Phase Retrieval from Gabor and Wavelet Magnitudes

Phase retrieval: exact finite criteria and two obstructions to uniform guarantees

Definitions and mathematical statement

Shared notation.

Standard number systems are denoted by $\mathbb N,\mathbb Z,\mathbb Q,\mathbb R,\mathbb C$; index ranges, computational models and specialized notation follow the cited problem source and the definitions below.

A frame $(\phi_\lambda)_{\lambda\in\Lambda}$ in a Hilbert space $H$ satisfies $A\|f\|^2\leq\sum_\lambda|\langle f,\phi_\lambda\rangle|^2\leq B\|f\|^2$ for constants $0<A\leq B<\infty$. Magnitudes cannot distinguish $f$ from $cf$ for $|c|=1$, so uniqueness is understood modulo this global phase. Stability bounds reconstruction error in this quotient by measurement error. Gabor systems use translations and modulations of a window; wavelet systems use translations and dilations.

Statement recorded in the supplied source.

What uniqueness and stability guarantees hold for recovering $f\in H$ from magnitude measurements $(|\langle f,\phi_\lambda\rangle|)_{\lambda\in\Lambda}$, particularly for Gabor or wavelet frames? [S2]

Scope qualification.

The frame, signal class, measurement norm, and local or uniform form of stability are parameters of the question; no universal stable inverse for every frame is asserted.

Source status.

The supplied catalogue labels this record partially solved. Its source-status metadata was checked against the supplied snapshot on 2026-10-11. This import does not independently establish current open status or mathematical correctness. The catalogue literature assessment is dated 2026-08-22.

Short English statement

Determine when time-frequency or wavelet magnitude data uniquely and stably determine a signal up to global phase.

Sources

• S1 Ulam AI and the UnsolvedMath Contributors, UnsolvedMath: A Curated Collection of Open Mathematics Problems, record 30004928 (OWR-8415360-006). Catalogue curation and metadata: CC BY 4.0. The supplied source question is retained; definitions and the English summary are editorial additions. https://huggingface.co/datasets/ulamai/UnsolvedMath.
• S2 Applied Harmonic Analysis and Data Science $2022$, Oberwolfach Reports, DOI 10.4171/owr/2021/55, pp. 3007--3066 https://doi.org/10.4171/owr/2021/55.
• S3 Applied Harmonic Analysis and Data Science, Oberwolfach Reports 55/2021, primary report PDF https://ems.press/content/serial-article-files/46933. Underlying Oberwolfach Reports material retains its source-specific licence, including CC BY-SA 4.0 where applicable; see THIRD_PARTY_NOTICES.md.

Verbatim supplied catalogue question

What uniqueness and stability guarantees hold for recovering $f\in H$ from magnitude measurements $(|\langle f,\phi_\lambda\rangle|)_{\lambda\in\Lambda}$, particularly for Gabor or wavelet frames?

The source's parameter-dependent question

The primary report, printed pages3025--3027, distinguishes signal spaces, windows, sampling sets, uniqueness modulo phase, and stability. It already reports both positive results on restricted signal classes and nonuniqueness for sampled Gaussian Gabor measurements on all of $L^2$. We do not replace those choices by a universal assertion about frames. We attempt an elementary span criterion and a global stability argument, then identify the limitations through explicit examples.

A complete finite real criterion

For a finite family $\phi_1,\ldots,\phi_M\in\mathbb R^d$, phase retrieval up to sign holds exactly when, for every subset $I\subseteq\{1,\ldots,M\}$, either its vectors or the vectors in its complement span $\mathbb R^d$. If both spans are proper, choose nonzero $u$ perpendicular to the first and nonzero $v$ perpendicular to the second. Then $f=u+v$ and $g=u-v$ have equal measurement magnitudes: on the first subset only the $v$ term contributes, with opposite signs, and on the complement only the $u$ term contributes. They are not equal up to sign, since either equality would force $u=0$ or $v=0$. Thus the condition is necessary. Conversely if $f,g$ have equal real magnitudes, then for every index \[ \langle f-g,\phi_i\rangle\langle f+g,\phi_i\rangle=0. \] Take $I$ to consist of those indices where the first factor is zero. On its complement the second is zero. The spanning condition forces $f-g=0$ or $f+g=0$, proving sufficiency.

This real argument fails for complex phases. For instance $e_1,e_2,e_1+e_2$ have the real spanning property in dimension two, but complex vectors $(1,i)$ and $(1,-i)$ have the same three magnitudes and are not global phase multiples: the first coordinate would force the multiplier to be one. The factorization into two real linear factors is exactly the step that fails.

A continuous Gabor frame with explicit nonuniqueness

Take a nonzero bounded window $g$ supported in $[-R,R]$, $R>0$, and two unit signals $u,v$ with compact supports separated by more than $2R$. No translate of the window intersects both supports. Thus for every time-frequency point, at most one of $V_gu,V_gv$ is nonzero, and \[ |V_g(u+v)(x,\omega)|=|V_g(u-v)(x,\omega)|. \] The signals are orthogonal by their disjoint supports. For any $|c|=1$, \[ \|(u+v)-c(u-v)\|_2^2 =|1-c|^2+|1+c|^2=4, \] so they are not phase-equivalent.

