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21
Property Testing and its connection to Learning and Approximation
"... We study the question of determining whether an unknown function has a particular property or is fflfar from any function with that property. A property testing algorithm is given a sample of the value of the function on instances drawn according to some distribution, and possibly may query the fun ..."
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Cited by 428 (62 self)
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We study the question of determining whether an unknown function has a particular property or is fflfar from any function with that property. A property testing algorithm is given a sample of the value of the function on instances drawn according to some distribution, and possibly may query the function on instances of its choice. First, we establish some connections between property testing and problems in learning theory. Next, we focus on testing graph properties, and devise algorithms to test whether a graph has properties such as being kcolorable or having a aeclique (clique of density ae w.r.t the vertex set). Our graph property testing algorithms are probabilistic and make assertions which are correct with high probability, utilizing only poly(1=ffl) edgequeries into the graph, where ffl is the distance parameter. Moreover, the property testing algorithms can be used to efficiently (i.e., in time linear in the number of vertices) construct partitions of the graph which corre...
Learning polynomials with queries: The highly noisy case
, 1995
"... Given a function f mapping nvariate inputs from a finite Kearns et. al. [21] (see also [27, 28, 22]). In the setting of agfieldFintoF, we consider the task of reconstructing a list nostic learning, the learner is to make no assumptions regarding of allnvariate degreedpolynomials which agree withf ..."
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Cited by 88 (18 self)
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Given a function f mapping nvariate inputs from a finite Kearns et. al. [21] (see also [27, 28, 22]). In the setting of agfieldFintoF, we consider the task of reconstructing a list nostic learning, the learner is to make no assumptions regarding of allnvariate degreedpolynomials which agree withfon a the natural phenomena underlying the input/output relationship tiny but nonnegligible fraction, , of the input space. We give a of the function, and the goal of the learner is to come up with a randomized algorithm for solving this task which accessesfas a simple explanation which best fits the examples. Therefore the black box and runs in time polynomial in1;nand exponential in best explanation may account for only part of the phenomena. d, provided is(pd=jFj). For the special case whend=1, In some situations, when the phenomena appears very irregular, we solve this problem for jFj>0. In this case the providing an explanation which fits only part of it is better than nothing. Interestingly, Kearns et. al. did not consider the use of running time of our algorithm is bounded by a polynomial queries (but rather examples drawn from an arbitrary distribuand exponential ind. Our algorithm generalizes a previously tion) as they were skeptical that queries could be of any help. known algorithm, due to Goldreich and Levin, that solves this We show that queries do seem to help (see below). task for the case whenF=GF(2)(andd=1).
Property Testing
 Handbook of Randomized Computing, Vol. II
, 2000
"... this technical aspect (as in the boundeddegree model the closest graph having the property must have at most dN edges and degree bound d as well). ..."
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Cited by 74 (11 self)
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this technical aspect (as in the boundeddegree model the closest graph having the property must have at most dN edges and degree bound d as well).
Testing of Clustering
 In Proc. 41th Annu. IEEE Sympos. Found. Comput. Sci
, 2000
"... A set X of points in ! d is (k; b)clusterable if X can be partitioned into k subsets (clusters) so that the diameter (alternatively, the radius) of each cluster is at most b. We present algorithms that by sampling from a set X , distinguish between the case that X is (k; b)clusterable and the ca ..."
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Cited by 58 (14 self)
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A set X of points in ! d is (k; b)clusterable if X can be partitioned into k subsets (clusters) so that the diameter (alternatively, the radius) of each cluster is at most b. We present algorithms that by sampling from a set X , distinguish between the case that X is (k; b)clusterable and the case that X is fflfar from being (k; b 0 )clusterable for any given 0 ! ffl 1 and for b 0 b. In fflfar from being (k; b 0 )clusterable we mean that more than ffl \Delta jX j points should be removed from X so that it becomes (k; b 0 )clusterable. We give algorithms for a variety of cost measures that use a sample of size independent of jX j, and polynomial in k and 1=ffl. Our algorithms can also be used to find approximately good clusterings. Namely, these are clusterings of all but an fflfraction of the points in X that have optimal (or close to optimal) cost. The benefit of our algorithms is that they construct an implicit representation of such clusterings in time independ...
Combinatorial Property Testing (a survey)
 In: Randomization Methods in Algorithm Design
, 1998
"... We consider the question of determining whether a given object has a predetermined property or is "far" from any object having the property. Specifically, objects are modeled by functions, and distance between functions is measured as the fraction of the domain on which the functions diffe ..."
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Cited by 42 (1 self)
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We consider the question of determining whether a given object has a predetermined property or is "far" from any object having the property. Specifically, objects are modeled by functions, and distance between functions is measured as the fraction of the domain on which the functions differ. We consider (randomized) algorithms which may query the function at arguments of their choice, and seek algorithms which query the function at relatively few places. We focus on combinatorial properties, and specifically on graph properties. The two standard representations of graphs  by adjacency matrices and by incidence lists  yield two different models for testing graph properties. In the first model, most appropriate for dense graphs, distance between Nvertex graphs is measured as the fraction of edges on which the graphs disagree over N 2 . In the second model, most appropriate for boundeddegree graphs, distance between Nvertex ddegree graphs is measured as the fraction of edges on ...
