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Transport of Algebraic Structure to Latent Embeddings
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Proposes structural transport nets that learn an invertible map from a pretrained latent space to a designed Euclidean "mirrored algebra," so latent operations inherit the source algebra's laws, and validates this on distributive lattices of sets built from INR embeddings.
Problem
Latent embeddings of inputs living in an abstract algebraic structure (e.g., sets under union/intersection) do not respect that structure's operations and laws, so learned latent operations are inconsistent with laws such as associativity.
Approach
Define operations on a Euclidean mirrored space that satisfy the desired laws, then learn a NICE-style bijection from the latent space to that mirror; latent operations are induced by conjugation, so they inherit the mirrored laws exactly. Also proves complementation (Boolean lattice) cannot be transported continuously to R^l, so only the distributive lattice is targeted.
Key results
- The Riesz mirrored algebra (min/max), which satisfies all 8 distributive lattice laws, achieves the best learned set-operation IoU and is perfectly self-consistent across equivalent terms.
- Learned operation performance is strongly tied to the number of satisfied source laws; algebras violating most laws perform worst, with a monotone increasing trend.
- Direct latent parameterizations are far less self-consistent under law-equivalent rewrites, with bottom-quartile consistency dropping sharply after just two law applications.
Measured effects reported in the paper
| What was changed | Where | Measured | Effect |
|---|---|---|---|
| Use mirrored operations that satisfy more distributive-lattice laws | Structural transport nets · Synthetic set-operation terms | IoU | Helped |
| Use the Riesz mirrored algebra (elementwise max for intersection and min for union) | Structural transport nets · Synthetic set-operation terms | IoU | Helped |
| Use flexible direct latent-space parameterizations instead of transported algebras with a similar number of satisfied laws | Set-operation learning · Synthetic set-operation terms | IoU | Helped |
| Use the Riesz mirrored algebra rather than learned baselines | Set-operation learning · Equivalent set-operation terms | Self-consistency IoU | Helped |
| Increase the number of symbolic law applications | Learned baselines · Equivalent set-operation terms | Self-consistency IoU | Hurt |
| Increase random-term length | Structural transport nets using candidate mirrored algebras · Synthetic set-operation terms | IoU | Hurt |
| Satisfy all desired distributive-lattice laws rather than only some or none | Structural transport nets · Equivalent set-operation terms | Self-consistency IoU | Helped |
| Increase absorption-law satisfaction while holding distributivity fixed | Structural transport nets · Synthetic set-operation terms | IoU | No clear effect |
Surprising findings
- Transporting the full Boolean lattice (with continuous complementation) into R^l is impossible, by a fixed-point argument for continuous involutions.
- Symmetric direct parameterization (commutative only) sometimes has higher self-consistency than most transported mirrored algebras that satisfy more laws.
Limitations
- Mirrored space must have the same dimension as the latent space, since transport relies on invertibility of the bijection.
- Only the distributive lattice (no complementation) is transportable; the Boolean lattice is infeasible for continuous operations.
- Experiments are limited to a synthetic 2D set setting with a fixed pretrained encoder and decoder.
Experimental setup
Datasets: synthetic random 2D point-defined subsets of [-1,1]^2 (10^4 sets)
Benchmarks: synthetic random set-operation terms on 2D subsets
Metrics: IoU self-consistency IoU binary cross-entropy
Baselines: direct MLP latent parameterization (concatenation) symmetric MLP latent parameterization h(g(z1)+g(z2))
Builds on: NICE SIREN inr2vec DeepSets
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