IDEA: Intrinsic Dimension Estimation Algorithm (original) (raw)
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On Local Intrinsic Dimension Estimation and Its Applications
In this paper, we present multiple novel applications for local intrinsic dimension estimation. There has been much work done on estimating the global dimension of a data set, typically for the purposes of dimensionality reduction. We show that by estimating dimension locally, we are able to extend the uses of dimension estimation to many applications, which are not possible with global dimension estimation. Additionally, we show that local dimension estimation can be used to obtain a better global dimension estimate, alleviating the negative bias that is common to all known dimension estimation algorithms. We illustrate local dimension estimation's uses towards additional applications, such as learning on statistical manifolds, network anomaly detection, clustering, and image segmentation.
Novel high intrinsic dimensionality estimators
Machine Learning, 2012
Recently, a great deal of research work has been devoted to the development of algorithms to estimate the intrinsic dimensionality (id) of a given dataset, that is the minimum number of parameters needed to represent the data without information loss. id estimation is important for the following reasons: the capacity and the generalization capability of discriminant methods depend on it; id is a necessary information for any dimensionality reduction technique; in neural network design the number of hidden units in the encoding middle layer should be chosen according to the id of data; the id value is strongly related to the model order in a time series, that is crucial to obtain reliable time series predictions. Although many estimation techniques have been proposed in the literature, most of them fail on noisy data, or compute underestimated values when the id is sufficiently high. In this paper, after reviewing some of the most important id estimators related to our work, we provide a theoretical motivation of the bias that causes the underestimation effect, and we present two id estimators based on the statistical properties of manifold neighborhoods, which have been developed in order to reduce this effect. We exhaustively evaluate the proposed techniques on synthetic and real datasets, by employing an objective evaluation measure to compare their performance with those achieved by state of the art algorithms; the results show that the proposed methods are promising, and produce reliable estimates also in the difficult case of datasets drawn from non-linearly embedded manifolds, characterized by high id.
Optimized intrinsic dimension estimator using nearest neighbor graphs
2010 IEEE International Conference on Acoustics, Speech and Signal Processing, 2010
We develop an approach to intrinsic dimension estimation based on k-nearest neighbor (kNN) distances. The dimension estimator is derived using a general theory on functionals of kNN density estimates. This enables us to predict the performance of the dimension estimation algorithm. In addition, it allows for optimization of free parameters in the algorithm. We validate our theory through simulations and compare our estimator to previous kNN based dimensionality estimation approaches.
Intrinsic Dimension Estimation: Relevant Techniques and a Benchmark Framework
Mathematical Problems in Engineering, 2015
When dealing with datasets comprising high-dimensional points, it is usually advantageous to discover some data structure. A fundamental information needed to this aim is the minimum number of parameters required to describe the data while minimizing the information loss. This number, usually called intrinsic dimension, can be interpreted as the dimension of the manifold from which the input data are supposed to be drawn. Due to its usefulness in many theoretical and practical problems, in the last decades the concept of intrinsic dimension has gained considerable attention in the scientific community, motivating the large number of intrinsic dimensionality estimators proposed in the literature. However, the problem is still open since most techniques cannot efficiently deal with datasets drawn from manifolds of high intrinsic dimension and nonlinearly embedded in higher dimensional spaces. This paper surveys some of the most interesting, widespread used, and advanced state-of-the-a...
IAEME PUBLICATION, 2020
Dimensionality reduction is an essential phase in data mining. Intrinsic dimension is used to determine the attributes which covers the entire data set. Various techniques are involved in calculating intrinsic dimension. Here the intrinsic dimension for diabetes medical dataset is to be done through box-counting dimension and correlation dimension. Analysis shows that correlation dimension is more efficient than box counting method in terms of size of the dataset. User defined distance for calculating fractal dimension reduces the reliability of correlation dimension method, so log-log pairs of a data set is used in the correlation dimension. The sample size and number of redundant variables influence the computation of correlation dimension. Implementation of box counting and correlation dimension for diabetes data sets confirm the effectiveness of intrinsic dimension estimation with log-log pairs plot.
