Canonical Piecewise-linear Representation Research Articles

The hinging hyperplanes: An alternative nonparametric representation of a production function.

In this paper we propose hinging hyperplanes (HHs) as a flexible nonparametric representation of a concave or an S-shaped production function. We derive the HHs using expressions with focus on the distinction between hinge location and the bending along each hinge. We argue that the HHs approximation can be estimated using a fixed endogenous determined partitioning of the input space. Assuming a homothetic production function allows us to separate the S-shape scaling law and the underlying core function. We propose an estimation procedure where two HHs function approximations of the core function and the scaling law are estimated simultaneously. A closed form expression of the inverse of the piecewise linear inverse scaling law is proposed and proved.We stress the known result that the HHs formulation is equivalent to the Canonical Piecewise-Linear representation of a piecewise linear function and exploit the result that the HHs formulations for the core function and for the scaling law provide global representations of all piecewise linear functional forms. A simulation study evaluates the performance of the proposed estimation procedure of an S-shaped production function.

The complete canonical piecewise-linear representation: functional form for minimal degenerate intersections

In this paper, we develop a functional form for canonical piecewise-linear (CPWL) functions defined over a minimal degenerate intersection of order k. This type of intersection is known to be the building block of CPWL representations. Accordingly, the representation proposed is the main component in a general CPWL representation and constitutes the first functional form that is valid in a domain that has arbitrary dimension and is subdivided by a generic partition.

Canonical piecewise-linear networks

In this paper, mapping networks will be considered from the viewpoint of the piecewise-linear (PWL) approximation. The so-called canonical representation plays a kernel role in the PWL representation theory. While this theory has been researched intensively in the contents of mathematics and circuit simulations, little has been seen in the research area about the theoretical aspect of neural networks. This paper modifies this theory and applies it as a mathematical support for mapping networks. The main modification is a "higher-level" generalization of the canonical representation with proofs of its availability in the set of PWL functions. The modified theory will first be used to study the canonical PWL feature of the popular multilayer perceptron-like (MLPL) networks. Second, it will be seen that the generalized canonical representation is itself suitable for a network implementation, which is called the standard canonical PWL network. More generally, the family of (generalized) canonical PWL networks is defined as those which may take the canonical PWL representation as a mathematical model. This family is large and practically meaningful. The standard canonical PWL networks may be taken as representatives in the family. The modification of the PWL representation theory as well as the introduction of this theory in the theoretical study of mapping networks, which provide a new concept of mapping networks, i.e., the canonical PWL network family, may be regarded as the main contributions of the paper.

The complete canonical piecewise-linear representation. I. The geometry of the domain space

Continuous piecewise-linear functions from R/sup n/ to R/sup m/ are analyzed in terms of the dimensions of their domain space and of degenerate kth-order intersections of region boundaries. The theory developed demonstrates how these two quantities are connected. Moreover, the exact number of independent parameters is demonstrated for boundary configurations containing degenerate intersections of arbitrary orders. >

A generalized canonical piecewise-linear representation

An extension of the well-known canonical representation for continuous piecewise-linear functions is introduced. This form is capable of describing all piecewise-linear functions in two dimensions, being no longer subject to any restrictions. Moreover, it is shown that just one nesting of absolute-value functions is sufficient to cover the whole class.< <ETX xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">&gt;</ETX>

Canonical piecewise-linear representation

Every continuous piecewise-linear function of one variable f:R/sup 1/ to R/sup 1/ has a unique canonical piecewise-linear representation. However, only a subclass of higher-dimensional piecewise-linear functions f:R/sup n/ to R/sup n/, n>1, has a canonical piecewise-linear representation. It is proved that the necessary and sufficient conditions for the existence of a canonical piecewise-linear representation is that fpossess a consistent variation property. The geometrical constraints imposed by this property are analyzed and discussed in detail along with many examples. >

Canonical piecewise-linear modeling

To take advantage of the remarkable computational efficiency of the canonical piecewise-linear approach for dc nonlinear electronic circuit analysis, the devices must be modeled by a canonical piecewise-linear model. This paper presents a unified parameter optimization algorithm for constructing such models. This algorithm is then applied to derive prototype canonical piecewise-linear models of p-n junction diodes, bipolar transistors, MOSFET's, and GaAs FET's. The canonical piecewise-linear model can be regarded as a universal model since its form remains unchanged for all devices. Only the coefficients differ from one device to another. For large-scale circuits, the canonical piecewise-linear representation has a decisive advantage over other representations in regard to the number of memory locations needed to specify the equations.