This paper focuses on the development of randomized approaches for building deep neural networks. A supervisory mechanism is proposed to constrain the random assignment of the hidden parameters (i.e., all biases and weights within the hidden layers). Full-rank oriented criterion is suggested and utilized as a termination condition to determine the number of nodes for each hidden layer, and a pre-defined error tolerance is used as a global indicator to decide the depth of the learner model. The read-out weights attached with all direct links from each hidden layer to the output layer are incrementally evaluated by the least squares method. Such a class of randomized leaner models with deep architecture is termed as deep stochastic configuration networks (DeepSCNs), of which the universal approximation property is verified with rigorous proof. Given abundant samples from a continuous distribution, DeepSCNs can speedily produce a learning representation, that is, a collection of random basis functions with the cascaded inputs together with the read-out weights. Simulation results with comparisons on function approximation align with the theoretical findings. Deep Stochastic Configuration Networks: Universal Approximation and Learning Representation

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