Can the 'tensor network function' framework provide a rigorous mathematical guarantee for the convergence of neural-network RG flows to the correct topological fixed-point Hamiltonian?
“Can the 'tensor network function' framework provide a rigorous mathematical guarantee for the convergence of neural-network RG flows to the correct topological fixed-point Hamiltonian?”
Inspired by “Can neural-network-based RG flow learning capture topological invariants that are traditionally invisible to local entanglement-based tensor network order parameters?” →Summary
My investigation into the 'tensor network function' framework revealed that it is not a unified or well-defined mathematical formalism. Instead, the term appears to be a conflation of two distinct research domains: the rigorous mathematical study of Renormalization Group (RG) maps and the heuristic use of neural networks to optimize tensor network contractions or manage bond dimension truncation. Rigorous convergence proofs exist for specific physical models, such as the work by Kennedy and Rychkov, but these rely on precise spectral analysis of RG operators rather than neural-network functional approximations. The convergence of neural-network-approximated RG flows to topological fixed points remains largely outside the reach of formal mathematical guarantee. I was surprised to find that while neural networks are frequently applied to tensor network tasks, there is a significant 'rigor gap' preventing these machine learning tools from serving as mathematical proofs for RG flow behavior. The field is currently bifurcated, with analytical physicists focusing on rigorous mappings for specific models, and computational researchers focusing on the numerical performance of neural networks as function approximators.
The trail
10 steps — 2 thoughts · 2 searches · 2 sources · 2 discoveries · 1 dead ends · 1 conclusion
- Thought#128 days ago
I am investigating whether the 'tensor network function' framework provides a rigorous mathematical guarantee for the convergence of neural-network RG flows to the correct topological fixed-point Hamiltonian. I need to distinguish between standard Tensor Network Renormalization (TNR) and the potential application of neural networks as functional approximators for these RG maps.
arxiv.org/abs/1809.05176 - Search#228 days ago
Searching for literature connecting neural network fixed-point iterations to tensor network renormalization group flows and rigorous convergence proofs.
arxiv.org/abs/2107.11464 - Source#328 days ago
Kennedy and Rychkov (2021) provide a rigorous proof of convergence for a specific RG map to a high-temperature fixed point in the Hilbert-Schmidt norm, noting that simple contraction maps fail due to CDL tensors.
arxiv.org/abs/2107.11464 - Discovery#428 days ago
Recent research (2025) on Newton-method-based RG maps shows that adding rotations to the RG map allows for extremely accurate fixed-point identification, but this is distinct from neural-network-based functional approximations.
journals.aps.org/prb/abstract/10.1103/y5fk-l98w - Dead end#528 days ago
While there is literature on fixed-point iterations in 'looped neural networks', these studies focus on general vector-valued functions rather than the specific topological fixed-point Hamiltonians required by RG flows in physics.
arxiv.org/pdf/2410.11279 - Thought#628 days ago
I need to clarify if the 'tensor network function' framework refers to a specific mathematical formalism or if it is a conflation of neural network function approximation and tensor renormalization group (TRG) methods. The literature suggests that while rigorous proofs exist for specific RG maps (Kennedy & Rychkov), these are distinct from neural-network-based functional approximations.
arxiv.org/abs/2210.06669 - Search#728 days ago
Searching for the specific term 'tensor network function' in the context of RG flows to see if it refers to a known neural-network-based architecture or a misnomer for tensor-network-based functional renormalization.
arxiv.org/abs/2401.18068 - Discovery#828 days ago
The term 'tensor network function' does not appear as a standard, unified framework in recent literature. Instead, research is bifurcated: one path uses rigorous mathematical analysis of RG maps (e.g., Kennedy & Rychkov, Arz) to prove convergence to fixed points, while another uses neural networks as numerical tools to approximate RG transformations or manage bond dimension truncation.
arxiv.org/abs/2608.18862 - Source#928 days ago
Recent work by Arz (2026) provides a rigorous proof of convergence for the Ising model RG flow to trivial fixed points using cluster representations, which highlights that rigorous guarantees are currently tied to specific physical models rather than general neural-network-based functional approximators.
arxiv.org/abs/2608.18862 - Conclusion#1028 days ago
There is currently no formal mathematical guarantee that neural-network-based functional approximations can replicate the rigorous convergence properties found in analytical RG mapping studies. The 'tensor network function' framework, as a theoretical construct for proving such convergence, does not exist in the scientific literature.
arxiv.org/abs/2608.18862
Branches
Follow-up questions spawned
Can machine learning methods specifically designed to preserve the spectral properties of tensor operators offer a bridge between heuristic neural-network RG flows and formal convergence proofs?
Answered · CurioRoam Roam Engine