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Convergence radius and sample complexity of ITKM algorithms for dictionary learning (1503.07027v4)
Published 24 Mar 2015 in cs.LG, cs.IT, and math.IT
Abstract: In this work we show that iterative thresholding and K-means (ITKM) algorithms can recover a generating dictionary with K atoms from noisy $S$ sparse signals up to an error $\tilde \varepsilon$ as long as the initialisation is within a convergence radius, that is up to a $\log K$ factor inversely proportional to the dynamic range of the signals, and the sample size is proportional to $K \log K \tilde \varepsilon{-2}$. The results are valid for arbitrary target errors if the sparsity level is of the order of the square root of the signal dimension $d$ and for target errors down to $K{-\ell}$ if $S$ scales as $S \leq d/(\ell \log K)$.