In this paper, we consider the question of determining whether a function f has property P or is e-far from any function with property P. A property testing algorithm is given a sample of the value of f on instances drawn according to some distribution. In some cases, it is also allowed to query f on instances of its choice. We study this question for different properties and establish some connections to problems in learning theory and approximation.In particular, we focus our attention on testing graph properties. Given access to a graph G in the form of being able to query whether an edge exists or not between a pair of vertices, we devise algorithms to test whether the underlying graph has properties such as being bipartite, k-Colorable, or having a p-Clique (clique of density p with respect to the vertex set). Our graph property testing algorithms are probabilistic and make assertions that are correct with high probability, while making a number of queries that is independent of the size of the graph. 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 that correspond to the property being tested, if it holds for the input graph.

Property Testing and its connection to Learning and Approximation

Boolean functions are perhaps the most basic objects of study in theoretical computer science. They also arise in other areas of mathematics, including combinatorics, statistical physics, and mathematical social choice. The field of analysis of Boolean functions seeks to understand them via their Fourier transform and other analytic methods. This text gives a thorough overview of the field, beginning with the most basic definitions and proceeding to advanced topics such as hypercontractivity and isoperimetry. Each chapter includes a "highlight application" such as Arrow's theorem from economics, the Goldreich-Levin algorithm from cryptography/learning theory, Hstad's NP-hardness of approximation results, and "sharp threshold" theorems for random graph properties. The book includes roughly 450 exercises and can be used as the basis of a one-semester graduate course. It should appeal to advanced undergraduates, graduate students, and researchers in computer science theory and related mathematical fields.

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Analysis of Boolean Functions

Analysis of Boolean Functions: List of Notation

Expression.- Spectral Clustering Gene Ontology Terms to Group Genes by Function.- Dynamic De-Novo Prediction of microRNAs Associated with Cell Conditions: A Search Pruned by Expression.- Clustering Gene Expression Series with Prior Knowledge.- A Linear Time Biclustering Algorithm for Time Series Gene Expression Data.- Time-Window Analysis of Developmental Gene Expression Data with Multiple Genetic Backgrounds.- Phylogeny.- A Lookahead Branch-and-Bound Algorithm for the Maximum Quartet Consistency Problem.- Computing the Quartet Distance Between Trees of Arbitrary Degree.- Using Semi-definite Programming to Enhance Supertree Resolvability.- An Efficient Reduction from Constrained to Unconstrained Maximum Agreement Subtree.- Pattern Identification in Biogeography.- On the Complexity of Several Haplotyping Problems.- A Hidden Markov Technique for Haplotype Reconstruction.- Algorithms for Imperfect Phylogeny Haplotyping (IPPH) with a Single Homoplasy or Recombination Event.- Networks.- A Faster Algorithm for Detecting Network Motifs.- Reaction Motifs in Metabolic Networks.- Reconstructing Metabolic Networks Using Interval Analysis.- Genome Rearrangements.- A 1.375-Approximation Algorithm for Sorting by Transpositions.- A New Tight Upper Bound on the Transposition Distance.- Perfect Sorting by Reversals Is Not Always Difficult.- Minimum Recombination Histories by Branch and Bound.- Sequences.- A Unifying Framework for Seed Sensitivity and Its Application to Subset Seeds.- Generalized Planted (l,d)-Motif Problem with Negative Set.- Alignment of Tandem Repeats with Excision, Duplication, Substitution and Indels (EDSI).- The Peres-Shields Order Estimator for Fixed and Variable Length Markov Models with Applications to DNA Sequence Similarity.- Multiple Structural RNA Alignment with Lagrangian Relaxation.- Faster Algorithms for Optimal Multiple Sequence Alignment Based on Pairwise Comparisons.- Ortholog Clustering on a Multipartite Graph.- Linear Time Algorithm for Parsing RNA Secondary Structure.- A Compressed Format for Collections of Phylogenetic Trees and Improved Consensus Performance.- Structure.- Optimal Protein Threading by Cost-Splitting.- Efficient Parameterized Algorithm for Biopolymer Structure-Sequence Alignment.- Rotamer-Pair Energy Calculations Using a Trie Data Structure.- Improved Maintenance of Molecular Surfaces Using Dynamic Graph Connectivity.- The Main Structural Regularities of the Sandwich Proteins.- Discovery of Protein Substructures in EM Maps.

