Sample-Based High-Dimensional Convexity Testing

In the problem of high-dimensional convexity testing, there is an unknown set S in the n-dimensional Euclidean space which is promised to be either convex or c-far from every convex body with respect to the standard multivariate normal distribution. The job of a testing algorithm is then to distinguish between these two cases while making as few inspections of the set S as possible. In this work we consider sample-based testing algorithms, in which the testing algorithm only has access to labeled samples (x,S(x)) where each x is independently drawn from the normal distribution. We give nearly matching sample complexity upper and lower bounds for both one-sided and two-sided convexity testing algorithms in this framework. For constant c, our results show that the sample complexity of one-sided convexity testing is exponential in n, while for two-sided convexity testing it is exponential in the square root of n

Testing Convexity and Acyclicity, and New Constructions for Dense Graph Embeddings

Property testing, especially that of geometric and graph properties, is an ongoing area of research. In this thesis, we present a result from each of the two areas. For the problem of convexity testing in high dimensions, we give nearly matching upper and lower bounds for the sample complexity of algorithms have one-sided and two-sided error, where algorithms only have access to labeled samples independently drawn from the standard multivariate Gaussian. In the realm of graph property testing, we give an improved lower bound for testing acyclicity in directed graphs of bounded degree. Central to the area of topological graph theory is the genus parameter, but the complexity of determining the genus of a graph is poorly understood when graphs become nearly complete. We summarize recent progress in understanding the space of minimum genus embeddings of such dense graphs. In particular, we classify all possible face distributions realizable by minimum genus embeddings of complete graphs, present new constructions for genus embeddings of the complete graphs, and find unified constructions for minimum triangulations of surfaces

Earthmover Resilience and Testing in Ordered Structures

One of the main challenges in property testing is to characterize those properties that are testable with a constant number of queries. For unordered structures such as graphs and hypergraphs this task has been mostly settled. However, for ordered structures such as strings, images, and ordered graphs, the characterization problem seems very difficult in general. In this paper, we identify a wide class of properties of ordered structures - the earthmover resilient (ER) properties - and show that the "good behavior" of such properties allows us to obtain general testability results that are similar to (and more general than) those of unordered graphs. A property P is ER if, roughly speaking, slight changes in the order of the elements in an object satisfying P cannot make this object far from P. The class of ER properties includes, e.g., all unordered graph properties, many natural visual properties of images, such as convexity, and all hereditary properties of ordered graphs and images. A special case of our results implies, building on a recent result of Alon and the authors, that the distance of a given image or ordered graph from any hereditary property can be estimated (with good probability) up to a constant additive error, using a constant number of queries

A structural theorem for local algorithms with applications to coding, testing, and privacy

We prove a general structural theorem for a wide family of local algorithms, which includes property testers, local decoders, and PCPs of proximity. Namely, we show that the structure of every algorithm that makes $q$ adaptive queries and satisfies a natural robustness condition admits a sample-based algorithm with $n^{1- 1/O(q^2 \log^2 q)}$ sample complexity, following the definition of Goldreich and Ron (TOCT 2016). We prove that this transformation is nearly optimal. Our theorem also admits a scheme for constructing privacy-preserving local algorithms. %Along the way, we obtain a sunflower-based combinatorial representation of robust local algorithms. Using the unified view that our structural theorem provides, we obtain results regarding various types of local algorithms, including the following. - We strengthen the state-of-the-art lower bound for relaxed locally decodable codes, obtaining an \emph{exponential} improvement on the dependency in query complexity; this resolves an open problem raised by Gur and Lachish (SICOMP 2021). - We show that any (constant-query) testable property admits a sample-based tester with sublinear sample complexity; this resolves a problem left open in a work of Fischer, Lachish, and Vasudev (FOCS 2015), bypassing an exponential blowup caused by previous techniques in the case of adaptive testers. - We prove that the known separation between proofs of proximity and testers is essentially maximal; this resolves a problem left open by Gur and Rothblum (ECCC 2013, Computational Complexity 2018) regarding sublinear-time delegation of computation. Our techniques strongly rely on relaxed sunflower lemmas and the Hajnal–Szemer\'{e}di theorem

A structural theorem for local algorithms with applications to coding, testing, and privacy

We prove a general structural theorem for a wide family of local algorithms, which includes property testers, local decoders, and PCPs of proximity. Namely, we show that the structure of every algorithm that makes q adaptive queries and satisfies a natural robustness condition admits a sample-based algorithm with n 1−1/O(q 2 log2 q) sample complexity, following the definition of Goldreich and Ron (TOCT 2016). We prove that this transformation is nearly optimal. Our theorem also admits a scheme for constructing privacy-preserving local algorithms. Using the unified view that our structural theorem provides, we obtain results regarding various types of local algorithms, including the following. We strengthen the state-of-the-art lower bound for relaxed locally decodable codes, obtaining an exponential improvement on the dependency in query complexity; this resolves an open problem raised by Gur and Lachish (SODA 2020). We show that any (constant-query) testable property admits a sample-based tester with sublinear sample complexity; this resolves a problem left open in a work of Fischer, Lachish, and Vasudev (FOCS 2015) by extending their main result to adaptive testers. We prove that the known separation between proofs of proximity and testers is essentially maximal; this resolves a problem left open by Gur and Rothblum (ECCC 2013, Computational Complexity 2018) regarding sublinear-time delegation of computation. Our techniques strongly rely on relaxed sunflower lemmas and the Hajnal–Szemer´edi theorem

