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 #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 #123556. 	[Title] BACKPROPAGATION SEPARATES WHERE PERCEPTRONS DO [Abstract] Feedforward nets with sigmoidal activation functions are often designed by minimizing a cost criterion. It has been pointed out before that this technique may be outperformed by the classical perceptron learning rule, at least on some problems. In this paper, we show that no such pathologies can arise if the error criterion is of a threshold LMS type, i.e., is zero for values "beyond" the desired target values. More precisely, we show that if the data are linearly separable, and one considers nets with no hidden neurons, then an error function as above cannot have any local minima that are not global. Simulations of networks with hidden units are consistent with these results, in that often data which can be classified when minimizing a threshold LMS criterion may fail to be classified when using instead a simple LMS cost. In addition, the proof gives the following stronger result, under the stated hypotheses: the continuous gradient adjustment procedure is such that from any initial weight configuration a separating set of weights is obtained in finite time. This is a precise analogue of the Perceptron Learning Theorem. The results are then compared with the more classical pattern recognition problem of threshold LMS with linear activations, where no spurious local minima exist even for nonseparable data: here it is shown that even if using the threshold criterion, such bad local minima may occur, if the data are not separable and sigmoids are used. 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 #14807. 	[Title] Exponentially many local minima for single neurons [Abstract] We show that for a single neuron with the logistic function as the transfer function the number of local minima of the error function based on the square loss can grow exponentially in the dimension. 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 #289088 has the title `Critical Points for Least-Squares Problems Involving Certain Analytic Functions, with Applications to Sigmoidal Nets  
` and the 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.
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Question: Which category should Paper #289088 be classified as ?
You can select one from ['Neural_Networks', 'Case_Based', 'Theory', 'Reinforcement_Learning', 'Probabilistic_Methods', 'Rule_Learning', 'Genetic_Algorithms'].


Paper #289088 has the title `Critical Points for Least-Squares Problems Involving Certain Analytic Functions, with Applications to Sigmoidal Nets  
` and the 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.
`.
Question: Which category should Paper #289088 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