Low-degree tests at large distances

We define tests of boolean functions which distinguish between linear (or quadratic) polynomials, and functions which are very far, in an appropriate sense, from these polynomials. The tests have optimal or nearly optimal trade-offs between soundness and the number of queries. In particular, we show that functions with small Gowers uniformity norms behave ``randomly'' with respect to hypergraph linearity tests. A central step in our analysis of quadraticity tests is the proof of an inverse theorem for the third Gowers uniformity norm of boolean functions. The last result has also a coding theory application. It is possible to estimate efficiently the distance from the second-order Reed-Muller code on inputs lying far beyond its list-decoding radius

A Hypergraph Dictatorship Test with Perfect Completeness

A hypergraph dictatorship test is first introduced by Samorodnitsky and Trevisan and serves as a key component in their unique games based \PCP construction. Such a test has oracle access to a collection of functions and determines whether all the functions are the same dictatorship, or all their low degree influences are $o(1).$ Their test makes $q\geq3$ queries and has amortized query complexity $1+O(\frac{\log q}{q})$ but has an inherent loss of perfect completeness. In this paper we give an adaptive hypergraph dictatorship test that achieves both perfect completeness and amortized query complexity $1+O(\frac{\log q}{q})$.Comment: Some minor correction

Sublinear-Time Computation in the Presence of Online Erasures

We initiate the study of sublinear-time algorithms that access their input via an online adversarial erasure oracle. After answering each query to the input object, such an oracle can erase $t$ input values. Our goal is to understand the complexity of basic computational tasks in extremely adversarial situations, where the algorithm's access to data is blocked during the execution of the algorithm in response to its actions. Specifically, we focus on property testing in the model with online erasures. We show that two fundamental properties of functions, linearity and quadraticity, can be tested for constant $t$ with asymptotically the same complexity as in the standard property testing model. For linearity testing, we prove tight bounds in terms of $t$, showing that the query complexity is $\Theta(\log t)$. In contrast to linearity and quadraticity, some other properties, including sortedness and the Lipschitz property of sequences, cannot be tested at all, even for $t=1$. Our investigation leads to a deeper understanding of the structure of violations of linearity and other widely studied properties. We also consider implications of our results for algorithms that are resilient to online adversarial corruptions instead of erasures

Property Testing with Online Adversaries

The online manipulation-resilient testing model, proposed by Kalemaj, Raskhodnikova and Varma (ITCS 2022 and Theory of Computing 2023), studies property testing in situations where access to the input degrades continuously and adversarially. Specifically, after each query made by the tester is answered, the adversary can intervene and either erase or corrupt $t$ data points. In this work, we investigate a more nuanced version of the online model in order to overcome old and new impossibility results for the original model. We start by presenting an optimal tester for linearity and a lower bound for low-degree testing of Boolean functions in the original model. We overcome the lower bound by allowing batch queries, where the tester gets a group of queries answered between manipulations of the data. Our batch size is small enough so that function values for a single batch on their own give no information about whether the function is of low degree. Finally, to overcome the impossibility results of Kalemaj et al. for sortedness and the Lipschitz property of sequences, we extend the model to include $t<1$, i.e., adversaries that make less than one erasure per query. For sortedness, we characterize the rate of erasures for which online testing can be performed, exhibiting a sharp transition from optimal query complexity to impossibility of testability (with any number of queries). Our online tester works for a general class of local properties of sequences. One feature of our results is that we get new (and in some cases, simpler) optimal algorithms for several properties in the standard property testing model.Comment: To be published in 15th Innovations in Theoretical Computer Science (ITCS 2024

The Gowers norm in the testing of Boolean functions

Thesis (Ph. D.)--Massachusetts Institute of Technology, Dept. of Mathematics, 2009.This electronic version was submitted by the student author. The certified thesis is available in the Institute Archives and Special Collections.Includes bibliographical references (p. 65-68).A property tester is a fast, randomized algorithm that reads only a few entries of the input, and based on the values of these entries, it distinguishes whether the input has a certain property or is "different" from any input having this property. Furthermore, we say that a property tester has completeness c and soundness s if it accepts all inputs having the property with probability at least c and accepts "different" inputs with probability at most s + o(1). In this thesis we present two property testers for boolean functions on the boolean cube f0; 1gn. We summarize our contribution as follows. We present a new dictatorship test that determines whether the function is a dictator (of the form f(x) = xi for some coordinate i), or a function that is an "anti-dictator." Our test is "adaptive," makes q queries, has completeness 1, and soundness O(q3) 2??q. Previously, a dictatorship test that has soundness (q + 1) . 2-q is achieved by Samorodnitsky and Trevisan, but their test has completeness strictly less than 1. Furthermore, the previously best known dictatorship test from the PCP literature with completeness 1 has soundness ... . Our contribution lies in achieving perfect completeness and low sound- ness simultaneously. We consider properties of functions that are invariant under linear transformations of the boolean cube. Previous works, such as linearity testing and low-degree testing, have focused on linear properties.(cont.) The one exception is a test due to Green for "triangle freeness": a function f satisfies this property if f(x); f(y); f(x + y) do not all equal 1, for any pair x; y 2 f0; 1gn. We extend this test to a more systematic study and consider non-linear properties that are described by a single forbidden pattern. Specifically, let M denote an r by k matrix over f0; 1g. We say that a function f is M-free if there are no ~x = (x1,...,xk), where x1,...,xk 2 f0; 1gn such that f(x1),...,f(xk) = 1 and M~x = ~0. If M can be represented by an underlying graph, we can analyze a test that determines whether a function is M-free or \far" from one. Our test makes k queries, has completeness 1, and soundness bounded away from 1. The technique from our work leads to alternate proofs that some previously studied linear properties are testable, albeit with worse parameters. Our results, though quite different in terms of context, are connected by similar techniques. Our analysis of the algorithms relies on the machinery of the Gowers uniformity norm, a recent and powerful tool in additive combinatorics.by Victor Yen-Wen Chen.Ph.D