Complexity of Sublinear Algorithms for Convexity in Higher Dimensions

Convexity plays a prominent role in both mathematics and computer science. It is defined for sets and functions, and many related problems can be solved efficiently if the set/function is convex. In this thesis, we focus on three problems related to convexity where we don't have that guarantee. The first problem we consider is the decision problem: is a given unknown set convex? We study it under the framework of property testing. In property testing, instead of distinguishing objects that satisfy the property from the ones that do not, we distinguish objects that satisfy the property from the ones that are ``far" from satisfying the property. We approach the problem of testing convex sets by studying a closely related problem of robust characterizations of convex sets, in which we study different characterizations of convex sets and determine their robustness. We say a characterization is robust, if sets that are ``far" from convex are also ``far" from satisfying the characterization. We examine the robustness of three characterizations of convex sets: line characterization, convex hull characterization and supporting hyperplane characterization. Furthermore, we discuss the implications of the robustness/non-robustness of these characterizations on the testing problem. The second problem we consider is the decision problem: is a given unknown function over the hypergrid [n]^d convex? The function is given via a valuation oracle, which we query to get the function value. We study this problem also under the framework of property testing. We refer to an algorithm that performs the testing task as non-adaptive if it submits all its queries before looking at the function value on any of the points, and adaptive otherwise. We show that any non-adaptive algorithm that tests convexity of functions f: [n]^d → ℝ for d ≥ 2, has query complexity that is linear in n and exponential in d. To understand if adaptivity helps, we consider the problem of testing convexity of functions over a stripe, [3] ⨉ [n], and show that there exists an adaptive testing algorithm that does exponentially better than any non-adaptive one. The third problem we consider is minimizing convex functions over the hypergrid when given access to the comparison oracle to the function on the points in the hypergrid. The comparison oracle to a function takes as input two points and determines which of them has the smaller function value. The discrete (non-convex) nature of the domain makes the problem challenging, since we do not have the property that local minimum implies global minimum, which is crucial for the minimization of convex functions over continuous domains. We, however, show that there is a minimization algorithm in two dimensions with polynomial time and query complexity, and for any constant dimension greater than two, there is a minimization algorithm with quasi-polynomial time and query complexity

Testing, Learning, Sampling, Sketching

We study several problems about sublinear algorithms, presented in two parts. Part I: Property testing and learning. There are two main goals of research in property testing and learning theory. The first is to understand the relationship between testing and learning, and the second is to develop efficient testing and learning algorithms. We present results towards both goals. - An oft-repeated motivation for property testing algorithms is to help with model selection in learning: to efficiently check whether the chosen hypothesis class (i.e. learning model) will successfully learn the target function. We present in this thesis a proof that, for many of the most useful and natural hypothesis classes (including halfspaces, polynomial threshold functions, intersections of halfspaces, etc.), the sample complexity of testing in the distribution-free model is nearly equal to that of learning. This shows that testing does not give a significant advantage in model selection in this setting. - We present a simple and general technique for transforming testing and learning algorithms designed for the uniform distribution over {0, 1}^d or [n]^d into algorithms that work for arbitrary product distributions over R d . This leads to an improvement and simplification of state-of-the-art results for testing monotonicity, learning intersections of halfspaces, learning polynomial threshold functions, and others. Part II. Adjacency and distance sketching for graphs. We initiate the thorough study of adjacency and distance sketching for classes of graphs. Two open problems in sublinear algorithms are: 1) to understand the power of randomization in communication; and 2) to characterize the sketchable distance metrics. We observe that constant-cost randomized communication is equivalent to adjacency sketching in a hereditary graph class, which in turn implies the existence of an efficient adjacency labeling scheme, the subject of a major open problem in structural graph theory. Therefore characterizing the adjacency sketchable graph classes (i.e. the constant-cost communication problems) is the probabilistic equivalent of this open problem, and an essential step towards understanding the power of randomization in communication. This thesis gives the first results towards a combined theory of these problems and uses this connection to obtain optimal adjacency labels for subgraphs of Cartesian products, resolving some questions from the literature. More generally, we begin to develop a theory of graph sketching for problems that generalize adjacency, including different notions of distance sketching. This connects the well-studied areas of distance sketching in sublinear algorithms, and distance labeling in structural graph theory