Factorized and composite linear inversion

Let $X,Y,Z$ be real or complex Hilbert spaces over one scalar field, let $F:X\to Y$ and $G:Y\to Z$ be compact linear maps, and set $A:=G\circ F:X\to Z$. For spectral regularization families $R_{\alpha }^G:Z\to Y$ for $G^{†}$ and $R_{\gamma }^F:Y\to X$ for $F^{†}$, and a parameter rule $(\alpha ,\gamma ):(0,\mathrm{\infty })\times Z\to (0,\mathrm{\infty }{)}^2$, define $T_{\delta }(z^{\delta }):=R_{\gamma (\delta ,z^{\delta })}^F(R_{\alpha (\delta ,z^{\delta })}^G{z}^{\delta })$ for $z^{\delta }\in Z$. The order is fixed by the factorization: first regularize $G$, then $F$; no general identity $A^{†}=F^{†}{G}^{†}$ is part of this definition.

Compact Hilbert inverse model

Compact spectral filters and source conditions

Brandt and Horn, §2.1 and Theorem 1, pp. 2–3

Clason, Chapter 5, §5.1, pp. 44–50

The upper row is the forward map: applying $F$ and then $G$ gives $A=G\circ F$. The lower row shows the estimator's domains and codomains: data in $Z$ first pass through $R_{\alpha }^G$ to $Y$, then through $R_{\gamma }^F$ to $X$. Read that row from right to left; the diagram omits the parameter rule's dependence on $\delta$ and $z^{\delta }$.

For noisy data $z^{\delta }$, the lower row is $z^{\delta }\mapsto R_{\alpha }^G{z}^{\delta }\mapsto R_{\gamma }^F{R}_{\alpha }^G{z}^{\delta }$. The diagram records these domains and orders only; no proof about convergence or pseudoinverses relies on it.

Assume Countable Choice. Let $X,Y,Z$ be real or complex Hilbert spaces over one scalar field. Let compact linear maps $F:X\to Y$ and $G:Y\to Z$ have a common singular system indexed by a finite or countable set $J$: $(x_j{)}_{j\in J}$, $(y_j{)}_{j\in J}$, and $(z_j{)}_{j\in J}$ are orthonormal bases of $\overline{R(F^{\ast })}$, $\overline{R(F)}=\overline{R(G^{\ast })}$, and $\overline{R(G)}$, and $F{x}_j={\lambda }_j{y}_j$, $F^{\ast }{y}_j={\lambda }_j{x}_j$, $G{y}_j={\mu }_j{z}_j$, and $G^{\ast }{z}_j={\mu }_j{y}_j$ with $0<{\lambda }_j,{\mu }_j\le 1$. Put $A:=G\circ F$. Let ${\nu }_0>2$, $1<\nu \le {\nu }_0-1$, $\rho >0$, $x^{†}=\mathrm{\mid }A{\mathrm{\mid }}^{\nu }w$ with $\mathrm{\parallel }w{\mathrm{\parallel }}_X\le \rho$, and $z:=A{x}^{†}\mathrm{\ne }0$; set $y^{†}:=F{x}^{†}$. The common system gives the compatible minimum-norm factorization on