2013/08/13 by Dmitry Batenkov, D. Batenkov, Omer Friedland +6 · 2 citations
Computer Science · Engineering · Mathematics · #65D15 #65T40 #Algebraic number #Bounded function #Classical Analysis and ODEs (math.CA) #Combinatorics #Curse of dimensionality #Discrete mathematics #Entropy (arrow of time) #FOS: Mathematics #Image and Signal Denoising Methods #Mathematical analysis #Mathematics #Numerical methods in inverse problems #Physics #Piecewise #Sparse and Compressive Sensing Techniques #Statistics #math.CA #msc:65D15 #msc:65T40
paper · pdf · doi:10.48550/arxiv.1308.2781
published in arXiv (Cornell University) (Cornell University)
arxiv created 2013/08/13 · openalex publication_date 2013/08/13 · arxiv updated 2013/08/14 · openalex created_date 2025/10/10 · openalex updated_date 2026/08/06
Let Q be a relatively compact subset in a Hilbert space V. For a given \e>0 let N(\e,Q) be the minimal number of linear measurements, sufficient to reconstruct any x ∈ Q with the accuracy \e. We call N(\e,Q) a sampling \e-entropy of Q. Using Dimensionality Reduction, as provided by the Johnson-Lindenstrauss lemma, we show that, in an appropriate probabilistic setting, N(\e,Q) is bounded from above by the Kolmogorov's \e-entropy H(\e,Q), defined as H(\e,Q)=log M(\e,Q), with M(\e,Q) being the minimal number of \e-balls covering Q. As the main application, we show that piecewise smooth (piecewise analytic) functions in one and several variables can be sampled with essentially the same accuracy rate as their regular counterparts. For univariate piecewise Ck-smooth functions this result, which settles the so-called Eckhoff conjecture, was recently established in \citeBat via a deterministic "algebraic reconstruction" algorithm.