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

Abstract

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.

Robust and Scalable Learning of Complex Intrinsic Dataset Geometry via ElPiGraph

“…In this particular case, applying non-linear dimensionality reduction (e.g., tSNE) could make the ‘hidden’ branch more visible in 2D at the cost of distorting the underlying data geometry. Nevertheless, this situation can be reproduced with any data dimensionality technique (for examples, see [ 56 ]).…”

“…In order to compare these algorithms with ElPiGraph, we used previously published LizardBrain generator of noisy branching data point clouds [ 56 ]. Briefly, it generates data points in a unit m -dimensional hypercube around a set of non-linear (e.g., parabolic) branches, such that each next branch starts from a randomly selected point on one of the previously generated branches in a random direction.…”

Fractional Norms and Quasinorms Do Not Help to Overcome the Curse of Dimensionality

“…All these methods use an object embedded in the data space. They are called Injective Methods [ 68 ]. In addition, a family of Projective Methods was developed.…”

“…These methods do not construct a data approximator, but project the dataspace onto a space of lower dimension with preservation of similarity or dissimilarity of objects. A brief review of modern injective and projective methods can be found in [ 68 ].…”

“…There are many approaches to non-linear evaluation of data dimensionality with various non-linear data approximants: manifolds, graphs or cell complexes [ 63 , 65 , 66 , 68 ]. The technology of neural network autoencoders is also efficient and very popular but its theoretical background is still under discussion [ 71 ].…”