function machine




A function machine is a generalization of neural networks to potentially infinite dimensional layers, motivated by the study of universal approximation of operators and functionals over abstract Banach spaces.


A function machine’s layers is defined by the type of layers it is composed with.


Let K d,K dK \subset \mathbb{R}^d, K' \subset \mathbb{R}^{d'} both compact, TT is some affine map.

  • (Operator layer) When T: 1(K,μ) 1(K,μ)T:\mathcal{L}^1(K, \mu)\to \mathcal{L}^1(K', \mu), T=T 𝔬T = T^\mathfrak{o} is said to be an operator layer if there is some continuous family of measures (W tμ)(W_t \ll \mu) indexed by tKt \in K' and a function b 1(K,μ)b \in \mathcal{L}^1(K, \mu) such that:
T 𝔬:f(t KfdW t+b(t))T^\mathfrak{o}: f \mapsto \left(t \mapsto \int_K f dW_t + b(t)\right)
  • (Functional layer) When T: 1(K,μ) dT:\mathcal{L}^1(K, \mu) \to \mathbb{R}^{d'}, T=T 𝔣T = T^\mathfrak{f} is said to be a functional layer if there is some measure WμW \ll \mu and a vector b db \in \mathbb{R}^d such that:
T 𝔣:ffdW+bT^\mathfrak{f}: f \mapsto \int f dW + b
  • (Basis layer) When T: d 1(K,μ)T:\mathbb{R}^d \to \mathcal{L}^1(K',\mu), T=T 𝔟T=T^\mathfrak{b} is said to be a basis layer if there is some function w:K dw:K'\to\mathbb{R}^d and b 1(K,μ)b \in \mathcal{L}^1(K', \mu) such that:

    T 𝔟:y(t i=1 dy i[w(t)] i+b(t))T^\mathfrak{b} : y \mapsto \left(t \mapsto \sum_{i=1}^d y_i[w(t)]_i + b(t) \right)
  • (Fully-connected layer) When the inputs and outputs of a layer are finite dimensional, we yield the standard fully-connected layer, denoted T 𝔫T^\mathfrak{n}

One can now, for instance, define an 1-hidden layer operator network to be the composition of two operator layers:

F[f]=(T 𝔬gT 𝔬)[f]=sb(s)+ Kg( KfdW t+b(t))dW sF[f] = (T^\mathfrak{o}\circ g \circ T^\mathfrak{o})[f] = s \mapsto b'(s) + \int_{K''}g\left(\int_K f dW_t + b(t) \right)dW_s

where (W tμ) tK(W_t \ll \mu)_{t \in K''}, (W sμ) sK(W_s\ll\mu)_{s \in K'} are families of absolutely continuous measures w.r.t. the Lebesgue measure μ\mu, bb and bb' are measurable bias functions over compact domains KK'' and KK' respectively, and gg an activation function.


  • William Guss, Deep Function Machines: Generalized Neural Networks for Topological Layer Expression,(arXiv:1612.04799

  • William Guss, Ruslan Salakhutdinov, On Universal Approximation by Neural Networks with Uniform Guarantees on Approximation of Infinite Dimensional Maps,(arXiv:1910.01545

Last revised on January 14, 2020 at 17:15:44. See the history of this page for a list of all contributions to it.