We show that for random bit strings, Up(n), with probability, p = 12 , the firstorder quantifier depth D(Up(n)) needed to distinguish non-isomorphic structures is Θ(lg lg n), with high probability. Further, we show that, with high probability, for random ordered graphs, G≤,p(n) with edge probabiltiy p = 12 , D(G≤,p(n)) = Θ(log∗ n), contrasting with the results of random (non-ordered) graphs, Gp...

We consider the setting of linear regression in high dimension. We focus on the problem of constructing adaptive and honest confidence sets for the sparse parameter θ, i.e. we want to construct a confidence set for θ that contains θ with high probability, and that is as small as possible. The l2 diameter of a such confidence set should depend on the sparsity S of θ the larger S, the wider the c...

The forest-harvesting and road-construction planning problem basically consists of managing land designated for timber production and divided into harvest cells. For each time period in the given time horizon one must decide which cells to cut and what access roads to build in order to maximize expected net profit under a risk manageable scheme to control the negative impact of the solutions in...

We consider a problem introduced by Mossel and Ross [Shotgun assembly of labeled graphs, arXiv:1504.07682]. Suppose a random n×n jigsaw puzzle is constructed by independently and uniformly choosing the shape of each “jig” from q possibilities. We are given the shuffled pieces. Then, depending on q, what is the probability that we can reassemble the puzzle uniquely? We say that two solutions of ...

It is proved that the shape of the typical cell of a stationary and isotropic Poisson random hyperplane tessellation is, with high probability, close to the shape of a ball if the kth intrinsic volume (k ≥ 2) of the typical cell is large. The shape of typical cells of large diameter is close to the shape of a segment.

We present the first result for kernel regression where the procedure adapts locally at a point x to both the unknown local dimension of the metric space X and the unknown Hölder-continuity of the regression function at x. The result holds with high probability simultaneously at all points x in a general metric space X of unknown structure.

For normed isotropic convex bodies in R n we investigate the behaviour of the (n ? 1)-dimensional volume of intersections with hyperplanes orthogonal to a xed direction, considered as a function of the distance of the hyperplane to the origin. It is a conjecture that for arbitrary normed isotropic convex bodies and random directions this function { with high probability { is close to a Gaussian...

A random jigsaw puzzle is constructed by arranging n2 square pieces into an n× n grid and assigning to each edge of a piece one of q available colours uniformly at random, with the restriction that touching edges receive the same colour. We show that if q = o(n) then with high probability such a puzzle does not have a unique solution, while if q ≥ n1+ε for any constant ε > 0 then the solution i...

For a k-ary predicate P , a random instance of CSP(P ) with n variables and m constraints is unsatisfiable with high probability when m ≥ O(n). The natural algorithmic task in this regime is refutation: finding a proof that a given random instance is unsatisfiable. Recent work of Allen et al. suggests that the difficulty of refuting CSP(P ) using an SDP is determined by a parameter cmplx(P ), t...