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Multiscale Photon-Limited Spectral Image Reconstruction

Kalyani Krishnamurthy, Maxim Raginsky, Rebecca Willett
2010 SIAM Journal of Imaging Sciences  
The key feature of this approach is that the partition cells are anisotropic across the spatial and spectral dimensions so that the method adapts to varying degrees of spatial and spectral smoothness,  ...  Specifically, our method searches over estimates defined over a family of recursive dyadic partitions in both the spatial and spectral domains, and finds the one that maximizes a penalized log likelihood  ...  The class of candidate estimators, Γ M,N , corresponds to functions which are piecewise constant spatially and piecewise polynomial spectrally, where the breakpoints between the constant and polynomial  ... 
doi:10.1137/090756259 fatcat:hue4lx73bbft5ied74fiksyef4

Centralized and Distributed Semiparametric Compression of Piecewise Smooth Functions

Varit Chaisinthop, Pier Luigi Dragotti
2011 IEEE Transactions on Signal Processing  
This thesis introduces novel wavelet-based semi-parametric centralized and distributed compression methods for a class of piecewise smooth functions.  ...  Our analysis is centered around a class of 1-D piecewise smooth functions.  ...  piecewise polynomial functions.  ... 
doi:10.1109/tsp.2011.2144590 fatcat:ykxtmrvhm5dnvhgdej5ltwwuqe

Additive Models, Trees, and Related Methods [chapter]

Trevor Hastie, Jerome Friedman, Robert Tibshirani
2001 Springer Series in Statistics  
These techniques each assume a (different) structured form for the unknown regression function, and by doing so they finesse the curse of dimensionality.  ...  We describe five related techniques: generalized additive models, trees, multivariate adaptive regression splines, the patient rule induction method, and hierarchical mixtures of experts.  ...  Each such partition generates a pair of piecewise constant basis functionsindicator functions for the two sets of categories.  ... 
doi:10.1007/978-0-387-21606-5_9 fatcat:4xwbpyjmiraldgm7wrkg6mk5im

Additive Models, Trees, and Related Methods [chapter]

Trevor Hastie, Robert Tibshirani, Jerome Friedman
2008 Springer Series in Statistics  
These techniques each assume a (different) structured form for the unknown regression function, and by doing so they finesse the curse of dimensionality.  ...  We describe five related techniques: generalized additive models, trees, multivariate adaptive regression splines, the patient rule induction method, and hierarchical mixtures of experts.  ...  Each such partition generates a pair of piecewise constant basis functionsindicator functions for the two sets of categories.  ... 
doi:10.1007/b94608_9 fatcat:2bpfwi3jtngubgnjeylxrbbuii

Additive Models, Trees, and Related Methods [chapter]

Trevor Hastie, Robert Tibshirani, Jerome Friedman
2008 Springer Series in Statistics  
These techniques each assume a (different) structured form for the unknown regression function, and by doing so they finesse the curse of dimensionality.  ...  We describe five related techniques: generalized additive models, trees, multivariate adaptive regression splines, the patient rule induction method, and hierarchical mixtures of experts.  ...  Each such partition generates a pair of piecewise constant basis functionsindicator functions for the two sets of categories.  ... 
doi:10.1007/978-0-387-84858-7_9 fatcat:poh55dehjbgmdhg445cysxuuq4

Level Lines Selection with Variational Models for Segmentation and Encoding

Coloma Ballester, Vicent Caselles, Laura Igual, Luis Garrido
2006 Journal of Mathematical Imaging and Vision  
The Tree of Shapes offers a compact and structured representation of the family of level lines of an image.  ...  To segment an image, we minimize the simplified Mumford-Shah energy functional on the set of partitions represented in this hierarchy.  ...  Acknowledgments The first and second authors acknowledge partial support by the Departament d'Universitats, Recerca i Societat de la Informació de la Generalitat de Catalunya and by PNPGC project, reference  ... 
doi:10.1007/s10851-006-7252-0 fatcat:h6biw4jpkbbndko5ah7pkpdhue

Fully adaptive multiresolution finite volume schemes for conservation laws

Albert Cohen, Sidi Mahmoud Kaber, Siegfried Müller, Marie Postel
2001 Mathematics of Computation  
One can enlarge Λ n ε into a graded treẽ Λ n+1 ε which contains both Λ n ε and Λ n+1 ε so that, if U n+1 J Of course, B n J is now the numerical flux balance computed from U n J which differs from V n  ...  Several solutions to this problem are proposed, analyzed, and compared in terms of accuracy and complexity.  ...  In contrast, it is known that the limit functions associated with (32) have C r smoothness for all r < 1. 2.3.3. Tree-structured compression.  ... 
doi:10.1090/s0025-5718-01-01391-6 fatcat:kezkzb7z7veyvhftpzodzcjulu

The Dynamics of Influence Systems [article]

