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Complexity of pattern classes and the Lipschitz property

Amiran Ambroladze, Emilio Parrado-Hernández, John Shawe-Taylor
2007 Theoretical Computer Science  
One of the most important properties for these complexities is their Lipschitz property: a composition of a class of functions with a fixed Lipschitz function may increase its complexity by at most twice  ...  In this paper we give a detailed proof of the Lipschitz property for the general case of a symmetric complexity measure that includes the Rademacher and Gaussian complexities as special cases.  ...  Acknowledgements This work was partially supported by the IST Programme of the European Community under the PASCAL Network of Excellence IST2002-506788 and by the UK Government under the EPSRC project  ... 
doi:10.1016/j.tcs.2007.03.047 fatcat:2oyjkl4yl5glhfxqljhwudiugu

Complexity of Pattern Classes and Lipschitz Property [chapter]

Amiran Ambroladze, John Shawe-Taylor
2004 Lecture Notes in Computer Science  
One of the most important properties for these complexities is their Lipschitz property: a composition of a class of functions with a fixed Lipschitz function may increase its complexity by at most twice  ...  Rademacher and Gaussian complexities are successfully used in learning theory for measuring the capacity of the class of functions to be learned.  ...  It bounds the error of pattern functions in terms of their empirical fit and the Rademacher complexity of the class.  ... 
doi:10.1007/978-3-540-30215-5_15 fatcat:llsuymmzevf5lkp2z5um4ol6ge

Mappings and Spaces, 2

Stephen W. Semmes
2011 Pure and Applied Mathematics Quarterly  
This paper is concerned with analysis on metric spaces in a variety of settings and with several kinds of structure.  ...  If f is a complex-valued C-Lipschitz function of order α on M and a is a complex number, then a f is (|a| C)-Lipschitz function of order α.  ...  The Dixmier trace is an asymptotic version of the trace that applies to slightly more than the usual trace class operators and vanishes on trace class operators.  ... 
doi:10.4310/pamq.2011.v7.n2.a9 fatcat:s3lnuf2wwveypiit5eod4yccjq

On the identification of monotone Wiener systems

Kristiaan Pelckmans
2010 49th IEEE Conference on Decision and Control (CDC)  
This paper studies the MINLIP estimator for the identification of Wiener systems consisting of a sequence of a linear FIR dynamical model, and a monotonically increasing static function.  ...  complexity control.  ...  There should be a sample z where the Lipschitz constant L 0 is attained, and in the -neighborhood of this sample z the Lipschitz-property shouldn't decay too fast, e.g. in the neighborhood of z the function  ... 
doi:10.1109/cdc.2010.5717902 dblp:conf/cdc/Pelckmans10 fatcat:iii3rpewxnexrcndqm4d2jf3fi

Random Matrix Theory Proves that Deep Learning Representations of GAN-data Behave as Gaussian Mixtures [article]

Mohamed El Amine Seddik, Cosme Louart, Mohamed Tamaazousti, Romain Couillet
2020 arXiv   pre-print
This paper shows that deep learning (DL) representations of data produced by generative adversarial nets (GANs) are random vectors which fall within the class of so-called concentrated random vectors.  ...  Further exploiting the fact that Gram matrices, of the type G = X^T X with X=[x_1,...  ...  fall within this class of random vectors, since the mapping x → φ(x) is Lipschitz.  ... 
arXiv:2001.08370v1 fatcat:wgcref2f5fc3bphorc4mv3zs7m

Deterministic approaches for solving practical black-box global optimization problems

Dmitri E. Kvasov, Yaroslav D. Sergeyev
2015 Advances in Engineering Software  
Their efficiency is shown on solving both the classes of random test problems and some practical engineering tasks.  ...  In many important design problems, some decisions should be made by finding the global optimum of a multiextremal objective function subject to a set of constrains.  ...  Acknowledgements The authors work was supported by the project 14.B37.21.0878 of the Ministry of Education and Science of the Russian Federation and by the grant 1960.2012.9 awarded by the President of  ... 
doi:10.1016/j.advengsoft.2014.09.014 fatcat:na3ehlnorzaydmank42ndcpek4

Degree of Sequentiality of Weighted Automata [chapter]

Laure Daviaud, Ismaël Jecker, Pierre-Alain Reynier, Didier Villevalois
2017 Lecture Notes in Computer Science  
property, a pattern on the automaton (a new twinning property) and a subclass of cost register automata.  ...  We call this minimum the degree of sequentiality of (the relation realized by) the WA.  ...  property for transducers, and is called Lipschitz property of order k (Lip k for short); second, a pattern property of transducers, called branching twinning property of order k (BTP k for short).  ... 
doi:10.1007/978-3-662-54458-7_13 fatcat:io62uuujlbgcpirftv3zhybdfa

What training reveals about neural network complexity [article]

