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.

Planning and control in stochastic domains with imperfect information

Abstract: Partially observable Markov decision processes (POMDPs) can be used to model complex control problems that include both action outcome uncertainty and imperfect observability. A control problem within the POMDP framework is expressed as a dynamic optimization problem with a value function that combines costs or rewards from multiple steps. Although the POMDP framework is more expressive than other simpler frameworks, like Markov decision processes (MDP), its associated optimization methods are more demanding computationally and only very small problems can be solved exactly in practice. The thesis focuses on two possible approaches that can he used to solve larger problems: approximation methods and exploitation of additional problem structure. First, a number of new efficient approximation methods and improvements of existing algorithms are proposed. These include (1) the fast informed bound method based on approximate dynamic programming updates that lead to piecewise linear and convex value functions with a constant number of linear vectors, (2) a grid-based point interpolation method that supports variable grids, (3) an incremental version of the linear vector method that updates value function derivatives, as well as (4) various heuristics for selecting grid-points. The new and existing methods are experimentally tested and compared on a set of three infinite discounted horizon problems of different complexity. The experimental results show that methods that preserve the shape of the value function over updates, such as the newly designed incremental linear vector and fast informed bound methods, tend to outperform other methods on the control performance test. Second, the thesis presents a number of techniques for exploiting additional structure in the model of complex control problems. These are studied as applied to a medical therapy planning problem--the management of patients with chronic ischemic heart disease. The new extensions proposed include factored and hierarchically structured models that combine the advantages of the POMDP and MDP frameworks and cut down the size and complexity of the information state space.

Piecewise linear differential equations and integrate-and-fire neurons: insights from two-dimensional membrane models.

Abstract: We derive and study two-dimensional generalizations of integrate-and-fire models which can be found from a piecewise linear idealization of the FitzHugh-Nagumo or Morris-Lecar model. These models give rise to new properties not present in one-dimensional integrate-and-fire models. A detailed analytical study of the models is presented. In particular, (i) for the piecewise linear FitzHugh-Nagumo model, we determine analytically the bistability regime between stationary solutions and oscillations, that is, typical for class-II models. (ii) In the piecewise Morris-Lecar model, we find a noncanonical class-I transition from a stationary state to oscillations with logarithmic dependence similar to that found for leaky integrate-and-fire models. (iii) Furthermore, we establish a relation to the recently proposed resonate-and-fire model and show that a short input current pulse can trigger several spikes.

Bifurcation of limit cycles by perturbing a piecewise linear Hamiltonian system with a homoclinic loop

Abstract: In this paper, we study limit cycle bifurcations for a kind of non-smooth polynomial differential systems by perturbing a piecewise linear Hamiltonian system with the center at the origin and a homoclinic loop around the origin. By using the first Melnikov function of piecewise near-Hamiltonian systems, we give lower bounds of the maximal number of limit cycles in Hopf and homoclinic bifurcations, and derive an upper bound of the number of limit cycles that bifurcate from the periodic annulus between the center and the homoclinic loop up to the first order in e . In the case when the degree of perturbing terms is low, we obtain a precise result on the number of zeros of the first Melnikov function.

Linear and Cubic Box Splines for the Body Centered Cubic Lattice

Abstract: In this paper we derive piecewise linear and piecewise cubic box spline reconstruction filters for data sampled on the body centered cubic (BCC) lattice. We analytically derive a time domain representation of these reconstruction filters and using the Fourier slice-projection theorem we derive their frequency responses. The quality of these filters, when used in reconstructing BCC sampled volumetric data, is discussed and is demonstrated with a raycaster. Moreover, to demonstrate the superiority of the BCC sampling, the resulting reconstructions are compared with those produced from similar filters applied to data sampled on the Cartesian lattice.

A digital hardware pulse-mode neuron with piecewise linear activation function

Abstract: This paper proposes a new type of digital pulse-mode neuron that employs piecewise-linear function as its activation function. The neuron is implemented on field programmable gate array (FPGA) and tested by experiments. As well as theoretical analysis, the experimental results show that the piecewise-linear function of the proposed neuron is programmable and robust against the change in the number of input signals. To demonstrate the effect of piecewise-linear activation function, pulse-mode multilayer neural network with on-chip learning is implemented on FPGA with the proposed neuron, and its learning performance is verified by experiments. By approximating the sigmoid function by the piecewise-linear function, the convergence rate of the learning and generalization capability are improved.