Neural Representation with Spherical Harmonics

• Continuous neural representation with spherical harmonics is a framework that uses an orthonormal basis to effectively model and interpolate functions on the sphere.
• It employs spectral transforms and learnable weights to achieve high accuracy, rotation-equivariance, and stability near coordinate singularities.
• Hybrid architectures merge neural parameterizations with traditional SH methods, enabling efficient operator learning and precise interpolation in spatial applications.

Continuous neural representation with spherical harmonics refers to the construction of neural function approximators on the sphere $\mathbb{S}^2$ (and higher-dimensional spheres) using the spherical-harmonic basis for both theoretical and practical advances. This paradigm produces neural models capable of learning, representing, and interpolating functions that are continuous, globally valid, and potentially equivariant under $\mathrm{SO(3)}$ (rotations). Spherical harmonics offer a mathematically principled, completeness-guaranteed, and non-redundant basis for $L^2$-integrable functions on the sphere, enabling neural architectures to natively respect spherical geometry, avoid coordinate artifacts (e.g., pole singularities), and achieve high accuracy for spatial and spatio-angular data.

1. Mathematical Foundation of Spherical Harmonics in Neural Representations

Spherical harmonics, denoted $Y_\ell^m(\theta, \phi)$ (complex form) or $Y_{\ell, m}(\theta, \phi)$ (real form), form an orthonormal basis for square-integrable functions on $\mathbb{S}^2$. Each basis function is indexed by degree $\ell \ge 0$ and order $m \in [-\ell, \ell]$: $$Y_\ell^m(\theta, \phi) = \sqrt{\frac{2\ell+1}{4\pi} \frac{(\ell - m)!}{(\ell + m)!}} P_\ell^m(\cos\theta) e^{i m \phi},$$ with the associated Legendre polynomial $P_\ell^m$. Orthonormality holds: $$\int_{\mathbb{S}^2} Y_\ell^m(\Omega) Y_{\ell' m'}^*(\Omega) d\Omega = \delta_{\ell\ell'} \delta_{mm'}.$$ Any $f \in L^2(\mathbb{S}^2)$ can be expanded as

$$f(\theta,\phi) = \sum_{\ell=0}^L \sum_{m=-\ell}^{\ell} \alpha_{\ell m} Y_{\ell m}(\theta,\phi),$$

where $\alpha_{\ell m}$ are SH coefficients, and $L$ is the truncation.

Numerically robust evaluation of $Y_{\ell m}$ and their derivatives is critical for neural networks; implementations such as sphericart provide $O(\ell_\text{max}^2)$ complexity for each sample and stable Cartesian recurrences for gradients (Bigi et al., 2023).

2. Spherical Harmonic Transforms and Spectral Neural Architectures

A continuous neural representation on the sphere is constructed by alternating between the spatial domain and the frequency (SH) domain. The forward SHT projects sample values onto SH coefficients (typically by least squares). In matrix form, for $P$ sampled directions $\{\Omega_p\}$: $$\mathbf{a} = (\mathbf{Y}^T \mathbf{Y})^{-1}\mathbf{Y}^T \mathbf{H},$$ where $\mathbf{Y} \in \mathbb{R}^{P \times (L+1)^2}$ contains $Y_{\ell m}(\Omega_p)$. The inverse SHT (ISHT) maps coefficients to sampled values: $\widehat{\mathbf{H}} = \mathbf{Y} \mathbf{a}.$ Spectral convolution is realized by filtering (Hadamard multiplication) in SH space with learnable spectral weights, followed by ISHT (Chen et al., 2023). For example, with a zonal kernel (only $m=0$ coefficients nonzero), per-coefficient modulation implements rotation-equivariant convolution.

Clebsch–Gordan Nets perform all operations—including quadratic nonlinearity—entirely in the SH domain using the CG transform for tensor product decomposition, achieving exact $\mathrm{SO(3)}$ equivariance by construction (Kondor et al., 2018).

