On the Representational Power of Restricted Boltzmann Machines for Symmetric Functions and Boolean Functions
Bibliographic record
Abstract
Restricted Boltzmann machines (RBMs) are used to build deep-belief networks that are widely thought to be one of the first effective deep learning neural networks. This paper studies the ability of RBMs to represent distributions over (0, 1in via softplus/hardplus RBM networks. It is shown that any distribution whose density depends on the number of 1's in their input can be approximated with arbitrarily high accuracy by an RBM of size 2n + 1, which improves the result of a previous study by reducing the size from n2to 2n + 1. A theorem for representing partially symmetric Boolean functions by softplus RBM networks is established. Accordingly, the representational power of RBMs for distributions whose mass represents the Boolean functions is investigated in comparison with that of threshold circuits and polynomial threshold functions. It is shown that a distribution over [0, 1]nwhose mass represents a Boolean function can be computed with a given margin δ by an RBM of size and parameters bounded by polynomials in n, if and only if it can be computed by a depth-2 threshold circuit with size and parameters bounded by polynomials in n.
Fetched live from OpenAlex and de-inverted. Abstracts are not stored in this database: the inverted indexes are 8.6 GB of the frame’s 9.3 GB of text, and the host has 13 GB free.
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 | 0.003 | 0.021 |
| Meta-epidemiology (narrow) | 0.001 | 0.001 |
| Meta-epidemiology (broad) | 0.001 | 0.001 |
| Bibliometrics | 0.001 | 0.001 |
| Science and technology studies | 0.001 | 0.004 |
| Scholarly communication | 0.002 | 0.008 |
| Open science | 0.002 | 0.003 |
| Research integrity | 0.002 | 0.004 |
| 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".