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

Approximation Algorithms for NP-Hard Problems

The workshop was concerned with the most important recent developments in the area of efficient approximation algorithms for NP-hard optimization problems as well as with new techniques for proving intrinsic lower bounds for efficient approximation

Toward constructing a psychosocial model of career wellbeing for the South African working adult

In this research, a cross-sectional quantitative survey was conducted on a convenience sample of working adults (N = 550) from different race, gender, age, qualification, job level and tenure groups in various South African organisations, with the intention of developing a psychosocial model of career wellbeing for the working adult in the South African context. To identify the elements and nature of the model, the relationship dynamics between the constructs of occupational passion, psychological career resources, psychosocial career preoccupations and the outcome of career satisfaction were examined. The mediating effect of psychological career resources and psychosocial career preoccupations on the relationship dynamics between occupational passion and career satisfaction was determined. The moderating effect of certain sociodemographic variables (race, gender, age, qualification, job level, and tenure) on the relationship dynamics between the research constructs was measured. In addition, an evaluation of the differences manifested by individuals from various sociodemographic backgrounds (race, gender, age, qualification, job level, and tenure) regarding the research constructs added to an understanding of the manifested model. Correlation and inferential statistical analyses (multi-level mediation modelling, regression analysis and tests for significant mean differences) indicated that career management practices should consider harmonious passion to be an important intrinsic motivational antecedent in explaining the variance in individuals’ career satisfaction as it can facilitate the development of important psychosocial resources. These resources include flexible career preferences, well-crafted career plans and actions to achieve career goals, career harmonisers (i.e. self-esteem, behavioural adaptability and emotional literacy) and a strong need to be upskilled and employable (career adaptation needs). These elements manifested as the core elements of the psychosocial career wellbeing profile. Job level and race were further indicated as important sociodemographic variables in explaining levels of career satisfaction. Differences between race, gender, age, qualification, job level and tenure groups, for the constructs of occupational passion, psychological career resources, psychosocial career preoccupations and career satisfaction, also need to be considered in the career wellbeing profile. Theoretically, the results advanced career theory by empirically validating the core elements of the career wellbeing profile. These may be applied to inform career management practices and consequently enhance the career wellbeing of working adults.Industrial and Organisational PsychologyPh. D. (Psychology (Industrial and Organisational Psychology)

Reduction in mesenchymal stem cell numbers in premature aging DNA repair deficient TTD mice

Background: Mice carrying mutations in DNA repair genes often show signs of accelerated ageing and therefore can be used as a model system to study age related diseases like osteoporosis. It has been shown that TTD mice, carrying a mutation in the nucleotide excision repair gene XPD (xeroderma pigmentosa group D), display features of ageing related osteoporosis as well as adipose tissue hypoplasia. Since both cell types involved, osteoblasts as well as adipocytes, arise from the same mesenchymal stem cell population, the aim of the current project was to study the number, proliferation and differentiation potential of these cells in TTD compared to wild type (WT) mice. This might provide us with useful information concerning the mechanism behind age-related osteoporosis and the loss of adipose tissue.Methods: Bone marrow from old TTD and WT mice was cultured under osteogenic or adipogenic conditions and analysed for alkaline phosphatase activity (ALP), mineralisation (osteoblast) and lipid deposition (adipocyte).Results: Under osteogenic conditions the number of ALP-positive colonies after 9 and 14 days of culture was significantly decreased (p=0.02) in TTD compared to WT mice. The rate at which new ALP-positive colonies are formed between day 9 and day 14 of culture has not changed between TTD and WT mice, indicating that the decrease in colony number is not due to a delay in differentiation. Mineralisation of ALP-positive colonies did not seem to be affected, with a borderline significant decrease on day 14 at the onset of mineralisation but no significant changes on day 21 of culture. Lipid deposition was strongly reduced in TTD compared to WT mice (p=0.01) after 35 days of culture.Conclusions: The observed reduction in osteoblast and adipocyte differentiation indicates a reduction of mesenchymal stem cell numbers in TTD mice. This reduction in mesenchymal stem cell numbers and the corresponding decline in osteoblast differentiation could explain the premature osteoporotic features observed in TTD mice. In line with this, the reduction of mesenchymal stem cells and adipocyte differentiation may underlie the adipose tissue hypoplasia observed in TTD mice