3. Operator Learning and Green’s Functions Formulation

Modern spherical neural operator architectures employ spectral integral operators derived from Green’s function expansions. Any Green’s function $G(\Omega, \Omega')$ solving $D G(\cdot, \Omega') = \delta(\cdot, \Omega')$ can be decomposed spectrally: $$G(\Omega, \Omega') = \sum_{\ell=0}^L \sum_{m=-\ell}^{\ell} \frac{1}{\lambda_\ell} Y_{\ell m}(\Omega) Y_{\ell m}^*(\Omega'),$$ with eigenvalues $\lambda_\ell$ of $D$. This enables operator learning via

$$(\mathcal{G} f)(\Omega) = \sum_{\ell, m} \rho_{\ell m} \hat{f}_{\ell m} Y_{\ell m}(\Omega)$$

with trainable spectral weights $\rho_{\ell m}$, generalizing classical convolution (which restricts to per-$\ell$). GSNO (Generalized Spherical Neural Operator) blocks introduce relative (SO(3)-equivariant) and absolute (position-dependent) spectral filtering, enabling controlled modeling of anisotropy and spatially localized features. The GSHNet architecture implements a U-Net style spectral hierarchy, handling multi-scale signal structure via band-limited SH representations and up/down-sampling in the frequency domain (Tang et al., 11 Dec 2025).

4. Neural Parameterizations of Spherical Harmonic Coefficients

Instead of training a separate SH series per spatial or spatio-angular location, coordinate-based neural parameterizations yield continuous representations. The Neural Spherical Harmonics (NeSH) approach uses an MLP whose input is a positional encoding of coordinates $x \in \mathbb{R}^3$; output are all $a_{\ell m}(x)$ up to degree $L$. The reconstructed signal is

$$S(x, u) = \sum_{\ell,m} a_{\ell m}(x) Y_{\ell m}(u)$$

for $u \in \mathbb{S}^2$. Training proceeds on measured data $(x_i, u_i, S_i)$ using a data-fidelity term (e.g., smooth L1) and sparsity regularization on coefficients. Once trained, the network enables continuous upsampling in both space and angle, denoising, and improved coherence, as shown for diffusion MRI signals (Hendriks et al., 2023).

5. Hybrid Architectures and Spherical Harmonics as Positional Embeddings

Another prominent design encodes input spherical coordinates $(\theta, \phi)$ via truncated SH expansions (vector $\mathrm{SH}_L(\theta,\phi)\in\mathbb{R}^{(L+1)^2}$). Feeding this vector into a neural network—for example, a Sinusoidal Representation Network (SIREN) block—yields a representation expressible as a composition of globally smooth, spectrally controlled basis functions and learned nonlinear transformations. Empirical results in geospatial tasks, synthetic benchmarks, and multi-task regression demonstrate that SH embedding (either with only a linear layer or with a shallow SIREN) achieves state-of-the-art generalization and stability near the poles, outperforming rectangular Fourier (DFS) encodings (Rußwurm et al., 2023). The spatial resolution is tunable by $L$, and SIREN layers extend frequency expressiveness beyond linear SH models.

6. Theoretical Approximation and Learning Properties

Function approximation on the sphere via neural networks is strongly linked to the properties of truncated SH spaces and their reproducing kernels. For functions in the Sobolev space $W^r_\infty(\mathbb{S}^{d-1})$, deep convolutional-spline architectures achieve optimal uniform rates by approximating the SH expansion with convolutionally constructed features (Toeplitz filters implement $\langle y_i, x\rangle$), followed by shallow nonlinear modules mapping $t\mapsto$ polynomial kernel features (Fang et al., 2020). The total number of parameters scales linearly with network depth for a given approximation error $\varepsilon$. Additive ridge functions (i.e., $f(x) = \sum_j g_j(\langle y_j,x\rangle)$) can be efficiently approximated in this framework, matching univariate complexity per component.

7. Computational and Implementation Considerations

• Real-valued SH are preferred for many neural implementations due to numerical advantages; vectorized, stable Cartesian tensor recursions improve both performance and gradient propagation (Bigi et al., 2023).
• GPU-optimized libraries (e.g., sphericart) permit efficient large-batch evaluation and automatic differentiation.
• Computational cost per point is $O(L^2)$; batching and parallelization are necessary for high $L$ or large datasets.
• When used as neural layers or features, SH are combined with learnable spectral weights or as input to MLP blocks; parameter count and expressiveness are tunably traded off via $L$ and network width/depth.
• Best practices include using scaled (solid) harmonics, integrating with radial basis filters for point cloud or graph networks, and exploiting batched SHT/ISHT operations for convolutional networks.

In summary, continuous neural representation with spherical harmonics is a versatile, theoretically grounded, and practically effective framework for constructing spherical neural networks and neural operators. SH-based models deliver precise, equivariant, and pole-stable representations, support fast computation, and exhibit universal approximation properties, making them core to advances in spherical machine learning, operator learning, and spherical scientific applications (Chen et al., 2023, Tang et al., 11 Dec 2025, Hendriks et al., 2023, Rußwurm et al., 2023, Kondor et al., 2018, Fang et al., 2020, Bigi et al., 2023).