human: Given a node-centered hypergraph: <hypergraph>, where nodes represent papers and hyperedges represent groups of papers sharing a common citing paper.

Details of the hypergraph:
Center node: "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]).

Connected through hyperedge HE-1 (induced by source paper "LINEAR SYSTEMS WITH SIGN-OBSERVATIONS"):
  with members:
    - "Sample Complexity for Learning Recurrent Perceptron Mappings"
    - "Interconnected Automata and Linear Systems: A Theoretical Framework in Discrete-Time In Hybrid Systems III: Verification"
    - "BACKPROPAGATION SEPARATES WHERE PERCEPTRONS DO"
    - "Observability of Linear Systems with Saturated Outputs"

Question: Please tell me which class the center node belongs to. The 7 classes are: Neural_Networks, Reinforcement_Learning, Probabilistic_Methods, Theory, Genetic_Algorithms, Case_Based, Rule_Learning. Directly output the class name.

gpt: Neural_Networks