Related papers: Bounds for the VC Dimension of 1NN Prototype Sets

In statistical setting of the pattern recognition problem the number of examples required to approximate an unknown labelling function is linear in the VC dimension of the target learning class. In this work we consider the question whether…

In this paper we construct a model for the classifying space, BVCG, of a crystallographic group G of rank n relative to the family VC of virtually-cyclic subgroups of G. The model is used to show that there exists no other model for the…

This work studies the maximum possible sign rank of $N \times N$ sign matrices with a given VC dimension $d$. For $d=1$, this maximum is {three}. For $d=2$, this maximum is $\tilde{\Theta}(N^{1/2})$. For $d >2$, similar but slightly less…

We discuss the VC-dimension of a class of multiples of integers and primes (equivalently indicator functions) and demonstrate connections to prime counting functions. Additionally, we prove limit theorems for the behavior of an empirical…

We recast the problem of calculating Vapnik-Chervonenkis (VC) density into one of counting types, and thereby calculate bounds (often optimal) on the VC density for some weakly o-minimal, weakly quasi-o-minimal, and $P$-minimal theories.

We investigate the Probably Approximately Correct (PAC) property of scenario decision algorithms, which refers to their ability to produce decisions with an arbitrarily low risk of violating unknown safety constraints, provided a sufficient…

There is an increasing body of evidence suggesting that exact nearest neighbour search in high-dimensional spaces is affected by the curse of dimensionality at a fundamental level. Does it necessarily mean that the same is true for k…

Explaining the unreasonable effectiveness of deep learning has eluded researchers around the globe. Various authors have described multiple metrics to evaluate the capacity of deep architectures. In this paper, we allude to the radius…

We refine upper bounds for the classical exponents of uniform approximation for a linear form on the Veronese curve in dimension from $3$ to $9$. For dimension three, this in particular shows that a bound previously obtained by two…

This paper presents a new similarity measure to be used for general tasks including supervised learning, which is represented by the K-nearest neighbor classifier (KNN). The proposed similarity measure is invariant to large differences in…

We introduce a new nearest-prototype classifier, the prototype vector machine (PVM). It arises from a combinatorial optimization problem which we cast as a variant of the set cover problem. We propose two algorithms for approximating its…

We propose a new sufficient dimension reduction approach designed deliberately for high-dimensional classification. This novel method is named maximal mean variance (MMV), inspired by the mean variance index first proposed by Cui, Li and…

This paper focuses on the relation between computational learning theory and resource-bounded dimension. We intend to establish close connections between the learnability/nonlearnability of a concept class and its corresponding size in…

Complex-valued neural networks (CVNNs) have been widely applied to various fields, especially signal processing and image recognition. However, few works focus on the generalization of CVNNs, albeit it is vital to ensure the performance of…

The design, analysis and application of a volumetric convolutional neural network (VCNN) are studied in this work. Although many CNNs have been proposed in the literature, their design is empirical. In the design of the VCNN, we propose a…

We study consistency properties of surrogate loss functions for general multiclass learning problems, defined by a general multiclass loss matrix. We extend the notion of classification calibration, which has been studied for binary and…

A critical set in an $n \times n$ Latin square is a minimal set of entries that uniquely identifies it among all Latin squares of the same size. It is conjectured by Nelder in 1979, and later independently by Mahmoodian, and Bate and van…

A hypothesis class admits a sample compression scheme, if for every sample labeled by a hypothesis from the class, it is possible to retain only a small subsample, using which the labels on the entire sample can be inferred. The size of the…

We study the problems of testing and learning high-dimensional discrete convex sets. The simplest high-dimensional discrete domain where convexity is a non-trivial property is the ternary hypercube, $\{-1,0,1\}^n$. The goal of this work is…

Consider the problem of learning a large number of response functions simultaneously based on the same input variables. The training data consist of a single independent random sample of the input variables drawn from a common distribution…