# Homomorphic Tempo Conditioning

- ID: 1309
- Canonical URL: https://synthcore.org/idea/1309/homomorphic-tempo-conditioning
- API JSON: https://synthcore.org/api/idea/1309.json
- API Markdown: https://synthcore.org/api/idea/1309.md
- Verification status: unverified
- Source: [arXiv:2607.23210](https://arxiv.org/abs/2607.23210)
- Category: architecture
- Solves: accuracy, sample-efficiency, scalability
- ML areas: transformer, ssm, rnn, embedding, attention
- Math tags: algebra, representation-theory, dynamical-systems
- Ratings: usefulness 6/10; difficulty 4/10; novelty 5/10

## Idea description

Condition a temporal neural network on a tempo or dilation ratio through a homomorphism from multiplicative positive scales to additive latent shifts. A ratio composed from several scale changes then produces the sum of their learned effects, allowing interpolation and extrapolation to rates absent from training instead of using an independent embedding per rate.

## Mathematical statement

The paper assigns a time-scale ratio \(\lambda\) a constant pitch interval \(\phi(\lambda)\), independent of time, and requires compositional consistency: applying ratios \(\lambda_1\) and \(\lambda_2\) should equal applying their product, so \(\phi(\lambda_1\lambda_2)=\phi(\lambda_1)+\phi(\lambda_2)\). For a positive rational \(r=\prod_p p^{e_p}\), where the product is over primes \(p\) and \(e_p\in\mathbb{Z}\) is the prime-factorization exponent, the discrete construction is \(\phi(r)=\sum_p e_p c_p\), where \(c_p\) is the learned interval associated with prime \(p\). The continuous logarithmic construction is \(\phi(\lambda)=k\log\lambda\), with \(\lambda>0\) and scalar or vector coefficient \(k\); it satisfies the homomorphism law because \(\log(ab)=\log a+\log b\). In a neural network, replace pitch interval by a vector shift \(h(\lambda)=W\log\lambda\), where \(W\in\mathbb{R}^{d}\) and \(d\) is the hidden dimension. The transferable asset is multiplicative-scale composition represented as additive conditioning with only \(O(d)\) parameters rather than one embedding per tempo.

## Key formulas

- $$\phi(\lambda_1\lambda_2)=\phi(\lambda_1)+\phi(\lambda_2),$$
- $$r=\prod_{p\ \mathrm{prime}}p^{e_p},\qquad \phi(r)=\sum_{p\ \mathrm{prime}}e_p c_p,$$
- $$\phi(\lambda)=k\log\lambda,\qquad \phi(\lambda_1\lambda_2)=k\log(\lambda_1\lambda_2)=\phi(\lambda_1)+\phi(\lambda_2),$$
- $$h_t^{\mathrm{cond}}=h_t+W\log\lambda\quad\text{or}\quad h_t^{\mathrm{cond}}=h_t+\sum_p e_p(r)c_p.$$

## Implementation notes

(1) Integrate this at the input or state-conditioning point of a small temporal Transformer or SSM. Each example should carry a positive relative tempo or time-dilation ratio \(\lambda\) between a reference sequence and its target sampling rate. Compute \(g(\lambda)=W\log\lambda\), then add this vector to every token representation, or add it to the query/key projections so temporal interactions are explicitly rate-conditioned. For a discrete experiment, factor rational rates and use \(g(r)=\sum_{p\in P}e_p(r)c_p\) for a finite prime set \(P\). (2) Pseudocode: input tokens `x[1:T]` and ratio `lambda`; compute `z = W * log(lambda)`; compute `h = encoder(x) + z[None, :]`; run the normal Transformer or SSM; predict the target sequence. Sample rates during training from a set of rational values and optionally from a log-uniform continuous distribution. (3) The paper supplies the homomorphic form and exact composition law; estimate \(W\) or the prime vectors \(c_p\) by backpropagation. Empirically test composition by checking whether conditioning for two successive resamplings equals conditioning for their product. (4) First experiment: train a 4-layer causal Transformer or diagonal SSM on synthetic chirps and speech-like sequences with rates \(\{1/2,2/3,1,3/2,2\}\), withholding \(3/4\) and \(4/3\). Compare no conditioning, ordinary categorical rate embeddings, and logarithmic conditioning at equal parameter count. Measure reconstruction or next-token loss on seen and withheld rates, interpolation smoothness, and error after composing two resamplings. Success is lower unseen-rate loss without additional inference FLOPs, ideally at least 10% lower held-out-rate error than categorical embeddings.

## Disclaimer

AI-generated research hypothesis, automatically tested. Not peer-reviewed.