Introduction Monotone distributions Conclusions Testing Monotone Continuous Distributions on High-dimensional Real Cubes Michal Adamaszek DIMAP, University of Warwick Joint work with Artur Czumaj and Christian Sohler Michal Adamaszek Testing Monotone Continuous Distributions
Introduction Monotone distributions Conclusions Testing probability distributions Test if a probability distribution has a given property P . Distribution is accessed by drawing random samples. Michal Adamaszek Testing Monotone Continuous Distributions
Introduction Monotone distributions Conclusions Testing probability distributions Test if a probability distribution has a given property P . Distribution is accessed by drawing random samples. Goal: distinguish between distributions with the property P , distributions which are far from P Michal Adamaszek Testing Monotone Continuous Distributions
Introduction Monotone distributions Conclusions Testing probability distributions Test if a probability distribution has a given property P . Distribution is accessed by drawing random samples. Goal: distinguish between distributions with the property P , distributions which are far from P minimizing the number of samples and with error probability ≤ 1 / 3. Michal Adamaszek Testing Monotone Continuous Distributions
Introduction Monotone distributions Conclusions Testing probability distributions Test if a probability distribution has a given property P . Distribution is accessed by drawing random samples. Goal: distinguish between distributions with the property P , distributions which are far from P minimizing the number of samples and with error probability ≤ 1 / 3. Examples: is the distribution uniform? is it equal to a fixed distr.? are two distributions identical? are they independent? estimate support size etc... Michal Adamaszek Testing Monotone Continuous Distributions
Introduction Monotone distributions Conclusions Classical/typical results Is a distribution on k points uniform √ ˜ O ( k ) samples. Are two distributions on k points close in L 1 -norm ˜ O ( k 2 / 3 ) samples. Is a distribution on { 0 , 1 , . . . , k } close to monotone √ ˜ O ( k ) samples. Is a distribution on [ k ] × [ k ] a product of its marginals ˜ O ( k ) samples. Batu, Fischer, Fortnow, Kumar, Rubinfeld, Smith, White et al. Michal Adamaszek Testing Monotone Continuous Distributions
Introduction Monotone distributions Conclusions Infinite domains Michal Adamaszek Testing Monotone Continuous Distributions
Introduction Monotone distributions Conclusions Infinite domains Ω = [0 , 1] n Michal Adamaszek Testing Monotone Continuous Distributions
Introduction Monotone distributions Conclusions Infinite domains Ω = [0 , 1] n continuous distributions with density f so that � Pr f [ A ] = f d µ. A Michal Adamaszek Testing Monotone Continuous Distributions
Introduction Monotone distributions Conclusions Infinite domains Ω = [0 , 1] n continuous distributions with density f so that � Pr f [ A ] = f d µ. A distributions with atoms � f + p i δ x i Michal Adamaszek Testing Monotone Continuous Distributions
Introduction Monotone distributions Conclusions Non-testable properties Michal Adamaszek Testing Monotone Continuous Distributions
Introduction Monotone distributions Conclusions Non-testable properties Is a distribution continuous, or purely discrete? Michal Adamaszek Testing Monotone Continuous Distributions
Introduction Monotone distributions Conclusions Non-testable properties Is a distribution continuous, or purely discrete? Is a continuous distribution uniform or is it ǫ -far from uniform in the L 1 metric? Michal Adamaszek Testing Monotone Continuous Distributions
Introduction Monotone distributions Conclusions Non-testable properties Is a distribution continuous, or purely discrete? Is a continuous distribution uniform or is it ǫ -far from uniform in the L 1 metric? “fatten” a discrete distribution on M random points, √ up to ∼ M draws this looks like a random distribution, but is very L 1 -far from uniform. Michal Adamaszek Testing Monotone Continuous Distributions
Introduction Monotone distributions Conclusions A testable property - discreteness on M points For arbitrary Ω distinguish between M � f = p i δ x i i =1 for some x 1 , . . . , x M , Pr f [ A ] < 1 − ǫ for any set A ⊂ Ω of size M . Michal Adamaszek Testing Monotone Continuous Distributions
Introduction Monotone distributions Conclusions A testable property - discreteness on M points Tester for discreteness on M points: Take 2 M /ǫ random samples If there are ≤ M distinct values accept. If there are > M distinct values reject. Michal Adamaszek Testing Monotone Continuous Distributions
