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About

Intrinsic Green Learning (IGL) is a Python library implementing the task-conditioned intrinsic-dimensionality discovery framework introduced in Quemy, A. (2026). Intrinsic Green's Learning: Supervised Learning on Manifolds via Inverse PDE. ICLR 2026 Workshop on AI and PDE.

Author

Alexandre Quemy (@aquemy, ORCID 0000-0002-5865-6403) — research and implementation. The library is developed under Hother and released under the MIT license.

Where IGL sits in the landscape

Intrinsic-dimensionality estimation has historically split into two camps:

  • Purely geometric estimators (Levina–Bickel, TwoNN, MLE-based approaches in scikit-dimension) look at the input data alone and ignore what the user actually wants to predict.
  • Implicit bottlenecks in deep networks (autoencoders, information-bottleneck methods, variational approaches) use a low-dimensional latent but don't report its size without a separate analysis pass.

IGL takes a third position: solve the supervised task and read off the effective dimension in the same training run. The discovered d_eff is a property of (input, task), not of the input alone — the same dataset resolves into different effective dimensions for a classifier, a regressor, and an autoencoder, and the hierarchy \(d_{\text{cls}} \le d_{\text{reg}} \le d_{\text{recon}}\) falls out naturally.

Theoretical underpinnings

The library realises the inverse-PDE formulation from the paper above:

  • The latent-space operator \(L\) is encoded by the Green's-function kernel \(G(z, s)\), which can be expressed either as a local product of per-dimension kernels (the default GreenKernel) or via its spectral expansion \(G(z, s) = \sum_n \phi_n(z)\,\phi_n(s) / \lambda_n\) (the SpectralKernel plus closed-form Fourier / Chebyshev / Legendre / Hermite / Laguerre bases, or learned Laplace–Beltrami / user-supplied graph bases).
  • The non-trivial null space of \(L\) — the modes the Green's- function expansion cannot reach (e.g. the constant function under a Neumann Laplacian) — is fitted as un-regularised columns of the design matrix via the kernel-agnostic NullSpaceBasis protocol.
  • Variable Projection with random Matryoshka truncation trains the encoder while sampling the latent-dimension cutoff \(k\), so a single trained model exposes a quality-vs-k dimension curve that reads off d_eff at deployment time.

Concrete derivations and worked examples live under Concepts; the per-symbol API reference under Reference.

Status

IGL is active research-grade software. The API follows strict typing discipline (basedpyright --strict, ≥99% test coverage with property-based fuzzing on the numerical core), but minor versions may introduce breaking changes as the framework evolves. The 0.x line is appropriate for research and experimentation; production users should pin specific minor versions.

Citing

See the Bibliography section of the project README for the recommended citation, or the CITATION.cff file for the structured form (GitHub's "Cite this repository" widget reads it automatically).