Andreas Loukas, Marinos Poiitis, Stefanie Jegelka
2021 arXiv   pre-print
We first observe that the Lipschitz constant close to the training data affects various aspects of the parameter trajectory, with more complex networks having a longer trajectory, bigger variance, and  ...  Our analysis provides evidence for BTH by relating the NN's Lipschitz constant at different regions of the input space with the behavior of the stochastic training procedure.  ...  Acknowledgments and Disclosure of Funding We are thankful to the anonymous reviewers, Giorgos Bouritsas, Martin Jaggi, Nikos Karalias, Mattia Atzeni, and Jean-Baptiste Cordonnier for engaging in fruitful  ... 
arXiv:2106.04186v2 fatcat:hzyj5uepenguljzxmyig55ar2q

Mappings and spaces, 2 [article]

Stephen Semmes
2006 arXiv   pre-print
This paper is concerned with analysis on metric spaces in a variety of settings and with several kinds of structure.  ...  C-Lipschitz function of order α on M and a is a complex number, then a f is (|a| C)-Lipschitz function of order α. If f 1 , f 2 are also bounded, then the product f 1 f 2 is Lipschitz of order α too.  ...  The composition of a Lipschitz mapping of order α and a Lipschitz mapping of order β is Lipschitz of order α β. Lipschitz mappings of any order are uniformly continuous.  ... 
arXiv:math/0612265v2 fatcat:ah2jlnkt7bcczfcnuskmfkmlz4

Worst case complexity of direct search

L. N. Vicente
2012 EURO Journal on Computational Optimization  
In this paper we prove that the broad class of direct-search methods of directional type based on imposing sufficient decrease to accept new iterates shares the worst case complexity bound of steepest  ...  of the objective function, and are proved for such directional direct-search methods when a sufficient decrease condition based on the size of the steps is imposed to accept new iterates.  ...  Acknowledgments The author would like to thank Michael L. Overton and Mert Gurbuzbalaban (New York University) for interesting discussions on the topic of the paper.  ... 
doi:10.1007/s13675-012-0003-7 fatcat:ppbipplr4fhrxkxibzs4igss4y

About Multigrid Convergence of Some Length Estimators [chapter]

Loïc Mazo, Étienne Baudrier
2014 Lecture Notes in Computer Science  
For this class, the pattern size depends on the resolution but not on the digitized function.  ...  An interesting property for curve length digital estimators is the convergence toward the continuous length and the associate convergence speed when the digitization step h tends to 0.  ...  Conclusion In this article, we have studied some convergence properties of a class of semilocal length estimators in the concave and the general cases.  ... 
doi:10.1007/978-3-319-09955-2_18 fatcat:hvwfz2ylx5c6xmdn5e5mxtfn5q

Supervised Learning with General Risk Functionals [article]

Liu Leqi, Audrey Huang, Zachary C. Lipton, Kamyar Azizzadenesheli
2022 arXiv   pre-print
Standard uniform convergence results bound the generalization gap of the expected loss over a hypothesis class.  ...  While prior works specialize in uniform convergence of particular functionals, our work provides uniform convergence for a general class of H\"older risk functionals for which the closeness in the Cumulative  ...  structure of CDF estimation and propose a new notion of permutation complexity that captures the complexity of the hypothesis class used for CDF estimation.  ... 
arXiv:2206.13648v1 fatcat:bz5pudwfs5do7mrczqgo3iy6am


2018 Uludağ University Journal of The Faculty of Engineering  
Therefore, the equilibrium and stability properties of the neural network model considered in this paper can be easily verified by exploiting some basic properties of some certain classes of matrices.  ...  This paper deals with the problem of the global asymptotic stability of the class of Takagi-Sugeno Fuzzy Cohen-Grossberg neural networks with multiple time delays.  ...  to the Lipschitz activation functions and derived new sufficient condition for the global asymptotic stability of the equilibrium point for this class of neural networks.  ... 
doi:10.17482/uumfd.406443 fatcat:uz6bp5zqyvdqvi3zbou2vb5734

Rademacher and Gaussian Complexities: Risk Bounds and Structural Results [chapter]

Peter L. Bartlett, Shahar Mendelson
2001 Lecture Notes in Computer Science  
We investigate the use of certain data-dependent estimates of the complexity of a function class, called Rademacher and Gaussian complexities.  ...  of the complexity of the basis classes.  ...  Acknowledgments An earlier version of this paper appeared in COLT'01 (Bartlett and Mendelson, 2001) .  ... 
doi:10.1007/3-540-44581-1_15 fatcat:vh4ohkuy5vgv3mwhelpcssoppu

Coarse Homotopy Groups [article]

Paul D. Mitchener
2020 arXiv   pre-print
The main result is that the coarse homotopy groups of cone of a compact simplicial complex coincide with the usual homotopy groups of the underlying compact simplicial complex.  ...  We prove that coarse homotopy indeed is an equivalence relation, and this in the most general context of abstract coarse structures. We introduce (in a geometric way) coarse homotopy groups.  ...  Here, the first inequality is the triangle inequality, the second the Lipschitz property of f | A and f | B and the third the geodesic property of γ.  ... 
arXiv:1811.10096v3 fatcat:3rhqvflcpjcnbnixpfr3blmrxm
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