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

2021
*
arXiv
*
pre-print

Theoretical analyses

arXiv:2111.03317v1
fatcat:aqw4ljok6rhpdfikuxekqn2q5e
*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. ...##
###
Pseudo-likelihood-based M-estimation of random graphs with dependent edges and parameter vectors of increasing dimension
[article]

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? ...

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

2020
*
arXiv
*
pre-print

Our result

arXiv:1905.01282v3
fatcat:nj7m5bu64zcipp4p4ts3vw3zkm
*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] . ...##
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Logarithmic reduction of the level of randomness in some probabilistic geometric constructions

2006
*
Journal of Functional Analysis
*

The main tool we use

doi:10.1016/j.jfa.2005.11.003
fatcat:2bl3m5eeovhpvboyoylq743kny
*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. ...##
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Random Geometric Graph: Some recent developments and perspectives
[article]

2022
*
arXiv
*
pre-print

The

arXiv:2203.15351v1
fatcat:sr4skrktvvdyhegc5ppnv7nodu
*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. ...##
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Some Fundamental Theorems in Mathematics
[article]

2022
*
arXiv
*
pre-print

An expository hitchhikers guide to

arXiv:1807.08416v4
fatcat:lw7lbsxyznfrnaozilxapihmdy
*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. ...##
###
Some applications of Laplace eigenvalues of graphs
[chapter]

1997
*
Graph Symmetry
*

In the last decade important relations between Laplace eigenvalues and eigenvectors of

doi:10.1007/978-94-015-8937-6_6
fatcat:n56ftyqt5reuhe6erwtacgzcqy
*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.) ...##
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Some observations on high-dimensional partial differential equations with Barron data
[article]

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*. ...

##
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Scalable Approximate Inference and Some Applications
[article]

2020
*
arXiv
*
pre-print

The main goal

arXiv:2003.03515v1
fatcat:whrdn6b23rdlzavojj6uklqs5q
*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. ...##
###
Some Recent Advances in Multiscale Geometric Analysis of Point Clouds
[chapter]

2011
*
Wavelets and Multiscale Analysis
*

We present three applications: the first

doi:10.1007/978-0-8176-8095-4_10
fatcat:yt3adfccbfcdxby526rt6a3poa
*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*. ...##
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Commentary: Introductory Comments to Some Applied Papers by David R. Brillinger, by Tore Schweder and Haiganoush Preisler
[chapter]

2011
*
Selected Works of David Brillinger
*

Matrix models

doi:10.1007/978-1-4614-1344-8_18
fatcat:j42pubtifvhyfbosnfxf5o2dgq
*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. ...##
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Stein's Method Meets Computational Statistics: A Review of Some Recent Developments
[article]

2022
*
arXiv
*
pre-print

The goal of this survey

arXiv:2105.03481v2
fatcat:muraepfjz5g6bht6vdarvismkq
*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. ...##
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Some geometric aspects of graphs and their eigenfunctions

1993
*
Duke mathematical journal
*

We also perform

doi:10.1215/s0012-7094-93-06921-9
fatcat:jbqp6eghqfgetkfy3g7wxpg44y
*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 ...##
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Some Insights into the Geometry and Training of Neural Networks
[article]

2016
*
arXiv
*
pre-print

This sheds more light

arXiv:1605.00329v1
fatcat:ero6i5k46rbmvigbw3dii6ugmm
*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 . ...##
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Global Convergence of Stochastic Gradient Descent for Some Non-convex Matrix Problems
[article]

2015
*
arXiv
*
pre-print

In this paper, we exhibit a step size scheme

arXiv:1411.1134v3
fatcat:425moxdp25hotmwuwfjq3mrtcq
*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}. ...
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