Robust cosmological inference from non-linear scales with k-th nearest neighbor statistics

We present the methodology for deriving accurate and reliable cosmological constraints from non-linear scales (<50Mpc/h) with k-th nearest neighbor (kNN) statistics. We detail our methods for choosing robust minimum scale cuts and validating galaxy-halo connection models. Using cross-validation, we identify the galaxy-halo model that ensures both good fits and unbiased predictions across diverse summary statistics. We demonstrate that we can model kNNs effectively down to transverse scales of rp ~ 3Mpc/h and achieve precise and unbiased constraints on the matter density and clustering amplitude, leading to a 2% constraint on sigma_8. Our simulation-based model pipeline is resilient to varied model systematics, spanning simulation codes, halo finding, and cosmology priors. We demonstrate the effectiveness of this approach through an application to the Beyond-2p mock challenge. We propose further explorations to test more complex galaxy-halo connection models and tackle potential observational systematics.Comment: 18 pages, 16 figures, submitted to MNRAS, comments welcom

Bilu-Linial Stable Instances of Max Cut and Minimum Multiway Cut

We investigate the notion of stability proposed by Bilu and Linial. We obtain an exact polynomial-time algorithm for $\gamma$-stable Max Cut instances with $\gamma \geq c\sqrt{\log n}\log\log n$ for some absolute constant $c > 0$. Our algorithm is robust: it never returns an incorrect answer; if the instance is $\gamma$-stable, it finds the maximum cut, otherwise, it either finds the maximum cut or certifies that the instance is not $\gamma$-stable. We prove that there is no robust polynomial-time algorithm for $\gamma$-stable instances of Max Cut when $\gamma < \alpha_{SC}(n/2)$, where $\alpha_{SC}$ is the best approximation factor for Sparsest Cut with non-uniform demands. Our algorithm is based on semidefinite programming. We show that the standard SDP relaxation for Max Cut (with $\ell_2^2$ triangle inequalities) is integral if $\gamma \geq D_{\ell_2^2\to \ell_1}(n)$, where $D_{\ell_2^2\to \ell_1}(n)$ is the least distortion with which every $n$ point metric space of negative type embeds into $\ell_1$. On the negative side, we show that the SDP relaxation is not integral when $\gamma < D_{\ell_2^2\to \ell_1}(n/2)$. Moreover, there is no tractable convex relaxation for $\gamma$-stable instances of Max Cut when $\gamma < \alpha_{SC}(n/2)$. That suggests that solving $\gamma$-stable instances with $\gamma =o(\sqrt{\log n})$ might be difficult or impossible. Our results significantly improve previously known results. The best previously known algorithm for $\gamma$-stable instances of Max Cut required that $\gamma \geq c\sqrt{n}$ (for some $c > 0$) [Bilu, Daniely, Linial, and Saks]. No hardness results were known for the problem. Additionally, we present an algorithm for 4-stable instances of Minimum Multiway Cut. We also study a relaxed notion of weak stability.Comment: 24 page

Iterative graph cuts for image segmentation with a nonlinear statistical shape prior

Shape-based regularization has proven to be a useful method for delineating objects within noisy images where one has prior knowledge of the shape of the targeted object. When a collection of possible shapes is available, the specification of a shape prior using kernel density estimation is a natural technique. Unfortunately, energy functionals arising from kernel density estimation are of a form that makes them impossible to directly minimize using efficient optimization algorithms such as graph cuts. Our main contribution is to show how one may recast the energy functional into a form that is minimizable iteratively and efficiently using graph cuts.Comment: Revision submitted to JMIV (02/24/13

Spanning trees with few branch vertices

A branch vertex in a tree is a vertex of degree at least three. We prove that, for all $s\geq 1$, every connected graph on $n$ vertices with minimum degree at least $(\frac{1}{s+3}+o(1))n$ contains a spanning tree having at most $s$ branch vertices. Asymptotically, this is best possible and solves, in less general form, a problem of Flandrin, Kaiser, Ku\u{z}el, Li and Ryj\'a\u{c}ek, which was originally motivated by an optimization problem in the design of optical networks.Comment: 20 pages, 2 figures, to appear in SIAM J. of Discrete Mat

New results about multi-band uncertainty in Robust Optimization

"The Price of Robustness" by Bertsimas and Sim represented a breakthrough in the development of a tractable robust counterpart of Linear Programming Problems. However, the central modeling assumption that the deviation band of each uncertain parameter is single may be too limitative in practice: experience indeed suggests that the deviations distribute also internally to the single band, so that getting a higher resolution by partitioning the band into multiple sub-bands seems advisable. The critical aim of our work is to close the knowledge gap about the adoption of a multi-band uncertainty set in Robust Optimization: a general definition and intensive theoretical study of a multi-band model are actually still missing. Our new developments have been also strongly inspired and encouraged by our industrial partners, which have been interested in getting a better modeling of arbitrary distributions, built on historical data of the uncertainty affecting the considered real-world problems. In this paper, we study the robust counterpart of a Linear Programming Problem with uncertain coefficient matrix, when a multi-band uncertainty set is considered. We first show that the robust counterpart corresponds to a compact LP formulation. Then we investigate the problem of separating cuts imposing robustness and we show that the separation can be efficiently operated by solving a min-cost flow problem. Finally, we test the performance of our new approach to Robust Optimization on realistic instances of a Wireless Network Design Problem subject to uncertainty.Comment: 15 pages. The present paper is a revised version of the one appeared in the Proceedings of SEA 201

Non-Uniform Robust Network Design in Planar Graphs

Robust optimization is concerned with constructing solutions that remain feasible also when a limited number of resources is removed from the solution. Most studies of robust combinatorial optimization to date made the assumption that every resource is equally vulnerable, and that the set of scenarios is implicitly given by a single budget constraint. This paper studies a robustness model of a different kind. We focus on \textbf{bulk-robustness}, a model recently introduced~\cite{bulk} for addressing the need to model non-uniform failure patterns in systems. We significantly extend the techniques used in~\cite{bulk} to design approximation algorithm for bulk-robust network design problems in planar graphs. Our techniques use an augmentation framework, combined with linear programming (LP) rounding that depends on a planar embedding of the input graph. A connection to cut covering problems and the dominating set problem in circle graphs is established. Our methods use few of the specifics of bulk-robust optimization, hence it is conceivable that they can be adapted to solve other robust network design problems.Comment: 17 pages, 2 figure