131 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
Testing Linear-Invariant Non-Linear Properties
We consider the task of testing properties of Boolean functions that are
invariant under linear transformations of the Boolean cube. Previous work in
property testing, including the linearity test and the test for Reed-Muller
codes, has mostly focused on such tasks for linear properties. The one
exception is a test due to Green for "triangle freeness": a function
f:\cube^{n}\to\cube satisfies this property if do not all
equal 1, for any pair x,y\in\cube^{n}.
Here we extend this test to a more systematic study of testing for
linear-invariant non-linear properties. We consider properties that are
described by a single forbidden pattern (and its linear transformations), i.e.,
a property is given by points v_{1},...,v_{k}\in\cube^{k} and
f:\cube^{n}\to\cube satisfies the property that if for all linear maps
L:\cube^{k}\to\cube^{n} it is the case that do
not all equal 1. We show that this property is testable if the underlying
matroid specified by is a graphic matroid. This extends
Green's result to an infinite class of new properties.
Our techniques extend those of Green and in particular we establish a link
between the notion of "1-complexity linear systems" of Green and Tao, and
graphic matroids, to derive the results.Comment: This is the full version; conference version appeared in the
proceedings of STACS 200
Symmetries in algebraic Property Testing
Thesis (Ph. D.)--Massachusetts Institute of Technology, Dept. of Electrical Engineering and Computer Science, 2010.Cataloged from PDF version of thesis.Includes bibliographical references (p. 94-100).Modern computational tasks often involve large amounts of data, and efficiency is a very desirable feature of such algorithms. Local algorithms are especially attractive, since they can imply global properties by only inspecting a small window into the data. In Property Testing, a local algorithm should perform the task of distinguishing objects satisfying a given property from objects that require many modifications in order to satisfy the property. A special place in Property Testing is held by algebraic properties: they are some of the first properties to be tested, and have been heavily used in the PCP and LTC literature. We focus on conditions under which algebraic properties are testable, following the general goal of providing a more unified treatment of these properties. In particular, we explore the notion of symmetry in relation to testing, a direction initiated by Kaufman and Sudan. We investigate the interplay between local testing, symmetry and dual structure in linear codes, by showing both positive and negative results. On the negative side, we exhibit a counterexample to a conjecture proposed by Alon, Kaufman, Krivelevich, Litsyn, and Ron aimed at providing general sufficient conditions for testing. We show that a single codeword of small weight in the dual family together with the property of being invariant under a 2-transitive group of permutations do not necessarily imply testing. On the positive side, we exhibit a large class of codes whose duals possess a strong structural property ('the single orbit property'). Namely, they can be specified by a single codeword of small weight and the group of invariances of the code. Hence we show that sparsity and invariance under the affine group of permutations are sufficient conditions for a notion of very structured testing. These findings also reveal a new characterization of the extensively studied BCH codes. As a by-product, we obtain a more explicit description of structured tests for the special family of BCH codes of design distance 5.by Elena Grigorescu.Ph.D
Succinct Representation of Codes with Applications to Testing
Motivated by questions in property testing, we search for linear
error-correcting codes that have the "single local orbit" property: i.e., they
are specified by a single local constraint and its translations under the
symmetry group of the code. We show that the dual of every "sparse" binary code
whose coordinates are indexed by elements of F_{2^n} for prime n, and whose
symmetry group includes the group of non-singular affine transformations of
F_{2^n} has the single local orbit property. (A code is said to be "sparse" if
it contains polynomially many codewords in its block length.) In particular
this class includes the dual-BCH codes for whose duals (i.e., for BCH codes)
simple bases were not known. Our result gives the first short (O(n)-bit, as
opposed to the natural exp(n)-bit) description of a low-weight basis for BCH
codes. The interest in the "single local orbit" property comes from the recent
result of Kaufman and Sudan (STOC 2008) that shows that the duals of codes that
have the single local orbit property under the affine symmetry group are
locally testable. When combined with our main result, this shows that all
sparse affine-invariant codes over the coordinates F_{2^n} for prime n are
locally testable. If, in addition to n being prime, if 2^n-1 is also prime
(i.e., 2^n-1 is a Mersenne prime), then we get that every sparse cyclic code
also has the single local orbit. In particular this implies that BCH codes of
Mersenne prime length are generated by a single low-weight codeword and its
cyclic shifts
Covariance properties under natural image transformations for the generalized Gaussian derivative model for visual receptive fields
This paper presents a theory for how geometric image transformations can be
handled by a first layer of linear receptive fields, in terms of true
covariance properties, which, in turn, enable geometric invariance properties
at higher levels in the visual hierarchy. Specifically, we develop this theory
for a generalized Gaussian derivative model for visual receptive fields, which
is derived in an axiomatic manner from first principles, that reflect symmetry
properties of the environment, complemented by structural assumptions to
guarantee internally consistent treatment of image structures over multiple
spatio-temporal scales.
It is shown how the studied generalized Gaussian derivative model for visual
receptive fields obeys true covariance properties under spatial scaling
transformations, spatial affine transformations, Galilean transformations and
temporal scaling transformations, implying that a vision system, based on image
and video measurements in terms of the receptive fields according to this
model, can to first order of approximation handle the image and video
deformations between multiple views of objects delimited by smooth surfaces, as
well as between multiple views of spatio-temporal events, under varying
relative motions between the objects and events in the world and the observer.
We conclude by describing implications of the presented theory for biological
vision, regarding connections between the variabilities of the shapes of
biological visual receptive fields and the variabilities of spatial and
spatio-temporal image structures under natural image transformations.Comment: 38 pages, 14 figure
Relaxed Local Correctability from Local Testing
We cement the intuitive connection between relaxed local correctability and
local testing by presenting a concrete framework for building a relaxed locally
correctable code from any family of linear locally testable codes with
sufficiently high rate. When instantiated using the locally testable codes of
Dinur et al. (STOC 2022), this framework yields the first asymptotically good
relaxed locally correctable and decodable codes with polylogarithmic query
complexity, which finally closes the superpolynomial gap between query lower
and upper bounds. Our construction combines high-rate locally testable codes of
various sizes to produce a code that is locally testable at every scale: we can
gradually "zoom in" to any desired codeword index, and a local tester at each
step certifies that the next, smaller restriction of the input has low error.
Our codes asymptotically inherit the rate and distance of any locally
testable code used in the final step of the construction. Therefore, our
technique also yields nonexplicit relaxed locally correctable codes with
polylogarithmic query complexity that have rate and distance approaching the
Gilbert-Varshamov bound.Comment: 18 page
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