Algorithms in Bioinformatics: 5th International Workshop, WABI 2005, Mallorca, Spain, October 3-6, 2005, Proceedings (Lecture Notes in Computer Science / Lecture Notes in Bioinformatics)

1 Introduction.- 2 Experiments, Deficiencies, Distances v.- 2.1 Comparing Risk Functions.- 2.2 Deficiency and Distance between Experiments.- 2.3 Likelihood Ratios and Blackwell's Representation.- 2.4 Further Remarks on the Convergence of Distri butions of Likelihood Ratios.- 2.5 Historical Remarks.- 3 Contiguity - Hellinger Transforms.- 3.1 Contiguity.- 3.2 Hellinger Distances, Hellinger Transforms.- 3.3 Historical Remarks.- 4 Gaussian Shift and Poisson Experiments.- 4.1 Introduction.- 4.2 Gaussian Experiments.- 4.3 Poisson Experiments.- 4.4 Historical Remarks.- 5 Limit Laws for Likelihood Ratios.- 5.1 Introduction.- 5.2 Auxiliary Results.- 5.2.1 Lindeberg's Procedure.- 5.2.2 Levy Splittings.- 5.2.3 Paul Levy's Symmetrization Inequalities.- 5.2.4 Conditions for Shift-Compactness.- 5.2.5 A Central Limit Theorem for Infinitesimal Arrays.- 5.2.6 The Special Case of Gaussian Limits.- 5.2.7 Peano Differentiable Functions.- 5.3 Limits for Binary Experiments.- 5.4 Gaussian Limits.- 5.5 Historical Remarks.- 6 Local Asymptotic Normality.- 6.1 Introduction.- 6.2 Locally Asymptotically Quadratic Families.- 6.3 A Method of Construction of Estimates.- 6.4 Some Local Bayes Properties.- 6.5 Invariance and Regularity.- 6.6 The LAMN and LAN Conditions.- 6.7 Additional Remarks on the LAN Conditions.- 6.8 Wald's Tests and Confidence Ellipsoids.- 6.9 Possible Extensions.- 6.10 Historical Remarks.- 7 Independent, Identically Distributed Observations.- 7.1 Introduction.- 7.2 The Standard i.i.d. Case: Differentiability in Quadratic Mean.- 7.3 Some Examples.- 7.4 Some Nonparametric Considerations.- 7.5 Bounds on the Risk of Estimates.- 7.6 Some Cases Where the Number of Observations Is Random.- 7.7 Historical Remarks.- 8 On Bayes Procedures.- 8.1 Introduction.- 8.2 Bayes Procedures Behave Nicely.- 8.3 The Bernstein-von Mises Phenomenon.- 8.4 A Bernstein-von Mises Result for the i.i.d. Case.- 8.5 Bayes Procedures Behave Miserably.- 8.6 Historical Remarks.- Author Index.

Asymptotics in Statistics–Some Basic Concepts

Given a Boolean function $f$, the $f$-isomorphism testing problem requires a randomized algorithm to distinguish functions that are identical to $f$ up to relabeling of the input variables from functions that are far from being so. An important open question in property testing is to determine for which functions $f$ we can test $f$-isomorphism with a constant number of queries. Despite much recent attention to this question, essentially only two classes of functions were known to be efficiently isomorphism testable: symmetric functions and juntas. We unify and extend these results by showing that all partially symmetric functions---functions invariant to the reordering of all but a constant number of their variables---are efficiently isomorphism testable. This class of functions, first introduced by Shannon, includes symmetric functions, juntas, and many other functions as well. We conjecture that these functions are essentially the only functions efficiently isomorphism-testable. To prove our main resul...

