On the complexity of partial order properties

The recognition complexity of ordered set properties is considered, i.e. how many questions have to be asked to decide if an unknown ordered set has a prescribed property. We prove a lower bound of Ω (n²) for properties that are characterized by forbidden substructures of fixed size. For the properties being connected, and having exactly k comparable paris we show that the recogintion complexity is (n:2); the complexity of interval orders is exactly (n:2) -1. Non-trivial upper bounds are given for being a lattice, containing a chain of length k ≥ 2 and having width k

On multivariate quantiles under partial orders

This paper focuses on generalizing quantiles from the ordering point of view. We propose the concept of partial quantiles, which are based on a given partial order. We establish that partial quantiles are equivariant under order-preserving transformations of the data, robust to outliers, characterize the probability distribution if the partial order is sufficiently rich, generalize the concept of efficient frontier, and can measure dispersion from the partial order perspective. We also study several statistical aspects of partial quantiles. We provide estimators, associated rates of convergence, and asymptotic distributions that hold uniformly over a continuum of quantile indices. Furthermore, we provide procedures that can restore monotonicity properties that might have been disturbed by estimation error, establish computational complexity bounds, and point out a concentration of measure phenomenon (the latter under independence and the componentwise natural order). Finally, we illustrate the concepts by discussing several theoretical examples and simulations. Empirical applications to compare intake nutrients within diets, to evaluate the performance of investment funds, and to study the impact of policies on tobacco awareness are also presented to illustrate the concepts and their use.Comment: Published in at http://dx.doi.org/10.1214/10-AOS863 the Annals of Statistics (http://www.imstat.org/aos/) by the Institute of Mathematical Statistics (http://www.imstat.org

Adaptive Low-Rank Methods for Problems on Sobolev Spaces with Error Control in L2L_2

Low-rank tensor methods for the approximate solution of second-order elliptic partial differential equations in high dimensions have recently attracted significant attention. A critical issue is to rigorously bound the error of such approximations, not with respect to a fixed finite dimensional discrete background problem, but with respect to the exact solution of the continuous problem. While the energy norm offers a natural error measure corresponding to the underlying operator considered as an isomorphism from the energy space onto its dual, this norm requires a careful treatment in its interplay with the tensor structure of the problem. In this paper we build on our previous work on energy norm-convergent subspace-based tensor schemes contriving, however, a modified formulation which now enforces convergence only in $L_2$. In order to still be able to exploit the mapping properties of elliptic operators, a crucial ingredient of our approach is the development and analysis of a suitable asymmetric preconditioning scheme. We provide estimates for the computational complexity of the resulting method in terms of the solution error and study the practical performance of the scheme in numerical experiments. In both regards, we find that controlling solution errors in this weaker norm leads to substantial simplifications and to a reduction of the actual numerical work required for a certain error tolerance.Comment: 26 pages, 7 figure

Algebraic Properties of Polar Codes From a New Polynomial Formalism

Polar codes form a very powerful family of codes with a low complexity decoding algorithm that attain many information theoretic limits in error correction and source coding. These codes are closely related to Reed-Muller codes because both can be described with the same algebraic formalism, namely they are generated by evaluations of monomials. However, finding the right set of generating monomials for a polar code which optimises the decoding performances is a hard task and channel dependent. The purpose of this paper is to reveal some universal properties of these monomials. We will namely prove that there is a way to define a nontrivial (partial) order on monomials so that the monomials generating a polar code devised fo a binary-input symmetric channel always form a decreasing set. This property turns out to have rather deep consequences on the structure of the polar code. Indeed, the permutation group of a decreasing monomial code contains a large group called lower triangular affine group. Furthermore, the codewords of minimum weight correspond exactly to the orbits of the minimum weight codewords that are obtained from (evaluations) of monomials of the generating set. In particular, it gives an efficient way of counting the number of minimum weight codewords of a decreasing monomial code and henceforth of a polar code.Comment: 14 pages * A reference to the work of Bernhard Geiger has been added (arXiv:1506.05231) * Lemma 3 has been changed a little bit in order to prove that Proposition 7.1 in arXiv:1506.05231 holds for any binary input symmetric channe

Higher-order principal component analysis for the approximation of tensors in tree-based low-rank formats

This paper is concerned with the approximation of tensors using tree-based tensor formats, which are tensor networks whose graphs are dimension partition trees. We consider Hilbert tensor spaces of multivariate functions defined on a product set equipped with a probability measure. This includes the case of multidimensional arrays corresponding to finite product sets. We propose and analyse an algorithm for the construction of an approximation using only point evaluations of a multivariate function, or evaluations of some entries of a multidimensional array. The algorithm is a variant of higher-order singular value decomposition which constructs a hierarchy of subspaces associated with the different nodes of the tree and a corresponding hierarchy of interpolation operators. Optimal subspaces are estimated using empirical principal component analysis of interpolations of partial random evaluations of the function. The algorithm is able to provide an approximation in any tree-based format with either a prescribed rank or a prescribed relative error, with a number of evaluations of the order of the storage complexity of the approximation format. Under some assumptions on the estimation of principal components, we prove that the algorithm provides either a quasi-optimal approximation with a given rank, or an approximation satisfying the prescribed relative error, up to constants depending on the tree and the properties of interpolation operators. The analysis takes into account the discretization errors for the approximation of infinite-dimensional tensors. Several numerical examples illustrate the main results and the behavior of the algorithm for the approximation of high-dimensional functions using hierarchical Tucker or tensor train tensor formats, and the approximation of univariate functions using tensorization

A Complete Axiomatization of Real-Time Processes

Once strictly the province of assembly-language programmers, real-time computing has developed into an area of important theoretical interest. Real-time computing incorporates all of the theoretical problems encountered in concurrent processing and introduces the additional complexity of accounting for the temporal behavior of processes. In this paper we investigate two problems in the theory of real-time processes: defining realistic semantic models and developing proof systems for real-time processes. We present here a semantic domain for real-time processes that captures the temporal constraints of concurrent programs. A partial ordering based on process containment is defined and shown to be a complete partial order on the domain. The domain is used to define the denotational semantics of a CSP-like language that incorporates pure time delay. An axiomatization of process containment is presented and shown to be complete for finite terms in this language. The axiomatization is useful for proving properties of real-time processes and deriving their temporal behavior