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Learning on Random Balls is Sufficient for Estimating (Some) Graph Parameters [article]

Takanori Maehara, Hoang NT
2021 arXiv   pre-print
Theoretical analyses for graph learning methods often assume a complete observation of the input graph.  ...  Our theoretical framework contributes a theoretical validation of mini-batch learning on graphs and leads to new learning-theoretic results on generalization bounds as well as size-generalizability without  ...  HN is partially supported by the Japanese Government MEXT SGU Scholarship No. 205144.  ... 
arXiv:2111.03317v1 fatcat:aqw4ljok6rhpdfikuxekqn2q5e

Pseudo-likelihood-based M-estimation of random graphs with dependent edges and parameter vectors of increasing dimension [article]

Jonathan R. Stewart, Michael Schweinberger
2021 arXiv   pre-print
for parameter vectors of increasing dimension based on a single observation of dependent random variables.  ...  We demonstrate that scalable estimation of random graph models with dependent edges is possible, by establishing the first consistency results and convergence rates for pseudo-likelihood-based M-estimators  ...  How can one estimate random graph models based on a single observation of a random graph with dependent edges and parameter vectors of increasing dimension?  ... 
arXiv:2012.07167v3 fatcat:2dx4huxrjne3rit6kscwcjd7p4

Learning Some Popular Gaussian Graphical Models without Condition Number Bounds [article]

Jonathan Kelner, Frederic Koehler, Raghu Meka, Ankur Moitra
2020 arXiv   pre-print
Our result for structure recovery in walk-summable GGMs is derived from a more general result for efficient sparse linear regression in walk-summable models without any norm dependencies.  ...  Graphical Lasso, CLIME) that provably recover the graph structure with a logarithmic number of samples, they assume various conditions that require the precision matrix to be in some sense well-conditioned  ...  Several papers have been written on faster implementations of the graphical lasso, e.g. the Big & Quic estimator of [23] .  ... 
arXiv:1905.01282v3 fatcat:nj7m5bu64zcipp4p4ts3vw3zkm

Logarithmic reduction of the level of randomness in some probabilistic geometric constructions

S. Artstein-Avidan, V.D. Milman
2006 Journal of Functional Analysis  
The main tool we use is random walks on expander graphs.  ...  It is an intriguing question whether some of them could be realized explicitly.  ...  Acknowledgments We thank Avi Wigderson for introducing to us the subject of derandomization using random walks on expander graphs, and for fruitful conversations.  ... 
doi:10.1016/j.jfa.2005.11.003 fatcat:2bl3m5eeovhpvboyoylq743kny

Random Geometric Graph: Some recent developments and perspectives [article]

Quentin Duchemin
2022 arXiv   pre-print
The Random Geometric Graph (RGG) is a random graph model for network data with an underlying spatial representation.  ...  We also explain how this model differs from classical community based random graph models and we review recent works that try to take the best of both worlds.  ...  This task is known as manifold learning in the Machine learning community.  ... 
arXiv:2203.15351v1 fatcat:sr4skrktvvdyhegc5ppnv7nodu

Some Fundamental Theorems in Mathematics [article]

Oliver Knill
2022 arXiv   pre-print
An expository hitchhikers guide to some theorems in mathematics.  ...  One does not see the state x(t) of the system but some output y(t) = Cx(t) + Du(t). The filter then "filters out" or "learns" the best estimate x * (t) from the observed data y(t).  ...  Let us say, a functional on discrete random variables is additive if it is of the form H(X) = x f (p x ) for some continuous function f for which f (t)/t is monotone.  ... 
arXiv:1807.08416v4 fatcat:lw7lbsxyznfrnaozilxapihmdy

Some applications of Laplace eigenvalues of graphs [chapter]

Bojan Mohar
1997 Graph Symmetry  
In the last decade important relations between Laplace eigenvalues and eigenvectors of graphs and several other graph parameters were discovered.  ...  In these notes we present some of these results and discuss their consequences.  ...  The basic idea of the randomized algorithms for approximating the volume of the convex body K is as follows. Let B be a ball contained in K. (Usually, such a ball is part of the input.)  ... 
doi:10.1007/978-94-015-8937-6_6 fatcat:n56ftyqt5reuhe6erwtacgzcqy

Some observations on high-dimensional partial differential equations with Barron data [article]

Weinan E, Stephan Wojtowytsch
2021 arXiv   pre-print
on the unit sphere (for ReLU activation), and (3) any sufficiently smooth function on R d .  ...  We give some examples of functions in Barron space or not in Barron space below. 3. 1 . 1 A counterexample on the unit ball.  ... 
arXiv:2012.01484v3 fatcat:bb4g4xtuofgu3gtwprkgabjcwe

Scalable Approximate Inference and Some Applications [article]

