The Representer Theorem by Michael Unser has recently unveiled explicit connections between deep NNs (using ReLUs as nonlinearities I believe) and splines.

One core idea is that both the linear operators and ReLUs can be seen as piecewise linear functions (linear splines), so all forward and backward operations, and the resulting representations are closed in that set.

This allows to connect DNNs with L1 methods, and therefore with some of the advances in Compressed Sensing from the last decades.

Not only that, it also leads to novel architectures with promising properties, like B-spline networks:

We develop an efficient computational solution to train deep neural networks (DNN) with free-form activation functions. To make the problem well-posed, we augment the cost functional of the DNN by adding an appropriate shape regularization: the sum of the second-order total variations of the trainable nonlinearities. The representer theorem for DNNs tells us that the optimal activation functions are adaptive piecewise-linear splines, which allows us to recast the problem as a parametric optimization. The challenging point is that the corresponding basis functions (ReLUs) are poorly conditioned and that the determination of their number and positioning is also part of the problem. We circumvent the difficulty by using an equivalent B-spline basis to encode the activation functions and by expressing the regularization as an l1-penalty. This results in the specification of parametric activation function modules that can be implemented and optimized efficiently on standard development platforms. We present experimental results that demonstrate the benefit of our approach.