# Symmetry-Quotiented Local Correlation Encoder

- ID: 1009
- Canonical URL: https://synthcore.org/idea/1009/symmetry-quotiented-local-correlation-encoder
- API JSON: https://synthcore.org/api/idea/1009.json
- API Markdown: https://synthcore.org/api/idea/1009.md
- Verification status: beats_baseline
- Source: [arXiv:2607.16481](https://arxiv.org/abs/2607.16481)
- Category: architecture
- Solves: stability, accuracy, sample-efficiency
- ML areas: training, cnn, vae, regularization
- Math tags: statistical-mechanics, geometry, linear-algebra, dynamical-systems
- Ratings: usefulness 7/10; difficulty 4/10; novelty 5/10

## Idea description

Replace raw molecular orientation vectors with local scalar features invariant under common three-dimensional rotations and the apolar transformation u_i -> -u_i. Feed these channels to a CNN autoencoder, VAE, or contrastive encoder so that configurations on the same physical symmetry orbit have identical inputs or latent codes. This should improve unsupervised phase discovery without supplying order-parameter labels.

## Mathematical statement

Let u_i be a unit orientation at lattice site i and let P2(x)=(3x^2-1)/2. For a displacement shell r, define c_i(r) as the average of P2(u_i dot u_j) over neighbors j in shell r. A common rotation R preserves each dot product because (R u_i) dot (R u_j)=u_i dot u_j. An apolar sign change changes a dot product only by a sign, and P2 is even, so the local feature is invariant. The feature tensor retains spatial correlations while removing arbitrary orientation-frame information. For evaluation, define Q=(1/N) sum_i (3 u_i u_i^T-I_3)/2 and S=lambda_max(Q), where N is the number of lattice sites, I_3 is the three-dimensional identity, and lambda_max is the largest eigenvalue.

## Key formulas

- $$P_2(x)=\frac{1}{2}(3x^2-1),\qquad P_2(-x)=P_2(x)$$
- $$c_i(r)=\frac{1}{|\mathcal{N}_r(i)|}\sum_{j\in\mathcal{N}_r(i)}P_2(\mathbf{u}_i\cdot\mathbf{u}_j)$$
- $$Q=\frac{1}{N}\sum_{i=1}^{N}\frac{1}{2}\left(3\mathbf{u}_i\mathbf{u}_i^{\mathsf T}-I_3\right),\qquad S=\lambda_{\max}(Q)$$
- $$d_{\mathrm{inv}}(X,Y)=\|\phi(X)-\phi(Y)\|_2,\qquad \phi(X)=\{c_i(r)\}_{i,r},\qquad d_{\mathrm{inv}}(X,gX)=0$$

## Implementation notes

1. Integration point: before the neural network, construct scalar lattice channels from each configuration X. For shells r=1,...,R, compute c_i(r) using periodic neighbor lists; start with R=2 or 3. Pass the R-channel tensor to a small three-dimensional CNN autoencoder or VAE. A second option is to concatenate these channels with raw inputs and enforce invariant latent consistency. 2. Pseudocode: `for X in batch: for r in shells: C[r,i] = mean((3*dot(u[i],u[j])**2 - 1)/2 for j in neighbors(i,r)); z = Encoder(C); C_hat = Decoder(z); loss = reconstruction_loss(C_hat,C) + beta*contrastive_loss(z)`. For the contrastive term, create a second view by applying one random global rotation and independent sign flips, then minimize squared latent distance between the two views. 3. Computed exactly from the paper mechanism: dot products, P2 values, shell averages, and transformed-copy invariance. Estimated empirically: latent phase separation, histogram bimodality, and correlation with the independently computed order parameter S. 4. First cheap experiment: generate Lebwohl-Lasher configurations with Metropolis Monte Carlo on L=12 or L=16 cubic lattices over temperatures around the isotropic-nematic transition. Compare a raw-vector CNN autoencoder, the invariant-correlation encoder, and a raw encoder with global-rotation augmentation, keeping latent dimension and optimizer fixed. The predicted signature is that the invariant representation has distance below 1e-6 between transformed copies, while the raw representation does not. Near the weak first-order transition, invariant latent or reconstruction-error histograms should become bimodal in the same temperature window where the independently measured S histogram is bimodal. The leading invariant latent coordinate should have an absolute Pearson correlation with S at least 0.2 larger than the raw baseline.

## Verification

- Status: beats_baseline
- Mechanism evidence: yes
- Mechanism confirmed: yes
- Practical verdict: helps
- Verdict: Built the symmetry-quotiented local P2 correlation encoder and numerically verified all three mechanism predictions. Across flip probabilities 0–1 and shell counts 1–4, transformed configurations had zero measured invariant-feature error, while raw-vector RMS differences remained about 0.79–0.87; P2 evenness also had zero error. In the toy phase experiment, the invariant autoencoder had lower reconstruction MSE (0.0233 vs 0.3236) and much stronger correlation with independently computed S (0.998 vs 0.143), although this is a small synthetic MLP-autoencoder test rather than a full Monte Carlo/CNN study.

### Mechanism check

- Verdict: Built the symmetry-quotiented local P2 correlation encoder and numerically verified all three mechanism predictions. Across flip probabilities 0–1 and shell counts 1–4, transformed configurations had zero measured invariant-feature error, while raw-vector RMS differences remained about 0.79–0.87; P2 evenness also had zero error. In the toy phase experiment, the invariant autoencoder had lower reconstruction MSE (0.0233 vs 0.3236) and much stronger correlation with independently computed S (0.998 vs 0.143), although this is a small synthetic MLP-autoencoder test rather than a full Monte Carlo/CNN study.
- Confidence: 9/10
- Limitations: The data are synthetic isotropic/nematic orientation clouds, not Lebwohl–Lasher Monte Carlo configurations near a weak first-order transition. The comparison uses a fully connected autoencoder rather than a CNN, has one fixed seed and small sample count, and does not test contrastive training, temperature histograms, FLOPs, or scaling to larger lattices.

### Practical benchmark

- Paired seeds: 8
- Baseline mean: 0
- Idea mean: 0
- p-value: 0.0081
- Paired wins: 8/8
- Benchmark verdict: idea better (significant)

## Artifacts

- [bench_results.json](https://synthcore.org/code/415/bench_results.json)
- [experiment.py](https://synthcore.org/code/415/experiment.py)
- [orientation_track.py](https://synthcore.org/code/415/orientation_track.py)
- [report.md](https://synthcore.org/code/415/report.md)
- [report_bench_2026-08-31T113941.md](https://synthcore.org/code/415/report_bench_2026-08-31T113941.md)
- [results.json](https://synthcore.org/code/415/results.json)
- [run_bench.py](https://synthcore.org/code/415/run_bench.py)
- [Download all files as ZIP](https://synthcore.org/download/415)

## Disclaimer

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