this exact datum: $A^{†}z=F^{†}{G}^{†}z=x^{†}$ and $G^{†}z=F{x}^{†}=y^{†}$. For $H=F,G$, let $R_{\eta }^H=f_{\eta }^H(H^{\ast }H)H^{\ast }$ and $p_{\eta }^H(t):=1-t{f}_{\eta }^H(t)$. Assume finite constants $C_H,S_H,K_H>0$ satisfy $\mathrm{\parallel }H{R}_{\eta }^H\mathrm{\parallel }\le C_H$, $\mathrm{\parallel }I-H{R}_{\eta }^H\mathrm{\parallel }\le S_H$, and $\mathrm{\parallel }{R}_{\eta }^H\mathrm{\parallel }\le K_H{\eta }^{-1\mathrm{/}2}$ for every $\eta >0$; and, for each $0<s\le {\nu }_0$, ${\sup }_{j\in J}{s}_{H,j}^s\mathrm{\mid }{p}_{\eta }^H(s_{H,j}^2)\mathrm{\mid }\le D_H(s){\eta }^{s\mathrm{/}2}$ for finite $D_H(s)$, where $s_{F,j}={\lambda }_j$ and $s_{G,j}={\mu }_j$. Fix ${\tau }_G>S_G$ and ${\tau }_F>S_F{C}_G$. Let ${\alpha }_k,{\gamma }_{\mathrm{\ell }}>0$ be decreasing sequences with successive ratios in $[\theta ,1)$ for one fixed $0<\theta <1$. For noisy data $z^{\delta }$ with $\mathrm{\parallel }{z}^{\delta }-z{\mathrm{\parallel }}_Z\le \delta$, choose first indices $k,\mathrm{\ell }$ (when they exist) such that $\mathrm{\parallel }G{R}_{{\alpha }_k}^G{z}^{\delta }-z^{\delta }{\mathrm{\parallel }}_Z\le {\tau }_G\delta <\mathrm{\parallel }G{R}_{{\alpha }_i}^G{z}^{\delta }-z^{\delta }{\mathrm{\parallel }}_Z$ for every $i<k$, and $\mathrm{\parallel }A{R}_{{\gamma }_{\mathrm{\ell }}}^F{R}_{{\alpha }_k}^G{z}^{\delta }-G{R}_{{\alpha }_k}^G{z}^{\delta }{\mathrm{\parallel }}_Z\le {\tau }_F\delta <\mathrm{\parallel }A{R}_{{\gamma }_i}^F{R}_{{\alpha }_k}^G{z}^{\delta }-G{R}_{{\alpha }_k}^G{z}^{\delta }{\mathrm{\parallel }}_Z$ for every $i<\mathrm{\ell }$. For any sequence $\delta \downarrow 0$ for which both selected indices exist for all sufficiently small $\delta$, there are constants $C_1,C_2$ independent of $\delta$ and $\rho$ such that, for all sufficiently small $\delta$, $\mathrm{\parallel }{x}^{†}-R_{{\gamma }_{\mathrm{\ell }}}^F{R}_{{\alpha }_k}^G{z}^{\delta }{\mathrm{\parallel }}_X\le C_1{\rho }^{1\mathrm{/}(\nu +1)}{\delta }^{\nu \mathrm{/}(\nu +1)}+C_2{\rho }^{2\mathrm{/}(\nu +1)}{\delta }^{(\nu -1)\mathrm{/}(\nu +1)}\mathrm{.}$ In particular, for fixed $\rho$ this two-stage reconstruction converges to $x^{†}$. The statement is conditional on existence of both stopping indices and makes no claim that $(G\circ F{)}^{†}=F^{†}{G}^{†}$ for arbitrary factorizations.

