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Latent Functional Maps: a spectral framework for representation alignment
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Introduces Latent Functional Maps, which transfer spectral functional maps from 3D geometry processing to neural latent spaces, enabling similarity measurement, correspondence finding, and zero-shot stitching and retrieval across models.
Problem
Modeling the relation between distinct neural representational spaces (for comparison, alignment, stitching and retrieval) is challenging; existing sample-level alignment methods are complex and less interpretable.
Approach
Build k-NN graphs on latent samples, compute graph Laplacian eigenbases, and solve for a small functional map matrix C between function spaces using descriptors (anchor distances, labels, or heat kernel signatures); the map yields an interpretable similarity measure and point-to-point correspondences that feed off-the-shelf alignment methods such as orthogonal mapping.
Key results
- On word embedding retrieval, LFM+Ortho reaches MRR 0.99 with only 5 anchors, versus 0.97 for Ortho at 300 anchors.
- LFM+Ortho consistently outperforms the Ortho baseline in zero-shot stitching across six datasets, especially with few anchors (<=50).
- The LFM similarity stays stable under linear-separability-preserving perturbations where CKA degrades.
Measured effects reported in the paper
| What was changed | Where | Measured | Effect |
|---|---|---|---|
| Use LFM Space instead of the direct baseline methods for word embedding retrieval with 5 anchors | Word embedding retrieval · Word embedding retrieval (18K words) | MRR: 0.01 -> 0.85 | Helped |
| Use LFM+Ortho instead of Ortho for word embedding retrieval with 5 anchors | Word embedding retrieval · Word embedding retrieval (18K words) | MRR: 0.01 -> 0.99 | Helped |
| Increase the number of anchors for Ortho | Word embedding retrieval · Word embedding retrieval (18K words) | MRR: 0.01 -> 0.97 | Helped |
| Use LFM+Procrustes instead of Relatives for CUB retrieval with 5 anchors | CUB retrieval · CUB | MRR: 0.01 -> 0.07 | Helped |
| Use LFM+Ortho instead of Ortho for CUB retrieval with 5 anchors | CUB retrieval · CUB | MRR: 0.00 -> 0.05 | Helped |
| Use LFM+Ortho instead of Ortho for CUB retrieval with 250 anchors | CUB retrieval · CUB | MRR: 0.33 -> 0.34 | No clear effect |
| Increase the number of eigenvectors used to compute the functional map (ablation) | LFM retrieval methods · Word embedding retrieval | MRR | Helped |
| Use angular distance instead of L2 distance to construct the graph (ablation) | LFM · MNIST retrieval under increasing noise | MRR | Helped |
| Use geodesic distance or cosine distance descriptors rather than the other tested descriptors (ablation) | LFM · MNIST retrieval | MRR | Helped |
Surprising findings
- The LFM similarity remains high under translations that preserve linear separability, where CKA degrades, which the authors present as a CKA sensitivity issue.
- Label-based descriptors give strong stitching performance without any anchor correspondences.
Limitations
- Performance is sensitive to the number of eigenvectors, and choosing it remains open.
- Assumes full correspondence between domains; partial mappings are not handled.
- Performance in fully unsupervised settings is still limited.
Experimental setup
Datasets: CIFAR-10 CIFAR100 ImageNet CUB MNIST AgNews FashionMNIST FastText Word2Vec
Benchmarks: CIFAR100 coarse CIFAR100 fine ImageNet CUB MNIST AgNews FashionMNIST CIFAR-10 word embedding retrieval (FastText-Word2Vec)
Metrics: accuracy mean reciprocal rank (MRR) similarity score reconstruction quality
Baselines: CKA CCA Relative representations Orthogonal transformation (Ortho) Procrustes Linear Affine
Builds on: functional maps relative representations Laplacian commutativity regularizer heat kernel signature spectral upsampling
Closest papers by meaning
- Connecting Neural Models Latent Geometries with Relative Geodesic Representations (NeurIPS 2025 · similarity 0.90)
- Functional Alignment Can Mislead: Examining Model Stitching (ICML 2025 · similarity 0.90)
- Towards a learning theory of representation alignment (ICLR 2025 · similarity 0.88)
- From Bricks to Bridges: Product of Invariances to Enhance Latent Space Communication (ICLR 2024 · similarity 0.88)
- Mapping the Multiverse of Latent Representations (ICML 2024 · similarity 0.87)
- Geometry-Aware Similarity Metrics for Neural Representations on Riemannian and Statistical Manifolds (NeurIPS 2026 · similarity 0.87)
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