Non-unique games over compact groups and orientation estimation in cryo-EM
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Bibliographic record
Abstract
Abstract Let <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline" overflow="scroll"> <mml:mi mathvariant="script">G</mml:mi> </mml:math> be a compact group and let <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline" overflow="scroll"> <mml:msub> <mml:mrow> <mml:mi>f</mml:mi> </mml:mrow> <mml:mrow> <mml:mi>i</mml:mi> <mml:mi>j</mml:mi> </mml:mrow> </mml:msub> <mml:mo>∈</mml:mo> <mml:mi>C</mml:mi> <mml:mrow> <mml:mo stretchy="false">(</mml:mo> <mml:mrow> <mml:mi mathvariant="script">G</mml:mi> </mml:mrow> <mml:mo stretchy="false">)</mml:mo> </mml:mrow> </mml:math> . We define the non-unique games (NUG) problem as finding <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline" overflow="scroll"> <mml:msub> <mml:mrow> <mml:mi>g</mml:mi> </mml:mrow> <mml:mrow> <mml:mn>1</mml:mn> </mml:mrow> </mml:msub> <mml:mo>,</mml:mo> <mml:mo>…</mml:mo> <mml:mo>,</mml:mo> <mml:mspace width="thinmathspace"/> <mml:msub> <mml:mrow> <mml:mi>g</mml:mi> </mml:mrow> <mml:mrow> <mml:mi>n</mml:mi> </mml:mrow> </mml:msub> <mml:mo>∈</mml:mo> <mml:mi mathvariant="script">G</mml:mi> </mml:math> to minimize <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline" overflow="scroll"> <mml:msubsup> <mml:mrow> <mml:mo>∑</mml:mo> </mml:mrow> <mml:mrow> <mml:mi>i</mml:mi> <mml:mo>,</mml:mo> <mml:mi>j</mml:mi> <mml:mo>=</mml:mo> <mml:mn>1</mml:mn> </mml:mrow> <mml:mrow> <mml:mi>n</mml:mi> </mml:mrow> </mml:msubsup> <mml:msub> <mml:mrow> <mml:mi>f</mml:mi> </mml:mrow> <mml:mrow> <mml:mi>i</mml:mi> <mml:mi>j</mml:mi> </mml:mrow> </mml:msub> <mml:mfenced close=")" open="("> <mml:mrow> <mml:msub> <mml:mrow> <mml:mi>g</mml:mi> </mml:mrow> <mml:mrow> <mml:mi>i</mml:mi> </mml:mrow> </mml:msub> <mml:msubsup> <mml:mrow> <mml:mi>g</mml:mi> </mml:mrow> <mml:mrow> <mml:mi>j</mml:mi> </mml:mrow> <mml:mrow> <mml:mo>−</mml:mo> <mml:mn>1</mml:mn> </mml:mrow> </mml:msubsup> </mml:mrow> </mml:mfenced> </mml:math> . We introduce a convex relaxation of the NUG problem to a semidefinite program (SDP) by taking the Fourier transform of f ij over <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline" overflow="scroll"> <mml:mi mathvariant="script">G</mml:mi> </mml:math> . The NUG framework can be seen as a generalization of the little Grothendieck problem over the orthogonal group and the unique games problem and includes many practically relevant problems, such as the maximum likelihood estimator to registering bandlimited functions over the unit sphere in d -dimensions and orientation estimation of noisy cryo-electron microscopy (cryo-EM) projection images. We implement an SDP solver for the NUG cryo-EM problem using the alternating direction method of multipliers (ADMM). Numerical study with synthetic datasets indicate that while our ADMM solver is slower than existing methods, it can estimate the rotations more accurately, especially at low signal-to-noise ratio (SNR).
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Full frame distilled prediction
Teacher imitationNot calibrated prevalence, not ground truth. Human validation pending. Learned from the 10,348 direct Codex labels and 10,348 direct Gemma labels. Candidate is the union of thresholded teacher heads; consensus is their intersection. These outputs are machine_predicted_unvalidated and are not human labels or direct frontier model labels.
Codex and Gemma teacher scores by category
| Category | Codex | Gemma |
|---|---|---|
| Metaresearch | 0.000 | 0.000 |
| Meta-epidemiology (narrow) | 0.000 | 0.000 |
| Meta-epidemiology (broad) | 0.000 | 0.000 |
| Bibliometrics | 0.000 | 0.000 |
| Science and technology studies | 0.000 | 0.000 |
| Scholarly communication | 0.000 | 0.000 |
| Open science | 0.000 | 0.001 |
| Research integrity | 0.000 | 0.001 |
| Insufficient payload (model declined to judge) | 0.000 | 0.000 |
Machine scores (provisional)
The two teacher heads of the student model, read on this work. A score orders the frame for review; it never asserts a category, and the validation status ships verbatim with every row.
Baseline scores from an immature model (maturity gate not passed, 7 training rounds). Scores rank; they never assert a category.
score_only:v0-immature-baseline · verbatim from the scoring run: score_only means the number may rank works, and no category label ships from it