Above are papers related to the following paper: [Paper_ID] Paper #384428. 	[Title] Algebraic Transformations of Objective Functions [Abstract] Many neural networks can be derived as optimization dynamics for suitable objective functions. We show that such networks can be designed by repeated transformations of one objective into another with the same fixpoints. We exhibit a collection of algebraic transformations which reduce network cost and increase the set of objective functions that are neurally implementable. The transformations include simplification of products of expressions, functions of one or two expressions, and sparse matrix products (all of which may be interpreted as Legendre transformations); also the minimum and maximum of a set of expressions. These transformations introduce new interneurons which force the network to seek a saddle point rather than a minimum. Other transformations allow control of the network dynamics, by reconciling the Lagrangian formalism with the need for fixpoints. We apply the transformations to simplify a number of structured neural networks, beginning with the standard reduction of the winner-take-all network from O(N 2 ) connections to O(N ). Also susceptible are inexact graph-matching, random dot matching, convolutions and coordinate transformations, and sorting. Simulations show that fixpoint-preserving transformations may be applied repeatedly and elaborately, and the example networks still robustly converge. Please summarise all the given information.

Above are papers related to the following paper: [Paper_ID] Paper #1120019. 	[Title] Minimax and Hamiltonian Dynamics of Excitatory-Inhibitory Networks [Abstract] A Lyapunov function for excitatory-inhibitory networks is constructed. The construction assumes symmetric interactions within excitatory and inhibitory populations of neurons, and antisymmetric interactions between populations. The Lyapunov function yields sufficient conditions for the global asymptotic stability of fixed points. If these conditions are violated, limit cycles may be stable. The relations of the Lyapunov function to optimization theory and classical mechanics are revealed by The dynamics of a neural network with symmetric interactions provably converges to fixed points under very general assumptions[1, 2]. This mathematical result helped to establish the paradigm of neural computation with fixed point attractors[3]. But in reality, interactions between neurons in the brain are asymmetric. Furthermore, the dynamical behaviors seen in the brain are not confined to fixed point attractors, but also include oscillations and complex nonperiodic behavior. These other types of dynamics can be realized by asymmetric networks, and may be useful for neural computation. For these reasons, it is important to understand the global behavior of asymmetric neural networks. The interaction between an excitatory neuron and an inhibitory neuron is clearly asymmetric. Here we consider a class of networks that incorporates this fundamental asymmetry of the brain's microcircuitry. Networks of this class have distinct populations of excitatory and inhibitory neurons, with antisymmetric interactions minimax and dissipative Hamiltonian forms of the network dynamics. Please summarise all the given information.

Paper #384428 has the title `Algebraic Transformations of Objective Functions  
` and the abstract `Many neural networks can be derived as optimization dynamics for suitable objective functions. We show that such networks can be designed by repeated transformations of one objective into another with the same fixpoints. We exhibit a collection of algebraic transformations which reduce network cost and increase the set of objective functions that are neurally implementable. The transformations include simplification of products of expressions, functions of one or two expressions, and sparse matrix products (all of which may be interpreted as Legendre transformations); also the minimum and maximum of a set of expressions. These transformations introduce new interneurons which force the network to seek a saddle point rather than a minimum. Other transformations allow control of the network dynamics, by reconciling the Lagrangian formalism with the need for fixpoints. We apply the transformations to simplify a number of structured neural networks, beginning with the standard reduction of the winner-take-all network from O(N 2 ) connections to O(N ). Also susceptible are inexact graph-matching, random dot matching, convolutions and coordinate transformations, and sorting. Simulations show that fixpoint-preserving transformations may be applied repeatedly and elaborately, and the example networks still robustly converge. 
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Question: Which category should Paper #384428 be classified as ?
You can select one from ['Neural_Networks', 'Case_Based', 'Theory', 'Reinforcement_Learning', 'Probabilistic_Methods', 'Rule_Learning', 'Genetic_Algorithms'].


Paper #384428 has the title `Algebraic Transformations of Objective Functions  
` and the abstract `Many neural networks can be derived as optimization dynamics for suitable objective functions. We show that such networks can be designed by repeated transformations of one objective into another with the same fixpoints. We exhibit a collection of algebraic transformations which reduce network cost and increase the set of objective functions that are neurally implementable. The transformations include simplification of products of expressions, functions of one or two expressions, and sparse matrix products (all of which may be interpreted as Legendre transformations); also the minimum and maximum of a set of expressions. These transformations introduce new interneurons which force the network to seek a saddle point rather than a minimum. Other transformations allow control of the network dynamics, by reconciling the Lagrangian formalism with the need for fixpoints. We apply the transformations to simplify a number of structured neural networks, beginning with the standard reduction of the winner-take-all network from O(N 2 ) connections to O(N ). Also susceptible are inexact graph-matching, random dot matching, convolutions and coordinate transformations, and sorting. Simulations show that fixpoint-preserving transformations may be applied repeatedly and elaborately, and the example networks still robustly converge. 
`.
Question: Which category should Paper #384428 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