18,955 research outputs found
Some closure features of locally testable affine-invariant properties
Thesis (S.M.)--Massachusetts Institute of Technology, Dept. of Electrical Engineering and Computer Science, 2013.Cataloged from PDF version of thesis.Includes bibliographical references (p. 31-32).We prove that the class of locally testable affine-invariant properties is closed under sums, intersections and "lifts". The sum and intersection are two natural operations on linear spaces of functions, where the sum of two properties is simply their sum as a vector space. The "lift" is a less well-studied property, which creates some interesting affine-invariant properties over large domains, from properties over smaller domains. Previously such results were known for "single-orbit characterized" affine-invariant properties, which are known to be a subclass of locally testable ones, and are potentially a strict subclass. The fact that the intersection of locally-testable affine-invariant properties are locally testable could have been derived from previously known general results on closure of property testing under set-theoretic operations, but was not explicitly observed before. The closure under sum and lifts is implied by an affirmative answer to a central question attempting to characterize locally testable affine-invariant properties, but the status of that question remains wide open. Affine-invariant properties are clean abstractions of commonly studied, and extensively used, algebraic properties such linearity and low-degree. Thus far it is not known what makes affine-invariant properties locally testable - no characterizations are known, and till this work it was not clear if they satisfied any closure properties. This work shows that the class of locally testable affine-invariant properties are closed under some very natural operations. Our techniques use ones previously developed for the study of "single-orbit characterized" properties, but manage to apply them to the potentially more general class of all locally testable ones via a simple connection that may be of broad interest in the study of affine-invariant properties.by Alan Xinyu Guo.S.M
Generalizing univariate signed rank statistics for testing and estimating a multivariate location parameter.
We generalize signed rank statistics to dimensions higher than one. This results in a class of orthogonally invariant and distribution free tests that can be used for testing spherical symmetry/location parameter. The corresponding estimator is orthogonally equivariant. Both the test and estimator can be chosen with asymptotic efficiency 1. The breakdown point of the estimator depends only on the scores, not on the dimension of the data. For elliptical distributions, we obtain an affine invariant test with the same asymptotic properties, if the signed rank statistic is applied to standardized data. We also present a method for computing the estimator numerically, and consider a real data example and some simulations. Finally, an application to detection of time-varying signals in spherically symmetric noise is given.Affine invariant tests; Asymptotic normality; Breakdown point; distribution free tests;
Every locally characterized affine-invariant property is testable
Let F = F_p for any fixed prime p >= 2. An affine-invariant property is a
property of functions on F^n that is closed under taking affine transformations
of the domain. We prove that all affine-invariant property having local
characterizations are testable. In fact, we show a proximity-oblivious test for
any such property P, meaning that there is a test that, given an input function
f, makes a constant number of queries to f, always accepts if f satisfies P,
and rejects with positive probability if the distance between f and P is
nonzero. More generally, we show that any affine-invariant property that is
closed under taking restrictions to subspaces and has bounded complexity is
testable.
We also prove that any property that can be described as the property of
decomposing into a known structure of low-degree polynomials is locally
characterized and is, hence, testable. For example, whether a function is a
product of two degree-d polynomials, whether a function splits into a product
of d linear polynomials, and whether a function has low rank are all examples
of degree-structural properties and are therefore locally characterized.
Our results depend on a new Gowers inverse theorem by Tao and Ziegler for low
characteristic fields that decomposes any polynomial with large Gowers norm
into a function of low-degree non-classical polynomials. We establish a new
equidistribution result for high rank non-classical polynomials that drives the
proofs of both the testability results and the local characterization of
degree-structural properties
A Characterization of Locally Testable Affine-Invariant Properties via Decomposition Theorems
Let be a property of function for
a fixed prime . An algorithm is called a tester for if, given
a query access to the input function , with high probability, it accepts
when satisfies and rejects when is "far" from satisfying
. In this paper, we give a characterization of affine-invariant
properties that are (two-sided error) testable with a constant number of
queries. The characterization is stated in terms of decomposition theorems,
which roughly claim that any function can be decomposed into a structured part
that is a function of a constant number of polynomials, and a pseudo-random
part whose Gowers norm is small. We first give an algorithm that tests whether
the structured part of the input function has a specific form. Then we show
that an affine-invariant property is testable with a constant number of queries
if and only if it can be reduced to the problem of testing whether the
structured part of the input function is close to one of a constant number of
candidates.Comment: 27 pages, appearing in STOC 2014. arXiv admin note: text overlap with
arXiv:1306.0649, arXiv:1212.3849 by other author
Testing Low Complexity Affine-Invariant Properties
Invariance with respect to linear or affine transformations of the domain is
arguably the most common symmetry exhibited by natural algebraic properties. In
this work, we show that any low complexity affine-invariant property of
multivariate functions over finite fields is testable with a constant number of
queries. This immediately reproves, for instance, that the Reed-Muller code
over F_p of degree d < p is testable, with an argument that uses no detailed
algebraic information about polynomials except that low degree is preserved by
composition with affine maps.
The complexity of an affine-invariant property P refers to the maximum
complexity, as defined by Green and Tao (Ann. Math. 2008), of the sets of
linear forms used to characterize P. A more precise statement of our main
result is that for any fixed prime p >=2 and fixed integer R >= 2, any
affine-invariant property P of functions f: F_p^n -> [R] is testable, assuming
the complexity of the property is less than p. Our proof involves developing
analogs of graph-theoretic techniques in an algebraic setting, using tools from
higher-order Fourier analysis.Comment: 38 pages, appears in SODA '1
Rank-based optimal tests of the adequacy of an elliptic VARMA model
We are deriving optimal rank-based tests for the adequacy of a vector
autoregressive-moving average (VARMA) model with elliptically contoured
innovation density. These tests are based on the ranks of pseudo-Mahalanobis
distances and on normed residuals computed from Tyler's [Ann. Statist. 15
(1987) 234-251] scatter matrix; they generalize the univariate signed rank
procedures proposed by Hallin and Puri [J. Multivariate Anal. 39 (1991) 1-29].
Two types of optimality properties are considered, both in the local and
asymptotic sense, a la Le Cam: (a) (fixed-score procedures) local asymptotic
minimaxity at selected radial densities, and (b) (estimated-score procedures)
local asymptotic minimaxity uniform over a class F of radial densities.
Contrary to their classical counterparts, based on cross-covariance matrices,
these tests remain valid under arbitrary elliptically symmetric innovation
densities, including those with infinite variance and heavy-tails. We show that
the AREs of our fixed-score procedures, with respect to traditional (Gaussian)
methods, are the same as for the tests of randomness proposed in Hallin and
Paindaveine [Bernoulli 8 (2002b) 787-815]. The multivariate serial extensions
of the classical Chernoff-Savage and Hodges-Lehmann results obtained there thus
also hold here; in particular, the van der Waerden versions of our tests are
uniformly more powerful than those based on cross-covariances. As for our
estimated-score procedures, they are fully adaptive, hence, uniformly optimal
over the class of innovation densities satisfying the required technical
assumptions.Comment: Published at http://dx.doi.org/10.1214/009053604000000724 in the
Annals of Statistics (http://www.imstat.org/aos/) by the Institute of
Mathematical Statistics (http://www.imstat.org
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