Intrinsic Dimension Estimation Using Packing Numbers

By Balázs Kégl. 2002

281 citations recorded by the source corpus.

Subfields: Representation & Self-Supervised Learning

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In Representation & Self-Supervised Learning

Citations in the corpus

Measuring the Intrinsic Dimension of Objective Landscapes (2018, methodology): “Finally, note that although our approach is related to a rich body of work on estimating the “intrinsic dimension of a dataset” (Camastra & Vinciarelli, 2002; Kégl, 2003; Fukunaga & Olsen, 1971; Levina & Bickel, 2005; Tenenbaum et al., 2000), it differs in a few respects.”
Topological Data Analysis (2016, background): “Some examples include Little et al. (2011); Lombardi et al. (2011); Hein & Audibert (2005); Levina & Bickel (2004); Kégl (2002); Costa & Hero (2004).”
Data dimensionality estimation methods: a survey (2003, methodology): “Although efficient algorithms [49],[50], [51] have been proposed, the BoxCounting dimension can be computed only for low-dimensional sets because the algorithmic complexity grows exponentially with the set dimensionality.”
Nearest-Neighbor Searching and Metric Space Dimensions (2005, methodology): “Kégl[K0́3] proposes using an upper bound on the packing number and a simple form of line-fitting as an estimate of the box dimension, although not using the greedy -net algorithm discussed in Subsection 4.”
Maximum Likelihood Estimation of Intrinsic Dimension (2004, background): “…of sample size and dimension is not well understood or studied beyond the obvious “curse of dimensionality”; the statistical properties of the estimators, such as bias and variance, have not been looked at (with the exception of [15]); and comparisons between methods are not always presented.”
Incremental nonlinear dimensionality reduction by manifold learning (2006, background): “This can be estimated by analyzing either the spectrum of the target inner product matrix or the residue of the low rank approximation as in [46], or by other methods to estimate the intrinsic dimensionality of a manifold [9], [10], [12], [27], [31], [39].”
Intrinsic dimension estimation: Advances and open problems (2016, methodology): “the multiscaling is motivated by ID dependence on the data scale [51,72,100].”
Geodesic entropic graphs for dimension and entropy estimation in manifold learning (2004, methodology): “Other methods have been proposed based on local geometric techniques, e.g., estimation of local neighborhoods [6] or fractal dimension [17] and estimating packing numbers [5] of the manifold.”
Convergence of Laplacian Eigenmaps (2006, methodology): “Some of the recent work includes estimating geometric invariants of the manifold, such as homology [35, 27], dimensionality [24, 20], geodesic distances [7], and comparing point clouds using Gromov-Hausdorff distance [22].”
Principal manifolds and nonlinear dimensionality reduction via tangent space alignment (2004, methodology): “More sophisticated algorithms have been proposed: Kegl considered an approach using packing numbers [12], and Costa and Hero considered asymptotically consistent estimates of d using geodesic minimal spanning trees [6].”