Above are papers related to the following paper: [Paper_ID] Paper #689152. 	[Title] Shattering all sets of k points in "general position" requires (k 1)=2 parameters [Abstract] For classes of concepts defined by certain classes of analytic functions depending on n parameters, there are nonempty open sets of samples of length 2n + 2 which cannot be shattered. A slighly weaker result is also proved for piecewise-analytic functions. The special case of neural networks is discussed. Please summarise all the given information.

Above are papers related to the following paper: [Paper_ID] Paper #289088. 	[Title] Critical Points for Least-Squares Problems Involving Certain Analytic Functions, with Applications to Sigmoidal Nets [Abstract] This paper deals with nonlinear least-squares problems involving the fitting to data of parameterized analytic functions. For generic regression data, a general result establishes the countability, and under stronger assumptions finiteness, of the set of functions giving rise to critical points of the quadratic loss function. In the special case of what are usually called "single-hidden layer neural networks," which are built upon the standard sigmoidal activation tanh(x) (or equivalently (1 + e x ) 1 ), a rough upper bound for this cardinality is provided as well. Please summarise all the given information.

Above are papers related to the following paper: [Paper_ID] Paper #335042. 	[Title] An Efficient Implementation of Sigmoidal Neural Nets in Temporal Coding with Noisy Spiking Neurons [Abstract] We show that networks of relatively realistic mathematical models for biological neurons can in principle simulate arbitrary feedforward sigmoidal neural nets in a way which has previously not been considered. This new approach is based on temporal coding by single spikes (respectively by the timing of synchronous firing in pools of neurons), rather than on the traditional interpretation of analog variables in terms of firing rates. The resulting new simulation is substantially faster and hence more consistent with experimental results about the maximal speed of information processing in cortical neural systems. As a consequence we can show that networks of noisy spiking neurons are "universal approximators" in the sense that they can approximate with regard to temporal coding any given continuous function of several variables. This result holds for a fairly large class of schemes for coding analog variables by firing times of spiking neurons. Our new proposal for the possible organization of computations in networks of spiking neurons systems has some interesting consequences for the type of learning rules that would be needed to explain the self-organization of such networks. Finally, our fast and noise-robust implementation of sigmoidal neural nets via temporal coding points to possible new ways of implementing feedforward and recurrent sigmoidal neural nets with pulse stream VLSI. Please summarise all the given information.

Above are papers related to the following paper: [Paper_ID] Paper #105856. 	[Title] Vapnik-Chervonenkis Dimension of Recurrent Neural Networks [Abstract] Most of the work on the Vapnik-Chervonenkis dimension of neural networks has been focused on feedforward networks. However, recurrent networks are also widely used in learning applications, in particular when time is a relevant parameter. This paper provides lower and upper bounds for the VC dimension of such networks. Several types of activation functions are discussed, including threshold, polynomial, piecewise-polynomial and sigmoidal functions. The bounds depend on two independent parameters: the number w of weights in the network, and the length k of the input sequence. In contrast, for feedforward networks, VC dimension bounds can be expressed as a function of w only. An important difference between recurrent and feedforward nets is that a fixed recurrent net can receive inputs of arbitrary length. Therefore we are particularly interested in the case k w. Ignoring multiplicative constants, the main results say roughly the following: * For architectures with activation = any fixed nonlinear polynomial, the VC dimension is wk. * For architectures with activation = any fixed piecewise polynomial, the VC dimension is between wk and w 2 k. * For architectures with activation = H (threshold nets), the VC dimension is between w log(k=w) and minfwk log wk; w 2 + w log wkg. * For the standard sigmoid (x) = 1=(1 + e x ), the VC dimension is between wk and w 4 k 2 . An earlier version of this paper has appeared in Proc. 3rd European Workshop on Computational Learning Theory, LNCS 1208, pages 223-237, Springer, 1997. Please summarise all the given information.