Testing Basic Boolean Formulae
 SIAM J. Disc. Math
, 2002
"... We consider the problem of determining whether a given function f : f0; 1g belongs to a certain class of Boolean functions F or whether it is far from the class. More precisely, given query access to the function f and given a distance parameter , we would like to decide whether f 2 F or whethe ..."
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Cited by 37 (6 self)
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We consider the problem of determining whether a given function f : f0; 1g belongs to a certain class of Boolean functions F or whether it is far from the class. More precisely, given query access to the function f and given a distance parameter , we would like to decide whether f 2 F or whether it diers from every g 2 F on more than an fraction of the domain elements. The classes of functions we consider are singleton (\dictatorship") functions, monomials, and monotone DNF functions with a bounded number of terms. In all cases we provide algorithms whose query complexity is independent of n (the number of function variables), and linear in 1=.
Algebraic Property Testing: The Role of Invariance
, 2007
"... We argue that the symmetries of a property being tested play a central role in property testing. We support this assertion in the context of algebraic functions, by examining properties of functions mapping a vector space K n over a field K to a subfield F. We consider Flinear properties that are i ..."
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Cited by 35 (16 self)
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We argue that the symmetries of a property being tested play a central role in property testing. We support this assertion in the context of algebraic functions, by examining properties of functions mapping a vector space K n over a field K to a subfield F. We consider Flinear properties that are invariant under linear transformations of the domain and prove that an O(1)local “characterization ” is a necessary and sufficient condition for O(1)local testability when K  = O(1). (A local characterization of a property is a definition of a property in terms of local constraints satisfied by functions exhibiting a property.) For the subclass of properties that are invariant under affine transformations of the domain, we prove that the existence of a single O(1)local constraint implies O(1)local testability. These results generalize and extend the class of algebraic properties, most notably linearity and lowdegreeness, that were previously known to be testable. In particular, the extensions include properties satisfied by functions of degree linear in n that turn out to be O(1)locally testable. Our results are proved by introducing a new notion that we term “formal characterizations”. Roughly this corresponds to characterizations that are given by a single local constraint and its permutations under linear transformations of the domain. Our main testing result shows that local formal characterizations
Testing problems with sublearning sample complexity
 In Proceedings of the Eleventh Annual ACM Conference on Computational Learning Theory
, 1998
"... We study the problem of determining, for a class of functions ¡ , whether an unknown target function ¢ is contained in ¡ or is “far ” from any function in ¡. Thus, in contrast to problems of learning, where we must construct a good approximation to ¢ in ¡ on the basis of sample data, in problems of ..."
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Cited by 17 (11 self)
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We study the problem of determining, for a class of functions ¡ , whether an unknown target function ¢ is contained in ¡ or is “far ” from any function in ¡. Thus, in contrast to problems of learning, where we must construct a good approximation to ¢ in ¡ on the basis of sample data, in problems of testing we are only required to determine the existence of a good approximation. Our main results demonstrate that, over the domain £ ¤ ¥ ¦ § ¨ for constant © , the number of examples required for testing grows only as � � � ��� � � � (where � is any small constant), for both decision trees of size � and a special class of neural networks with � hidden units. This is in contrast to the � � � � examples required for learning these same classes. Our tests are based on combinatorial constructions demonstrating that these classes can be approximated by small classes of coarse partitions of space, and rely on repeated application of the wellknown Birthday Paradox. � Supported by an ONR Science Scholar Fellowship at the Bunting Institute. 1
Testing Acyclicity of Directed Graphs in Sublinear Time
 In Proceedings of ICALP
, 2000
"... This paper initiates the study of testing properties of directed graphs. In particular, the paper considers the most basic property of directed graphs  acyclicity. Because the choice of representation affects the choice of algorithm, the two main representations of graphs are studied. For the a ..."
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Cited by 13 (5 self)
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This paper initiates the study of testing properties of directed graphs. In particular, the paper considers the most basic property of directed graphs  acyclicity. Because the choice of representation affects the choice of algorithm, the two main representations of graphs are studied. For the adjacencymatrix representation, most appropriate for dense graphs, a testing algorithm is developed that requires query and time complexity of ~ O(1=ffl 2 ), where ffl is a distance parameter independent of the size of the graph. The algorithm, which can probe the adjacency matrix of the graph, accepts every graph that is acyclic, and rejects, with probability at least 2=3, every graph whose adjacency matrix should be modified in at least ffl fraction of its entries so that it becomes acyclic. For the incidence list representation, most appropriate for sparse graphs, an \Omega\Gamma jVj 1=3 ) lower bound is proved on the number of queries and the time required for testing, where V...
Distributionfree property testing
 Proc. RANDOM
, 2003
"... I would like to thank Eyal for his endless guidance, devotion and encouragement, and above all for his support and sense of humor. I will like to thank my wonderful family that accompanies me in every road I take, and unconditionally supports me in every decision I make. Especially, I like to thank ..."
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Cited by 12 (0 self)
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I would like to thank Eyal for his endless guidance, devotion and encouragement, and above all for his support and sense of humor. I will like to thank my wonderful family that accompanies me in every road I take, and unconditionally supports me in every decision I make. Especially, I like to thank my husband Yuval, my partner for the journey of life, for hearing me all the time and sometimes even listening. The generous financial help of the Technion is gratefully acknowledged.