# A faster subquadratic algorithm for finding outlier correlations

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**A faster subquadratic algorithm for finding outlier correlations.** / Karppa, Matti; Kaski, Petteri; Kohonen, Jukka.

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*ACM Transactions on Algorithms*, Vuosikerta. 14, Nro 3, 31, Sivut 1-26. https://doi.org/10.1145/3174804

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*ACM Transactions on Algorithms*,

*14*(3), 1-26. [31]. https://doi.org/10.1145/3174804

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TY - JOUR

T1 - A faster subquadratic algorithm for finding outlier correlations

AU - Karppa, Matti

AU - Kaski, Petteri

AU - Kohonen, Jukka

PY - 2018/7/1

Y1 - 2018/7/1

N2 - We study the problem of detecting outlier pairs of strongly correlated variables among a collection of n variables with otherwise weak pairwise correlations. After normalization, this task amounts to the geometric task where we are given as input a set of n vectors with unit Euclidean norm and dimension d, and for some constants 0 < τ < ρ < 1, we are asked to find all the outlier pairs of vectors whose inner product is at least ρ in absolute value, subject to the promise that all but at most q pairs of vectors have inner product at most τ in absolute value. Improving on an algorithm of Valiant [FOCS 2012; J. ACM 2015], we present a randomized algorithm that for Boolean inputs ({−1, 1}-valued data normalized to unit Euclidean length) runs in time (Equation presented), where 0 < γ < 1 is a constant tradeoff parameter and M(μ, ν) is the exponent to multiply an [nμ] × [nν ] matrix with an [nν] × [nμ] matrix and Δ = 1/(1 − logτ ρ). As corollaries we obtain randomized algorithms that run in time (Equation presented) and in time (Equation presented), where 2 ≤ ω < 2.38 is the exponent for square matrix multiplication and 0.3 < α ≤ 1 is the exponent for rectangular matrix multiplication. The notation Õ (·) hides polylogarithmic factors in n and d whose degree may depend on ρ and τ. We present further corollaries for the light bulb problem and for learning sparse Boolean functions.

AB - We study the problem of detecting outlier pairs of strongly correlated variables among a collection of n variables with otherwise weak pairwise correlations. After normalization, this task amounts to the geometric task where we are given as input a set of n vectors with unit Euclidean norm and dimension d, and for some constants 0 < τ < ρ < 1, we are asked to find all the outlier pairs of vectors whose inner product is at least ρ in absolute value, subject to the promise that all but at most q pairs of vectors have inner product at most τ in absolute value. Improving on an algorithm of Valiant [FOCS 2012; J. ACM 2015], we present a randomized algorithm that for Boolean inputs ({−1, 1}-valued data normalized to unit Euclidean length) runs in time (Equation presented), where 0 < γ < 1 is a constant tradeoff parameter and M(μ, ν) is the exponent to multiply an [nμ] × [nν ] matrix with an [nν] × [nμ] matrix and Δ = 1/(1 − logτ ρ). As corollaries we obtain randomized algorithms that run in time (Equation presented) and in time (Equation presented), where 2 ≤ ω < 2.38 is the exponent for square matrix multiplication and 0.3 < α ≤ 1 is the exponent for rectangular matrix multiplication. The notation Õ (·) hides polylogarithmic factors in n and d whose degree may depend on ρ and τ. We present further corollaries for the light bulb problem and for learning sparse Boolean functions.

KW - Correlation

KW - Fast matrix multiplication

KW - Light bulb problem

KW - Rectangular matrix multiplication

KW - Similarity search

UR - http://www.scopus.com/inward/record.url?scp=85052553224&partnerID=8YFLogxK

U2 - 10.1145/3174804

DO - 10.1145/3174804

M3 - Article

VL - 14

SP - 1

EP - 26

JO - ACM Transactions on Algorithms

JF - ACM Transactions on Algorithms

SN - 1549-6325

IS - 3

M1 - 31

ER -

ID: 28318629