High-Order Neural Networks are Equivalent to Ordinary Neural Networks
Bibliographic record
Abstract
Equivalence of computational systems can assist in obtaining abstract systems, and thus enable better understanding of issues related their design and performance. For more than four decades, artificial neural networks have been used in many scientific applications to solve classification problems as well as other problems. Since the time of their introduction, multilayer feedforward neural network referred as Ordinary Neural Network (ONN), that contains only summation activation (Sigma) neurons, and multilayer feedforward High-order Neural Network (HONN), that contains Sigma neurons, and product activation (Pi) neurons, have been treated in the literature as different entities. In this work, we studied whether HONNs are mathematically equivalent to ONNs. We have proved that every HONN could be converted to some equivalent ONN. In most cases, one just needs to modify the neuronal transfer function of the Pi neuron to convert it to a Sigma neuron. The theorems that we have derived clearly show that the original HONN and its corresponding equivalent ONN would give exactly the same output, which means; they can both be used to perform exactly the same functionality. We also derived equivalence theorems for several other non-standard neural networks, for example, recurrent HONNs and HONNs with translated multiplicative neurons. This work rejects the hypothesis that HONNs and ONNs are different entities, a conclusion that might initiate a new research frontier in artificial neural network research.
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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 | 0.008 |
| Meta-epidemiology (narrow) | 0.001 | 0.000 |
| Meta-epidemiology (broad) | 0.001 | 0.001 |
| Bibliometrics | 0.001 | 0.000 |
| Science and technology studies | 0.001 | 0.004 |
| Scholarly communication | 0.002 | 0.005 |
| Open science | 0.001 | 0.002 |
| Research integrity | 0.001 | 0.003 |
| Insufficient payload (model declined to judge) | 0.005 | 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".