Sparse Approximation of a Kernel Mean

Cortés, Efrén Cruz; Scott, Clayton

doi:10.48550/arxiv.1503.00323

Cited by 3 publications

(5 citation statements)

References 40 publications

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“…Cortes and Scott [12] provide another approach to the sparse kernel mean problem. They run Gonzalez's algorithms [21] for k-center on the points P ∈ R d (iteratively add points to Q, always choosing the furthest point from any in Q) and terminate when the furthest distance to the nearest point in Q is Θ(ε).…”

Section: Known Results On Kde Coresetsmentioning

confidence: 99%

“…Restrictions Algorithm Joshi et al [28] d/ε 2 bounded VC random sample Fasy et al [17] (d/ε 2 ) log(d∆/ε) Lipschitz random sample Lopaz-Paz et al [31] 1/ε 2 characteristic kernels random sample Chen et al [10] 1/(εr P ) characteristic kernels iterative Bach et al [3] (1/r 2 P ) log(1/ε) characteristic kernels iterative Bach et al [3] 1/ε 2 characteristic kernels, weighted iterative Lacsote-Julien et al [29] 1/ε 2 characteristic kernels iterative Harvey and Samadi [23] (1/ε) √ n log 2.5 (n) characteristic kernels iterative Cortez and Scott [12] k 0 (≤ (∆/ε) d ) Lipschitz; d is constant iterative Phillips [37] (1/ε)…”

Section: Background On Kernels and Related Coresetsmentioning

confidence: 99%

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Near-Optimal Coresets of Kernel Density Estimates

Phillips

Tai

2019

Discrete Comput Geom

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We construct near-optimal coresets for kernel density estimates for points in R d when the kernel is positive definite. Specifically we show a polynomial time construction for a coreset of size O( √ d/ε · log 1/ε), and we show a near-matching lower bound of size Ω(min{ √ d/ε, 1/ε 2 }). When d ≥ 1/ε 2 , it is known that the size of coreset can be O(1/ε 2 ). The upper bound is a polynomial-in-(1/ε) improvement when d ∈ [3, 1/ε 2 ) and the lower bound is the first known lower bound to depend on d for this problem. Moreover, the upper bound restriction that the kernel is positive definite is significant in that it applies to a wide-variety of kernels, specifically those most important for machine learning. This includes kernels for information distances and the sinc kernel which can be negative.

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Section: Known Results On Kde Coresetsmentioning

confidence: 99%

Section: Background On Kernels and Related Coresetsmentioning

confidence: 99%

Near-Optimal Coresets of Kernel Density Estimates

Phillips

Tai

2019

Discrete Comput Geom

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“…Runtime Restrictions Joshi et al [21] (1/ε 2 )(d + log(1/δ)) |Q| samples centrally symmetric, positive Fasy et al [9] (d/ε 2 ) log(d∆/εδ) |Q| samples .. Gretton et al [15] (1/ε 4 ) log(1/δ) |Q| samples characteristic kernels Phillips [27] (1/εσ) 2d/(d+2) log d/(d+2) (1/εσ) n/ε 2 (1/σ)-Lipschitz, d is constant Phillips [27] 1/ε n log n d = 1 Chen et al [4] 1/(εr P ) n/(εr P ) characteristic kernels Bach et al [2] (1/r 2 P ) log(1/ε) n log(1/ε)/r 2 P characteristic kernels Bach et al [2] 1/ε 2 n/ε 2 characteristic kernels, weighted Harvey and Samadi [17] (1/ε) √ n log 2.5 (n) poly(n, 1/ε, d) characteristic kernels Cortez and Scott [6] k…”

Section: Paper Coreset Sizementioning

confidence: 99%

“…Cortes and Scott [6] provide another approach to the sparse kernel mean problem. They run Gonzalez's algorithms [14] for k-center on the points P ∈ R d (iteratively add points to Q, always choosing the furthest point from any in Q) and terminate when the furthest distance to the nearest point in Q is Θ(ε).…”

Section: Paper Coreset Sizementioning

confidence: 99%

Improved Coresets for Kernel Density Estimates

Phillips

Tai

2018

Proceedings of the Twenty-Ninth Annual ACM-SIAM Symposium on Discrete Algorithms

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We study the construction of coresets for kernel density estimates. That is we show how to approximate the kernel density estimate described by a large point set with another kernel density estimate with a much smaller point set. For characteristic kernels (including Gaussian and Laplace kernels), our approximation preserves the L ∞ error between kernel density estimates within error ε, with coreset size 4/ε 2 , but no other aspects of the data, including the dimension, the diameter of the point set, or the bandwidth of the kernel common to other approximations. When the dimension is unrestricted, we show this bound is tight for these kernels as well as a much broader set.This work provides a careful analysis of the iterative Frank-Wolfe algorithm adapted to this context, an algorithm called kernel herding. This analysis unites a broad line of work that spans statistics, machine learning, and geometry.When the dimension d is constant, we demonstrate much tighter bounds on the size of the coreset specifically for Gaussian kernels, showing that it is bounded by the size of the coreset for axis-aligned rectangles. Currently the best known constructive bound is O( 1 ε log d 1 ε ), and non-constructively, this can be improved by log 1 ε . This improves the best constant dimension bounds polynomially for d ≥ 3.

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“…Some attention has focused on calculating the kernel density estimates given by a large underlying training dataset [15,22]. As an active area of computer vision, there as been a particular interest on two-dimensional problems [4,20,23]. These often utilize some variation of the fast Gauss transform of Greengard and Strain [9].…”

Section: Prior Workmentioning

confidence: 99%

Sparse Density Representations for Simultaneous Inference on Large Spatial Datasets

Arnold

2015

Preprint

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Large spatial datasets often represent a number of spatial point processes generated by distinct entities or classes of events. When crossed with covariates, such as discrete time buckets, this can quickly result in a data set with millions of individual density estimates. Applications that require simultaneous access to a substantial subset of these estimates become resource constrained when densities are stored in complex and incompatible formats. We present a method for representing spatial densities along the nodes of sparsely populated trees. Fast algorithms are provided for performing set operations and queries on the resulting compact tree structures. The speed and simplicity of the approach is demonstrated on both real and simulated spatial data.

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Sparse Approximation of a Kernel Mean

Cited by 3 publications

References 40 publications

Near-Optimal Coresets of Kernel Density Estimates

Near-Optimal Coresets of Kernel Density Estimates

Improved Coresets for Kernel Density Estimates

Sparse Density Representations for Simultaneous Inference on Large Spatial Datasets

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