Composite estimator and order

Compact Hilbert inverse model

Power source ball and minimum-norm convention

Spectral interpolation inequality

Hilbert-space Moore–Penrose inverse

Hilbert adjoint convention

Compact-operator singular system

Spectral filter and residual conventions

Brandt and Horn, Theorem 6, proof, and conclusion on optimality

Clason, Theorem 5.11 and complete proof

Assume Countable Choice. Let $X,Y,Z$ be real or complex Hilbert spaces over one scalar field. Let compact linear maps $F:X\to Y$ and $G:Y\to Z$ have a common singular system indexed by a finite or countable set $J$: $(x_j{)}_{j\in J}$, $(y_j{)}_{j\in J}$, and $(z_j{)}_{j\in J}$ are orthonormal bases of $\overline{R(F^{\ast })}$, $\overline{R(F)}=\overline{R(G^{\ast })}$, and $\overline{R(G)}$, and $F{x}_j={\lambda }_j{y}_j$, $F^{\ast }{y}_j={\lambda }_j{x}_j$, $G{y}_j={\mu }_j{z}_j$, and $G^{\ast }{z}_j={\mu }_j{y}_j$ with $0<{\lambda }_j,{\mu }_j\le 1$. Put $A:=G\circ F$. Let ${\nu }_0>2$, $1<\nu \le {\nu }_0-1$, $\rho >0$, $w\in X$ with $\mathrm{\parallel }w{\mathrm{\parallel }}_X\le \rho$, $x^{†}:=\mathrm{\mid }A{\mathrm{\mid }}^{\nu }w$ where $\mathrm{\mid }A\mathrm{\mid }=(A^{\ast }A{)}^{1\mathrm{/}2}$, and $z:=A{x}^{†}\mathrm{\ne }0$. The common system makes $x^{†}$ and $F{x}^{†}$ the minimum-norm solutions at their respective stages. For $H=F,G$, let $R_{\eta }^H:=f_{\eta }^H(H^{\ast }H)H^{\ast }$ and $p_{\eta }^H(t):=1-t{f}_{\eta }^H(t)$. Assume finite constants $C_H,S_H,K_H>0$ satisfy $\mathrm{\parallel }H{R}_{\eta }^H\mathrm{\parallel }\le C_H$, $\mathrm{\parallel }I-H{R}_{\eta }^H\mathrm{\parallel }\le S_H$, and $\mathrm{\parallel }{R}_{\eta }^H\mathrm{\parallel }\le K_H{\eta }^{-1\mathrm{/}2}$ for every $\eta >0$; and, for every $0<s\le {\nu }_0$, assume a finite constant $D_H(s)$ satisfies ${\sup }_{j\in J}{s}_{H,j}^s\mathrm{\mid }{p}_{\eta }^H(s_{H,j}^2)\mathrm{\mid }\le D_H(s){\eta }^{s\mathrm{/}2}$ for every $\eta >0$, where $s_{F,j}={\lambda }_j$ and $s_{G,j}={\mu }_j$. Fix ${\tau }_G>S_G$ and ${\tau }_F>S_F{C}_G$. Let $({\alpha }_k{)}_{k\ge 1}$ and $({\gamma }_{\mathrm{\ell }}{)}_{\mathrm{\ell }\ge 1}$ be decreasing positive sequences with successive ratios in $[\theta ,1)$ for one fixed $0<\theta <1$. For noisy data $z^{\delta }$ with $\mathrm{\parallel }{z}^{\delta }-z{\mathrm{\parallel }}_Z\le \delta$, choose the first indices $k,\mathrm{\ell }$, when they exist, such that $\mathrm{\parallel }G{R}_{{\alpha }_k}^G{z}^{\delta }-z^{\delta }{\mathrm{\parallel }}_Z\le {\tau }_G\delta <\mathrm{\parallel }G{R}_{{\alpha }_i}^G{z}^{\delta }-z^{\delta }{\mathrm{\parallel }}_Z$ for every $i<k$, and $\mathrm{\parallel }A{R}_{{\gamma }_{\mathrm{\ell }}}^F{R}_{{\alpha }_k}^G{z}^{\delta }-G{R}_{{\alpha }_k}^G{z}^{\delta }{\mathrm{\parallel }}_Z\le {\tau }_F\delta <\mathrm{\parallel }A{R}_{{\gamma }_i}^F{R}_{{\alpha }_k}^G{z}^{\delta }-G{R}_{{\alpha }_k}^G{z}^{\delta }{\mathrm{\parallel }}_Z$ for every $i<\mathrm{\ell }$. For fixed factors, filters, parameter sequences and radius $\rho$, define $E_{\delta }$ as the supremum of $\mathrm{\parallel }{R}_{{\gamma }_{\mathrm{\ell }}}^F{R}_{{\alpha }_k}^G{z}^{\delta }-x^{†}{\mathrm{\parallel }}_X$ over all $w$ and all $z^{\delta }$ satisfying these source, noise, nonzero-data, and stopping-index conditions. Determine the sharp asymptotic order of $E_{\delta }$ as $\delta \downarrow 0$ whenever this admissible set is nonempty. In particular, is the exponent $(\nu -1)\mathrm{/}(\nu +1)$ in the upper bound $C_1{\rho }^{1\mathrm{/}(\nu +1)}{\delta }^{\nu \mathrm{/}(\nu +1)}+C_2{\rho }^{2\mathrm{/}(\nu +1)}{\delta }^{(\nu -1)\mathrm{/}(\nu +1)}$ optimal for fixed $\rho$, or can it be improved uniformly over this composite class?

Brandt and Horn, Theorem 6 and conclusion, printed pp. 6–9 and 17

As of 2026-10-21, the arXiv record lists v1 only. In §4, Brandt and Horn explicitly leave open whether their a-priori convergence speed is optimal. They separately call the double-discrepancy principle's finite-qualification result suboptimal and state its rate, but do not pose the DDP exponent's sharpness as that explicit open question or give a matching lower bound in Theorem 6. This item retains the DDP sharpness question as its own formulation rather than attributing it to the authors. Under the common singular system, source conditions, finite filter qualification, and both residual rules, the local theorem supplies a conditional two-term upper bound; the source's displayed radius exponent is inconsistent with its proof. Targeted searches for the exact arXiv identifier, title, and composite double-discrepancy optimality found no later resolution as of this date. No lower bound matching the two-stage rate is asserted here.

This formulation asks about the deterministic worst-case rate of the specified two-stage residual-selection rule; it is not a claim about optimality among all reconstruction procedures.