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1On the Identifiability of Overcomplete Dictionaries via the Minimisation Principle Underlying KSVD
, 2013
"... This article gives theoretical insights into the performance of KSVD, 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 ..."
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This article gives theoretical insights into the performance of KSVD, 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 KSVD 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 KSVD criterion if the coefficient distribution exhibits sufficient decay. This decay can be characterised by the coherence of the dictionary and the `1norm 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 KSVD criterion within distance O(KN−1/4) to the generating dictionary. Index Terms dictionary learning, sparse coding, KSVD, finite sample size, sampling complexity, dictionary identification, minimisation criterion, sparse representation 1
Spectra of symmetrized shuffling operators
"... Abstract. For a finite real reflection group W and a Worbit O of flats in its reflection arrangement – or equivalently a conjugacy class of its parabolic subgroups – we introduce a statistic noninvO(w) on w in W that counts the number of “Ononinversions ” of w. This generalizes the classical (non ..."
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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 selfadjoint and positive semidefinite, via two explicit factorizations into a
Akademisk avhandling för teknisk doktorsexamen vid
, 1994
"... mcmxciv This thesis deals with combinatorics in connection with Coxeter groups, finitely generated but not necessarily finite. The representation theory of groups as nonsingular matrices over a field is of immense theoretical importance, but also basic for computational group theory, where the group ..."
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of the thesis can be summarized as follows. • We prove that for all Coxeter graphs constructed from an npath 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
P o
, 2007
"... 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 finitevolume 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 lowmoment ..."
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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 finitevolume 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 roundreduced and side channel analysis of the full PRINTCipher48 ..."
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Algebraic cryptanalysis of the roundreduced and side channel analysis of the full PRINTCipher48
RICE UNIVERSITY Regime Change: Sampling Rate vs. BitDepth in Compressive Sensing
, 2011
"... The compressive sensing (CS) framework aims to ease the burden on analogtodigital converters (ADCs) by exploiting inherent structure in natural and manmade signals. It has been demonstrated that structured signals can be acquired with just a small number of linear measurements, on the order of t ..."
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. We develop a new theoretical framework to analyze this extreme case and develop new algorithms for signal reconstruction from such coarsely quantized measurements. The 1bit CS framework leads us to scenarios where it may be more appropriate to reduce bitdepth instead of sampling rate. We find
A DysonSchwinger study of the fourgluon vertex
"... Abstract: We present a selfconsistent calculation of the fourgluon vertex of Landau gauge YangMills theory from a truncated DysonSchwinger 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 ..."
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Abstract: We present a selfconsistent calculation of the fourgluon vertex of Landau gauge YangMills theory from a truncated DysonSchwinger 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 DysonSchwinger calculations that are in good agreement with lattice data. All quantities are therefore fixed and no higher Green functions enter within this truncation. Our selfconsistent solution resolves the full momentum dependence of the vertex but is limited to the treelevel 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 treelevel 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.
Results 1  10
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