Group-invariant moments under tomographic projections
Pith reviewed 2026-05-10 17:27 UTC · model grok-4.3
The pith
The d-th moments of random tomographic projections determine the full d-th order rotation-invariant moments of the original object whenever d is at most the projection dimension.
A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.
Core claim
Whenever d ≤ m, the d-th order moment of the projected data Y = P(R · f), where R is Haar-uniform in SO(n) and P projects onto an m-dimensional subspace, determines the full d-th order Haar-orbit moment of f independently of the ambient dimension n. An explicit algorithmic procedure recovers the latter from the former. As a consequence, any identifiability result for the unprojected model based on d-th order group-invariant moment extends directly to the tomographic setting at the same moment order. In particular, for n=3, m=2, and d=2, the covariance of the 2D projection images determines the second-order rotationally invariant moment of the underlying 3D object.
What carries the argument
The d-th order Haar-orbit moment of f, recovered from the d-th order moment of the random projections Y via an explicit invertible mapping that holds for all d ≤ m.
If this is right
- Any identifiability result based on d-th order group-invariant moments in the unprojected model carries over unchanged to the tomographic setting.
- For n=3, m=2 and d=2 the covariance of 2D projections determines the second-order rotationally invariant moment of a 3D object.
- An explicit algorithm computes the Haar-orbit moments directly from the observed projected moments.
- The determination is independent of the original ambient dimension n.
Where Pith is reading between the lines
- The result suggests that moment-based reconstruction methods in imaging can operate at lower orders than the ambient dimension would suggest.
- Similar moment-preservation properties may hold for other linear operators or group actions beyond orthogonal projections.
- Combining the recovery procedure with existing moment-inversion algorithms could reduce sample complexity for 3D reconstruction tasks.
Load-bearing premise
The rotations must be distributed uniformly according to the Haar measure on the rotation group, and the moment order d must not exceed the projection dimension m.
What would settle it
A pair of distinct functions f and g such that the d-th moments of their random projections agree for some d ≤ m, yet their Haar-orbit moments differ.
Figures
read the original abstract
Let $f:\mathbb{R}^n\to\mathbb{R}$ be an unknown object, and suppose the observations are tomographic projections of randomly rotated copies of $f$ of the form $Y = P(R\cdot f)$, where $R$ is Haar-uniform in $\mathrm{SO}(n)$ and $P$ is the projection onto an $m$-dimensional subspace, so that $Y:\mathbb{R}^m\to\mathbb{R}$. We prove that, whenever $d\le m$, the $d$-th order moment of the projected data determines the full $d$-th order Haar-orbit moment of $f$, independently of the ambient dimension $n$. We further provide an explicit algorithmic procedure for recovering the latter from the former. As a consequence, any identifiability result for the unprojected model based on $d$-th order group-invariant moment extends directly to the tomographic setting at the same moment order. In particular, for $n=3$, $m=2$, and $d=2$, our result recovers a classical result in the cryo-EM literature: the covariance of the 2D projection images determines the second order rotationally invariant moment of the underlying 3D object.
Editorial analysis
A structured set of objections, weighed in public.
Referee Report
Summary. The paper proves that for an unknown function f: R^n → R, the d-th order moments of its m-dimensional tomographic projections Y = P(R · f), where R is Haar-uniform in SO(n), determine the d-th order Haar-orbit moments of f whenever d ≤ m, independently of the ambient dimension n. It provides an explicit algorithmic procedure to recover the latter from the former. As a consequence, identifiability results based on d-th order group-invariant moments for the unprojected model extend directly to the tomographic setting. The result recovers the classical cryo-EM fact that the covariance of 2D projections determines the second-order rotationally invariant moment of the 3D object when n=3, m=2, d=2.
Significance. If the central determination result holds, the work is significant for cryo-EM and related tomographic reconstruction problems. It shows that projection does not increase the moment order needed for recovering rotationally invariant information, with an explicit recovery algorithm that is independent of n. This directly strengthens moment-based identifiability arguments in the literature and provides a concrete computational bridge between projected data and group-invariant tensors.
minor comments (3)
- [§2] §2 (or the section defining the moment maps): clarify whether the d-th order moments are taken with respect to the standard Lebesgue measure on the projected domain or with respect to the induced measure from the projection; this affects the explicit form of the recovery map.
- [Algorithm section] Algorithm 1 (or the algorithmic procedure section): the description of the recovery step should include a brief complexity analysis or reference to the linear algebra operations involved, since the procedure is claimed to be explicit and practical.
- [Introduction] Introduction, paragraph on cryo-EM: the citation to the classical result should be paired with a one-sentence statement of how the new theorem specializes exactly to that case (e.g., via the identification of the second-moment tensor).
Simulated Author's Rebuttal
We thank the referee for the positive summary of our manuscript, the recognition of its significance for cryo-EM and related tomographic problems, and the recommendation for minor revision. The referee's description accurately captures the central result on the determination of d-th order Haar-invariant moments from m-dimensional projections when d ≤ m.
Circularity Check
No circularity; direct injectivity proof on moment maps
full rationale
The paper establishes a mathematical determination result: for d ≤ m the d-th moment of Y = P(R·f) recovers the d-th Haar-orbit moment of f via an explicit algorithm, shown by proving injectivity of the moment map on the space of group-invariant tensors. This follows from linear algebra and dimension counting in the target space of P, without any fitted parameters, self-definitional reductions, or load-bearing self-citations. The consequence for identifiability results is a straightforward extension, not a circular renaming or ansatz. The derivation is self-contained against external benchmarks and does not reduce any claimed prediction to its inputs by construction.
Axiom & Free-Parameter Ledger
axioms (2)
- domain assumption Rotations R are distributed according to the Haar measure on SO(n)
- domain assumption P is the orthogonal projection onto an m-dimensional subspace
Reference graph
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