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Optimal Learning via the Fourier Transform for Sums of Independent Integer Random Variables
[article]

2015
*
arXiv
*
pre-print

We study

arXiv:1505.00662v2
fatcat:xf7kh3r5lfcmblxlnsrqwnqnym
*the*structure and learnability*of**sums**of**independent**integer**random**variables*(SIIRVs). ...*For*k ∈Z_+, a k-SIIRV*of*order n ∈Z_+ is*the*probability distribution*of**the**sum**of*n*independent**random**variables*each supported on {0, 1, ..., k-1}. ... Introduction Motivation and Background We study*sums**of**independent**integer**random**variables*: Definition.*For*k ∈ Z + , a k-IRV is any*random**variable*supported on {0, 1, . . . , k −1}. ...##
###
Efficiently Learning Fourier Sparse Set Functions

2019
*
Neural Information Processing Systems
*

Our algorithm can also efficiently

dblp:conf/nips/AmrollahiZKK19
fatcat:er6akandzzbbdddqfjvm55skv4
*learn*(*sums**of*) decision trees*of*small depth.*The*algorithm exploits techniques from*the*sparse*Fourier**transform*literature and is easily implementable. ... This implies that sparse graphs with k edges can,*for**the*first time, be*learned*from O(k log n) observations*of*cut values and in linear time in*the*number*of*vertices. ...*Optimization*is performed on p*variables*which results in Ω(n 2 ) runtime*for*graphs and Ω(n d ) time*for**the*general order d sparse recovery case. ...##
###
Low-rank Characteristic Tensor Density Estimation Part I: Foundations
[article]

2021
*
arXiv
*
pre-print

Any multivariate density can be represented by its characteristic function,

arXiv:2008.12315v2
fatcat:xrnmikomofenrbjvn2dkr6h73m
*via**the**Fourier**transform*. ... This tensor can be naturally estimated from observed realizations*of**the**random*vector*of*interest,*via*sample averaging. ... A Characteristic Function Approach*The*characteristic function*of*a*random**variable*X is*the**Fourier**transform**of*its density, and it can be interpreted as an expectation:*the**Fourier**transform*at frequency ...##
###
Testing for Families of Distributions via the Fourier Transform

2018
*
Neural Information Processing Systems
*

We apply our

dblp:conf/nips/StewartDC18
fatcat:w6lkjhjerjcktdre77btkarid4
*Fourier*-based framework to obtain near sample-*optimal*and computationally efficient testers*for**the*following fundamental distribution families:*Sums**of**Independent**Integer**Random**Variables*...*The*main contribution*of*this work is a simple and general testing technique that is applicable to all distribution families whose*Fourier*spectrum satisfies a certain approximate sparsity property. ... Our Results Our first result is a nearly sample-*optimal*testing algorithm*for**sums**of**independent**integer**random**variables*(SIIRVs). ...##
###
Continuous Kernel Learning
[chapter]

2016
*
Lecture Notes in Computer Science
*

Kernel

doi:10.1007/978-3-319-46227-1_41
fatcat:7rmfjf5srjbqrd35u2wqqsbxiy
*learning*is*the*problem*of*determining*the*best kernel (either from a dictionary*of*fixed kernels, or from a smooth space*of*kernel representations)*for*a given task. ... In this paper, we describe a new approach to kernel*learning*that establishes connections between*the**Fourier*-analytic representation*of*kernels arising out*of*Bochner's theorem and a specific kind*of*... [10] , in contrast,*optimize**Fourier*embeddings, but decompose each ω i into a parameter σ i multiplied by a nonlinear function*of*a uniform*random**variable*to represent*the*sample. ...##
###
Constant depth circuits, Fourier transform, and learnability

1993
*
Journal of the ACM
*

*The*algorithm observes

*the*behavior

*of*an AC'" function on O(nPO'Y'Og(n)) randomly chosen inputs, and derives a good approximation

*for*

*the*

*Fourier*

*transform*

*of*

*the*function. ... Perhaps

*the*most interesting application is an O(n POIYIOg(n ')-time algorithm

*for*

*learning*functions in ACO. ... We would like to thank

*the*anonymous referees whose comments helped to both improve and simplify

*the*presentation. ...

##
###
Book reports

2005
*
Computers and Mathematics with Applications
*

Computers and Mathematics with Applications 49 (2005) 1585-1622 www.elsevier.com/locate/camwa BOOK REPORTS

doi:10.1016/j.camwa.2005.04.001
fatcat:melc2iqigrh6tn37ldct5k73jm
*The*Book Reports section is a regular feature*of*Computers ~ Mathematics with Applications. ... 2.10*The*probability,*for**integers*,*of*being relatively prime. 2.11 Bernoulli*random*walks considered at some stopping time. 2.12 cosh, sinh,*the**Fourier**transform*and conditional*independence*. 2.13 cosh ... . 5.15 An almost sure convergence result*for**sums**of*stable*random**variables*. ...##
###
Variable Elimination in the Fourier Domain
[article]

2016
*
arXiv
*
pre-print

We explore a different type

arXiv:1508.04032v2
fatcat:suh5talwibbgxix33olmxilq44
*of*compact representation based on discrete*Fourier*representations, complementing*the*classical approach based on conditional*independencies*. ... We demonstrate*the*significance*of*this approach by applying it to*the**variable*elimination algorithm. ... Hadamard-*Fourier**Transformation*Hadamard-*Fourier**transformation*has attracted a lot*of*attention in PAC*Learning*Theory. ...##
###
Learning Mixtures of Smooth Product Distributions: Identifiability and Algorithm
[article]