Introduction Monotone distributions Conclusions A testable property - discreteness on M points Tester for discreteness on M points: Take 2 M /ǫ random samples If there are ≤ M distinct values accept. If there are > M distinct values reject. Lower bound Ω( M 1 − o (1) ) follows from bounds for estimating distribution support size (eg. Raskhodnikova et al’09, Valiant’08). Match these bounds? Michal Adamaszek Testing Monotone Continuous Distributions
Introduction Monotone distributions Conclusions Monotone distributions and uniformity Find a class of distributions for which being uniform is testable. Michal Adamaszek Testing Monotone Continuous Distributions
Introduction Monotone distributions Conclusions Monotone distributions and uniformity Find a class of distributions for which being uniform is testable. Ω = [0 , 1] n The density f is monotone if f ( x ) ≤ f ( y ) whenever x i ≤ y i for all i . Michal Adamaszek Testing Monotone Continuous Distributions
Introduction Monotone distributions Conclusions Monotone distributions and uniformity Find a class of distributions for which being uniform is testable. Ω = [0 , 1] n The density f is monotone if f ( x ) ≤ f ( y ) whenever x i ≤ y i for all i . Given a distribution with monotone density f , Is f the uniform distribution U ? Or is it ǫ -far from U in the L 1 metric d ( f , g ) = 1 � | f − g | . 2 Ω Michal Adamaszek Testing Monotone Continuous Distributions
Introduction Monotone distributions Conclusions Discrete vs. continuous cubes Rubinfeld, Servedio’05 Testing uniformity of monotone distributions on the boolean cube { 0 , 1 } n with L 1 distance Is possible with O ( n log( n /ǫ ) /ǫ 2 ) samples. Requires Ω( n / log 2 n ) samples. Michal Adamaszek Testing Monotone Continuous Distributions
Introduction Monotone distributions Conclusions Discrete vs. continuous cubes Rubinfeld, Servedio’05 Testing uniformity of monotone distributions on the boolean cube { 0 , 1 } n with L 1 distance Is possible with O ( n log( n /ǫ ) /ǫ 2 ) samples. Requires Ω( n / log 2 n ) samples. Michal Adamaszek Testing Monotone Continuous Distributions
Introduction Monotone distributions Conclusions 1 / 4 1 / 3 1 / 4 1 / 3 1 / 6 1 / 4 1 / 6 1 / 4 Michal Adamaszek Testing Monotone Continuous Distributions
Introduction Monotone distributions Conclusions 1 / 4 1 / 3 1 / 4 1 / 3 1 / 6 1 / 4 1 / 6 1 / 4 lower bound → lower bound upper bound ← upper bound Michal Adamaszek Testing Monotone Continuous Distributions
Introduction Monotone distributions Conclusions Discrete vs. continuous cubes Rubinfeld, Servedio’05 Testing uniformity of monotone distributions on the boolean cube { 0 , 1 } n with L 1 distance Is possible with O ( n log( n /ǫ ) /ǫ 2 ) samples. Requires Ω( n / log 2 n ) samples. Michal Adamaszek Testing Monotone Continuous Distributions
Introduction Monotone distributions Conclusions Discrete vs. continuous cubes Rubinfeld, Servedio’05 Testing uniformity of monotone distributions on the boolean cube { 0 , 1 } n with L 1 distance Is possible with O ( n log( n /ǫ ) /ǫ 2 ) samples. Requires Ω( n / log 2 n ) samples. Our result Testing uniformity of monotone distributions on the real cube [0 , 1] n with L 1 distance Is possible with O ( n /ǫ 2 ) samples. Michal Adamaszek Testing Monotone Continuous Distributions
Introduction Monotone distributions Conclusions Tester Idea: estimate � x � 1 = x 1 + x 2 + . . . + x n . Michal Adamaszek Testing Monotone Continuous Distributions
Introduction Monotone distributions Conclusions Tester Idea: estimate � x � 1 = x 1 + x 2 + . . . + x n . If U is the uniform distribution then E U [ � x � 1 ] = n 2 . Michal Adamaszek Testing Monotone Continuous Distributions
Introduction Monotone distributions Conclusions Tester Idea: estimate � x � 1 = x 1 + x 2 + . . . + x n . If U is the uniform distribution then E U [ � x � 1 ] = n 2 . Theorem If f is a monotone distribution, ǫ -far from uniform then E f [ � x � 1 ] ≥ n 2 + ǫ 2 . Michal Adamaszek Testing Monotone Continuous Distributions
Introduction Monotone distributions Conclusions Tester Ω = [0 , 1] n , f - unknown monotone distribution. Draw C samples x 1 , . . . , x C , E = 1 ˜ � � x i � 1 . C If ˜ E > n 2 + ǫ 4 say ǫ far from uniform. If ˜ E ≤ n 2 + ǫ 4 say uniform. C = 40 n /ǫ 2 is good (use Feige’s inequality). Michal Adamaszek Testing Monotone Continuous Distributions
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