Press’s conjectured near-optimal iterative inverse complexity

Construct an iterative inverse for the full approximate discrete Radon transform with complexity O(N^2 log^q N) for some 2≤q≤3, ideally achieving the optimal q=2 complexity O(N^2 log^2 N log ε^{-1}) for accuracy ε, and prove the corresponding complexity bound.

Background

Press’s multigrid method was reported to reduce the error by a fixed factor per O(N2 log2 N) operations, suggesting an overall complexity of O(N2 log3 N) to reach a prescribed accuracy. The conjecture asks whether an iterative inverse with a complexity exponent between 2 and 3 exists, with q=2 regarded as optimal.

The paper notes that these complexity bounds had not been proved. The oblique inverses constructed in the paper do not fully resolve this conjecture because their cost contains an additional factor related to the smallest singular value of the ADRT.

References

Press observed that his full multigrid inverse reduces the error by a fixed factor per $O(N2\log2N)$ operations, reaching a given accuracy in $O(N2\log3N)$ work, and conjectured that an iterative inverse of complexity $O(N2\logqN)$ with $2\le q\le3$ exists, $q=2$ being optimal, i.e.\ $O(N2\log2N\,\log\varepsilon{-1})$ for accuracy $\varepsilon$ ; see the accounts in \S1 and \S1, where SPIFE attains $q=2$ for data in the range. These bounds were not proved.

— Explicit Oblique Inverses of the Approximate Discrete Radon Transform  (2610.23363 - Rim, 16 Oct 2026) in Section 1, subsection “Press’s conjecture”