This article gives theoretical insights into the performance of K-SVD, a dictionary learning algorithm that has gained significant popularity in practical applications. The particular question studied here is when a dictionary Φ ∈ Rd×K can be recovered as local minimum of the minimisation criterion underlying K-SVD from a set of N training signals yn = Φxn. A theoretical analysis of the problem leads to two types of identifiability results assuming the training signals are generated from a tight frame with coefficients drawn from a random symmetric distribution. First asymptotic results showing, that in expectation the generating dictionary can be recovered exactly as a local minimum of the K-SVD criterion if the coefficient distribution exhibits sufficient decay. This decay can be characterised by the coherence of the dictionary and the `1-norm of the coefficients. Based on the asymptotic results it is further demonstrated that given a finite number of training samples N, such that N / logN = O(K3d), except with probability O(N−Kd) there is a local minimum of the K-SVD criterion within distance O(KN−1/4) to the generating dictionary. Index Terms dictionary learning, sparse coding, K-SVD, finite sample size, sampling complexity, dictionary identification, minimisation criterion, sparse representation 1

generally, for any real arrangement of linear hyperplanes. At this level of generality, one finds that, after appropriate scaling, νO corresponds to a Markov chain on the chambers of the arrangement. We show that νO is self-adjoint and positive semidefinite, via two explicit factorizations into a

of the thesis can be summarized as follows. • We prove that for all Coxeter graphs constructed from an n-path of unlabelled edges by adding a new labelled edge and a new vertex (sometimes two new edges and vertices), there is a permutational representation of the corresponding group. Group elements correspond

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We study the SU(2) gluon and ghost propagators in Landau gauge on lattices up to a size of 1124. A comparison with the SU(3) case is made and finite-volume effects are then investigated. We find that for a large range of momenta the SU(2) and SU(3) propagators are remarkably alike. In the low

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Algebraic cryptanalysis of the round-reduced and side channel analysis of the full PRINTCipher-48

. We develop a new theoretical framework to analyze this extreme case and develop new algorithms for signal reconstruction from such coarsely quantized measurements. The 1-bit CS framework leads us to scenarios where it may be more appropriate to reduce bit-depth instead of sampling rate. We find

Abstract: We present a self-consistent calculation of the four-gluon vertex of Landau gauge Yang-Mills theory from a truncated Dyson-Schwinger equation. The equation contains the leading diagrams in the ultraviolet and is solved using as the only input results for lower Green functions from previous Dyson-Schwinger calculations that are in good agreement with lattice data. All quantities are therefore fixed and no higher Green functions enter within this truncation. Our self-consistent solution resolves the full momentum dependence of the vertex but is limited to the tree-level tensor structure at the moment. Calculations of selected dressing functions for other tensor structures from this solution are used to exemplify that they are suppressed compared to the tree-level structure except for possible logarithmic enhancements in the deep infrared. Our results furthermore allow to extract a qualitative fit for the vertex and a running coupling.