SOLIT parameter choice on a geometric-variance grid

Assume Countable Choice. Let $T:X\to Y$ be an injective compact Hilbert–Schmidt operator of infinite rank between separable real Hilbert spaces, let $S=T^{\ast }T$, and let $(s_j,v_j,u_j)$ be a fixed positive singular system with ${\lambda }_j=s_j^2$. Let $Z$ be a real isonormal Gaussian process on $Y$, meaning that $g\mapsto Z(g)$ is linear into $L^2$, is centered, has covariance $\mathbb{E}[Z(g)Z(h)]=⟨g,h{⟩}_Y$, and has independent standard-normal values on every finite orthonormal tuple. In the weak observation experiment $Y_x(g)=⟨Tx,g{⟩}_Y+\sigma Z(g)$ with known noise standard deviation $\sigma >0$, use the ordered filter $q_{\alpha }$, the geometric-variance candidates ${\alpha }_0>\cdots >{\alpha }_{m_{\max }}$, and variances $v_m={\sigma }^2{\sum }_j{\lambda }_j{q}_{{\alpha }_m}({\lambda }_j{)}^2$ from the preceding definition. For each candidate index $m$, the regularized estimator ${\widehat{x}}_m$ is the $L^2$ limit of its finite singular-coordinate sums; its $j$th coordinate is ${\lambda }_j{q}_{{\alpha }_m}({\lambda }_j){\theta }_j+\sigma s_j{q}_{{\alpha }_m}({\lambda }_j){\xi }_j$, where ${\theta }_j=⟨x,v_j{⟩}_X$ and ${\xi }_j=Z(u_j)$.

Take as input the observed sequence $y=(y_j{)}_{j\ge 1}$, where $y_j:=Y_x(u_j)$. For each candidate $m$ and each $N=1,2,\dots$, compute

If these partial sums converge in $X$, define ${\widehat{x}}_m(y)$ to be their limit; otherwise set ${\widehat{x}}_m(y)=0_X$. This specifies an $X$-valued output for every input sequence. The quantities $v_{m_1,m_2}$ and $z_{m_1,m_2}(x)$ are determined by the known operator, filter, noise level and Gaussian coordinate law.

For $0\le m_1<m_2\le m_{\max }$, set

and let $N_{m_1,m_2}$ be the norm of the centered estimator difference divided by $\sigma$. For $x>0$, define its upper quantile by

Given fixed $\beta >0$ and $\gamma >0$, for $0\le m<m_{\max }$ put $x_m:=2(1+\gamma )\log (v_{m+1}\mathrm{/}{v}_0)$, and for each pair $m_1<m_2$ put ${\eta }_{m_1,m_2}:=\sigma z_{m_1,m_2}(x_{m_1})+\beta \sqrt{v_{m_1,m_2}}$. Accept an index $m_1$ exactly when $\mathrm{\parallel }{\widehat{x}}_{m_1}(y)-{\widehat{x}}_{m_2}(y){\mathrm{\parallel }}_X\le {\eta }_{m_1,m_2}$ for every $m_2$ with $m_1<m_2\le m_{\max }$, and output the smallest accepted index $\widehat{m}$ and parameter $\widehat{\alpha }={\alpha }_{\widehat{m}}$. For $m_1=m_{\max }$, accept without comparisons, including when $m_{\max }=0$. This rule uses only the observed sequence, the known operator and filter, the noise level, the fixed candidate grid, and $\beta ,\gamma$; it does not use $x$ or its source index $\phi$.

Ordered filters, source conditions, and candidate-grid variance

The positive-noise SVD sequence experiment

The weak isonormal observation process

Independent Gaussian coordinates of white noise

Well-definedness and bias–variance properties of the regularized estimators

Li and Werner, Adaptive minimax optimality in statistical inverse problems via SOLIT, §§3.2.1–3.2.2

Spokoiny and Willrich, Bootstrap tuning in ordered model selection, §§2.1–2.4

The arrows show the order of the finite procedure: form the regularized estimates, compare each candidate with every finer one using the two-index threshold, then return the smallest candidate that passes all of its comparisons. The diagram illustrates the algorithm only; the proofs of its properties use the coordinate definitions and inequalities.

The constant $\gamma$ controls the strength of the noise quantiles, while $\beta$ controls the bias tolerance. The quantile definition remains meaningful when a pairwise Gaussian difference is degenerate; then its upper quantile is zero.