Related papers: Coordinate recognition: General theory, Groups, an…

We give a general constructive proof for hierarchical coordinatizations (Lagrange Decompositions) of permutation groups. The generalization originates from the investigation of how the subgroup chains of finite permutation groups yield…

We investigate structural implications arising from the condition that a given directed graph does not interpret, in the sense of primitive positive interpretation with parameters or orbits, every finite structure. Our results generalize…

Few-shot recognition involves training an image classifier to distinguish novel concepts at test time using few examples (shot). Existing approaches generally assume that the shot number at test time is known in advance. This is not…

We develop a rigidity theory for bar-joint frameworks in Euclidean $d$-space in which specified classes of edges are allowed to change length in a coordinated fashion that requires differences of lengths to be preserved within each class.…

We report a series of robust empirical observations, demonstrating that deep Neural Networks learn the examples in both the training and test sets in a similar order. This phenomenon is observed in all the commonly used benchmarks we…

As the first part of the treatise on A General Theory of Concept Lattice (I-V), this work develops the general concept lattice for the problem concerning categorization of objects according to their properties. Unlike the conventional…

Let G be a group. Two elements x,y are said to be in the same z-class if their centralizers are conjugate in G. Let V be a vector space of dimension n over a field F of characteristic different from 2. Let B be a non-degenerate symmetric,…

One considers geometry with the intransitive equaivalence relation. Such a geometry is a physical geometry, i.e. it is described completely by the world function, which is a half of the squared distance function. The physical geometry…

Symmetries (transformations by group actions) are present in many datasets, and leveraging them holds considerable promise for improving predictions in machine learning. In this work, we aim to understand when and how deep networks -- with…

Recognition is the fundamental task of visual cognition, yet how to formalize the general recognition problem for computer vision remains an open issue. The problem is sometimes reduced to the simplest case of recognizing matching pairs,…

Graded rings provide a natural algebraic framework for encoding symmetry via decompositions into homogeneous components indexed by a group, together with multiplication rules reflecting the group operation. Among graded rings, strongly…

Suppose $R\rightarrow S$ is a faithfully flat ring map. The theory of twisted forms lets one compute, given an $R$-module $M$, how many isomorphism classes of $R$-modules $M^{\prime}$ satisfy $S\otimes_R M\cong S\otimes_R M^{\prime}$. This…

The sensitivity of image classifiers to small perturbations in the input is often viewed as a defect of their construction. We demonstrate that this sensitivity is a fundamental property of classifiers. For any arbitrary classifier over the…

Metric learning seeks to embed images of objects suchthat class-defined relations are captured by the embeddingspace. However, variability in images is not just due to different depicted object classes, but also depends on other latent…

We look at spaces of infinite-by-infinite matrices, and consider closed subsets that are stable under simultaneous row and column operations. We prove that up to symmetry, any of these closed subsets is defined by finitely many equations.

In standard classification, we typically treat class categories as independent of one-another. In many problems, however, we would be neglecting the natural relations that exist between categories, which are often dictated by an underlying…

This paper proposes a definition of recognizable transducers over monads and comonads, which bridges two important ongoing efforts in the current research on regularity. The first effort is the study of regular transductions, which extends…

The notion of a categorical quotient can be generalized since its standard categorical concept does not recover the expected quotients in certain categories. We present a more general formulation in the form of $\mathcal{F}$-quotients in a…

We establish discrete Korn type inequalities for particle systems within the general class of objective structures that represents a far reaching generalization of crystal lattice structures. For space filling configurations whose symmetry…

We present a generic construction of finite realisations of amalgamation patterns. An amalgamation pattern is specified by a finite collection of finite template structures together with a collection of partial isomorphisms between them. A…