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Separable Neural Networks: Approximation Theory, NTK Regime, and Preconditioned Gradient Descent

ICLR 2026 Conference SubmissionAnonymous Authors
OpenReview Score: 6.0Download Report PDF
Separable Neural NetworksApproximation TheoryPreconditioned Gradient DescentNeural Tangent Kernel
Abstract:

Separable neural networks (SepNNs) are emerging neural architectures that significantly reduce computational costs by factorizing a multivariate function into linear combinations of univariate functions, benefiting downstream applications such as implicit neural representations (INRs) and physics-informed neural networks (PINNs). However, fundamental theoretical analysis for SepNN, including detailed representation capacity and spectral bias characterization & alleviation, remains unexplored. This work makes three key contributions to theoretically understanding and improving SepNN. First, using Weierstrass-based approximation and universal approximation theory, we prove that SepNN can approximate any multivariate function with arbitrary precision, confirming its representation completeness. Second, we derive the neural tangent kernel (NTK) regimes for SepNN, showing that the NTK of infinite-width SepNN converges to a deterministic (or random) kernel under infinite (or fixed) decomposition rank, with corresponding convergence and spectral bias characterization. Third, we propose an efficient separable preconditioned gradient descent (SepPGD) for optimizing SepNN, which alleviates the spectral bias of SepNN by provably adjusting its NTK spectrum. The SepPGD enjoys an efficient O(nD)\mathcal{O}(nD) complexity for nDn^D training samples, which is much more efficient than previous neural network PGD methods. Extensive experiments for kernel ridge regression, image and surface representation using INRs, and numerical PDEs using PINNs validate the efficiency of SepNN and the effectiveness of SepPGD for alleviating spectral bias.

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Overview

Overall Novelty Assessment

The paper contributes three theoretical results for separable neural networks: a universal approximation theorem, neural tangent kernel regime characterization, and a preconditioned optimization algorithm. It resides in the 'Separable and Factorized Architecture Theory' leaf, which contains only three papers total including this work. This represents a sparse research direction within the broader taxonomy, suggesting that rigorous theoretical analysis of separable architectures remains relatively underdeveloped compared to empirical applications or domain-specific factorization methods found elsewhere in the tree.

The taxonomy reveals that most factorization research concentrates on practical applications rather than foundational theory. Neighboring branches include 'Depth Separation and Expressiveness' (one paper on depth advantages), 'Preconditioned and Second-Order Methods' (two papers on curvature-based optimization), and extensive application-focused subtopics spanning recommendation systems, neuroimaging, and edge deployment. The paper's theoretical focus on approximation guarantees and training dynamics bridges the sparse 'Approximation Theory' branch with the more populated 'Optimization Methods' branch, positioning it at an intersection where formal analysis meets algorithmic development.

Among thirty candidates examined, the universal approximation contribution shows one refutable candidate from ten examined, while the NTK regime analysis found no clear refutations across ten candidates. The preconditioned gradient descent algorithm also encountered one refutable candidate among ten examined. These statistics suggest that while some theoretical ground may overlap with prior work on approximation or optimization, the specific combination of separable architecture analysis, NTK characterization, and tailored preconditioning appears less thoroughly explored within the limited search scope. The NTK contribution appears particularly novel given zero refutations found.

Based on the limited thirty-candidate search, the work addresses a theoretically sparse area where formal guarantees for separable networks remain uncommon. The analysis does not cover exhaustive literature review across all optimization or approximation theory, so additional relevant work may exist outside the top semantic matches examined. The combination of three distinct theoretical contributions targeting a single architecture class suggests substantive effort to establish foundational understanding in an undertheorized domain.

