Filters








8,313 Hits in 6.3 sec

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

Ilias Diakonikolas, Daniel M. Kane, Alistair Stewart
2015 arXiv   pre-print
We study 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}.  ... 
arXiv:1505.00662v2 fatcat:xf7kh3r5lfcmblxlnsrqwnqnym

Efficiently Learning Fourier Sparse Set Functions

Andisheh Amrollahi, Amir Zandieh, Michael Kapralov, Andreas Krause
2019 Neural Information Processing Systems  
Our algorithm can also efficiently 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.  ... 
dblp:conf/nips/AmrollahiZKK19 fatcat:er6akandzzbbdddqfjvm55skv4

Low-rank Characteristic Tensor Density Estimation Part I: Foundations [article]

Magda Amiridi, Nikos Kargas, Nicholas D. Sidiropoulos
2021 arXiv   pre-print
Any multivariate density can be represented by its characteristic function, 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  ... 
arXiv:2008.12315v2 fatcat:xrnmikomofenrbjvn2dkr6h73m

Testing for Families of Distributions via the Fourier Transform

Alistair Stewart, Ilias Diakonikolas, Clément L. Canonne
2018 Neural Information Processing Systems  
We apply our 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).  ... 
dblp:conf/nips/StewartDC18 fatcat:w6lkjhjerjcktdre77btkarid4

Continuous Kernel Learning [chapter]

John Moeller, Vivek Srikumar, Sarathkrishna Swaminathan, Suresh Venkatasubramanian, Dustin Webb
2016 Lecture Notes in Computer Science  
Kernel 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.  ... 
doi:10.1007/978-3-319-46227-1_41 fatcat:7rmfjf5srjbqrd35u2wqqsbxiy

Constant depth circuits, Fourier transform, and learnability

Nathan Linial, Yishay Mansour, Noam Nisan
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.  ... 
doi:10.1145/174130.174138 fatcat:iwwlqkfxazgijjobyhhevsk4su

Book reports

2005 Computers and Mathematics with Applications  
Computers and Mathematics with Applications 49 (2005) 1585-1622 www.elsevier.com/locate/camwa BOOK REPORTS 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.  ... 
doi:10.1016/j.camwa.2005.04.001 fatcat:melc2iqigrh6tn37ldct5k73jm

Variable Elimination in the Fourier Domain [article]

Yexiang Xue, Stefano Ermon, Ronan Le Bras, Carla P. Gomes, Bart Selman
2016 arXiv   pre-print
We explore a different type 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.  ... 
arXiv:1508.04032v2 fatcat:suh5talwibbgxix33olmxilq44

Learning Mixtures of Smooth Product Distributions: Identifiability and Algorithm [article]

Nikos Kargas, Nicholas D. Sidiropoulos
2019 arXiv   pre-print
We study 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.  ... 
arXiv:1904.01156v1 fatcat:x527y4xn7vg75fjsxuyuoo2ypy

The fourier transform of poisson multinomial distributions and its algorithmic applications

Ilias Diakonikolas, Daniel M. Kane, Alistair Stewart
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 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  ... 
doi:10.1145/2897518.2897552 dblp:conf/stoc/DiakonikolasKS16 fatcat:qn6olejg3zhc3fiyzpb7fc7ycu

Embedding Hard Learning Problems Into Gaussian Space

Adam Klivans, Pravesh Kothari, Marc Herbstritt
2014 International Workshop on Approximation Algorithms for Combinatorial Optimization  
We give 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.  ... 
doi:10.4230/lipics.approx-random.2014.793 dblp:conf/approx/KlivansK14 fatcat:uogqzy45fbfjjfueqtqmd3dwc4

Properly Learning Poisson Binomial Distributions in Almost Polynomial Time [article]

Ilias Diakonikolas, Daniel M. Kane, Alistair Stewart
2015 arXiv   pre-print
A Poisson binomial distribution (PBD) 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.  ... 
arXiv:1511.04066v1 fatcat:okzvbgorpzftrljba7ed55s7wa

The Fourier Transform of Poisson Multinomial Distributions and its Algorithmic Applications [article]

Ilias Diakonikolas, Daniel M. Kane, Alistair Stewart
2016 arXiv   pre-print
An (n, k)-Poisson Multinomial Distribution (PMD) is a 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  ... 
arXiv:1511.03592v2 fatcat:httnhcqrm5el3aylitlslougri

Convolutional Factor Graphs as Probabilistic Models [article]

Yongyi Mao, Frank Kschischang, Brendan J. Frey
2012 arXiv   pre-print
This paper shows that CFGs are natural models 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  ... 
arXiv:1207.4136v1 fatcat:hjyihda6tzfl5j26xyyg2cdk74

An O*(2^n ) Algorithm for Graph Coloring and Other Partitioning Problems via Inclusion--Exclusion

Mikko Koivisto
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.  ... 
doi:10.1109/focs.2006.11 dblp:conf/focs/Koivisto06 fatcat:rjb5ch4azbew5brgekdihsg224
« Previous Showing results 1 — 15 out of 8,313 results