Topic
Piecewise
About: Piecewise is a research topic. Over the lifetime, 21064 publications have been published within this topic receiving 432096 citations. The topic is also known as: piecewise-defined function & hybrid function.
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TL;DR: In this article, a class of continuous time systems with part continuous, part discrete state is described by differential equations combined with multistable elements, where transitions of these elements between their discrete states are triggered by the continuous part of the state and not directly by inputs.
Abstract: A class of continuous time systems with part continuous, part discrete state is described by differential equations combined with multistable elements. Transitions of these elements between their discrete states are triggered by the continuous part of the state and not directly by inputs. The dynamic behavior of such systems, in response to piecewise continuous inputs, is defined under suitable assumptions. A general Mayer-type optimization problem is formulated. Conditions are given for a solution to be well-behaved, so that variational methods can be applied. Necessary conditions for optimality are stated and the jump conditions are interpreted geometrically.
261 citations
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TL;DR: In this article, Stein and Ulam showed that the shift transformation on a compact invariant subset of the space of one-way infinite sequences of symbols chosen from a finite set is continuous but not necessarily open.
Abstract: transformation. The main result includes a classical theorem of Poincare-Denjoy [2] on homeomorphisms of the circle onto itself. It also provides a partial answer to a question of Ulam's [3] concerning the possibility of piecewise linearising continuous transformations of the unit interval. This problem was also mentioned by Stein and Ulam in [4], together with the remark that necessary conditions can be given in terms of the trees of points, but that no meaningful sufficient conditions are known. In the same work a few special examples are examined. Our main theorem also has a bearing on certain transformations discussed by Renyi [5]. In ??2-4 we consider the shift transformation acting on a compact invariant subset of the space of one-way infinite sequences of symbols chosen from a finite set. The shift transformation on such a set is continuous but not necessarily open. If X, T are the compact invariant set and the shift transformation, respectively, we refer to (X, T) as a symbolic dynamical system [6]. For a symbolic dynamical system (X, T) we define a number called the absolute entropy(') which dominates the entropy of T with respect to each normalised T invariant Borel measure, and show that if T is regionally transitive then there is always one invariant measure with respect to which the entropy of T equals the absolute entropy of T. When T is open, (or equivalently, when (X, T) is an intrinsic Markov chain) this "maximal" measure is unique. A further theorem states that, under certain conditions, there exists a normalised Borel measure with respect to which T acts in a "linear" fashion. In ??5-6 we apply this latter theorem to certain transformations of the unit interval and obtain our main result, Theorem 5.
259 citations
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TL;DR: It is shown that, for a large class of piecewise smooth functions, the number of neurons needed by a shallow network to approximate a function is exponentially larger than the corresponding number of neuron needs by a deep network for a given degree of function approximation.
Abstract: Recently there has been much interest in understanding why deep neural networks are preferred to shallow networks. We show that, for a large class of piecewise smooth functions, the number of neurons needed by a shallow network to approximate a function is exponentially larger than the corresponding number of neurons needed by a deep network for a given degree of function approximation. First, we consider univariate functions on a bounded interval and require a neural network to achieve an approximation error of $\varepsilon$ uniformly over the interval. We show that shallow networks (i.e., networks whose depth does not depend on $\varepsilon$) require $\Omega(\text{poly}(1/\varepsilon))$ neurons while deep networks (i.e., networks whose depth grows with $1/\varepsilon$) require $\mathcal{O}(\text{polylog}(1/\varepsilon))$ neurons. We then extend these results to certain classes of important multivariate functions. Our results are derived for neural networks which use a combination of rectifier linear units (ReLUs) and binary step units, two of the most popular type of activation functions. Our analysis builds on a simple observation: the multiplication of two bits can be represented by a ReLU.
259 citations
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01 Jan 1994
TL;DR: An alternative model for mixtures of experts which uses a different parametric form for the gating network, trained by the EM algorithm, and which yields faster convergence.
Abstract: We propose an alternative model for mixtures of experts which uses a different parametric form for the gating network. The modified model is trained by the EM algorithm. In comparison with earlier models--trained by either EM or gradient ascent--there is no need to select a learning stepsize. We report simulation experiments which show that the new architecture yields faster convergence. We also apply the new model to two problem domains: piecewise nonlinear function approximation and the combination of multiple previously trained classifiers.
258 citations
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TL;DR: In this paper, it was shown that the adiabatic theorem holds for the ground state of atoms in quantized radiation field without the traditional gap condition, and that the general result gives no information on the rate at which the adabiabatic limit is approached, but with additional spectral information one can also estimate this rate.
Abstract: We prove the adiabatic theorem for quantum evolution without the traditional gap condition. All that this adiabatic theorem needs is a (piecewise) twice differentiable finite dimensional spectral projection. The result implies that the adiabatic theorem holds for the ground state of atoms in quantized radiation field. The general result we prove gives no information on the rate at which the adiabatic limit is approached. With additional spectral information one can also estimate this rate.
258 citations