[Paper Review] Generating valid Euclidean distance matrices

Moritz Hoffmann, Frank Noé|arXiv (Cornell University)|Oct 7, 2019

Machine Learning in Materials Science44 references37 citations

TL;DR

The paper introduces a neural architecture that generates valid Euclidean distance matrices (EDMs) with a predefined embedding dimension and uses a Wasserstein GAN with a permutation-invariant critic to model distributions of point clouds, demonstrated on C7O2H10 isomers from QM9.

ABSTRACT

Generating point clouds, e.g., molecular structures, in arbitrary rotations, translations, and enumerations remains a challenging task. Meanwhile, neural networks utilizing symmetry invariant layers have been shown to be able to optimize their training objective in a data-efficient way. In this spirit, we present an architecture which allows to produce valid Euclidean distance matrices, which by construction are already invariant under rotation and translation of the described object. Motivated by the goal to generate molecular structures in Cartesian space, we use this architecture to construct a Wasserstein GAN utilizing a permutation invariant critic network. This makes it possible to generate molecular structures in a one-shot fashion by producing Euclidean distance matrices which have a three-dimensional embedding.

Motivation & Objective

• Generate valid Euclidean distance matrices without explicit Cartesian coordinates to ensure rotation/translation invariance.
• Develop a GAN framework, specifically a Wasserstein GAN, to learn distributions of EDMs.
• Incorporate permutation invariance in the critic to handle point clouds as sets.
• Apply the method to QM9 isomers and assess generalization and chemical validity.

Proposed method

• Parameterize symmetric hollow matrices and enforce EDM validity via a loss that penalizes non-EDMs using the PSD condition with a J projection.
• Use a symmetric matrix L transformed by a non-negative function g to obtain a Gram matrix M, from which EDM D is reconstructed.
• Relate D to M via D_{ij}=M_{ii}+M_{jj}-2M_{ij} and enforce embedding dimension by limiting the rank of M through eigenvalue-based penalties.
• Train a Wasserstein GAN with a SchNet-based critic operating on pairwise distances to achieve permutation-invariant evaluation of EDMs.
• In the generator, enforce EDM structure by applying a softplus-based reparameterization to ensure positive semi-definiteness and control rank (embedding dimension).
• Extend the model to include atom type information and additional losses such as cross-entropy for types, harmonic repulsion, and EDM-specific penalties during training.

Experimental results

Research questions

• RQ1 Can a neural network generate valid EDMs with a predefined embedding dimension without direct Cartesian coordinates?
• RQ2 Can a WGAN with a permutation-invariant critic learn a distribution over EDMs corresponding to molecular geometries?
• RQ3 How well does the EDM-based generation generalize to unseen molecules and conformations within QM9?
• RQ4 What is the diversity of generated molecular topologies and conformations relative to the training data?
• RQ5 Do generated structures yield energetically reasonable configurations after relaxation?

Key findings

• The method produces EDMs that correspond to point clouds embedded in 3D space, yielding rotation- and translation-invariant representations.
• The WGAN with a SchNet critic can learn distributions of EDMs for a subset of QM9 molecules (C7O2H10 isomers) and generalize beyond the training set.
• Generated samples show qualitatively correct pairwise distance distributions and can be relaxed to energies within the QM9 energy range.
• A fraction of generated samples pass a basic validity test (bonding, valency) and can be represented by SMILES, indicating topology diversity.
• Conformation analysis reveals multiple unique conformations beyond those in the QM9 subset, including new topologies and conformations.
• Energies of relaxed generated structures fall within the same range as those in the QM9 database (approx. -1586 to -1581 eV).
• Generated examples include both close matches to QM9 structures and novel molecules, with some conformations not present in the database.