Gauge field theory and complex geometry

Motivated by Felix Klein's notion that geometry is governed by its group of symmetry transformations, Charles Ehresmann initiated the study of geometric structures on topological spaces locally modeled on a homogeneous space of a Lie group. These locally homogeneous spaces later formed the context of Thurston's 3-dimensional geometrization program. The basic problem is for a given topology S and a geometry X = G/H, to classify all the possible ways of introducing the local geometry of G/H into S. For example, a sphere admits no local Euclidean geometry: there is no metrically accurate Euclidean atlas of the earth. One develops a space whose points are equivalence classes of geometric structures on S, which itself exhibits a rich geometry and symmetries arising from the topological symmetries of S. In this talk I will survey several examples of the classification of locally homogeneous geometric structures on manifolds in low dimension, and how it leads to a general study of surface group representations. In particular geometric structures are a useful tool in understanding local and global properties of deformation spaces of representations of fundamental groups.

Locally homogeneous geometric manifolds

Mining and analysing streaming data is crucial for many applications, and this area of research has gained extensive attention over the past decade. However, there are several inherent problems that continue to challenge the hardware and the state-of-the art algorithmic solutions. Examples of such problems include the unbound size, varying speed and unknown data characteristics of arriving instances from a data stream. The aim of this research is to portray key challenges faced by algorithmic solutions for stream mining, particularly focusing on the prevalent issue of concept drift. A comprehensive discussion of concept drift and its inherent data challenges in the context of stream mining is presented, as is a critical, in-depth review of relevant literature. Current issues with the evaluative procedure for concept drift detectors is also explored, highlighting problems such as a lack of established base datasets and the impact of temporal dependence on concept drift detection. By exposing gaps in the current literature, this study suggests recommendations for future research which should aid in the progression of stream mining and concept drift detection algorithms.

/pdf/data-stream-mining-methods-and-challenges-for-handling-alczbzrdkd.pdf

Data stream mining: methods and challenges for handling concept drift

Practical applications involving big data, such as weather monitoring, identification of customer preferences, Internet log analysis, and sensors warnings require challenging data analysis, since these are examples of problems whose data are generated in streams and usually demand real-time analytics. Patterns in such data stream problems may change quickly. Consequently, machine learning models that operate in this context must be updated over time. This phenomenon is called concept drift in machine learning and data mining literature. Several different directions have been pursued to learn from data stream and to deal with concept drift. However, most drift detection methods consider that an instance's class label is available right after its prediction, since these methods work by monitoring the prediction results of a base classifier or an ensemble of classifiers. Nevertheless, this constraint is unrealistic in several practical problems. To cope with this constraint, some works are focused on proposing efficient unsupervised or semi-supervised concept drift detectors. While interesting and recent overview papers dedicated to supervised drift detectors have been published, the scenario is not the same in terms of unsupervised methods. Therefore, this work presents a comprehensive overview of approaches that tackle concept drift in classification problems in an unsupervised manner. Additional contribution includes a proposed taxonomy of state-of-the-art approaches for concept drift detection based on unsupervised strategies. This article is categorized under: Technologies > Classification Technologies > Machine Learning. © 2020 The Authors. WIREs Data Mining and Knowledge Discovery published by Wiley Periodicals LLC.

/pdf/an-overview-of-unsupervised-drift-detection-methods-36nvlb8r4c.pdf

An overview of unsupervised drift detection methods

Data stream mining has become an important research area over the past decade due to the increasing amount of data available today. Sources from various domains generate a near-limitless volume of data in temporal order. Such data are referred to as data streams, and are generally nonstationary as the characteristics of data evolves over time. This phenomenon is called concept drift, and is an issue of great importance in the literature, since it makes models obsolete by decreasing their predictive performance. In the presence of concept drift, it is necessary to adapt to change in data to build more robust and effective classifiers. Drift detectors are designed to run jointly with classification models, updating them when a significant change in data distribution is observed. In this paper, we present an implicit (unsupervised) algorithm called One-Class Drift Detector (OCDD), which uses a one-class learner with a sliding window to detect concept drift. We perform a comprehensive evaluation on mostly recent 17 prevalent concept drift detection methods and an adaptive classifier using 13 datasets. The results show that OCDD outperforms the other methods by producing models with better predictive performance on both real-world and synthetic datasets.

