Universal consistency of twin support vector machines

A classification problem aims at constructing a best classifier with the smallest risk. When the sample size approaches infinity, the learning algorithms for a classification problem are characterized by an asymptotical property, i.e., universal consistency. It plays a crucial role in measuring the construction of classification rules. A universal consistent algorithm ensures that the larger the sample size of the algorithm is, the more accurately the distribution of the samples could be reconstructed. Support vector machines (SVMs) are regarded as one of the most important models in binary classification problems. How to effectively extend SVMs to twin support vector machines (TWSVMs) so as to improve performance of classification has gained increasing interest in many research areas recently. Many variants for TWSVMs have been proposed and used in practice. Thus in this paper, we focus on the universal consistency of TWSVMs in a binary classification setting. We first give a general framework for TWSVM classifiers that unifies most of the variants of TWSVMs for binary classification problems. Based on it, we then investigate the universal consistency of TWSVMs. To do this, we give some useful definitions of risk, Bayes risk and universal consistency for TWSVMs. Theoretical results indicate that universal consistency is valid for various TWSVM classifiers under some certain conditions, including covering number, localized covering number and stability. For applications of our general framework, several variants of TWSVMs are considered.

Universal consistency of the k-NN rule in metric spaces and Nagata dimension

The k nearest neighbour learning rule (under the uniform distance tie breaking) is universally consistent in every metric space X that is sigma-finite dimensional in the sense of Nagata. This was pointed out by Cérou and Guyader (2006) as a consequence of the main result by those authors, combined with a theorem in real analysis sketched by D. Preiss (1971) (and elaborated in detail by Assouad and Quentin de Gromard (2006)). We show that it is possible to give a direct proof along the same lines as the original theorem of Charles J. Stone (1977) about the universal consistency of the k-NN classifier in the finite dimensional Euclidean space. The generalization is non-trivial because of the distance ties being more prevalent in the non-Euclidean setting, and on the way we investigate the relevant geometric properties of the metrics and the limitations of the Stone argument, by constructing various examples.

Biodiversity clock and conservation triangle: Integrative platform for biodiversity monitoring, evaluation, and preemptive conservation intervention

Business-as-usual is no more an option on the table for biodiversity conservation. Disruptive transformation at both policy and polity levels are pressing needs. The possibilities presented by the current wave of information and communication technology can act as travelators to meet the conservation targets. Here, we introduce twin concepts of biodiversity clock and conservation triangle that posit as convergence plane to seamlessly consolidate ongoing discrete efforts and convey real-time biodiversity information in a lucid schematic form. In its present form, the biodiversity clock depicts 12 ecological and 6 biophysical components. The universal consistency in clock-reading facilitates the biodiversity clock to be read and interpreted identically across the world. A ternary plot of the International Union of Conservation of Nature (IUCN) species conservation status is presented as the conservation triangle. Together, the biodiversity clock and the conservation triangle are invaluable in strategizing biodiversity conservation, post-2020. Leveraged smartly, they make possible pre-emptive intervention for biodiversity conservation.