ConstrICON final table entries:

Equivariance reg taylor expansion

The regularization of ICON is driven in some sense by the implicit inversion in the network. There is no such inversion in the W bipath loss, so there is no regularization of the underlying map. Instead, any regularization comes from the smoothness of the underlying map, and a penalty on the magnitude of the deviation on it? Or, possibly there is some penalty on the smoothness of the deviation if U is small.

L=∣∣Φ[W∘A,U∘B]−W−1∘Φ[A,B]∘U∣∣\mathcal{L} = ||\Phi[W \circ A, U \circ B] - W^{-1} \circ \Phi[A, B] \circ U|| Φ equivariant,Φ^=Φ+ϵn(x)\Phi ~ \text{equivariant}, \hat{\Phi} = \Phi + \epsilon n(x) L[Φ^]=ϵ∣∣∇W−1⋅(n(x)−n(U(x))∣∣\mathcal{L}[\hat{\Phi}] = \epsilon || \nabla W^{-1} \cdot (n(x) - n(U(x))||

Grad equivariance reg taylor expansion

L=∣∣∇[Φ[W∘A,U∘B]−W−1∘Φ[A,B]∘U]∣∣\mathcal{L} = ||\nabla[\Phi[W \circ A, U \circ B] - W^{-1} \circ \Phi[A, B] \circ U]||

On a hunch, assume W id

L=∣∣∇[Φ[A,U∘B]−Φ[A,B]∘U]∣∣\mathcal{L} = ||\nabla[\Phi[A, U \circ B] - \Phi[A, B] \circ U]|| L=∣∣∇[Φ[A,U∘B]+ϵ(n(x))−Φ[A,B]∘U]−ϵ(U(x))]∣∣\mathcal{L} = ||\nabla[\Phi[A, U \circ B] + \epsilon(n(x)) - \Phi[A, B] \circ U] - \epsilon(U(x))]|| L=∣∣∇[ϵn(x)−ϵn(U(x))]∣∣\mathcal{L} = ||\nabla[\epsilon n(x) - \epsilon n (U(x))]|| L=ϵ∣∣∇[n(x)−n(U(x))]∣∣\mathcal{L} = \epsilon||\nabla[n(x) - n(U(x))]||

Pretty directly a penalty on the gradient of the deviation, penalty on the second order derivative of the deviation if U small.

Research idea: black box optimize similarity by varying parameters of U

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