Log-sum-exp neural networks and posynomial models for convex and log-log-convex data
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We show that a one-layer feedforward neural network with exponential activation functions in the inner layer and logarithmic activation in the output neuron is a universal approximator of convex functions. Such a network represents a family of scaled log-sum exponential functions, here named LSET. The proof uses a dequantization argument from tropical geometry. Under a suitable exponential transformation LSE maps to a family of generalized posynomial functions GPOST, which we also show to be universal approximators for log-log-convex functions. The key feature of interest in the proposed approach is that, once a LSET network is trained on data, the resulting model is convex in the variables, which makes it readily amenable to efficient design based on convex optimization. Similarly, once a GPOST model is trained on data, it yields a posynomial model that can be efficiently optimized with respect to its variables by using Geometric Programming (GP). Many relevant phenomena in physics and engineering can indeed be modeled, either exactly or approximately, via convex or log-log-convex models. The proposed methodology is illustrated by two numerical examples in which LSET and GPOST models are used to first approximate data gathered from the simulations of two physical processes (the vibration from a vehicle suspension system, and the peak power generated by the combustion of propane), and to later optimize these models.
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