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arxiv: 2307.06835 · v3 · submitted 2023-07-13 · 🧮 math.FA · cs.IT· math.AG· math.IT

The generic crystallographic phase retrieval problem

Dan Edidin , Arun Suresh This is my paper

Pith reviewed 2026-05-24 07:56 UTC · model grok-4.3

classification 🧮 math.FA cs.ITmath.AGmath.IT
keywords phase retrievalpower spectrumsparse vectorsgeneric basisuniquenesscrystallographic phase retrievalFourier magnitudes
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The pith

If a vector is sparse in a generic basis then at sparsity level up to N/2 the generic such vector is uniquely determined up to sign by its power spectrum.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper studies the problem of recovering a real or complex vector from the squared magnitudes of its discrete Fourier transform, known as the power spectrum. It assumes the vector is sparse relative to a generic basis of the ambient space. The central result states that when the sparsity is at most roughly half the dimension, a generic sparse vector is the only one (up to global sign) that produces the observed power spectrum. A deterministic uniqueness statement holds at the lower sparsity level of roughly one quarter the dimension. The same conclusions are proved in the complex setting and extend prior work on that case.

Core claim

For a generic basis of R^N, if the sparsity level s satisfies s ≤ N/2 then the generic s-sparse vector is uniquely determined up to sign from its power spectrum; if s ≤ N/4 then every s-sparse vector is uniquely determined up to sign. Parallel statements hold for vectors in C^N.

What carries the argument

The power spectrum map sending an s-sparse vector in a generic basis to the vector of squared Fourier magnitudes, shown to be injective up to sign on a Zariski-open set when s ≤ N/2.

If this is right

  • Phase retrieval from intensity data alone becomes possible for generic sparse signals at sparsity levels twice as high as the deterministic threshold.
  • The N/4 bound supplies a uniform guarantee that works for every sparse vector rather than only generic ones.
  • The results supply a theoretical foundation for reconstruction algorithms that operate solely on power-spectrum data when the sparsity basis is known to be generic.
  • Analogous uniqueness holds in both the real and complex settings, broadening the range of signals to which the statements apply.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • Numerical checks on random bases could quickly confirm that the uniqueness threshold is attained in practice.
  • The same algebraic technique might yield stability bounds or noise-robust versions of the recovery statement.
  • Connections to other sparse-recovery settings, such as those using random projections, become natural once the power-spectrum map is shown to be injective on generic sparse vectors.

Load-bearing premise

The basis with respect to which the signal is sparse must be generic and therefore avoid a lower-dimensional algebraic degeneracy set.

What would settle it

An explicit generic basis together with two distinct s-sparse vectors (neither a scalar multiple of the other by -1) that produce identical power spectra at s = N/2 would falsify the generic uniqueness claim.

read the original abstract

In this paper we consider the problem of recovering a signal $x \in \mathbb{R}^N$ from its power spectrum assuming that the signal is sparse with respect to a generic basis for $\mathbb{R}^N$. Our main result is that if the sparsity level is at most $\sim\! N/2$ in this basis then the generic sparse vector is uniquely determined up to sign from its power spectrum. We also prove that if the sparsity level is $\sim\! N/4$ then every sparse vector is determined up to sign from its power spectrum. Analogous results are also obtained for the power spectrum of a vector in $\mathbb{C}^N$ which extend earlier results of Wang and Xu \cite{arXiv:1310.0873}.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

0 major / 2 minor

Summary. The paper studies the phase retrieval problem of recovering a vector x in R^N (or C^N) from its power spectrum, under the assumption that x is k-sparse with respect to a generic basis of the ambient space. The central claims are that when k ≲ N/2 the generic k-sparse vector is uniquely recoverable up to global sign, and when k ≲ N/4 every k-sparse vector is uniquely recoverable up to sign; analogous statements are proved in the complex case, extending earlier work of Wang and Xu.

Significance. If the results hold, they supply rigorous generic uniqueness theorems for a practically relevant variant of phase retrieval, obtained via dimension-counting arguments in algebraic geometry that avoid explicit parameter fitting and rely only on the avoidance of finitely many degeneracy loci. The extension to the complex setting and the improvement from generic to uniform recovery at the N/4 threshold are notable strengths.

minor comments (2)
  1. The precise constants hidden by the ∼ notation in the abstract and introduction should be stated explicitly (e.g., the largest integer k for which the statements hold) so that readers can compare the thresholds directly with earlier results.
  2. Notation for the power-spectrum map and the sparse set should be introduced once in a dedicated preliminary section rather than being redefined inline in multiple places.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for their positive assessment of the manuscript and for recommending acceptance.

Circularity Check

0 steps flagged

No significant circularity

full rationale

The paper establishes uniqueness theorems for sparse vectors from power spectra via algebraic dimension counting on the fibers of the power-spectrum map, showing that generic bases avoid explicit degeneracy loci. These steps rely on standard real/complex algebraic geometry and do not reduce to self-definitions, fitted parameters renamed as predictions, or load-bearing self-citations; the cited prior work (Wang-Xu) is external and the argument is self-contained against external benchmarks.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

Results rest on standard linear algebra and algebraic geometry notions of genericity; no free parameters or invented entities appear in the abstract.

axioms (1)
  • standard math Existence and density of generic bases in R^N and C^N with respect to which sparsity is defined
    Central to the statement that the basis is generic and the results apply to such bases.

pith-pipeline@v0.9.0 · 5652 in / 1171 out tokens · 41090 ms · 2026-05-24T07:56:02.964206+00:00 · methodology

discussion (0)

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Reference graph

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