Refinable functions are the pillar stones in approximation theory (e.g. biorthogonal wavelet bases, various fractal-like objects, B-splines, and subdivision schemes defined by a different refinable mask). In our recent paper [1], we showed that refinable functions are approximated by the outputs of deep ReLU Neural Networks with a fixed width and increasing depth with accuracy exponential in terms of their number of parameters. However, the classical refinable function definition challenges the theoretical analysis ibid. In this study, we introduce a new family of refinable functions by proposing a new refinable mask. In our theoretical analysis, we provide sufficient conditions on the refinable mask to guarantee L1-solution and to estimate the Holder exponent of continuity of the highest-order well-defined derivatives. Last, we show that the new functions are easier to be learned by neural networks, and prove that deep ReLU networks with a fixed width and increasing depth approximate the new refinable functions with accuracy exponential in terms of their number of parameters.

List of papers relevant papers:

    1) Neural Network Approximation of Refinable Functions
Daubechies, I., DeVore, R., Dym, N., Faigenbaum-Golovin, S., Kovalsky, S.Z., Lin, K.C., Park, J., Petrova, G., Sober, B.
IEEE Transactions on Information Theory, 69(1), 482-495, 2022.

    2) Folding Differently: Non-symmetric Refinable Functions with Implications to Neural Network Approximation
Faigenbaum-Golovin, S., Daubechies, I.,
In preparation.