arXiv subjects

Anton Kolleck

Publications and source records attributed to Anton Kolleck.

3 recordsLinked to original sources

arXiv2016

Carl's inequality for quasi-Banach spaces

We prove that for any two quasi-Banach spaces $X$ and $Y$ and any $α>0$ there exists a constant $γ_α>0$ such that $$ \sup_{1\le k\le n}k^αe_k(T)\le γ_α\sup_{1\le k\le n} k^αc_k(T) $$ holds for all linear and bounded operators $T:X\to Y$. Here $e_k(T)$ is the $k$-th entropy number of $T$ and $c_k(T)$ is the $k$-th Gelfand number of $T$. For Banach spaces $X$ and $Y$ this inequality is widely used and well-known as Carl's inequality. For general quasi-Banach spaces it is a new result.

math.FA↗

arXiv2015

Non-asymptotic Analysis of $\ell_1$-norm Support Vector Machines

Support Vector Machines (SVM) with $\ell_1$ penalty became a standard tool in analysis of highdimensional classification problems with sparsity constraints in many applications including bioinformatics and signal processing. Although SVM have been studied intensively in the literature, this paper has to our knowledge first non-asymptotic results on the performance of $\ell_1$-SVM in identification of sparse classifiers. We show that a $d$-dimensional $s$-sparse classification vector can be (with high probability) well approximated from only $O(s\log(d))$ Gaussian trials. The methods used in the proof include concentration of measure and probability in Banach spaces.

cs.IT↗

arXiv2014

On some aspects of approximation of ridge functions

We present effective algorithms for uniform approximation of multivariate functions satisfying some prescribed inner structure. We extend in several directions the analysis of recovery of ridge functions $f(x)=g(\langle a,x\rangle)$ as performed earlier by one of the authors and his coauthors. We consider ridge functions defined on the unit cube $[-1,1]^d$ as well as recovery of ridge functions defined on the unit ball from noisy measurements. We conclude with the study of functions of the type $f(x)=g(\|a-x\|_{l_2^d}^2)$.

math.FA↗