Smooth affine shear tight frames with MRA structure
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Bibliographic record
Abstract
Finding efficient representations is one of the most challenging and heavily sought problems in mathematics. Representation using shearlets recently receives a lot of attention due to their desirable properties in both theory and applications. Using the framework of frequency-based affine systems, in this paper we introduce and systematically study affine shear tight frames which include all known shearlet tight frames as special cases. Our results in this paper will resolve several key questions on shearlets. We provide a complete characterization for an affine shear tight frame and then use it to obtain smooth affine shear tight frames with all their generators in the Schwarz class. Though multiresolution analysis (MRA) is the foundation and key feature of wavelet analysis for fast numerical implementation of a wavelet transform, all the known shearlets so far do not possess any MRA structure and filter banks. In order to study affine shear tight frames with MRA structure, we introduce the notion of a sequence of affine shear tight frames and then we provide a complete characterization for it. Based on our characterizations, we present two different approaches, i.e., non-stationary and quasi-stationary, for the construction of sequences of affine shear tight frames with MRA structure such that all their generators are smooth (in the Schwarz class) and they have underlying filter banks. Consequently, their associated transforms can be efficiently implemented using filter banks similarly as a fast wavelet transform does.
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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) | 0.001 | 0.000 |
| Meta-epidemiology (broad) | 0.001 | 0.000 |
| Bibliometrics | 0.000 | 0.000 |
| Science and technology studies | 0.000 | 0.000 |
| Scholarly communication | 0.000 | 0.000 |
| Open science | 0.001 | 0.000 |
| Research integrity | 0.001 | 0.001 |
| Insufficient payload (model declined to judge) | 0.004 | 0.000 |
Machine scores (provisional)
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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