Intrinsic Dimension Estimation: Relevant Techniques and a Benchmark Framework

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-art methodologies. Unfortunately, since no benchmark database exists in this research field, an objective comparison among different techniques is not possible. Consequently, we suggest a benchmark framework and apply it to comparatively evaluate relevant state-of-the-art estimators

Retinal Vessel Segmentation using Tensor Voting

Medical imaging studies generate tremendous amounts of data that are reviewedmanually by physicians every day. Medical image segmentation aims to automate theprocess of extracting (segmenting) “interesting” structures from background structuresin the images, saving physicians time and opening the door to more sophisticatedanalysis such as automatically correlating studies over time. This work focuseson segmenting blood vessels (in particular the retinal vasculature), a task that requiresintegrating both local and global properties of the vasculature to produce goodquality segmentations. We use the Tensor Voting framework as it naturally groupsstructures together based on the consensus of locally voting segments. We investigateseveral ways of encoding the image data as tensors and compare our results quantitativelywith a publically available hand-labeled data set. We demonstrate competitiveperformance versus previously published techniques

Intrinsic Dimension Estimation: Relevant Techniques and a Benchmark Framework

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-art methodologies. Unfortunately, since no benchmark database exists in this research field, an objective comparison among different techniques is not possible. Consequently, we suggest a benchmark framework and apply it to comparatively evaluate relevant stateof-the-art estimators

Efficient Kernelization of Discriminative Dimensionality Reduction

Schulz A, Brinkrolf J, Hammer B. Efficient Kernelization of Discriminative Dimensionality Reduction. Neurocomputing. 2017;268(SI):34-41.Modern nonlinear dimensionality reduction (DR) techniques project high dimensional data to low dimensions for their visual inspection. Provided the intrinsic data dimensionality is larger than two, DR nec- essarily faces information loss and the problem becomes ill-posed. Dis- criminative dimensionality reduction (DiDi) offers one intuitive way to reduce this ambiguity: it allows a practitioner to identify what is relevant and what should be regarded as noise by means of intuitive auxiliary information such as class labels. One powerful DiDi method relies on a change of the data metric based on the Fisher information. This technique has been presented for vectorial data so far. The aim of this contribution is to extend the technique to more general data structures which are characterised in terms of pairwise similarities only by means of a kernelisation. We demonstrate that a computation of the Fisher metric is possible in kernel space, and that it can efficiently be integrated into modern DR technologies such as t-SNE or faster Barnes-Hut-SNE. We demonstrate the performance of the approach in a variety of benchmarks