are computed is linear in the intrinsic dimension of M. As an application, we present an algorithm for estimating the intrinsic dimension of M. 1
A Monte Carlo Implementation of the . . .
 MONTHLY WEATHER REVIEW
, 1999
"... Knowledge of the probability distribution of initial conditions is central to almost all practical studies of predictability and to improvements in stochastic prediction of the atmosphere. Traditionally, data assimilation for atmospheric predictability or prediction experiments has attempted to fi ..."
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Knowledge of the probability distribution of initial conditions is central to almost all practical studies of predictability and to improvements in stochastic prediction of the atmosphere. Traditionally, data assimilation for atmospheric predictability or prediction experiments has attempted to find a single "best" estimate of the initial state. Additional information about the initial condition probability distribution is then obtained primarily through heuristic techniques that attempt to generate representative perturbations around the best estimate. However,