Ordered spectral filters, source indices, and variance grids

Assume Countable Choice. Let $T:X\to Y$ be an injective compact linear operator of infinite rank between separable real Hilbert spaces (each with a countable dense subset), and assume $T$ is Hilbert–Schmidt, meaning that for its positive singular values $s_j$ the sum ${\sum }_{j\ge 1}{s}_j^2$ is finite. Put $S=T^{\ast }T$, ${\lambda }_j=s_j^2$, and ${\lambda }_1=\mathrm{\parallel }S\mathrm{\parallel }$, with the positive eigenvalues listed with multiplicity in nonincreasing order. Let $\phi :[0,\mathrm{\infty })\to [0,\mathrm{\infty })$ be continuous and strictly increasing with $\phi (0)=0$, and take its inverse on its range. Assume the paper's convexity condition on the spectral range: $x\mapsto x{\phi }^{-1}(\sqrt{x})$ is convex for $0\le x\le \phi ({\lambda }_1{)}^2$. Define the source class ${\mathcal{W}}_{\phi }(\rho ):=\{\phi (S)w:w\in X,\mathrm{\parallel }w{\mathrm{\parallel }}_X\le \rho \}$ for $\rho >0$; equivalently, it is the source ball with singular-value index $\psi (t)=\phi (t^2)$.

An ordered spectral filter is a family of functions $q_{\alpha }:[0,{\lambda }_1]\to [0,\mathrm{\infty })$ indexed by a set $\mathcal{A}\subset (0,\mathrm{\infty })$ with $0$ as an accumulation point, such that for constants $C_q^{\mathrm{\prime }}<\mathrm{\infty }$ and $C_q^{\mathrm{\prime }\mathrm{\prime }}=1$,

and $q_{\alpha }(x)\le q_{\tilde{\alpha }}(x)$ whenever $\tilde{\alpha }<\alpha$, for every $x\in [0,{\lambda }_1]$. The index function $\phi$ is a qualification of this filter if for some $C_{\phi }<\mathrm{\infty }$,

For $\alpha \in \mathcal{A}$, define the variance function

For $\sigma >0$, a decreasing candidate sequence $({\alpha }_m{)}_{m\in {\mathbb{N}}_0}$ in $\mathcal{A}$ has geometric variance growth if $V_{T,q}({\alpha }_0)>0$ and there are constants $1<{\theta }_1\le {\theta }_2<\mathrm{\infty }$ such that

Set $v_m:={\sigma }^2{V}_{T,q}({\alpha }_m)$ and

the operator norm of the estimator-noise covariance in the singular basis. The bound $q_{{\alpha }_m}({\lambda }_j)\le C_q^{\mathrm{\prime }}\mathrm{/}{\alpha }_m$ and ${\sum }_j{\lambda }_j<\mathrm{\infty }$ make each $u_m$ and $v_m$ finite. The finite candidate set ends at

The geometric lower bound $v_m\ge v_0{\theta }_1^m$ makes this index finite.

The singular-value source ball convention and spectral multiplier for an index function

Compact spectral multipliers, source powers, and regularizing filters

Positive singular systems and the compact SVD

Countable Choice for the upstream countable singular-system construction

Li and Werner, Adaptive minimax optimality in statistical inverse problems via SOLIT, §§3.1.1–3.2.1

Clason, Regularization of Inverse Problems, complete lecture notes, Chapter 5 §§5.1–5.2

The source class agrees with the earlier singular-value convention because its multiplier at a singular value $s_j$ is $\psi (s_j)=\phi (s_j^2)=\phi ({\lambda }_j)$. The variance function is finite for every fixed $\alpha \in \mathcal{A}$: the filter bound gives $q_{\alpha }({\lambda }_j)\le C_q^{\mathrm{\prime }}\mathrm{/}\alpha$ and ${\sum }_j{\lambda }_j<\mathrm{\infty }$.