Heuristic Framework for Multiscale Testing of the Multi-Manifold Hypothesis

Medina, F. Patricia; Ness, Linda; Weber, Melanie; Djima, Karamatou Yacoubou

doi:10.1007/978-3-030-11566-1_3

Cited by 4 publications

(3 citation statements)

References 40 publications

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“…Another example is LIDAR technology, which generates 3D point clouds in the form of the surrounding terrain (e.g., a bridge will result in a flat 2D surface for the road, 1D cables, etc.) (Medina et al, 2019).…”

Section: Multi-manifold and Manifold Alignment Learningmentioning

confidence: 99%

Lizard Brain: Tackling Locally Low-Dimensional Yet Globally Complex Organization of Multi-Dimensional Datasets

Bac

Zinovyev

2020

Front. Neurorobot.

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Machine learning deals with datasets characterized by high dimensionality. However, in many cases, the intrinsic dimensionality of the datasets is surprisingly low. For example, the dimensionality of a robot's perception space can be large and multi-modal but its variables can have more or less complex non-linear interdependencies. Thus multidimensional data point clouds can be effectively located in the vicinity of principal varieties possessing locally small dimensionality, but having a globally complicated organization which is sometimes difficult to represent with regular mathematical objects (such as manifolds). We review modern machine learning approaches for extracting low-dimensional geometries from multi-dimensional data and their applications in various scientific fields.

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Section: Multi-manifold and Manifold Alignment Learningmentioning

confidence: 99%

Lizard Brain: Tackling Locally Low-Dimensional Yet Globally Complex Organization of Multi-Dimensional Datasets

Bac

Zinovyev

2020

Front. Neurorobot.

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“…These distance measurements are transformed to measurements of actual three-dimensional points on the reflective target in object space. An introductory explanation of LiDAR technology is also available in [25]. There, a LiDAR image of the Golden Gate Bridge is decomposed into components with the same intrinsic dimension.…”

Section: Example: Pseudo-welding Curves For Lidar Data Counting Measuresmentioning

confidence: 99%

Product formalisms for measures on spaces with binary tree structures: representation, visualization, and multiscale noise

Bassu

Jones

Ness

et al. 2020

Forum of Mathematics, Sigma

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In this paper, we present a theoretical foundation for a representation of a data set as a measure in a very large hierarchically parametrized family of positive measures, whose parameters can be computed explicitly (rather than estimated by optimization), and illustrate its applicability to a wide range of data types. The preprocessing step then consists of representing data sets as simple measures. The theoretical foundation consists of a dyadic product formula representation lemma, and a visualization theorem. We also define an additive multiscale noise model that can be used to sample from dyadic measures and a more general multiplicative multiscale noise model that can be used to perturb continuous functions, Borel measures, and dyadic measures. The first two results are based on theorems in [15, 3, 1]. The representation uses the very simple concept of a dyadic tree and hence is widely applicable, easily understood, and easily computed. Since the data sample is represented as a measure, subsequent analysis can exploit statistical and measure theoretic concepts and theories. Because the representation uses the very simple concept of a dyadic tree defined on the universe of a data set, and the parameters are simply and explicitly computable and easily interpretable and visualizable, we hope that this approach will be broadly useful to mathematicians, statisticians, and computer scientists who are intrigued by or involved in data science, including its mathematical foundations.

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“…See the work in [8] and [7] for classification of a natural scene using support vector machines. We also refer the interested reading to [17] which multi-scale testing of a multi-manifold hypothesis where LiDAR data is used as a case study and intrinsic dimension is computed.…”

Section: Introductionmentioning

confidence: 99%

Machine Learning in LiDAR 3D point clouds

Medina¹,

Paffenroth²

2021

Preprint

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LiDAR point clouds contain measurements of complicated natural scenes and can be used to update digital elevation models, glacial monitoring, detecting faults and measuring uplift detecting, forest inventory, detect shoreline and beach volume changes, landslide risk analysis, habitat mapping and urban development, among others. A very important application is the classification of the 3D cloud into elementary classes. For example, it can be used to differentiate between vegetation, man-made structures and water. Our goal is to present a preliminary comparison study for classification of 3D point cloud LiDAR data that includes several types of feature engineering. In particular, we demonstrate that providing context by augmenting each point in the LiDAR point cloud with information about its neighboring points can improve the performance of downstream learning algorithms. We also experiment with several dimension reduction strategies, ranging from Principal Component Analysis (PCA) to neural network based auto-encoders, and demonstrate how they affect classification performance in LiDAR point clouds. For instance, we observe that combining feature engineering with a dimension reduction method such as PCA, there is an improvement in the accuracy of the classification with respect to doing a straightforward classification with the raw data.

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Heuristic Framework for Multiscale Testing of the Multi-Manifold Hypothesis

Cited by 4 publications

References 40 publications

Lizard Brain: Tackling Locally Low-Dimensional Yet Globally Complex Organization of Multi-Dimensional Datasets

Lizard Brain: Tackling Locally Low-Dimensional Yet Globally Complex Organization of Multi-Dimensional Datasets

Product formalisms for measures on spaces with binary tree structures: representation, visualization, and multiscale noise

Machine Learning in LiDAR 3D point clouds

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