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Register a new userConvex Hulls, Oracles, and Homology
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This paper presents a new algorithm for the convex hull problem, which is based on a reduction to a combinatorial decision problem POLYTOPE-COMPLETENESS-COMBINATORIAL, which in turn can be solved by a simplicial homology computation. Like other convex hull algorithms, our algorithm is polynomial (in the size of input plus output) for simplicial or simple input. We show that the ``no-case of POLYTOPE-COMPLETENESS-COMBINATORIAL has a certificate that can be checked in polynomial time (if integrity of the input is guaranteed).
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Let $X_1,ldots,X_n$ be independent random points that are distributed according to a probability measure on $mathbb{R}^d$ and let $P_n$ be the random convex hull generated by $X_1,ldots,X_n$ ($ngeq d+1$). Natural classes of probability distributions are characterized for which, by means of Blaschke-Petkantschin formulae from integral geometry, one can show that the mean facet number of $P_n$ is strictly monotonically increasing in $n$.
Let $xi_1,xi_2,ldots$ be a sequence of independent copies of a random vector in $mathbb R^d$ having an absolutely continuous distribution. Consider a random walk $S_i:=xi_1+cdots+xi_i$, and let $C_{n,d}:=text{conv}(0,S_1,S_2,ldots,S_n)$ be the convex hull of the first $n+1$ points it has visited. The polytope $C_{n,d}$ is called $k$-neighborly if for every indices $0leq i_0 <cdots < i_kleq n$ the convex hull of the $k+1$ points $S_{i_0},ldots, S_{i_k}$ is a $k$-dimensional face of $C_{n,d}$. We study the probability that $C_{n,d}$ is $k$-neighborly in various high-dimensional asymptotic regimes, i.e. when $n$, $d$, and possibly also $k$ diverge to $infty$. There is an explicit formula for the expected number of $k$-dimensional faces of $C_{n,d}$ which involves Stirling numbers of both kinds. Motivated by this formula, we introduce a distribution, called the Lah distribution, and study its properties. In particular, we provide a combinatorial interpretation of the Lah distribution in terms of random compositions and records, and explicitly compute its factorial moments. Limit theorems which we prove for the Lah distribution imply neighborliness properties of $C_{n,d}$. This yields a new class of random polytopes exhibiting phase transitions parallel to those discovered by Vershik and Sporyshev, Donoho and Tanner for random projections of regular simplices and crosspolytopes.
Let $K in R^d$ be a convex body, and assume that $L$ is a randomly rotated and shifted integer lattice. Let $K_L$ be the convex hull of the (random) points $K cap L$. The mean width $W(K_L)$ of $K_L$ is investigated. The asymptotic order of the mean width difference $W(l K)-W((l K)_L)$ is maximized by the order obtained by polytopes and minimized by the order for smooth convex sets as $l to infty$.
We study to what extent quantum algorithms can speed up solving convex optimization problems. Following the classical literature we assume access to a convex set via various oracles, and we examine the efficiency of reductions between the different oracles. In particular, we show how a separation oracle can be implemented using $tilde{O}(1)$ quantum queries to a membership oracle, which is an exponential quantum speed-up over the $Omega(n)$ membership queries that are needed classically. We show that a quantum computer can very efficiently compute an approximate subgradient of a convex Lipschitz function. Combining this with a simplification of recent classical work of Lee, Sidford, and Vempala gives our efficient separation oracle. This in turn implies, via a known algorithm, that $tilde{O}(n)$ quantum queries to a membership oracle suffice to implement an optimization oracle (the best known classical upper bound on the number of membership queries is quadratic). We also prove several lower bounds: $Omega(sqrt{n})$ quantum separation (or membership) queries are needed for optimization if the algorithm knows an interior point of the convex set, and $Omega(n)$ quantum separation queries are needed if it does not.
We consider the convex hull of the perturbed point process comprised of $n$ i.i.d. points, each distributed as the sum of a uniform point on the unit sphere $S^{d-1}$ and a uniform point in the $d$-dimensional ball centered at the origin and of radius $n^{alpha}, alpha in (-infty, infty)$. This model, inspired by the smoothed complexity analysis introduced in computational geometry cite{DGGT,ST}, is a perturbation of the classical random polytope. We show that the perturbed point process, after rescaling, converges in the scaling limit to one of five Poisson point processes according to whether $alpha$ belongs to one of five regimes. The intensity measure of the limit Poisson point process undergoes a transition at the values $alpha = frac{-2} {d -1}$ and $alpha = frac{2} {d + 1}$ and it gives rise to four rescalings for the $k$-face functional on perturbed data. These rescalings are used to establish explicit expectation asymptotics for the number of $k$-dimensional faces of the convex hull of either perturbed binomial or Poisson data. In the case of Poisson input, we establish explicit variance asymptotics and a central limit theorem for the number of $k$-dimensional faces. Finally it is shown that the rescaled boundary of the convex hull of the perturbed point process converges to the boundary of a parabolic hull process.
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