Narrow Results

Domain adaptation is an important subfield of transfer learning, it has been successfully applied in many applications of machine learning. Recently, significant theoretical and algorithmic advances have been achieved in domain adaptation. The theoretical analyses for domain adaptation are based on VC dimension and Rademacher complexity. There are also some covering number-based results, but most of these bounds are based on the results of Rademacher complexity, indirectly given by the relationship between covering number and Rademacher complexity. In this paper, we propose a theoretical analysis framework for domain adaptation, thus the error bound can be derived directly by covering number, which is an effective method for analyzing the generalization error in statistical learning theory. We derive generalization error bound for domain adaptation with a class of loss functions satisfying the assumptions. We also propose a mixup contrastive adversarial network for domain adaptation by introducing a mixup module for enhancing the alignment of the source and target domains during domain transfer, and a contrastive learning module for solving class-level alignment after domain transfer. Experimental results demonstrate the effectiveness of the proposed algorithm and the property of the theoretical results.

We investigate the Ramsey theory of continuous graph-structures on complete, separable metric spaces and apply the results to the problem of covering a plane by functions.

Let the homogeneity number of a pair-coloring c:[X]²→2 be the number of c-homogeneous subsets of X needed to cover X. We isolate two continuous pair-colorings on the Cantor space 2^ω, c_min and c_max, which satisfy and prove:

Theorem. (1) For every Polish space X and every continuous pair-coloringc:[X]²→2with,

(2) There is a model of set theory in whichand.

The consistency of and of follows from [20].

We prove that is equal to the covering number of (2^ω)² by graphs of Lipschitz functions and their reflections on the diagonal. An iteration of an optimal forcing notion associated to c_min gives:

Theorem.There is a model of set theory in which

(1) ℝ² is coverable byℵ₁graphs and reflections of graphs of continuous real functions;

(2) ℝ² is not coverable byℵ₁graphs and reflections of graphs of Lipschitz real functions.

Figure 1.1 in the introduction summarizes the ZFC results in Part I of the paper. The independence results in Part II show that any two rows in Fig. 1.1 can be separated if one excludes from row (3).

A group partition is a group cover in which the components have trivial pairwise intersection. Based on an idea of I. M. Isaacs in this paper we define the semi-partition of a group as a group cover in which the intersection of any three components is trivial. We examine some of the properties of groups with semi-partitions, including its relation to the covering number and partition number for groups.

The ranking problem aims at learning real-valued functions to order instances, which has attracted great interest in statistical learning theory. In this paper, we consider the regularized least squares ranking algorithm within the framework of reproducing kernel Hilbert space. In particular, we focus on analysis of the generalization error for this ranking algorithm, and improve the existing learning rates by virtue of an error decomposition technique from regression and Hoeffding’s decomposition for U-statistics.

Shift-invariant spaces play an important role in approximation theory, wavelet analysis, finite elements, etc. In this paper, we consider the stability and reconstruction algorithm of random sampling in multiply generated shift-invariant spaces $V^p(\mathrm{\Phi })$. Under some decay conditions of the generator $\mathrm{\Phi }$, we approximate $V^p(\mathrm{\Phi })$ with finite-dimensional subspaces and prove that with overwhelming probability, the stability of sampling set conditions holds uniformly for all functions in certain compact subsets of $V^p(\mathrm{\Phi })$ when the sampling size is sufficiently large. Moreover, we show that this stability problem is connected with properties of the random matrix generated by $\mathrm{\Phi }$. In the end, we give a reconstruction algorithm for the random sampling of functions in $V^p(\mathrm{\Phi })$.

A new multi-kernel regression learning algorithm is studied in this paper. In our setting, the hypothesis space is generated by two Mercer kernels, thus it has stronger approximation ability than the single kernel case. We provide the mathematical foundation for this regularized learning algorithm. We obtain satisfying capacity-dependent error bounds and learning rates by the covering number method.

In 2007, Zhi-Wei Sun defined a covering number to be a positive integer $L$ such that there exists a covering system of the integers where the moduli are distinct divisors of $L$ greater than 1. A covering number $L$ is called primitive if no proper divisor of $L$ is a covering number. Sun constructed an infinite set ${ℒ}$ of primitive covering numbers, and he conjectured that every primitive covering number must satisfy a certain condition. In this paper, for a given $L\in {ℒ}$, we derive a formula that gives the exact number of coverings that have $L$ as the least common multiple of the set $M$ of moduli, under certain restrictions on $M$. Additionally, we disprove Sun’s conjecture by constructing an infinite set of primitive covering numbers that do not satisfy his primitive covering number condition.