10 Lectures and 42 Open Problems — Angular Synchronization via Projected Power Method

Projected power for phases: exact noiseless behavior and two extrapolation failures

Definitions and mathematical statement

Shared notation.

Standard number systems are $\mathbb N,\mathbb Z,\mathbb Q,\mathbb R,\mathbb C$. Specialized notation and index ranges follow the cited source and the definitions below.

Angular synchronization estimates a vector $z\in\mathbb C^n$ with $|z_i|=1$ from noisy relative-phase measurements. In the standard source model $C=zz^*+\sigma W$, $W$ is Hermitian Gaussian noise. The optimization maximizes $x^*Cx$ subject to $|x_i|=1$, with a common global phase unidentifiable. A projected or generalized power iteration applies the source linear update and normalizes each nonzero coordinate to unit modulus; initialization, shifts and zero-coordinate conventions are part of the algorithm. High probability refers to the source noise law as $n$ grows.

Record scope and qualifications.

The catalogue title is retained as metadata. Named certificates, contracts, bounds or frameworks in that title are not asserted as results of the original source extract. The source catalogue labels the historical question solved. Zhong and Boumal's primary abstract states global convergence of their generalized power method, with linear rate, in a noise regime $\sigma=O(\sqrt{n/\log n})$. Their algorithm and initialization hypotheses must be used; this does not assert convergence of every unshifted iteration from every starting point. The original question and catalogue label are retained separately from the cited result.

Statement recorded in the supplied source.

Does the projected power method converge (with high probability) to the optimal solution of the angular synchronization problem with (small enough) gaussian noise? [S2]

Source status.

The supplied catalogue labels this source record solved; that metadata was checked against the supplied snapshot on 2026-10-11. The label is preserved for provenance and is not an independent verification of present open status or mathematical correctness. This is documentation of a source-labelled solved record, not a newly proposed unsolved problem.

Short English statement

Does the specified projected power iteration find a globally optimal phase-synchronization solution when Gaussian noise is sufficiently small?

Sources

• S1 Ulam AI and the UnsolvedMath Contributors, UnsolvedMath: A Curated Collection of Open Mathematics Problems, record 2801001 (AMR-027-1001). Catalogue curation and metadata: CC BY 4.0. The source extract is retained; definitions, English summaries and qualifications are editorial additions. https://huggingface.co/datasets/ulamai/UnsolvedMath.
• S2 AMR Open Problem Lists, source linked by catalogue record AMR-027-1001. https://afonsobandeira.wordpress.com/2015/12/09/42l10psynchronization/.
• S3 Yiqiao Zhong and Nicolas Boumal, Near-optimal bounds for phase synchronization, arXiv:1703.06605; SIAM Journal on Optimization, DOI 10.1137/17M1122025. Abstract inspected for the scoped convergence claim. https://arxiv.org/abs/1703.06605.

Verbatim supplied catalogue question

Does the projected power method converge (with high probability) to the optimal solution of the angular synchronization problem with (small enough) gaussian noise?

Attempt and results

The source asks for high-probability convergence under a Gaussian noise model. Initialization, shifts, and handling zero coordinates affect the algorithm. We analyze an explicitly stated unshifted update and then a shifted ascent argument. Neither is substituted for an unspecified universal algorithm. The optimal noisy estimator is not in general the latent vector.

Noiseless calculation and an initialization obstruction.

Let $C=zz^*$ with $|z_i|=1$. Define $\operatorname{phase}(w)_i=w_i/|w_i|$ when $w_i\ne0$, retaining the previous iterate's coordinate when $w_i=0$. For an iterate $x$ with unit coordinates, put $c=z^*x$. If $c\ne0$, then $Cx=zc$ and the next iterate is $z(c/|c|)$. Its objective is $n^2$, which is maximal since $|z^*x|\le\sum_i|\overline z_ix_i|=n$. Thus one iteration suffices from every nonorthogonal initialization. But if $c=0$, the update is zero everywhere and the stated convention retains $x$. For $n=2$, $z=(1,1)$ and $x=(1,-1)$, this is a stationary iterate of objective zero although the maximum is four. So no proof can assert convergence from every starting point, even at zero noise, for this update convention. This does not refute the source's probabilistic question with a suitable specified initialization.

Why noisy optimum is not exactly the latent vector.

Take $z=(1,1)$ and \[ C=\begin{pmatrix}1&1+i\epsilon\\1-i\epsilon&1\end{pmatrix},\qquad\epsilon>0. \] This is an arbitrarily small Hermitian perturbation of $zz^*$. Write $c=1+i\epsilon$ and choose $x=(1,\overline c/|c|)$. Direct substitution gives $x^*Cx=2+2|c|>4=z^*Cz$. Thus a claim of exact recovery of $z$ cannot be inferred from convergence to the global maximizer of the noisy problem. The displayed perturbation is deterministic and is not asserted to have positive probability as one exact Gaussian realization; its role is to falsify that deterministic identification of the two objectives.

A monotonicity proof and its missing global step.

Choose a real shift $\alpha$ so $B=C+\alpha I\succeq0$. The shift changes the objective only by the constant $\alpha n$ on unit-modulus vectors, but it changes the iteration. Set $y=\operatorname{phase}(Bx)$ with any unit-modulus choices at zero coordinates. Coordinatewise maximization gives $\operatorname{Re}(y^*Bx)\ge\operatorname{Re}(x^*Bx)=x^*Bx$. Writing $d=y-x$, \[ y^*By-x^*Bx=2\operatorname{Re}(d^*Bx)+d^*Bd\ge0. \] This proves objective ascent. It does not prove global convergence: for the noiseless example above, take any $\alpha>0$. The bad $x$ obeys $Bx=\alpha x$, so it is a fixed point of the shifted update too. Therefore the exact attempted implication "positive shift plus monotone objective implies convergence to a global optimum" is false. Its counterexample retains a strict objective gap. Controlling a specified initialization and its entire noisy trajectory is an additional task.

Context and adversarial audit

Zhong and Boumal's primary abstract, arXiv:1703.06605, was consulted on 11 October 2026. It reports global convergence with a linear rate for its generalized power method in the noise regime $\sigma=O(\sqrt{n/\log n})$. That reported theorem is not re-proved here, and its algorithmic hypotheses cannot be replaced by arbitrary starting points or shifts. The preserved catalogue solved status refers to literature, while this manuscript's unresolved status describes its own proof coverage. The calculations are elementary checks, not a new noisy convergence result. The audit checked global-phase equivalence, zero-coordinate conventions, the sign of the complex off-diagonal term, and the difference between objective monotonicity and global optimality. 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. 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. No proof for the Gaussian trajectory and a specified successful initialization is supplied; the cited literature theorem is reported rather than reproduced, and monotonicity alone cannot imply global convergence.

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.

• Yiqiao Zhong and Nicolas Boumal, Near-optimal bounds for phase synchronization, arXiv:1703.06605 $2017$; primary abstract inspected for reported noise regime and convergence scope. https://arxiv.org/abs/1703.06605.

Final status

Unresolved in this attempt. No proof for the Gaussian trajectory and a specified successful initialization is supplied; the cited literature theorem is reported rather than reproduced, and monotonicity alone cannot imply global convergence.