Algebraic Neural Networks: Stability to Deformations

Stability of Algebraic Neural Networks to Small Perturbations

“…Proof. See [13] From theorem 2 and taking into account Theorem 1, we can conclude that AlgNNs are stable to the perturbations considered, indeed the norm of the Fréchet derivative of the filter acting on the perturbation is bounded by the size of the perturbation. Notice also that the non commutativity between the operators S and T 1 does not change the functional form of the upper size in eqn.…”

“…u i are the eigenvectors of S, and , is the inner product. As proven in [13] the value of δ is upper bounded by the weighted difference between the eigenvectors of x and T 1 .…”

“…Particular cases of the algebraic models lead to signal processing frameworks well known in the literature. For instance using the polynomial algebra, which has a single generator, it is possible to derive discrete time signal processing (DTSP), graph signal processing (GSP) and graphon signal processing (WSP) considering different representations [9,13], i.e. different choices of M and ρ. DTSP uses a vector space (countably infinite dimensional) where the elements are sequences of square summable coefficients and the shift operator is the delay function, while in GSP the vector space is an N −dimensional vector space where N is the number of nodes in the graph and the shift operator could be the adjacency matrix of the graph while in WSP the vector space is the set of functions with finite energy in the interval [0, 1] and the shift is associated to an integral transform whose kernel is given by a graphon [9,13].…”

“…It is important to point out that ρ is not necessarily a homomorphism. As pointed out in [13] perturbations in the domain of the signals can be reframed in terms of this perturbation model.…”

“…Proof. See [13] Theorem 1 show how the filters act directly on the deformation. In particular, we can see from the therm D p (S) {T(S)} that the Fréchet derivative of the physical representation of the filter acts directly on T(S), and this is independent from the properties of T(S), i.e.…”