Efficient Design of Orthonormal Wavelet Bases for Signal Representation
Bibliographic record
Abstract
The efficient representation of a signal as a linear combination of elementary "atoms" or building blocks is central to much signal processing theory and many applications. Wavelets provide a powerful, flexible, and efficiently implementable class of such atoms. In this paper, we develop an efficient method for selecting an orthonormal wavelet that is matched to a given signal in the sense that the squared error between the signal and some finite resolution wavelet representation of it is minimized. Since the squared error is not an explicit function of the design parameters, some form of approximation of this objective is required if conventional optimization techniques are to be used. Previous approximations have resulted in nonconvex optimization problems, which require delicate management of local minima. In this paper, we employ an approximation that results in a design problem that can be transformed into a convex optimization problem and efficiently solved. Constraints on the smoothness of the wavelet can be efficiently incorporated into the design. We show that the error incurred in our approximation is bounded by a function that decays to zero as the number of vanishing moments of the wavelet grows. In our examples, we demonstrate that our method provides wavelet bases that yield substantially better performance than members of standard wavelet families and are competitive with those designed by more intricate nonconvex optimization methods.
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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 | 0.002 | 0.004 |
| Meta-epidemiology (narrow) | 0.001 | 0.001 |
| Meta-epidemiology (broad) | 0.001 | 0.000 |
| Bibliometrics | 0.001 | 0.001 |
| Science and technology studies | 0.000 | 0.001 |
| Scholarly communication | 0.001 | 0.001 |
| Open science | 0.001 | 0.001 |
| Research integrity | 0.001 | 0.001 |
| Insufficient payload (model declined to judge) | 0.001 | 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".