Nearest neighbor (NN) classification assumes locally constant class conditional probabilities, and suffers from bias in high dimensions with a small sample set. In this paper, we propose a novel cam weighted distance to ameliorate the curse of dimensionality. Different from the existing neighborhood-based methods which only analyze a small space emanating from the query sample, the proposed nea...

We continue Mizar formalization of General Topology according to the book [13] by Engelking. In the article the formalization of Section 1.1 is completed. Namely, the paper includes the formalization of theorems on correspondence of the basis and basis in a point, definitions of the character of a point and a topological space, a neighborhood system, and the weight of a topological space. The f...

We introduce the Larger than Life family of two-dimensional two-state cellular automata that generalize certain nearest neighbor outer totalistic cellular automaton rules to large neighborhoods. We describe linear and quadratic rescalings of John Conway’s celebrated Game of Life to these large neighborhood cellular automaton rules and present corresponding generalizations of Life’s famous glide...

A theory of granular computing is presented. Some classical example are examined to obtain the hint of new directions Obvious examples are dynamic programming, fuzzy numbers, infinitesimal number and access control model, (pre)topological spaces are examined; Localized multi-level granulation can be modeled by generalized topological spaces, called neighborhood systems. For most general granula...

In this paper, we extend the general topological completeness result [3] of the strong hybrid language H(E) with the global modality E to the weaker hybrid language H(@) with the nominals i and the satisfaction operators @i (Theorem 1). Furthermore, we also generalize this topological completeness result of H(@) to monotonic neighborhood frames (Theorem 2).

We consider the non-orthodox proof rules of hybrid logic from the viewpoint of topological semantics. Topological semantics is more general than Kripke semantics. We show that the hybrid proof rule BG is topologically not sound. Indeed, among all topological spaces the BG rule characterizes those that can be represented as a Kripke frame (i.e., the Alexandroff spaces). We also demonstrate that,...

We report the theoretical discovery of a class of 2D tight-binding models containing nearly flatbands with nonzero Chern numbers. In contrast with previous studies, where nonlocal hoppings are usually required, the Hamiltonians of our models only require short-range hopping and have the potential to be realized in cold atomic gases. Because of the similarity with 2D continuum Landau levels, the...

Analysis of multidimensional data is challenging. Topological invariants can be used to summarize essential features of such data sets. In this work, we propose to compute the Betti numbers from a generative model based on a simplicial complex learnt from the data. The algorithm called a Generative Simplicial Complex can extract topological features from multidimensional datasets with non-trivi...

A k nearest neighbor (kNN) classifier classifies a query instance to the most frequent class of its k nearest neighbors in the training instance space. For imbalanced class distribution, a query instance is often overwhelmed by majority class instances in its neighborhood and likely to be classified to the majority class. We propose to identify exemplar minority class training instances and gen...