This remains a frame example, not just an arbitrary deficient measurement system. With $V_gf(x,\omega)=\int f(t)\overline{g(t-x)}e^{-2\pi i\omega t}\,dt$, Plancherel in $\omega$ and Tonelli give \[ \int_{\mathbb R^2}|V_gf(x,\omega)|^2\,dx\,d\omega =\int_{\mathbb R}|f(t)|^2 \left(\int_{\mathbb R}|g(t-x)|^2dx\right)dt =\|g\|_2^2\|f\|_2^2. \] The product $f(t)g(t-x)$ is in $L^2$ for each $x$ by boundedness of $g$, justifying Plancherel; the equality extends with the usual Fourier $L^2$ interpretation. Hence the continuous Gabor system is a tight frame for linear measurements even though its magnitudes fail uniqueness. This does not address noncompact Gaussian windows or the source's bandlimited classes.

No uniform Lipschitz inverse for a countable infinite frame

Let $H$ be an infinite-dimensional real or complex Hilbert space and $(\phi_j)_{j\ge1}$ a frame with finite upper bound. We show that no dimension-free positive lower Lipschitz bound for the magnitude map holds on all unit vectors modulo phase. Fix a unit vector $u$ and $\eta>0$. Its analysis coefficients are square-summable, so a finite set $I$ satisfies \[ \sum_{j\notin I}|\langle u,\phi_j\rangle|^2<\eta^2. \] Choose a unit $v$ orthogonal to $u$ and all $\phi_j$ with $j\in I$; this is possible because their span is finite-dimensional and $H$ is infinite-dimensional. Set $f=u+v,g=u-v$. They have quotient distance two by the preceding orthogonality calculation. For complex numbers $a,b$, the reverse triangle inequality applied after cancelling either term gives \[ \left||a+b|-|a-b|\right|\le2\min(|a|,|b|). \] The measurement difference is zero on $I$ and at most $2|\langle u,\phi_j\rangle|$ off $I$. Therefore \[ \left\|\left(|\langle f,\phi_j\rangle| -|\langle g,\phi_j\rangle|\right)_j\right\|_{\ell_2}<2\eta. \] Dividing $f,g$ by $\sqrt2$ makes them unit vectors, with quotient distance $\sqrt2$ and measurement distance below $\sqrt2\eta$. Since $\eta$ is arbitrary, there is no uniform positive lower Lipschitz constant. The argument allows the frame to be phase-retrieving: lack of a uniform stable inverse does not by itself imply a pair with exactly equal magnitudes.

Audit and remaining target

The audit checked real versus complex scalar identities, continuous versus countable measurement norms, the window support separation, the nonzero quotient distance, and the finite-span orthogonal choice in infinite dimension. Plancherel and Tonelli are the standard foundational Fourier/measure identities used in the displayed frame computation. No local stability theorem or characterization for a particular sampled Gabor or wavelet frame is obtained. The source's full family of uniqueness and stability questions remains unresolved by these known-style auxiliary results. No numerical experiment or theorem prover was used.

Process and contributions

Mathematics and model-assisted adversarial self-audit: OpenAI gpt-6-astra, ultra reasoning, as dispatched for this research batch; exact serving revision not exposed. Codex (GPT-6; exact serving revision not exposed) materially assisted with orchestration and format review. No human mathematical authorship, mathematical edits, or independent human proof checking is asserted. The coordinating Codex (GPT-6; exact serving revision not exposed) also provided a separate model-assisted written-proof review of this manuscript. This is not independent human verification. Contribution prompt A was applied to the full definition and supplied source row, then prompt B was applied adversarially to the derived results. The item-specific audit above records the scope checks and rejected extrapolations. Local assembly and structural checks do not establish mathematical correctness. The source catalogue status is preserved in TOP_PROBLEM and is separate from this attempt status.

Integration review.

Codex (GPT-6; exact serving revision not exposed) reviewed the written arguments and expanded layout and source-reference wrappers. Repository metadata and the production web renderer were checked using Python 3.13 and JavaScript with MathJax 3.2.2. This model-assisted review is not independent human proof verification; no PDF compilation or formal proof checking is asserted.

Completion Estimate

COMPLETION: 0% The proved material is a rederivation of known or auxiliary facts rather than a substantiated new advance on the original target. The finite real criterion, compact-window ambiguity and global infinite-frame instability do not characterize the source Gabor/wavelet signal, window and sampling regimes or their local stability.

References

The complete original Sources subsection above is retained, including catalogue attribution and licence notices. The additional list records primary sources actually consulted for this attempt on 11 October 2026; uncited source links reproduced above were not all rechecked.

• OWR55/2021, Applied Harmonic Analysis and Data Science; complete relevant primary passages read in cached report, with printed pages identified in the manuscript. https://ems.press/content/serial-article-files/46933.

Final status

Unresolved in this attempt. The finite real criterion, compact-window ambiguity and global infinite-frame instability do not characterize the source Gabor/wavelet signal, window and sampling regimes or their local stability.