Intrinsic dimension estimation via nearest constrained subspace classifier
2014
We consider the problems of classification and intrinsic dimension estimation on image data. A new subspace based classifier is proposed for supervised classification or intrinsic dimension estimation. The distribution of the data in each class is modeled by a union of of a finite number of affine subspaces of the feature space. The affine subspaces have a common dimension, which is assumed to be much less than the dimension of the feature space. The subspaces are found using regression based on the 0-norm. The proposed method is a generalisation of classical NN (Nearest Neighbor), NFL (Nearest Feature Line) classifiers and has a close relationship to NS (Nearest Subspace) classifier. The proposed classifier with an accurately estimated dimension parameter generally outperforms its competitors in terms of classification accuracy. We also propose a fast version of the classifier using a neighborhood representation to reduce its computational complexity. Experiments on publicly available datasets corroborate these claims.
Intrinsic Dimensionality Estimation within Tight Localities: A Theoretical and Experimental Analysis
Cornell University - arXiv, 2022
Accurate estimation of Intrinsic Dimensionality (ID) is of crucial importance in many data mining and machine learning tasks, including dimensionality reduction, outlier detection, similarity search and subspace clustering. However, since their convergence generally requires sample sizes (that is, neighborhood sizes) on the order of hundreds of points, existing ID estimation methods may have only limited usefulness for applications in which the data consists of many natural groups of small size. In this paper, we propose a local ID estimation strategy stable even for 'tight' localities consisting of as few as 20 sample points. The estimator applies MLE techniques over all available pairwise distances among the members of the sample, based on a recent extreme-value-theoretic model of intrinsic dimensionality, the Local Intrinsic Dimension (LID). Our experimental results show that our proposed estimation technique can achieve notably smaller variance, while maintaining comparable levels of bias, at much smaller sample sizes than state-of-the-art estimators.
Optimized intrinsic dimension estimation using nearest neighbor graphs
We develop an approach to intrinsic dimension estimation based on k-nearest neighbor (kNN) distances. The dimension estimator is derived using a general theory on functionals of kNN density estimates. This enables us to predict the performance of the dimension estimation algorithm. In addition, it allows for optimization of free parameters in the algorithm. We validate our theory through simulations and compare our estimator to previous kNN based dimensionality estimation approaches.
Intrinsic Dimensionality Estimation within Tight Localities
Proceedings of the 2019 SIAM International Conference on Data Mining, 2019
Accurate estimation of Intrinsic Dimensionality (ID) is of crucial importance in many data mining and machine learning tasks, including dimensionality reduction, outlier detection, similarity search and subspace clustering. However, since their convergence generally requires sample sizes (that is, neighborhood sizes) on the order of hundreds of points, existing ID estimation methods may have only limited usefulness for applications in which the data consists of many natural groups of small size. In this paper, we propose a local ID estimation strategy stable even for 'tight' localities consisting of as few as 20 sample points. The estimator applies MLE techniques over all available pairwise distances among the members of the sample, based on a recent extreme-valuetheoretic model of intrinsic dimensionality, the Local Intrinsic Dimension (LID). Our experimental results show that our proposed estimation technique can achieve notably smaller variance, while maintaining comparable levels of bias, at much smaller sample sizes than state-of-the-art estimators.
Minimum Neighbor Distance Estimators of Intrinsic Dimension
Most of the machine learning techniques suffer the “curse of dimensionality” effect when applied to high dimensional data. To face this limitation, a common preprocessing step consists in employing a dimensionality reduction technique. In literature, a great deal of research work has been devoted to the development of algorithms performing this task. Often, these techniques require as parameter the number of dimensions to be retained; to this aim, they need to estimate the “intrinsic dimensionality” of the given dataset, which refers to the minimum number of degrees of freedom needed to capture all the information carried by the data. Although many estimation techniques have been proposed, most of them fail in case of noisy data or when the intrinsic dimensionality is too high. In this paper we present a family of estimators based on the probability density function of the normalized nearest neighbor distance. We evaluate the proposed techniques on both synthetic and real datasets comparing their performances with those obtained by state of the art algorithms; the achieved results prove that the proposed methods are promising.