A Note on the Richness of Convex Hulls of Vc Classes
نویسندگان
چکیده
We prove the existence of a class A of subsets of R of vc dimension 1 such that the symmetric convex hull F of the class of characteristic functions of sets in A is rich in the following sense. For any absolutely continuous probability measure μ on R, measurable set B ⊂ R and > 0, there exists a function f ∈ F such that the measure of the symmetric difference of B and the set where f is positive is less than . The question was motivated by the investigation of the theoretical properties of certain algorithms in machine learning. Let A be a class of sets in R and define the symmetric convex hull of A as the class of functions
منابع مشابه
Entropy Numbers of Linear Function Classes
This paper collects together a miscellany of results originally motivated by the analysis of the generalization performance of the “maximum-margin” algorithm due to Vapnik and others. The key feature of the paper is its operator-theoretic viewpoint. New bounds on covering numbers for classes related to Maximum Margin classes are derived directly without making use of a combinatorial dimension s...
متن کاملSome results on the polynomial numerical hulls of matrices
In this note we characterize polynomial numerical hulls of matrices $A in M_n$ such that$A^2$ is Hermitian. Also, we consider normal matrices $A in M_n$ whose $k^{th}$ power are semidefinite. For such matriceswe show that $V^k(A)=sigma(A)$.
متن کاملInvestigating Species Diversity of Family Brassicaceae along Khorramabad-Andimeshk Highway
To achieve sustainable development as well as the conservation of natural ecosystems and their biodiversity, it is necessary to study ecological foctors and their impacts on plant species diversity. The Khorramabad-Andimeshk highway was selected in the current study to investigate the diversity of the family Brassicaceae. The length of this highway is 100 km and its height ranged from 420 to 17...
متن کاملOn Noetherian Classes
Let I (α) ≤ a. It was Taylor–Cavalieri who first asked whether hulls can be computed. We show that |s|−3 = 0−1. Is it possible to construct hulls? U. Newton [11] improved upon the results of V. Wang by extending hyper-nonnegative domains.
متن کاملOn the Local Structure of Rank-one Convex Hulls
In this note we prove that if K is a compact set of m×n matrices containing an isolated point X with no rank-one connection into the convex hull of K \ {X}, then the rank-one convex hull separates as K = ( K \ {X} )rc ∪ {X}. This is an extension of a result of P. Pedregal, which holds for 2× 2 matrices.
متن کاملذخیره در منابع من
با ذخیره ی این منبع در منابع من، دسترسی به آن را برای استفاده های بعدی آسان تر کنید
عنوان ژورنال:
دوره شماره
صفحات -
تاریخ انتشار 2003