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Partially Symmetric Functions Are Efficiently Isomorphism Testable

Monotone Boolean functions, and the monotone Boolean circuits that compute them, have been intensively studied in complexity theory. In this paper we study the structure of Boolean functions in terms of the minimum number of negations in any circuit computing them, a complexity measure that interpolates between monotone functions and the class of all functions. We study this generalization of monotonicity from the vantage point of learning theory, establishing nearly matching upper and lower bounds on the uniform-distribution learnability of circuits in terms of the number of negations they contain. Our upper bounds are based on a new structural characterization of negation-limited circuits that extends a classical result of A.A. Markov. Our lower bounds, which employ Fourier-analytic tools from hardness amplification, give new results even for circuits with no negations (i.e. monotone functions).

https://drops.dagstuhl.de/opus/volltexte/2015/5321/pdf/31.pdf/

Learning circuits with few negations

We present a new methodology for proving distribution testing lower bounds, establishing a connection between distribution testing and the simultaneous message passing (SMP) communication model. Extending the framework of Blais, Brody, and Matulef [15], we show a simple way to reduce (private-coin) SMP problems to distribution testing problems. This method allows us to prove new distribution testing lower bounds, as well as to provide simple proofs of known lower bounds.Our main result is concerned with testing identity to a specific distribution, p, given as a parameter. In a recent and influential work, Valiant and Valiant [55] showed that the sample complexity of the aforementioned problem is closely related to the e2/3-quasinorm of p. We obtain alternative bounds on the complexity of this problem in terms of an arguably more intuitive measure and using simpler proofs. More specifically, we prove that the sample complexity is essentially determined by a fundamental operator in the theory of interpolation of Banach spaces, known as Peetre’s K-functional. We show that this quantity is closely related to the size of the effective support of p (loosely speaking, the number of supported elements that constitute the vast majority of the mass of p). This result, in turn, stems from an unexpected connection to functional analysis and refined concentration of measure inequalities, which arise naturally in our reduction.

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Distribution Testing Lower Bounds via Reductions from Communication Complexity

We prove two new results about the randomized query complexity of composed functions. First, we show that the randomized composition conjecture is false: there are families of partial Boolean functions $f$ and $g$ such that $R(f\circ g)\ll R(f) R(g)$. In fact, we show that the left hand side can be polynomially smaller than the right hand side (though in our construction, both sides are polylogarithmic in the input size of $f$). 
Second, we show that for all $f$ and $g$, $R(f\circ g)=\Omega(\mathop{noisyR}(f)\cdot R(g))$, where $\mathop{noisyR}(f)$ is a measure describing the cost of computing $f$ on noisy oracle inputs. We show that this composition theorem is the strongest possible of its type: for any measure $M(\cdot)$ satisfying $R(f\circ g)=\Omega(M(f)R(g))$ for all $f$ and $g$, it must hold that $\mathop{noisyR}(f)=\Omega(M(f))$ for all $f$. We also give a clean characterization of the measure $\mathop{noisyR}(f)$: it satisfies $\mathop{noisyR}(f)=\Theta(R(f\circ gapmaj_n)/R(gapmaj_n))$, where $n$ is the input size of $f$ and $gapmaj_n$ is the $\sqrt{n}$-gap majority function on $n$ bits.

A Tight Composition Theorem for the Randomized Query Complexity of Partial Functions.

A function f:l −1,1rn → l −1,1r is a k-junta if it depends on at most k of its variables. We consider the problem of tolerant testing of k-juntas, where the testing algorithm must accept any function that is e-close to some k-junta and reject any function that is e′-far from every k′-junta for some e′ = O(e) and k′ = O(k).Our first result is an algorithm that solves this problem with query complexity polynomial in k and 1/e. This result is obtained via a new polynomial-time approximation algorithm for submodular function minimization (SFM) under large cardinality constraints, which holds even when only given an approximate oracle access to the function.Our second result considers the case where k′ = k. We show how to obtain a smooth tradeoff between the amount of tolerance and the query complexity in this setting. Specifically, we design an algorithm that, given ρ ∈ (0,1), accepts any function that is e ρ/16-close to some k-junta and rejects any function that is e-far from every k-junta. The query complexity of the algorithm is O (k log k/e ρ (1-ρ)k.Finally, we show how to apply the second result to the problem of tolerant isomorphism testing between two unknown Boolean functions f and g. We give an algorithm for this problem whose query complexity only depends on the (unknown) smallest k such that either f or g is close to being a k-junta.

Eric Blais

Papers

Partially Symmetric Functions Are Efficiently Isomorphism Testable

Learning circuits with few negations

Distribution Testing Lower Bounds via Reductions from Communication Complexity

A Tight Composition Theorem for the Randomized Query Complexity of Partial Functions.

Tolerant Junta Testing and the Connection to Submodular Optimization and Function Isomorphism