Open AccessProceedings Article
Learning Minimum Volume Sets
Clayton Scott,Robert Nowak +1 more
- 05 Dec 2005
- Vol. 18, pp 1209-1216
TL;DR: In this article, the problem of estimating minimum volume sets based on independent samples distributed according to a probability measure P and a reference measure μ is addressed, where no other information is available regarding P, but the reference measure is assumed to be known.
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Abstract: Given a probability measure P and a reference measure μ, one is often interested in the minimum μ-measure set with P-measure at least α. Minimum volume sets of this type summarize the regions of greatest probability mass of P, and are useful for detecting anomalies and constructing confidence regions. This paper addresses the problem of estimating minimum volume sets based on independent samples distributed according to P. Other than these samples, no other information is available regarding P, but the reference measure μ is assumed to be known. We introduce rules for estimating minimum volume sets that parallel the empirical risk minimization and structural risk minimization principles in classification. As in classification, we show that the performances of our estimators are controlled by the rate of uniform convergence of empirical to true probabilities over the class from which the estimator is drawn. Thus we obtain finite sample size performance bounds in terms of VC dimension and related quantities. We also demonstrate strong universal consistency and an oracle inequality. Estimators based on histograms and dyadic partitions illustrate the proposed rules.
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Citations
A Unifying Review of Deep and Shallow Anomaly Detection
Lukas Ruff,Jacob R. Kauffmann,Robert A. Vandermeulen,Grégoire Montavon,Wojciech Samek,Marius Kloft,Thomas G. Dietterich,Klaus-Robert Müller +7 more
- 04 Feb 2021
TL;DR: Deep learning approaches to anomaly detection (AD) have recently improved the state of the art in detection performance on complex data sets, such as large collections of images or text as mentioned in this paper, and led to the introduction of a great variety of new methods.
A Unifying Review of Deep and Shallow Anomaly Detection
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TL;DR: This review aims to identify the common underlying principles and the assumptions that are often made implicitly by various methods in deep learning, and draws connections between classic “shallow” and novel deep approaches and shows how this relation might cross-fertilize or extend both directions.
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References
•Book
Elements of information theory
Thomas M. Cover,Joy A. Thomas +1 more
- 01 Jan 1991
TL;DR: The author examines the role of entropy, inequality, and randomness in the design of codes and the construction of codes in the rapidly changing environment.
Statistical learning theory
Vladimir Vapnik
- 01 Jan 1998
TL;DR: Presenting a method for determining the necessary and sufficient conditions for consistency of learning process, the author covers function estimates from small data pools, applying these estimations to real-life problems, and much more.
30.4K
•Book
Classification and regression trees
Leo Breiman
- 01 Jan 1983
TL;DR: The methodology used to construct tree structured rules is the focus of a monograph as mentioned in this paper, covering the use of trees as a data analysis method, and in a more mathematical framework, proving some of their fundamental properties.
22.7K
Estimating the Support of a High-Dimensional Distribution
TL;DR: In this paper, the authors propose a method to estimate a function f that is positive on S and negative on the complement of S. The functional form of f is given by a kernel expansion in terms of a potentially small subset of the training data; it is regularized by controlling the length of the weight vector in an associated feature space.