Bernard Chazelle
2012 arXiv   pre-print
Besides resolving the dynamics of a popular family of multiagent systems, the other contribution of this work is to introduce a new type of renormalization-based bifurcation analysis for multiagent systems  ...  Influence systems form a large class of multiagent systems designed to model how influence, broadly defined, spreads across a dynamic network.  ...  Acknowledgments I wish to thank Pascal Koiran and John Tsitsiklis for helpful conversations.  ... 
arXiv:1204.3946v2 fatcat:bn5i5jmmafhodhujfgdemckmzy

Verifiable Reinforcement Learning via Policy Extraction [article]

Osbert Bastani and Yewen Pu and Armando Solar-Lezama
2019 arXiv   pre-print
We propose VIPER, an algorithm that combines ideas from model compression and imitation learning to learn decision tree policies guided by a DNN policy (called the oracle) and its Q-function, and show  ...  loses, and (iii) learn a provably stable decision tree policy for cart-pole.  ...  Acknowledgments This work was funded by the Toyota Research Institute and NSF InTrans award 1665282.  ... 
arXiv:1805.08328v2 fatcat:wr2p22z4kzf3lisl3tdxyuvg64

Profile Entropy: A Fundamental Measure for the Learnability and Compressibility of Discrete Distributions [article]

Yi Hao, Alon Orlitsky
2020 arXiv   pre-print
We show that for samples of discrete distributions, profile entropy is a fundamental measure unifying the concepts of estimation, inference, and compression.  ...  the best estimator over any label-invariant distribution collection; c) serves as the limit of profile compression, for which we derive optimal near-linear-time block and sequential algorithms.  ...  In addition, the function is Lipschitz and hence for an absolute constant C and an arbitrary interval I := [a, b] ⊆ [0, 1], one can construct an explicit polynomial g(z) of degree at most d ∈ N, satisfying  ... 
arXiv:2002.11665v1 fatcat:igog7qivh5a3xezxwmlxw2455q

Metric spaces of shapes and applications: compression, curve matching and low-dimensional representation

Matt Feiszli, Sergey Kushnarev, Kathryn Leonard
2014 Geometry Imaging and Computing  
The first, a C 1 -type metric on classes of shapes with Lipschitz tangent angle, allows for estimates of massiveness such as ε-entropy.  ...  A Sobolev-type metric on piecewise C 2 curves allows for efficient curve matching based on a multiscale wavelet-like analysis.  ...  Discussion The first part of the paper discusses measures of massiveness (ε-entropy) of a C 1 -type of metric on the space of planar curves and a related concept of coding and compression of shapes.  ... 
doi:10.4310/gic.2014.v1.n2.a1 fatcat:brfr2q5r2rhftnicwz4e64lcqy

Diffusive Influence Systems

Bernard Chazelle
2015 SIAM journal on computing (Print)  
Besides resolving the dynamics of a widely used family of multiagent systems, we introduce a general renormalization method for the bifurcation analysis of multiagent systems.  ...  We build a general analytical framework which we then use to prove that, while Turing-complete, influence dynamics of the diffusive type is almost surely asymptotically periodic.  ...  Acknowledgments I wish to thank Nalini Anantharaman, Pascal Koiran, John Tsitsiklis, and the anonymous referees for their useful comments and suggestions.  ... 
doi:10.1137/120882640 fatcat:6wt3fclsfrgydjczbsr6c4q4na

Nonlinear approximation and its applications [chapter]

Ronald A. DeVore
2009 Multiscale, Nonlinear and Adaptive Approximation  
He looked like a high school student to me but he impressed everyone with his talk on whether polynomial operators could produce both polynomial and spectral orders of approximation.  ...  We became the best of friends and frequent collaborators.  ...  To build a compression for functions, we first choose our compression metric L p . We then agree on a minimal smoothness ε that we shall assume of the functions in L p .  ... 
doi:10.1007/978-3-642-03413-8_6 fatcat:asbbvpclzfhdlm4avypiasve3u

Learning with tree tensor networks: complexity estimates and model selection [article]

Bertrand Michel, Anthony Nouy
2021 arXiv   pre-print
Tree tensor networks, or tree-based tensor formats, are prominent model classes for the approximation of high-dimensional functions in computational and data science.  ...  ) by minimizing a penalized empirical risk, with a penalty depending on the complexity of the model class and derived from estimates of the metric entropy of tree tensor networks.  ...  The space V L is linearly identified with the space of piecewise constant functions on the uniform partition of [0, 1) into 2 L intervals.  ... 
arXiv:2007.01165v3 fatcat:7bdhgsccmbbmhpfzmx2c4jiflq

Orthogonal bandelet bases for geometric images approximation

Gabriel Peyré, Stéphane Mallat
2008 Communications on Pure and Applied Mathematics  
Fast algorithms and compression applications are described.  ...  This bandlet construction has a hierarchical structure over wavelet coefficients taking advantage of existing regularity among these coefficients.  ...  For compression applications, images are decomposed in a best bandlet basis and the resulting coefficients are quantized and entropy coded.  ... 
doi:10.1002/cpa.20242 fatcat:ggwvayom75dbholxqu532tysti
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