Note on the Rademacher-Walsh Polynomial Basis Functions
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Bibliographic record
Abstract
Over the years, one of the methods of choice to estimate probability density functions for a given random variable (defined on binary input space) has been the expansion of the estimation function in Rademacher-Walsh Polynomial basis functions. For a set of $L$ features (often considered as an ``$L$-dimensional binary vector''), the Rademacher-Walsh Polynomial approach requires $2^{L}$ basis functions. This can quickly become computationally complicated and notationally clumsy to handle whenever the value of $L$ is large. In current pattern recognition applications it is often the case that the value of $L$ can be 100 or more.In this paper we show that the expansion of the probability density function estimation in Rademacher-Walsh Polynomial basis functions is equivalent to the expansion of the estimation function in a set of Dirac kernel functions. The latter approach is not only able to eloquently allay the computational bottle--neck and notational awkwardness mentioned above, but may also be naturally neater and more ``elegant'' than the Rademacher-Walsh Polynomial basis function approach even when this latter approach is computationally feasible.
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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 | 0.003 | 0.001 |
| 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 | 0.001 |
| Open science | 0.002 | 0.000 |
| Research integrity | 0.000 | 0.001 |
| Insufficient payload (model declined to judge) | 0.000 | 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 it