Piecewise linear function

About: Piecewise linear function is a research topic. Over the lifetime, 8133 publications have been published within this topic receiving 161444 citations.

Hardness of embedding simplicial complexes in Rd

Abstract: Let EMBEDk→d be the following algorithmic problem: Given a finite simplicial complex K of dimension at most k, does there exist a (piecewise linear) embedding of K into Rd? Known results easily imply polynomiality of EMBEDk→2 (k = 1, 2; the case k = 1, d = 2 is graph planarity) and of EMBEDk→2k for all k ≥ 3 (even if k is not considered fixed).We show that the celebrated result of Novikov on the algorithmic unsolvability of recognizing the 5-sphere implies that EMBEDd→d and EMBED(d-1)→d are undecidable for each d ≥ 5. Our main result is NP-hardness of EMBED2→4 and, more generally, of EMBEDk→d for all k, d with d → 4 and d → k → (2d - 2)/3.

A general mechanism to generate three limit cycles in planar Filippov systems with two zones

Abstract: Discontinuous piecewise linear systems with two zones are considered. A general canonical form that includes all the possible configurations in planar linear systems is introduced and exploited. It is shown that the existence of a focus in one zone is sufficient to get three nested limit cycles, independently on the dynamics of the another linear zone. Perturbing a situation with only one hyperbolic limit cycle, two additional limit cycles are obtained by using an adequate parametric sector of the unfolding of a codimension-two focus-fold singularity.

The Knapsack Sharing Problem

Abstract: The knapsack sharing problem has a utility or tradeoff function for each variable and seeks to maximize the value of the smallest tradeoff function (a maximin objective function). A single constraint places an upper bound on the sum of the non-negative variables. We develop efficient algorithms for piecewise linear, nonlinear, and piecewise nonlinear tradeoff functions and for any knapsack sharing problem with integer variables. These algorithms for the knapsack sharing problem extend the sharing problem algorithm in a companion paper to any piecewise linear, nonlinear, or piecewise nonlinear tradeoff functions.

Statistical Compressed Sensing of Gaussian Mixture Models

Abstract: A novel framework of compressed sensing, namely statistical compressed sensing (SCS), that aims at efficiently sampling a collection of signals that follow a statistical distribution, and achieving accurate reconstruction on average, is introduced. SCS based on Gaussian models is investigated in depth. For signals that follow a single Gaussian model, with Gaussian or Bernoulli sensing matrices of O(k) measurements, considerably smaller than the O(k log(N/k)) required by conventional CS based on sparse models, where N is the signal dimension, and with an optimal decoder implemented via linear filtering, significantly faster than the pursuit decoders applied in conventional CS, the error of SCS is shown tightly upper bounded by a constant times the best k-term approximation error, with overwhelming probability. The failure probability is also significantly smaller than that of conventional sparsity-oriented CS. Stronger yet simpler results further show that for any sensing matrix, the error of Gaussian SCS is upper bounded by a constant times the best k-term approximation with probability one, and the bound constant can be efficiently calculated. For Gaussian mixture models (GMMs), that assume multiple Gaussian distributions and that each signal follows one of them with an unknown index, a piecewise linear estimator is introduced to decode SCS. The accuracy of model selection, at the heart of the piecewise linear decoder, is analyzed in terms of the properties of the Gaussian distributions and the number of sensing measurements. A maximization-maximization (Max-Max) algorithm that iteratively estimates the Gaussian models parameters, the signals model selection, and decodes the signals, is presented for GMM-based SCS. In real image sensing applications, GMM-based SCS is shown to lead to improved results compared to conventional CS, at a considerably lower computational cost.

The embedding capacity of 4-dimensional symplectic ellipsoids

Abstract: This paper calculates the function c(a) whose value at a is the inmum of the size of a ball that contains a symplectic image of the ellipsoidE(1;a). (Here a 1 is the ratio of the area of the large axis to that of the smaller axis.) The structure of the graph of c(a) is surprisingly rich. The volume constraint implies that c(a) is always greater than or equal to the square root of a, and it is not hard to see that this is equality for large a. However, for a less than the fourth power 4 of the golden ratio, c(a) is piecewise linear, with graph that alternately lies on a line through the origin and is horizontal. We prove this by showing that there are exceptional curves in blow ups of the complex projective plane whose homology classes are given by the continued fraction expansions of ratios of Fibonacci numbers. On the interval [ 4 ; 7] we nd c(a) = (a + 1)=3. For a 7, the function c(a) coincides with the square root except on a nite number of intervals where it is again piecewise linear. The embedding constraints coming from embedded contact homology give rise to another capacity function cECH which may be computed by counting lattice points in appropriate right angled triangles. According to Hutchings and Taubes, the functorial properties of embedded contact homology imply that cECH(a) c(a) for all a. We show here that cECH(a) c(a) for all a.