Jun Han
2020 arXiv   pre-print
The main goal is to estimate the expectation of interested functions w.r.t. a target distribution.  ...  Approximate inference in probability models is a fundamental task in machine learning.  ...  When observations D = {x i , y i } N i=1 are available, the task is to learn the parameter θ.  ... 
arXiv:2003.03515v1 fatcat:whrdn6b23rdlzavojj6uklqs5q

Some Recent Advances in Multiscale Geometric Analysis of Point Clouds [chapter]

Guangliang Chen, Anna V. Little, Mauro Maggioni, Lorenzo Rosasco
2011 Wavelets and Multiscale Analysis  
We present three applications: the first one to the estimation of intrinsic dimension of sampled manifolds, the second one to the construction of multiscale dictionaries, called geometric wavelets, for  ...  We discuss recent work based on multiscale geometric analysis for the study of large data sets that lie in high-dimensional spaces but have low-dimensional structure.  ...  By "curse of dimensionality" one usually means the large number of samples needed for estimating functions of many parameters.  ... 
doi:10.1007/978-0-8176-8095-4_10 fatcat:yt3adfccbfcdxby526rt6a3poa

Commentary: Introductory Comments to Some Applied Papers by David R. Brillinger, by Tore Schweder and Haiganoush Preisler [chapter]

Peter Guttorp, David Brillinger
2011 Selected Works of David Brillinger  
Matrix models for stage-structured populations like the sheep blowfly have become popular (Caswell (2000) Matrix Population Models: Construction, Analysis, and Interpretation, is cited some 1900 times)  ...  The flies were kept in a cage, and fed on a constant diet . The population experienced substantial fluctuations in size over the observational period.  ...  The pr esent pap er is an early st udy with vit al paramet ers, particularly the mortality rate, dep ending on population size and being affecte d by random vari ation.  ... 
doi:10.1007/978-1-4614-1344-8_18 fatcat:j42pubtifvhyfbosnfxf5o2dgq

Stein's Method Meets Computational Statistics: A Review of Some Recent Developments [article]

Andreas Anastasiou, Alessandro Barp, François-Xavier Briol, Bruno Ebner, Robert E. Gaunt, Fatemeh Ghaderinezhad, Jackson Gorham, Arthur Gretton, Christophe Ley, Qiang Liu, Lester Mackey, Chris. J. Oates (+2 others)
2022 arXiv   pre-print
The goal of this survey is to bring together some of these recent developments and, in doing so, to stimulate further research into the successful field of Stein's method and statistics.  ...  estimation and goodness-of-fit testing.  ...  FXB and CJO were supported by the Lloyds Register Foundation Programme on Data-Centric Engineering and The Alan Turing Institute under the EPSRC grant EP/N510129/1.  ... 
arXiv:2105.03481v2 fatcat:muraepfjz5g6bht6vdarvismkq

Some geometric aspects of graphs and their eigenfunctions

Joel Friedman
1993 Duke mathematical journal  
We also perform some numerical experiments suggesting that the fiber product can yield graphs with small second eigenvalue.  ...  Proposition 2.2 The Dirichlet eigenpairs of a graph with boundary, G, i.e. successive orthogonal minimizers of the R restricted to functions vanishing on the boundary of G, are the  ...  The results are listed in only known that the average spectral radius is ≤ 2 √ d − 1 + 2 log d + C for some constant C (and n sufficiently large depending on d) for most graphs (see [Fri88] ), and Alon's  ... 
doi:10.1215/s0012-7094-93-06921-9 fatcat:jbqp6eghqfgetkfy3g7wxpg44y

Some Insights into the Geometry and Training of Neural Networks [article]

Ewout van den Berg
2016 arXiv   pre-print
This sheds more light on the vanishing gradient problem, explains the need for regularization, and suggests an approach for subsampling training data to improve performance.  ...  These findings expose the connections between scaling of the weight parameters and the density of the training samples.  ...  More interesting perhaps is a lower bound on the approximation distance. For the unit ball B we have the following: Theorem 3.1. Let B denote the unit ball in R d .  ... 
arXiv:1605.00329v1 fatcat:ero6i5k46rbmvigbw3dii6ugmm

Global Convergence of Stochastic Gradient Descent for Some Non-convex Matrix Problems [article]

Christopher De Sa, Kunle Olukotun, Christopher Ré
2015 arXiv   pre-print
In this paper, we exhibit a step size scheme for SGD on a low-rank least-squares problem, and we prove that, under broad sampling conditions, our method converges globally from a random starting point  ...  Stochastic gradient descent (SGD) on a low-rank factorization is commonly employed to speed up matrix problems including matrix completion, subspace tracking, and SDP relaxation.  ...  For any graph with node set N and edge set E, the MAXCUT problem on the graph requires us to solve minimize (i,j)∈E y i y j subject to y i ∈ {−1, 1}.  ... 
arXiv:1411.1134v3 fatcat:425moxdp25hotmwuwfjq3mrtcq
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