Above are papers related to the following paper: [Paper_ID] Paper #264556. 	[Title] A Delay-Line Based Motion Detection Chip [Abstract] Inspired by a visual motion detection model for the rabbit retina and by a computational architecture used for early audition in the barn owl, we have designed a chip that employs a correlation model to report the one-dimensional field motion of a scene in real time. Using subthreshold analog VLSI techniques, we have fabricated and successfully tested a 8000 transistor chip using a standard MOSIS process. Please summarise all the given information.

Above are papers related to the following paper: [Paper_ID] Paper #753070. 	[Title] Similar Classifiers and VC Error Bounds [Abstract] We improve error bounds based on VC analysis for classes with sets of similar classifiers. We apply the new error bounds to separating planes and artificial neural networks. Key words machine learning, learning theory, generalization, Vapnik-Chervonenkis, separating planes, neural networks. Please summarise all the given information.

Above are papers related to the following paper: [Paper_ID] Paper #1120049. 	[Title] Dimension of Recurrent Neural Networks [Abstract] DIMACS Technical Report 96-56 December 1996 Please summarise all the given information.

Above are papers related to the following paper: [Paper_ID] Paper #50354. 	[Title] BACKPROPAGATION CAN GIVE RISE TO SPURIOUS LOCAL MINIMA EVEN FOR NETWORKS WITHOUT HIDDEN LAYERS [Abstract] We give an example of a neural net without hidden layers and with a sigmoid transfer function, together with a training set of binary vectors, for which the sum of the squared errors, regarded as a function of the weights, has a local minimum which is not a global minimum. The example consists of a set of 125 training instances, with four weights and a threshold to be learnt. We do not know if substantially smaller binary examples exist. Please summarise all the given information.

Above are papers related to the following paper: [Paper_ID] Paper #175548. 	[Title] Neural Networks with Quadratic VC Dimension [Abstract] This paper shows that neural networks which use continuous activation functions have VC dimension at least as large as the square of the number of weights w. This result settles a long-standing open question, namely whether the well-known O(w log w) bound, known for hard-threshold nets, also held for more general sigmoidal nets. Implications for the number of samples needed for valid generalization are discussed. Please summarise all the given information.

Above are papers related to the following paper: [Paper_ID] Paper #289085. 	[Title] Networks of Spiking Neurons: The Third Generation of Neural Network Models [Abstract] The computational power of formal models for networks of spiking neurons is compared with that of other neural network models based on McCulloch Pitts neurons (i.e. threshold gates) respectively sigmoidal gates. In particular it is shown that networks of spiking neurons are computationally more powerful than these other neural network models. A concrete biologically relevant function is exhibited which can be computed by a single spiking neuron (for biologically reasonable values of its parameters), but which requires hundreds of hidden units on a sigmoidal neural net. This article does not assume prior knowledge about spiking neurons, and it contains an extensive list of references to the currently available literature on computations in networks of spiking neurons and relevant results from neuro biology. Please summarise all the given information.

Paper #689152 has the title `Shattering all sets of k points in "general position" requires (k 1)=2 parameters  
` and the abstract `For classes of concepts defined by certain classes of analytic functions depending on n parameters, there are nonempty open sets of samples of length 2n + 2 which cannot be shattered. A slighly weaker result is also proved for piecewise-analytic functions. The special case of neural networks is discussed.
`.
Question: Which category should Paper #689152 be classified as ?
You can select one from ['Neural_Networks', 'Case_Based', 'Theory', 'Reinforcement_Learning', 'Probabilistic_Methods', 'Rule_Learning', 'Genetic_Algorithms'].


Paper #689152 has the title `Shattering all sets of k points in "general position" requires (k 1)=2 parameters  
` and the abstract `For classes of concepts defined by certain classes of analytic functions depending on n parameters, there are nonempty open sets of samples of length 2n + 2 which cannot be shattered. A slighly weaker result is also proved for piecewise-analytic functions. The special case of neural networks is discussed.
`.
Question: Which category should Paper #689152 be classified as ?
You can select one from ['Neural_Networks', 'Case_Based', 'Theory', 'Reinforcement_Learning', 'Probabilistic_Methods', 'Rule_Learning', 'Genetic_Algorithms'].

This paper should be classified into: Neural_Networks