Note on the Rademacher-Walsh Polynomial Basis Functions
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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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 | 0.005 | 0.022 |
| Meta-epidemiology (narrow) | 0.001 | 0.001 |
| Meta-epidemiology (broad) | 0.001 | 0.001 |
| Bibliometrics | 0.002 | 0.002 |
| Science and technology studies | 0.001 | 0.005 |
| Scholarly communication | 0.002 | 0.006 |
| Open science | 0.001 | 0.003 |
| Research integrity | 0.002 | 0.010 |
| Insufficient payload (model declined to judge) | 0.004 | 0.003 |
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".