Related papers: 2 Notes on Classes with Vapnik-Chervonenkis Dimens…
The fundamental result of Li, Long, and Srinivasan on approximations of set systems has become a key tool across several communities such as learning theory, algorithms, computational geometry, combinatorics and data analysis. The goal of…
Given a set $E \subset {\Bbb F}_q^3$, where ${\Bbb F}_q$ is the field with $q$ elements. Consider a set of "classifiers" ${\mathcal H}^3_t(E)=\{h_y: y \in E\}$, where $h_y(x)=1$ if $x \cdot y=t$, $x \in E$, and $0$ otherwise. We are going…
This article deals with the generalization performance of margin multi-category classifiers, when minimal learnability hypotheses are made. In that context, the derivation of a guaranteed risk is based on the handling of capacity measures…
In 2008, Kasiviswanathan et al. defined private learning as a combination of PAC learning and differential privacy. Informally, a private learner is applied to a collection of labeled individual information and outputs a hypothesis while…
We study the Vapnik-Chervonenkis (VC) dimension of the set of quadratic residues (i.e. squares) in finite fields, $\mathbb F_q$, when considered as a subset of the additive group. We conjecture that as $q \to \infty$, the squares have the…
Multi-distribution learning is a natural generalization of PAC learning to settings with multiple data distributions. There remains a significant gap between the known upper and lower bounds for PAC-learnable classes. In particular, though…
We consider the fundamental question of learnability of a hypotheses class in the supervised learning setting and in the general learning setting introduced by Vladimir Vapnik. We survey classic results characterizing learnability in term…
We prove three results on the dimension structure of complexity classes. 1. The Point-to-Set Principle, which has recently been used to prove several new theorems in fractal geometry, has resource-bounded instances. These instances…
We compute that the index set of PAC-learnable concept classes is $m$-complete $\Sigma^0_3$ within the set of indices for all concept classes of a reasonable form. All concept classes considered are computable enumerations of computable…
The logical analysis of data, LAD, is a technique that yields two-class classifiers based on Boolean functions having disjunctive normal form (DNF) representation. Although LAD algorithms employ optimization techniques, the resulting binary…
We consider the problem of determining which classes of functions can be tested more efficiently than they can be learned, in the distribution-free sample-based model that corresponds to the standard PAC learning setting. Our main result…
Given a set $X$ and a collection ${\mathcal H}$ of functions from $X$ to $\{0,1\}$, the VC-dimension measures the complexity of the hypothesis class $\mathcal{H}$ in the context of PAC learning. In recent years, this has been connected to…
We study a model of machine teaching where the teacher mapping is constructed from a size function on both concepts and examples. The main question in machine teaching is the minimum number of examples needed for any concept, the so-called…
The study of strategic or adversarial manipulation of testing data to fool a classifier has attracted much recent attention. Most previous works have focused on two extreme situations where any testing data point either is completely…
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…
Feature selection involes identifying the most relevant subset of input features, with a view to improving generalization of predictive models by reducing overfitting. Directly searching for the most relevant combination of attributes is…
We investigate the VC-dimension of the perceptron and simple two-layer networks like the committee- and the parity-machine with weights restricted to values $\pm1$. For binary inputs, the VC-dimension is determined by atypical pattern sets,…
In this paper, we shall compare two metrics in terms of orderly dependence, a notion developed in exponential vector space in the article 'Basis and Dimension of Exponential Vector Space' by Jayeeta Saha and Sandip Jana in Transactions of…
In 1979 Valiant introduced the complexity class VNP of p-definable families of polynomials, he defined the reduction notion known as p-projection and he proved that the permanent polynomial and the Hamiltonian cycle polynomial are…
We study the generalization capabilities of Group Convolutional Neural Networks (GCNNs) with ReLU activation function by deriving upper and lower bounds for their Vapnik-Chervonenkis (VC) dimension. Specifically, we analyze how factors such…