Concept learning using one-class classifiers for implicit drift detection in evolving data streams

Motivated by the Statistical Learning Theory (SLT), which provides a theoretical framework to ensure when supervised learning algorithms generalize input data, this manuscript relies on the Algorithmic Stability framework to prove learning bounds for the unsupervised concept drift detection on data streams. Based on such proof, we also designed the Plover algorithm to detect drifts using different measure functions, such as Statistical Moments and the Power Spectrum. In this way, the criterion for issuing data changes can also be adapted to better address the target task. From synthetic and real-world scenarios, we observed that each data stream may require a different measure function to identify concept drifts, according to the underlying characteristics of the corresponding application domain. In addition, we discussed about the differences of our approach against others from literature, and showed illustrative results confirming the usefulness of our proposal.

On learning guarantees to unsupervised concept drift detection on data streams

This paper is devoted to a coordinate-free approach to several classic geometries such as hyperbolic (real, complex, quaternionic), elliptic (spherical, Fubini-Study), and lorentzian (de Sitter, anti de Sitter) ones. These geometries carry a certain simple structure that is in some sense stronger than the riemannian structure. Their basic geometrical objects have linear nature and provide natural compactifications of classic spaces. The usual riemannian concepts are easily derivable from the strong structure and thus gain their coordinate-free form. Many examples illustrate fruitful features of the approach. The framework introduced here has already been shown to be adequate for solving problems concerning particular classic spaces.

Coordinate-Free Classic Geometries

We study complex hyperbolic disc bundles over closed orientable surfaces that arise from discrete and faithful representations H_n->PU(2,1), where H_n is the fundamental group of the orbifold S^2(2,...,2) and thus contains a surface group as a subgroup of index 2 or 4. The results obtained provide the first complex hyperbolic disc bundles M->{\Sigma} that: admit both real and complex hyperbolic structures; satisfy the equality 2(\chi+e)=3\tau; satisfy the inequality \chi/2 PU(2,1) with fractional Toledo invariant; where {\chi} is the Euler characteristic of \Sigma, e denotes the Euler number of M, and {\tau} stands for the Toledo invariant of M. To get a satisfactory explanation of the equality 2(\chi+e)=3\tau, we conjecture that there exists a holomorphic section in all our examples. 
In order to reduce the amount of calculations, we systematically explore coordinate-free methods.

Complex Hyperbolic Structures on Disc Bundles over Surfaces

We study oriented disc bundles M over a closed orientable surface $\Sigma$
that arise from certain discrete subgroups in PU(2,1) generated by reflections
in ultraparallel complex geodesics in the complex hyperbolic plane HC^2. The
results obtained allow us to construct the first examples of: Disc bundles M
over $\Sigma$ that satisfy the equality $2(\chi+e)=3\tau$; Disc bundles M over
$\Sigma$ that satisfy the inequality $\chi/2<e$; Disc bundles M over $\Sigma$
that admit both real hyperbolic and complex hyperbolic structures; Discrete and
faithful representations $\varrho:\pi_1\Sigma\to\PU(2,1)$ with fractional
Toledo invariant; and Nonhomeomorphic disc bundles M over the same $\Sigma$ and
with the same $\tau$, where $\chi$ stands for the Euler characteristic
$\chi(\Sigma)$ of $\Sigma$, e, for the Euler number e(M) of M, and $\tau$, for
the Toledo invariant of M. To get a satisfactory explanation of the equality
$2(\chi+e)=3\tau$, we conjecture that there exists a holomorphic section in all
our examples. Constructing examples is based on a new version of Poincar'e's Polyhedron
Theorem where requirements concerning the tessellation have a form which is as
local as possible. This version can be easily adapted to be applied in subtle
situations lacking the concept of convexity. A more important feature of the examples is that, in [Ana], we will
distinguish some examples having the same e, $\chi$, and $\tau$ with new
discrete invariants, which will lead to a detailed knowledge about the
corresponding Teichmuller space. In [AGu], we apply the introduced methods to construct a trivial bundle
carrying complex hyperbolic structure.

Carlos H. Grossi

Papers

On learning guarantees to unsupervised concept drift detection on data streams

Coordinate-Free Classic Geometries

Complex Hyperbolic Structures on Disc Bundles over Surfaces

Complex Hyperbolic Structures on Disc Bundles over Surfaces

Complex Hyperbolic Structures on Disc Bundles over Surfaces I. General Settings. A Series of Examples