Taxonomy

Core-task Taxonomy Papers
47
3
Claimed Contributions
30
Contribution Candidate Papers Compared
2
Refutable Paper

Research Landscape Overview

Core task: Theoretical analysis and optimization of separable neural networks. The field centers on understanding and exploiting factorized or modular structures in neural architectures, where complex computations decompose into simpler, interpretable components. The taxonomy reveals four main branches: Approximation Theory and Representation Capacity examines the expressive power and theoretical guarantees of separable forms, often drawing on tensor decomposition and low-rank factorization ideas (e.g., Global Tensor Optimality[36], Low Rank Factorization[28]). Optimization Methods and Training Dynamics investigates how gradient-based learning interacts with factorized parameterizations, including implicit regularization effects (Momentum Implicit Regularization[20]) and specialized curvature approximations (Kronecker Factored Curvature[1]). Architecture Design and Applications explores practical instantiations ranging from modular networks (ModuleNet[13], Modular Networks[8]) to neural architecture search strategies (SNAS[2], SM-NAS[33]) and domain-specific deployments. Interpretability and Uncertainty Quantification focuses on leveraging separability for explainability (Factorized Explainer[11], Parameter Space Interpretability[12]) and probabilistic reasoning. Within Approximation Theory and Representation Capacity, a particularly active line of work addresses the fundamental trade-offs between compactness and expressiveness in factorized representations. Some studies pursue tensor-based formulations to achieve global optimality guarantees (Global Tensor Optimality[36]), while others investigate how separable structures constrain or enable certain function classes. Separable Neural Networks[0] sits squarely in this theoretical branch, contributing rigorous analysis of separable architectures' representational limits and optimization landscapes. Its emphasis on provable properties distinguishes it from more empirically driven factorization schemes like Factorizing Knowledge[3] or Neural Matrix Factorization[4], which prioritize practical performance over formal guarantees. By clarifying when and why separable forms succeed or fail, this work helps bridge the gap between classical approximation theory and modern deep learning practice, informing both algorithm design and our understanding of implicit biases in factorized models.

Claimed Contributions

Universal approximation theorem for separable neural networks

The authors establish that separable neural networks (including CP, TT, and Tucker variants) can approximate any continuous multivariate function on compact sets to arbitrary accuracy. This result extends prior work limited to bivariate cases and uses a unified proof technique combining the Stone-Weierstrass theorem with universal approximation theory.

10 retrieved papers
Can Refute
Neural tangent kernel regimes for separable neural networks

The authors characterize the training dynamics of SepNNs by deriving their NTK under different asymptotic conditions. They prove that the NTK converges to a deterministic kernel when both width and rank approach infinity, and to a random kernel when width is infinite but rank is fixed, enabling analysis of convergence rates and spectral bias.

10 retrieved papers
Separable preconditioned gradient descent algorithm

The authors introduce SepPGD, a computationally efficient preconditioning method that adjusts the eigenvalue distribution of the NTK matrix to alleviate spectral bias in SepNNs. The method achieves O(nD) complexity for nD training samples by applying smaller preconditioners separately to factor networks, which is significantly more efficient than existing neural network preconditioning approaches.

10 retrieved papers
Can Refute

Core Task Comparisons

Comparisons with papers in the same taxonomy category

Contribution Analysis

Detailed comparisons for each claimed contribution

Contribution

Universal approximation theorem for separable neural networks

The authors establish that separable neural networks (including CP, TT, and Tucker variants) can approximate any continuous multivariate function on compact sets to arbitrary accuracy. This result extends prior work limited to bivariate cases and uses a unified proof technique combining the Stone-Weierstrass theorem with universal approximation theory.

Contribution

Neural tangent kernel regimes for separable neural networks

The authors characterize the training dynamics of SepNNs by deriving their NTK under different asymptotic conditions. They prove that the NTK converges to a deterministic kernel when both width and rank approach infinity, and to a random kernel when width is infinite but rank is fixed, enabling analysis of convergence rates and spectral bias.

Contribution

Separable preconditioned gradient descent algorithm

The authors introduce SepPGD, a computationally efficient preconditioning method that adjusts the eigenvalue distribution of the NTK matrix to alleviate spectral bias in SepNNs. The method achieves O(nD) complexity for nD training samples by applying smaller preconditioners separately to factor networks, which is significantly more efficient than existing neural network preconditioning approaches.

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