Above are papers related to the following paper: [Paper_ID] Paper #133553. 	[Title] Sample Complexity for Learning Recurrent Perceptron Mappings [Abstract] Recurrent perceptron classifiers generalize the classical perceptron model. They take into account those correlations and dependences among input coordinates which arise from linear digital filtering. This paper provides tight bounds on sample complexity associated to the fitting of such models to experimental data. 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 #273949. 	[Title] On the Effect of Analog Noise in Discrete-Time Analog Computations [Abstract] We introduce a model for analog computation with discrete time in the presence of analog noise that is flexible enough to cover the most important concrete cases, such as noisy analog neural nets and networks of spiking neurons. This model subsumes the classical model for digital computation in the presence of noise. We show that the presence of arbitrarily small amounts of analog noise reduces the power of analog computational models to that of finite automata, and we also prove a new type of upper bound for the Please summarise all the given information.

Above are papers related to the following paper: [Paper_ID] Paper #1109891. 	[Title] Interconnected Automata and Linear Systems: A Theoretical Framework in Discrete-Time In Hybrid Systems III: Verification [Abstract] This paper summarizes the definitions and several of the main results of an approach to hybrid systems, which combines finite automata and linear systems, developed by the author in the early 1980s. Some related more recent results are briefly mentioned as well. 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 #16485. 	[Title] POWER OF NEURAL NETS [Abstract] Report SYCON-91-11 ABSTRACT This paper deals with the simulation of Turing machines by neural networks. Such networks are made up of interconnections of synchronously evolving processors, each of which updates its state according to a "sigmoidal" linear combination of the previous states of all units. The main result states that one may simulate all Turing machines by nets, in linear time. In particular, it is possible to give a net made up of about 1,000 processors which computes a universal partial-recursive function. (This is an update of Report SYCON-91-08; new results include the simulation in linear time of binary-tape machines, as opposed to the unary alphabets used in the previous version.) Please summarise all the given information.

Above are papers related to the following paper: [Paper_ID] Paper #141160. 	[Title] LINEAR SYSTEMS WITH SIGN-OBSERVATIONS [Abstract] This paper deals with systems that are obtained from linear time-invariant continuous-or discrete-time devices followed by a function that just provides the sign of each output. Such systems appear naturally in the study of quantized observations as well as in signal processing and neural network theory. Results are given on observability, minimal realizations, and other system-theoretic concepts. Certain major differences exist with the linear case, and other results generalize in a surprisingly straightforward manner. Please summarise all the given information.

Above are papers related to the following paper: [Paper_ID] Paper #193931. 	[Title] What Size Neural Network Gives Optimal Generalization? Convergence Properties of Backpropagation [Abstract] Technical Report UMIACS-TR-96-22 and CS-TR-3617 Institute for Advanced Computer Studies University of Maryland College Park, MD 20742 Abstract One of the most important aspects of any machine learning paradigm is how it scales according to problem size and complexity. Using a task with known optimal training error, and a pre-specified maximum number of training updates, we investigate the convergence of the backpropagation algorithm with respect to a) the complexity of the required function approximation, b) the size of the network in relation to the size required for an optimal solution, and c) the degree of noise in the training data. In general, for a) the solution found is worse when the function to be approximated is more complex, for b) oversized networks can result in lower training and generalization error in certain cases, and for c) the use of committee or ensemble techniques can be more beneficial as the level of noise in the training data is increased. For the experiments we performed, we do not obtain the optimal solution in any case. We further support the observation that larger networks can produce better training and generalization error using a face recognition example where a network with many more parameters than training points generalizes better than smaller 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 #1131345. 	[Title] Lemma 2.3 The system is reachable and observable and realizes the same input/output behavior as [Abstract] Here we show a similar construction for multiple-output systems, with some modifications. Let = (A; B; C) s be a discrete-time sign-linear system with state space IR n and p outputs. Perform a change of ; where A 1 (n 1 fi n 1 ) is invertible and A 2 (n 2 fi n 2 ) is nilpotent. If (A; B) is a reachable pair and (A; C) is an observable pair, then is minimal in the sense that any other sign-linear system with the same input/output behavior has dimension at least n. But, if n 1 &lt; n, then det A = 0 and is not observable and hence not canonical. Let us find another system ~ (necessarily not sign-linear) which has the same input/output behavior as , but is canonical. Let i be the relative degree of the ith row of the Markov sequence A, and = minf i : i = 1; : : : ; pg. Let the initial state be x. There is a difference between the case when the smallest relative degree is greater or equal to n 2 and the case when &lt; n 2 . Roughly speaking, when n 2 the outputs of the sign-linear system give us information about sign (Cx), sign (CAx), : : : , sign (CA 1 x), which are the first outputs of the sys tem. After that, we can use the inputs and outputs to learn only about x 1 (the first n 1 components of x). When &lt; n 2 , we may be able to use some controls to learn more about x 2 (the last n 2 components of x) before time n 2 when the nilpotency of A 2 has finally Lemma 2.4 Two states x and z are indistinguishable for if and only if (x) = (z). Proof. In the case n 2 , we have only the equations x 1 = z 1 and the equality of the 's. The first ` output terms for are exactly the terms of . So these equalities are satisfied if and only if the first ` output terms coincide for x and z, for any input. Equality of everything but the first n 1 components is equivalent to the first n 2 output terms coinciding for x and z, since the jth row of the qth output, for initial state x, for example, is either sign (c j A q x) if j &gt; q, or sign (c j A q x + + A j j u q j +1 + ) if j q in which case we may use the control u q j +1 to identify c j A q x (using Remark 3.3 in [1]). Please summarise all the given information.

Above are papers related to the following paper: [Paper_ID] Paper #917493. 	[Title] Observability of Linear Systems with Saturated Outputs [Abstract] In this paper, we present necessary and sufficient conditions for observability of the class of output-saturated systems. These are linear systems whose output passes through a saturation function before it can be measured. Please summarise all the given information.

Above are papers related to the following paper: [Paper_ID] Paper #133563. 	[Title] NEURAL NETS AS SYSTEMS MODELS AND CONTROLLERS suitability of "neural nets" as models for dynamical [Abstract] This paper briefly surveys some recent results relevant 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 #133553 has the title `Sample Complexity for Learning Recurrent Perceptron Mappings  
` and the abstract `Recurrent perceptron classifiers generalize the classical perceptron model. They take into account those correlations and dependences among input coordinates which arise from linear digital filtering. This paper provides tight bounds on sample complexity associated to the fitting of such models to experimental data. 
`.
Question: Which category should Paper #133553 be classified as ?
You can select one from ['Neural_Networks', 'Case_Based', 'Theory', 'Reinforcement_Learning', 'Probabilistic_Methods', 'Rule_Learning', 'Genetic_Algorithms'].


Paper #133553 has the title `Sample Complexity for Learning Recurrent Perceptron Mappings  
` and the abstract `Recurrent perceptron classifiers generalize the classical perceptron model. They take into account those correlations and dependences among input coordinates which arise from linear digital filtering. This paper provides tight bounds on sample complexity associated to the fitting of such models to experimental data. 
`.
Question: Which category should Paper #133553 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