Riemannian Generative Decoder Eval

---
name: riemannian-generative-decoder-eval
description: Evaluates the ability of a decoder-only latent variable model to learn geometry-respecting latent spaces on Riemannian manifolds. It probes reconstruction fidelity, preservation of intrinsic data structures (cyclical, hierarchical, phylogenetic), and downstream predictive utility of the learned latents. Use when the user wants to benchmark on Cell cycle stages (scRNA-seq), Branching diffusion process (synthetic tree), Human mitochondrial DNA (hmtDNA), or asks about evaluating this task. Reports Pearson correlation.
metadata:
  skill_kind: dataset_eval
  source_arxiv: 2506.19133
  bibtex_key: [REDACTED]
  confidence: high
---

# riemannian-generative-decoder-eval

> Riemannian generative decoder — Bjerregaard et al. (2025) (arXiv:2506.19133, 2025)

## What this evaluates

Evaluates the ability of a decoder-only latent variable model to learn geometry-respecting latent spaces on Riemannian manifolds. It probes reconstruction fidelity, preservation of intrinsic data structures (cyclical, hierarchical, phylogenetic), and downstream predictive utility of the learned latents.

## Datasets

- **Cell cycle stages (scRNA-seq)** — total ?; splits: train (-1), val (-1), test (-1)
- **Branching diffusion process (synthetic tree)** — total ?; splits: train (-1), test (-1)
- **Human mitochondrial DNA (hmtDNA)** — total ?; splits: train (-1), test (-1)

## Metrics

- `Pearson correlation` **(primary)** — range: [-1, 1]
  - Linear correlation coefficient between ground-truth pairwise distances (e.g., cell cycle phase, tree path length, haplogroup distance) and latent geodesic distances on the chosen manifold.
- `Spearman correlation` — range: [-1, 1]
  - Rank-based correlation coefficient between ground-truth pairwise distances and latent geodesic distances, capturing monotonic but non-linear relationships.
- `MAE` — range: [0, ∞)
  - Mean Absolute Error measuring reconstruction fidelity via L1-norm between original data samples and decoder reconstructions.
- `MSE` — range: [0, ∞)
  - Mean Squared Error measuring reconstruction fidelity via L2-norm between original data samples and decoder reconstructions.
- `Accuracy` — range: [0, 1]
  - Classification accuracy of downstream models (logistic regression or XGBoost) trained on learned latents to predict categorical metadata (e.g., geographical region, haplogroup).

## Input / output format

**Input**: Raw biological or synthetic data samples (scRNA-seq gene counts, synthetic tree node features, hmtDNA sequences) fed into a decoder-only network to produce manifold-valued latents and reconstructions.

**Output**: Manifold-valued latent vectors and reconstructed data samples. Downstream classifiers are trained on the latents to predict categorical labels or continuous phases.

## Scoring recipe

```python
def evaluate(latents, data, targets, manifold):
    reconstructions = decoder(latents)
    mae = mean(abs(data - reconstructions))
    mse = mean((data - reconstructions)**2)
    true_dists = pairwise_distance(targets)
    latent_dists = manifold.geodesic_distance(latents)
    pearson = pearsonr(true_dists, latent_dists)
    spearman = spearmanr(true_dists, latent_dists)
    acc = train_classifier(latents, targets).score(test_latents, test_targets)
    return pearson, spearman, mae, mse, acc
```

## Common pitfalls

- Generalization performance is noted as generally of little significance for dimensionality reduction, so test set metrics may not reflect practical utility.
- Hyperbolic models rescale absolute path lengths by curvature, causing Spearman (rank) to improve while Pearson (linear) may not, requiring careful interpretation of correlation metrics.
- Noise scale σ heavily trades off local reconstruction accuracy against global geometric correlation; optimal σ varies by manifold and dataset.

## Evidence (verbatim from paper)

> Pearson/Spearman correlate phase distances to latent distances while MAE/MSE measure reconstruction by L1/L2-norm.

## Citation

```bibtex
@misc{bjerregaard2025riemannian,
  title={Riemannian generative decoder},
  author={Bjerregaard et al. (2025)},
  year={2025},
  note={arXiv:2506.19133}
}
```

- arXiv: 2506.19133