Applied Math Colloquium | Dr. José Ramón Pareja Monturiol,Tensorizing neural networks into matrix product states for interpretable representations

Tensor networks provide efficient and interpretable representations of high-dimensional objects, with matrix product states (MPS) playing a central role in quantum many-body physics. At the same time, neural networks (NNs) provide powerful but often black-box representations of complex functions and quantum states. In this talk, I will present a recently developed tensorization method that converts trained NNs, or more generally black-box high-dimensional functions, into MPS representations using only function evaluations and a set of relevant samples. The method builds on ideas from tensor sketching and cross interpolation and, rather than decomposing the parameters of a neural network layer by layer, directly reconstructs an MPS approximation of the function represented by the model.

In physical applications, this approach can be used to transform neural quantum states into MPS representations, making their learned structure accessible to standard tensor-network diagnostics such as bond dimensions, entanglement entropies, and local order parameters. As an example, I will show how the method recovers an MPS representation of the AKLT ground state and allows its symmetry-protected topological phase to be identified from the resulting tensors. Beyond interpretability, tensorization can also be used for model compression and as an initialization for subsequent MPS optimization. Finally, I will discuss ongoing directions towards extending the approach to higher-dimensional tensor-network geometries such as PEPS, and the challenges introduced by their internal loops.