The structure of balanced multivariate biorthogonal multiwavelets and dual multiframelets
Bibliographic record
Abstract
Multiwavelets and multiframelets are of interest in several applications such as numerical algorithms and signal processing, due to their desirable properties such as high smoothness and vanishing moments with relatively small supports of their generating functions and masks. In order to process and represent vector-valued discrete data efficiently and sparsely by a multiwavelet transform, a multiwavelet has to be prefiltered or balanced. Balanced orthonormal univariate multiwavelets and multivariate biorthogonal multiwavelets have been studied and constructed in the literature. Dual multiframelets include (bi)orthogonal multiwavelets as special cases, but their fundamental prefiltering and balancing property has not yet been investigated in the literature. In this paper we shall study the balancing property of multivariate multiframelets from the point of view of the discrete multiframelet transform. This approach, to our best knowledge, has not been considered so far in the literature even for multiwavelets, but it reveals the essential structure of prefiltering and the balancing property of multiwavelets and multiframelets. We prove that every biorthogonal multiwavelet can be prefiltered with the balancing order matching the order of its vanishing moments; that is, from every given compactly supported multivariate biorthogonal multiwavelet, one can always build another (essentially equivalent) compactly supported biorthogonal multiwavelets with its balancing order matching the order of the vanishing moments of the original one. More generally, we show that if a dual multiframelet can be prefiltered, then it can be equivalently transformed into a balanced dual multiframelet with the same balancing order. However, we notice that most available dual multiframelets in the literature cannot be simply prefiltered with its balancing order matching its order of vanishing moments and they must be designed to possess high balancing orders. The key ingredient of our approach is based on investigating some properties of the subdivision and transition operators acting on discrete vector polynomial sequences, which play a central role in a discrete multiframelet transform and are of interest in their own right. We also establish a new canonical form of a matrix mask, which greatly facilitates the investigation and construction of multiwavelets and multiframelets. In this paper, we obtain a complete criterion and the essential structure for balanced or prefiltered dual multiframelets in the most general setting. Our investigation of the balancing property of a multiframelet deepens our understanding of the multiframelet transform in signal processing and scientific computation.
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How this classification was reachedexpand
Full frame machine prediction
Teacher imitationNot calibrated prevalence, not ground truth. Human validation pending. The Gemma side is a direct model label for every work in the frame, read from the title-only record. The Codex side is a classifier learned from the 10,348 direct Codex labels and calibrated to design-weighted sample rates; fields without enough sample support carry no Codex call. Candidate is the union of the two sides; consensus is their intersection. These outputs are machine_predicted_unvalidated and are not human labels.
Distilled classifier scores by category (both heads)
| Category | Codex | Gemma |
|---|---|---|
| Metaresearch | 0.000 | 0.002 |
| Meta-epidemiology (narrow) | 0.001 | 0.000 |
| Meta-epidemiology (broad) | 0.000 | 0.000 |
| Bibliometrics | 0.001 | 0.001 |
| Science and technology studies | 0.000 | 0.001 |
| Scholarly communication | 0.001 | 0.002 |
| Open science | 0.000 | 0.001 |
| Research integrity | 0.001 | 0.001 |
| Insufficient payload (model declined to judge) | 0.003 | 0.001 |
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 itClassification
machine, unvalidatedMachine predicted; a candidate call from one source (direct Gemma or distilled Codex), not a consensus.
How this classification was reached, model by model and score by score, is at the end of the page under "How this classification was reached".