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Learning from distributions via support measure machines
 Advances in Neural Information Processing Systems 25
, 2012
"... This paper presents a kernelbased discriminative learning framework on probability measures. Rather than relying on large collections of vectorial training examples, our framework learns using a collection of probability distributions that have been constructed to meaningfully represent training da ..."
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Cited by 28 (10 self)
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data. By representing these probability distributions as mean embeddings in the reproducing kernel Hilbert space (RKHS), we are able to apply many standard kernelbased learning techniques in straightforward fashion. To accomplish this, we construct a generalization of the support vector machine (SVM
Supplementary Material to Learning from Distributions via Support Measure Machines
"... increasing function Ω: [0,+∞) → R, and a loss function ℓ: (P × R2)m → R ∪ {+∞}, any f ∈ H minimizing the regularized risk functional ℓ (P1, y1,EP1 [f],...,Pm, ym,EPm [f]) + Ω (‖f‖H) (1) admits a representation of the form f =∑mi=1 αiµPi for some αi ∈ R, i = 1,...,m. Proof. By virtue of Proposition ..."
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increasing function Ω: [0,+∞) → R, and a loss function ℓ: (P × R2)m → R ∪ {+∞}, any f ∈ H minimizing the regularized risk functional ℓ (P1, y1,EP1 [f],...,Pm, ym,EPm [f]) + Ω (‖f‖H) (1) admits a representation of the form f =∑mi=1 αiµPi for some αi ∈ R, i = 1,...,m. Proof. By virtue of Proposition 2 in [1], the linear functional EP[·] are bounded for all P ∈ P. Then, given P1,P2,...,Pm, any f ∈ H can be decomposed as f = fµ + f where fµ ∈ H lives in the span of µPi, i.e., fµ = ∑m i=1 αiµPi and f ⊥ ∈ H satisfying, for all j, 〈f⊥, µPj 〉 = 0. Hence, for all j, we have EPj [f] = EPj [fµ + f ⊥] = 〈fµ + f ⊥, µPj 〉 = 〈fµ, µPj 〉+ 〈f ⊥, µPj 〉 = 〈fµ, µPj 〉 which is independent of f⊥. As a result, the loss functional ℓ in (1) does not depend on f⊥. For the regularization functional Ω, since f ⊥ is orthogonal to ∑m i=1 αiµPi and Ω is strictly monotonically increasing, we have Ω(‖f‖) = Ω(‖fµ + f ⊥‖) = Ω( ‖fµ‖2 + ‖f⊥‖2) ≥ Ω(‖fµ‖) with equality if and only if f ⊥ = 0 and thus f = fµ. Consequently, any minimizer must take the form f = ∑m i=1 αiµPi = ∑m i=1 αiEPi [k(x, ·)].
Gaussian processes for machine learning
 in: Adaptive Computation and Machine Learning
, 2006
"... Abstract. We give a basic introduction to Gaussian Process regression models. We focus on understanding the role of the stochastic process and how it is used to define a distribution over functions. We present the simple equations for incorporating training data and examine how to learn the hyperpar ..."
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Cited by 631 (2 self)
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Abstract. We give a basic introduction to Gaussian Process regression models. We focus on understanding the role of the stochastic process and how it is used to define a distribution over functions. We present the simple equations for incorporating training data and examine how to learn
Ensemble Methods in Machine Learning
 MULTIPLE CLASSIFIER SYSTEMS, LBCS1857
, 2000
"... Ensemble methods are learning algorithms that construct a set of classifiers and then classify new data points by taking a (weighted) vote of their predictions. The original ensemble method is Bayesian averaging, but more recent algorithms include errorcorrecting output coding, Bagging, and boostin ..."
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Cited by 607 (3 self)
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Ensemble methods are learning algorithms that construct a set of classifiers and then classify new data points by taking a (weighted) vote of their predictions. The original ensemble method is Bayesian averaging, but more recent algorithms include errorcorrecting output coding, Bagging
Support vector machine active learning for image retrieval
, 2001
"... Relevance feedback is often a critical component when designing image databases. With these databases it is difficult to specify queries directly and explicitly. Relevance feedback interactively determinines a user’s desired output or query concept by asking the user whether certain proposed images ..."
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Cited by 448 (29 self)
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are relevant or not. For a relevance feedback algorithm to be effective, it must grasp a user’s query concept accurately and quickly, while also only asking the user to label a small number of images. We propose the use of a support vector machine active learning algorithm for conducting effective relevance
Sparse Bayesian Learning and the Relevance Vector Machine
, 2001
"... This paper introduces a general Bayesian framework for obtaining sparse solutions to regression and classication tasks utilising models linear in the parameters. Although this framework is fully general, we illustrate our approach with a particular specialisation that we denote the `relevance vec ..."
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Cited by 958 (5 self)
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vector machine' (RVM), a model of identical functional form to the popular and stateoftheart `support vector machine' (SVM). We demonstrate that by exploiting a probabilistic Bayesian learning framework, we can derive accurate prediction models which typically utilise dramatically fewer
Machine Learning in Automated Text Categorization
 ACM COMPUTING SURVEYS
, 2002
"... The automated categorization (or classification) of texts into predefined categories has witnessed a booming interest in the last ten years, due to the increased availability of documents in digital form and the ensuing need to organize them. In the research community the dominant approach to this p ..."
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Cited by 1658 (22 self)
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to this problem is based on machine learning techniques: a general inductive process automatically builds a classifier by learning, from a set of preclassified documents, the characteristics of the categories. The advantages of this approach over the knowledge engineering approach (consisting in the manual
A learning algorithm for Boltzmann machines
 Cognitive Science
, 1985
"... The computotionol power of massively parallel networks of simple processing elements resides in the communication bandwidth provided by the hardware connections between elements. These connections con allow a significant fraction of the knowledge of the system to be applied to an instance of a probl ..."
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Cited by 586 (13 self)
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to a general learning rule for modifying the connection strengths so as to incorporate knowledge obout o task domain in on efficient way. We describe some simple examples in which the learning algorithm creates internal representations thot ore demonstrobly the most efficient way of using
Training Support Vector Machines: an Application to Face Detection
, 1997
"... We investigate the application of Support Vector Machines (SVMs) in computer vision. SVM is a learning technique developed by V. Vapnik and his team (AT&T Bell Labs.) that can be seen as a new method for training polynomial, neural network, or Radial Basis Functions classifiers. The decision sur ..."
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Cited by 728 (1 self)
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We investigate the application of Support Vector Machines (SVMs) in computer vision. SVM is a learning technique developed by V. Vapnik and his team (AT&T Bell Labs.) that can be seen as a new method for training polynomial, neural network, or Radial Basis Functions classifiers. The decision
Estimating the Support of a HighDimensional Distribution
, 1999
"... Suppose you are given some dataset drawn from an underlying probability distribution P and you want to estimate a "simple" subset S of input space such that the probability that a test point drawn from P lies outside of S is bounded by some a priori specified between 0 and 1. We propo ..."
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Cited by 766 (29 self)
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Suppose you are given some dataset drawn from an underlying probability distribution P and you want to estimate a "simple" subset S of input space such that the probability that a test point drawn from P lies outside of S is bounded by some a priori specified between 0 and 1. We
Results 1  10
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2,284,132