Bounded Independence Fools Degree-2 Threshold Functions

Let x be a random vector coming from any k-wise independent distribution over {-1,1}^n. For an n-variate degree-2 polynomial p, we prove that E[sgn(p(x))] is determined up to an additive epsilon for k = poly(1/epsilon). This answers an open question of Diakonikolas et al. (FOCS 2009). Using standard constructions of k-wise independent distributions, we obtain a broad class of explicit generators that epsilon-fool the class of degree-2 threshold functions with seed length log(n)*poly(1/epsilon). Our approach is quite robust: it easily extends to yield that the intersection of any constant number of degree-2 threshold functions is epsilon-fooled by poly(1/epsilon)-wise independence. Our results also hold if the entries of x are k-wise independent standard normals, implying for example that bounded independence derandomizes the Goemans-Williamson hyperplane rounding scheme. To achieve our results, we introduce a technique we dub multivariate FT-mollification, a generalization of the univariate form introduced by Kane et al. (SODA 2010) in the context of streaming algorithms. Along the way we prove a generalized hypercontractive inequality for quadratic forms which takes the operator norm of the associated matrix into account. These techniques may be of independent interest.Comment: Using v1 numbering: removed Lemma G.5 from the Appendix (it was wrong). Net effect is that Theorem G.6 reduces the m^6 dependence of Theorem 8.1 to m^4, not m^

What determines self-employment? : a comparative study

This article consists of a comparative study of the incidence of self-employment (SE) between Greece, which has the highest rate of SE in the European Union and the United Kingdom, which has amongst the lowest. Data from the Greek and the UK Labour Force Surveys are used in order to assess how personal attributes of an individual have an impact on the incidence of SE. It is found that common patterns exist between these two countries. In particular, it is found that for both countries, males have greater odds of being self-employed than females, older people have greater odds than younger, individuals employed in the primary and tertiary sectors have greater odds than the ones employed in the secondary, and that individuals with primary or secondary education have greater odds of being self-employed than individuals holding higher degrees. The incidence of SE is also found to differ according to the occupation of the individual. On the other hand, the findings indicate that individuals, residing in London, have greater odds of being self-employed than individuals working outside UK's capital, whereas in Greece the pattern is reversed