F-INR: Functional Tensor Decomposition for Implicit Neural Representations

TL;DR

F-INR tackles the scalability of implicit neural representations by factorizing high-dimensional functions into axis-specific sub-networks learned via functional tensor decomposition. By supporting CP, TT, or Tucker modes with configurable ranks, it achieves substantial speedups (up to $20\times$) and fidelity gains (over $6.0$ dB PSNR) across image, geometry, NeRF, and physics tasks while remaining backend- and decomposition-agnostic. The approach preserves differentiability for gradient-based optimization, enabling physics-informed learning and PDE-constrained problems, and demonstrates broad applicability beyond traditional INRs. This functional factorization framework opens avenues for adaptive rank selection and advanced tensor formats, suggesting a scalable paradigm for high-dimensional signal modeling.

Abstract

Implicit Neural Representations (INRs) model signals as continuous, differentiable functions. However, monolithic INRs scale poorly with data dimensionality, leading to excessive training costs. We propose F-INR, a framework that addresses this limitation by factorizing a high-dimensional INR into a set of compact, axis-specific sub-networks based on functional tensor decomposition. These sub-networks learn low-dimensional functional components that are then combined via tensor operations. This factorization reduces computational complexity while additionally improving representational capacity. F-INR is both architecture- and decomposition-agnostic. It integrates with various existing INR backbones (e.g., SIREN, WIRE, FINER, Factor Fields) and tensor formats (e.g., CP, TT, Tucker), offering fine-grained control over the speed-accuracy trade-off via the tensor rank and mode. Our experiments show F-INR accelerates training by up to $20\times$ and improves fidelity by over \num{6.0} dB PSNR compared to state-of-the-art INRs. We validate these gains on diverse tasks, including image representation, 3D geometry reconstruction, and neural radiance fields. We further show F-INR's applicability to scientific computing by modeling complex physics simulations. Thus, F-INR provides a scalable, flexible, and efficient framework for high-dimensional signal modeling. Project page: https://f-inr.github.io

Figures (6)

• Figure 1: Efficient INRs via Functional Tensor Decomposition. INR models use a single, large network to predict one value (or a batch of values) at a time. Our approach decomposes the function into smaller networks, enabling full prediction in a single step with configurable tensor decomposition modes and compression ranks.
• Figure 2: Tensor Diagrams for Three Decompositions \ref{['fig:decomp_cp']}-\ref{['fig:decomp_tu']}. This schematic penrose1971Applicationskoldatensordecomps defines each circle as a component, and spokes determine its dimension. The spokes highlight the decomposition computation and how to recover the original tensor. In F-INR, each component is an individual neural network.
• Figure 3: Qualitative Image Results. F-INR improves both fidelity and training speed over the original backbones. We observe fidelity improvements of up to +7 dB (SIREN) and training speedups reaching $20\times$ for native backbones (WIRE) and $2.2\times$ for optimized CUDA implementations (ReLU). Error maps are magnified for visualization. See supplementary material for more results.
• Figure 4: Qualitative SDF Results. For Armadillo, F-INR models with strong inductive biases (WIRE saragadam2022wire, SIREN sitzmann2019siren, ReLU+PE mildenhall2020nerf) capture fine details with high fidelity. Others, like Tanh and FINER liu2024finer, preserve the macro-structure but fail to reconstruct high-frequency details, resulting in oversmoothed surfaces and lower IOU scores. More examples are provided in the supplementary material.
• Figure 5: Qualitative NeRF Results. Novel view renderings from our best-performing F-INR model (ReLU+HE with a TT rank of 16) on the Lego (top) and Drums (bottom) scenes. The results demonstrate the model's ability to reconstruct fine geometric details and complex view-dependent effects with high fidelity.