Feature and energy identities for homogeneous projections

On ${\mathbb{R}}^d\setminus \{0\}$, write $u=\mathrm{\mid }u\mathrm{\mid }\rho (u)$. Positive two-homogeneity gives $\mathrm{\Phi }(u)=\mathrm{\mid }u{\mathrm{\mid }}^2\mathrm{\Phi }(\rho (u))$ and $V(u)=\mathrm{\mid }u{\mathrm{\mid }}^{2}V(\rho (u))$; at $u=0$ both vanish. Integrating these identities and applying the definition of $h^2$ proves both displayed projection identities.

For weak convergence on the compact sphere, the masses $h^2({\mu }_n)({\mathbb{S}}^{d-1})$ are bounded. The continuous map $\mathrm{\Phi }{\mathrm{\mid }}_{{\mathbb{S}}^{d-1}}$ has compact image in $\mathcal{H}$. Given $ϵ>0$, choose finitely many vectors $g_j=\mathrm{\Phi }({\theta }_j)$ whose open $ϵ$-balls cover that image and put $U_j=\{\theta :\mathrm{\parallel }\mathrm{\Phi }(\theta )-g_j\mathrm{\parallel }<ϵ\}$. Define $r_j(\theta )=\mathrm{dist}(\theta ,{\mathbb{S}}^{d-1}\setminus U_j)$ when the complement is nonempty, and $r_j\equiv 1$ when $U_j={\mathbb{S}}^{d-1}$. The continuous functions ${\psi }_j=r_j\mathrm{/}({\sum }_i{r}_i)$ are well-defined because the finite open sets cover the sphere, and ${\psi }_j(\theta )>0$ only when $\theta \in U_j$. Therefore $\mathrm{\parallel }\mathrm{\Phi }(\theta )-{\sum }_j{\psi }_j(\theta )g_j\mathrm{\parallel }<ϵ$ uniformly. Weak convergence gives convergence of each scalar coefficient $\int {\psi }_{j}d{h}^2({\mu }_n)$, hence strong convergence of the approximating vector integrals. The uniform approximation error is at most $ϵ$ times the uniformly bounded masses, so $\int \mathrm{\Phi }d{h}^2({\mu }_n)\to \int \mathrm{\Phi }d\nu$ in norm. Since $V{\mathrm{\mid }}_{{\mathbb{S}}^{d-1}}$ is continuous, weak convergence also gives $\int Vd{h}^2({\mu }_n)\to \int Vd\nu$. Continuity of $R$ proves the energy limit.

If $\mathrm{\Theta }=\mathrm{\varnothing }$, the only finite measures on $\mathrm{\Theta }$ and $\mathbb{R}\times \mathrm{\Theta }$ are zero, so all partial feature moments and penalties vanish and all asserted convergence conclusions hold directly. Otherwise $\mathrm{\Theta }$ is nonempty. In this case, fix $z\in \mathcal{H}$ and apply [F2] to the bounded scalar function $\psi (\theta )=⟨z,\phi (\theta )⟩$. This gives the displayed feature identity for every $z$, hence equality of the Hilbert-valued feature moments because inner products determine vectors. For the penalty, $\mathrm{\mid }w\mathrm{\mid }-w=2w_{-}\ge 0$, and [F2] applied to $\tilde{V}$ gives $\int \mathrm{\mid }w\mathrm{\mid }\tilde{V}(\theta )d\mu -\int \tilde{V}d{h}^1(\mu )=2\int w_{-}\tilde{V}(\theta )d\mu \ge 0.$ If $h^1({\mu }_n)$ converges against bounded continuous scalar tests, testing against $⟨z,\phi (\cdot )⟩$ for each $z\in \mathcal{H}$ proves weak convergence of the feature moments. Under the additional AC, closed-Euclidean, separable-Hilbert assumptions in the Statement, [F4] improves this to norm convergence. For the claimed failure of penalty determination, take any nonempty $\mathrm{\Theta }$ and fix $\theta \in \mathrm{\Theta }$. Let ${\mu }_1$ assign mass one to $(1,\theta )$ and let ${\mu }_2$ assign mass two to $(1,\theta )$ and mass one to $(-1,\theta )$. Then both have $h^1$ equal to the measure assigning mass one to $\theta$, while their $\mathrm{\mid }w\mathrm{\mid }$ penalties for $\tilde{V}\equiv 1$ are $1$ and $3$, respectively. Thus $h^1$ convergence alone cannot imply penalty or energy convergence.