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 #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 #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 #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 #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.

Paper #917493 has the title `Observability of Linear Systems with Saturated Outputs  
` and the 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.
`.
Question: Which category should Paper #917493 be classified as ?
You can select one from ['Neural_Networks', 'Case_Based', 'Theory', 'Reinforcement_Learning', 'Probabilistic_Methods', 'Rule_Learning', 'Genetic_Algorithms'].


Paper #917493 has the title `Observability of Linear Systems with Saturated Outputs  
` and the 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.
`.
Question: Which category should Paper #917493 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