2019
*
arXiv
*
pre-print

We study

arXiv:1904.01156v1
fatcat:x527y4xn7vg75fjsxuyuoo2ypy
*the*problem*of**learning*a mixture model*of*non-parametric product distributions. ...*The*problem*of**learning*a mixture model is that*of*finding*the*component distributions along with*the*mixing weights using observed samples generated from*the*mixture. ...*The**Fourier**transform**of*a convolution is*the*point-wise product*of**Fourier**transforms*. ...##
###
The fourier transform of poisson multinomial distributions and its algorithmic applications

2016
*
Proceedings of the 48th Annual ACM SIGACT Symposium on Theory of Computing - STOC 2016
*

Formally, an (n, k)-Poisson Multinomial Distribution (PMD) is a

doi:10.1145/2897518.2897552
dblp:conf/stoc/DiakonikolasKS16
fatcat:qn6olejg3zhc3fiyzpb7fc7ycu
*random**variable**of**the*form X = n i=1 X i , where*the*X i 's are*independent**random*vectors supported on*the*set {e 1 , e 2 , . . . , e k ... We remark that our*learning*algorithm outputs a succinct description*of*its hypothesis H,*via*its Discrete*Fourier**Transform*(DFT), H, which is supported on a small size set. ...*The*high-level structure*of*our*learning*algorithm relies on*the*sparsity*of**the**Fourier**transform*, and is similar to*the*algorithm in our previous work [DKS15a]*for**learning**sums**of**independent**integer*...##
###
Embedding Hard Learning Problems Into Gaussian Space

2014
*
International Workshop on Approximation Algorithms for Combinatorial Optimization
*

We give

doi:10.4230/lipics.approx-random.2014.793
dblp:conf/approx/KlivansK14
fatcat:uogqzy45fbfjjfueqtqmd3dwc4
*the*first representation-*independent*hardness result*for*agnostically*learning*halfspaces with respect to*the*Gaussian distribution. ... As far as we are aware, this is*the*first representation-*independent*hardness result*for*supervised*learning*when*the*underlying distribution is restricted to be a Gaussian. ... We thank Chengang Wu*for*numerous discussions during*the*preliminary stages*of*this work. We thank*the*anonymous reviewers*for*pointing out*the*typos in a previous version*of*this paper. ...##
###
Properly Learning Poisson Binomial Distributions in Almost Polynomial Time
[article]

2015
*
arXiv
*
pre-print

A Poisson binomial distribution (PBD)

arXiv:1511.04066v1
fatcat:okzvbgorpzftrljba7ed55s7wa
*of*order n is*the*discrete probability distribution*of**the**sum**of*n mutually*independent*Bernoulli*random**variables*. ...*The*previously best known running time*for*properly*learning*PBDs was (1/ϵ)^O((1/ϵ)). As one*of*our main contributions, we provide a novel structural characterization*of*PBDs. ... Introduction*The*Poisson binomial distribution (PBD) is*the*discrete probability distribution*of*a*sum**of*mutually*independent*Bernoulli*random**variables*. ...##
###
The Fourier Transform of Poisson Multinomial Distributions and its Algorithmic Applications
[article]

2016
*
arXiv
*
pre-print

An (n, k)-Poisson Multinomial Distribution (PMD) is a

arXiv:1511.03592v2
fatcat:httnhcqrm5el3aylitlslougri
*random**variable**of**the*form X = ∑_i=1^n X_i, where*the*X_i's are*independent**random*vectors supported on*the*set*of*standard basis vectors in R^k. ... In this paper, we obtain a refined structural understanding*of*PMDs by analyzing their*Fourier**transform*. ...*The*high-level structure*of*our*learning*algorithm relies on*the*sparsity*of**the**Fourier**transform*, and is similar to*the*algorithm in our previous work [DKS15a]*for**learning**sums**of**independent**integer*...##
###
Convolutional Factor Graphs as Probabilistic Models
[article]

2012
*
arXiv
*
pre-print

This paper shows that CFGs are natural models

arXiv:1207.4136v1
fatcat:hjyihda6tzfl5j26xyyg2cdk74
*for*probability functions when summation*of**independent*latent*random**variables*is involved. ...*The*requirement*of*a linear*transformation*between latent*variables*(with certain*independence*restriction) and*the*bserved*variables*, to an extent, limits*the*modelling flexibility*of*CFGs. ... Exploiting*the*evaluation-marginalization duality and*the*convolution theorem*of**the**Fourier**transform*, a more efficient way*for*computing F X V (x V \E , x E ) can be performed*via**the*Fast*Fourier**Transform*...##
###
An O*(2^n ) Algorithm for Graph Coloring and Other Partitioning Problems via Inclusion--Exclusion

2006
*
2006 47th Annual IEEE Symposium on Foundations of Computer Science (FOCS'06)
*

*of*computing a partition

*sum*

*of*

*the*form (1)

*via*an

*integer*-coding technique and self-reducibility (

*for*similar techniques see, e.g., [29, 19] ). ... We use

*the*principle

*of*inclusion and exclusion, combined with polynomial time segmentation and fast Möbius

*transform*, to solve

*the*generic problem

*of*

*summing*or

*optimizing*over

*the*partitions

*of*n elements ... Acknowledgements I am grateful to Heikki Mannila

*for*valuable conversations on this work. ...

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