18,687 research outputs found
Near-Optimal Closeness Testing of Discrete Histogram Distributions
We investigate the problem of testing the equivalence between two discrete
histograms. A {\em -histogram} over is a probability distribution that
is piecewise constant over some set of intervals over . Histograms
have been extensively studied in computer science and statistics. Given a set
of samples from two -histogram distributions over , we want to
distinguish (with high probability) between the cases that and
. The main contribution of this paper is a new
algorithm for this testing problem and a nearly matching information-theoretic
lower bound. Specifically, the sample complexity of our algorithm matches our
lower bound up to a logarithmic factor, improving on previous work by
polynomial factors in the relevant parameters. Our algorithmic approach applies
in a more general setting and yields improved sample upper bounds for testing
closeness of other structured distributions as well
Optimal Algorithms for Testing Closeness of Discrete Distributions
We study the question of closeness testing for two discrete distributions.
More precisely, given samples from two distributions and over an
-element set, we wish to distinguish whether versus is at least
\eps-far from , in either or distance. Batu et al. gave
the first sub-linear time algorithms for these problems, which matched the
lower bounds of Valiant up to a logarithmic factor in , and a polynomial
factor of \eps.
In this work, we present simple (and new) testers for both the and
settings, with sample complexity that is information-theoretically
optimal, to constant factors, both in the dependence on , and the dependence
on \eps; for the testing problem we establish that the sample
complexity is $\Theta(\max\{n^{2/3}/\eps^{4/3}, n^{1/2}/\eps^2 \}).
Optimal Testing of Discrete Distributions with High Probability
We study the problem of testing discrete distributions with a focus on the
high probability regime. Specifically, given samples from one or more discrete
distributions, a property , and parameters , we want to distinguish {\em with probability at least } whether
these distributions satisfy or are -far from
in total variation distance. Most prior work in distribution
testing studied the constant confidence case (corresponding to ), and provided sample-optimal testers for a range of properties.
While one can always boost the confidence probability of any such tester by
black-box amplification, this generic boosting method typically leads to
sub-optimal sample bounds.
Here we study the following broad question: For a given property
, can we {\em characterize} the sample complexity of testing
as a function of all relevant problem parameters, including the
error probability ? Prior to this work, uniformity testing was the only
statistical task whose sample complexity had been characterized in this
setting. As our main results, we provide the first algorithms for closeness and
independence testing that are sample-optimal, within constant factors, as a
function of all relevant parameters. We also show matching
information-theoretic lower bounds on the sample complexity of these problems.
Our techniques naturally extend to give optimal testers for related problems.
To illustrate the generality of our methods, we give optimal algorithms for
testing collections of distributions and testing closeness with unequal sized
samples
Optimal Testing of Discrete Distributions with High Probability
We study the problem of testing discrete distributions with a focus on the high probability regime. Specifically, given samples from one or more discrete distributions, a property , and parameters , we want to distinguish {\em with probability at least } whether these distributions satisfy or are -far from in total variation distance. Most prior work in distribution testing studied the constant confidence case (corresponding to ), and provided sample-optimal testers for a range of properties. While one can always boost the confidence probability of any such tester by black-box amplification, this generic boosting method typically leads to sub-optimal sample bounds. Here we study the following broad question: For a given property , can we {\em characterize} the sample complexity of testing as a function of all relevant problem parameters, including the error probability ? Prior to this work, uniformity testing was the only statistical task whose sample complexity had been characterized in this setting. As our main results, we provide the first algorithms for closeness and independence testing that are sample-optimal, within constant factors, as a function of all relevant parameters. We also show matching information-theoretic lower bounds on the sample complexity of these problems. Our techniques naturally extend to give optimal testers for related problems. To illustrate the generality of our methods, we give optimal algorithms for testing collections of distributions and testing closeness with unequal sized samples
Two Party Distribution Testing: Communication and Security
We study the problem of discrete distribution testing in the two-party setting. For example, in the standard closeness testing problem, Alice and Bob each have t samples from, respectively, distributions a and b over [n], and they need to test whether a=b or a,b are epsilon-far (in the l_1 distance). This is in contrast to the well-studied one-party case, where the tester has unrestricted access to samples of both distributions. Despite being a natural constraint in applications, the two-party setting has previously evaded attention.
We address two fundamental aspects of the two-party setting: 1) what is the communication complexity, and 2) can it be accomplished securely, without Alice and Bob learning extra information about each other\u27s input. Besides closeness testing, we also study the independence testing problem, where Alice and Bob have t samples from distributions a and b respectively, which may be correlated; the question is whether a,b are independent or epsilon-far from being independent. Our contribution is three-fold: 1) We show how to gain communication efficiency given more samples, beyond the information-theoretic bound on t. The gain is polynomially better than what one would obtain via adapting one-party algorithms. 2) We prove tightness of our trade-off for the closeness testing, as well as that the independence testing requires tight Omega(sqrt{m}) communication for unbounded number of samples. These lower bounds are of independent interest as, to the best of our knowledge, these are the first 2-party communication lower bounds for testing problems, where the inputs are a set of i.i.d. samples. 3) We define the concept of secure distribution testing, and provide secure versions of the above protocols with an overhead that is only polynomial in the security parameter
Distributional Property Testing in a Quantum World
A fundamental problem in statistics and learning theory is to test properties of distributions. We show that quantum computers can solve such problems with significant speed-ups. We also introduce a novel access model for quantum distributions, enabling the coherent preparation of quantum samples, and propose a general framework that can naturally handle both classical and quantum distributions in a unified manner. Our framework generalizes and improves previous quantum algorithms for testing closeness between unknown distributions, testing independence between two distributions, and estimating the Shannon / von Neumann entropy of distributions. For classical distributions our algorithms significantly improve the precision dependence of some earlier results. We also show that in our framework procedures for classical distributions can be directly lifted to the more general case of quantum distributions, and thus obtain the first speed-ups for testing properties of density operators that can be accessed coherently rather than only via sampling
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