Density of Spherically-Embedded Stiefel and Grassmann Codes

The density of a code is the fraction of the coding space covered by packing balls centered around the codewords. This paper investigates the density of codes in the complex Stiefel and Grassmann manifolds equipped with the chordal distance. The choice of distance enables the treatment of the manifolds as subspaces of Euclidean hyperspheres. In this geometry, the densest packings are not necessarily equivalent to maximum-minimum-distance codes. Computing a code's density follows from computing: i) the normalized volume of a metric ball and ii) the kissing radius, the radius of the largest balls one can pack around the codewords without overlapping. First, the normalized volume of a metric ball is evaluated by asymptotic approximations. The volume of a small ball can be well-approximated by the volume of a locally-equivalent tangential ball. In order to properly normalize this approximation, the precise volumes of the manifolds induced by their spherical embedding are computed. For larger balls, a hyperspherical cap approximation is used, which is justified by a volume comparison theorem showing that the normalized volume of a ball in the Stiefel or Grassmann manifold is asymptotically equal to the normalized volume of a ball in its embedding sphere as the dimension grows to infinity. Then, bounds on the kissing radius are derived alongside corresponding bounds on the density. Unlike spherical codes or codes in flat spaces, the kissing radius of Grassmann or Stiefel codes cannot be exactly determined from its minimum distance. It is nonetheless possible to derive bounds on density as functions of the minimum distance. Stiefel and Grassmann codes have larger density than their image spherical codes when dimensions tend to infinity. Finally, the bounds on density lead to refinements of the standard Hamming bounds for Stiefel and Grassmann codes.Comment: Two-column version (24 pages, 6 figures, 4 tables). To appear in IEEE Transactions on Information Theor

Sherali-Adams gaps, flow-cover inequalities and generalized configurations for capacity-constrained Facility Location

Metric facility location is a well-studied problem for which linear programming methods have been used with great success in deriving approximation algorithms. The capacity-constrained generalizations, such as capacitated facility location (CFL) and lower-bounded facility location (LBFL), have proved notorious as far as LP-based approximation is concerned: while there are local-search-based constant-factor approximations, there is no known linear relaxation with constant integrality gap. According to Williamson and Shmoys devising a relaxation-based approximation for \cfl\ is among the top 10 open problems in approximation algorithms. This paper advances significantly the state-of-the-art on the effectiveness of linear programming for capacity-constrained facility location through a host of impossibility results for both CFL and LBFL. We show that the relaxations obtained from the natural LP at $\Omega(n)$ levels of the Sherali-Adams hierarchy have an unbounded gap, partially answering an open question of \cite{LiS13, AnBS13}. Here, $n$ denotes the number of facilities in the instance. Building on the ideas for this result, we prove that the standard CFL relaxation enriched with the generalized flow-cover valid inequalities \cite{AardalPW95} has also an unbounded gap. This disproves a long-standing conjecture of \cite{LeviSS12}. We finally introduce the family of proper relaxations which generalizes to its logical extreme the classic star relaxation and captures general configuration-style LPs. We characterize the behavior of proper relaxations for CFL and LBFL through a sharp threshold phenomenon.Comment: arXiv admin note: substantial text overlap with arXiv:1305.599

Optimal Rates of Statistical Seriation

Given a matrix the seriation problem consists in permuting its rows in such way that all its columns have the same shape, for example, they are monotone increasing. We propose a statistical approach to this problem where the matrix of interest is observed with noise and study the corresponding minimax rate of estimation of the matrices. Specifically, when the columns are either unimodal or monotone, we show that the least squares estimator is optimal up to logarithmic factors and adapts to matrices with a certain natural structure. Finally, we propose a computationally efficient estimator in the monotonic case and study its performance both theoretically and experimentally. Our work is at the intersection of shape constrained estimation and recent work that involves permutation learning, such as graph denoising and ranking.Comment: V2 corrects an error in Lemma A.1, v3 corrects appendix F on unimodal regression where the bounds now hold with polynomial probability rather than exponentia