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Fully-Differentiable Tensor Spherical Harmonics in JAX

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# Fully-Differentiable Tensor Spherical Harmonics in JAX

<img width="400" height="360" alt="image" src="https://github.com/user-attachments/assets/4437b0d6-aecc-491f-9d2d-684f8444aa15" />

Fitting a tensor field from noisy scattered measurements via gradient descent on tensor spherical harmonic coefficients.

`harmonyx` is an implementation of the *tensor spherical harmonics* in [JAX](https://github.com/jax-ml/jax), building on top of the [e3nn-jax](https://github.com/e3nn/e3nn-jax) framework.

```bash
uv pip install harmonyx
```

Spherical irrep (or tensor) signals are functions defined on the sphere $S^2$ which map each point to a $SO(3)$ irrep of type $s$. We can imagine decomposing each component of the output, giving $2s+1$ regular spherical harmonics. For each spherical harmonic degree $l$, we obtain a set of functions which transform as the tensor product rep $s \otimes l$ of $SO(3)$. By decomposing this tensor product rep into a direct sum of irreps (of type $j$), we obtain the definition of [tensor spherical harmonics](https://scipp-legacy.pbsci.ucsc.edu/~haber/archives/physics214_13/tensor_harmonics.pdf).

$$(Y_{j,m_j}^{\ell,s})_{m_s}=\sum_{m_\ell}C^{j,m_j}_{\ell,m_\ell,s,m_s}Y^{m_\ell}_{\ell}$$

**Note: This library uses the e3nn-jax conventions for defining the spherical harmonics and irreps. Hence, the Clebsch-Gordan coefficients and resulting harmonics will differ slightly from those used in quantum physics. Our [preprint](https://arxiv.org/abs/2602.21466) and the symbolic checks in `symbolic/` use the conventions defined in quantum physics.**

The usual (scalar) [spherical harmonics](https://en.wikipedia.org/wiki/Spherical_harmonics) simply correspond to $s = 0$, as $0 \otimes l \equiv l$ and we can drop the redundant labels of $j$ and $m_s$.

Thus, the tensor spherical harmonics are a generalization of the spherical harmonics to arbitrary spin $s \geq 0$. To specify a tensor spherical harmonic, we must specify  the overall transformation type $j$, output irrep type $s$, and spherical harmonic degree $l$.

For example, to create the tensor spherical harmonic corresponding to $s = 2, l = 1, j = 2, m_j = 0$:
```python
from harmonyx import TensorSphericalHarmonics
x = TensorSphericalHarmonics.tensor_spherical_harmonic(s=2, l=1, j=2, mj=0, parity=1)
```

## Manipulating Tensor Spherical Harmonics

To obtain the coefficients of the tensor spherical harmonics from a tensor field:
```python
x = TensorSphericalHarmonics.from_tensor_signal(
    tensor_signal, # e3nn.SphericalSignal with dimension [2s + 1, res_beta, res_alpha].
    s=2,
    lmax=10,
    parity=1
)
```
and to convert the coefficients back to a tensor field at a given resolution (here, a $100 \times 99$ Gauss-Legendre Grid):
```python
tensor_signal = x.to_tensor_signal(
    res_beta=100,
    res_alpha=99,
    quadrature="gausslegendre"
)
```

If you only want to evaluate the tensor spherical harmonic coefficients at a given set of points, you can do:
```python
points = jax.random.normal(jax.random.PRNGKey(0), shape=(10, 3))  # 10 random points
points = points / jnp.linalg.norm(points, axis=-1, keepdims=True)  # Normalize to lie on the sphere

values_at_points = x.at_points(points) # e3nn.IrrepsArray with shape [10, 2s + 1] containing the values of the tensor spherical harmonic at the given points.
```

To get the coefficient for a specific $(l, j, m_j)$ from the tensor spherical harmonic, you can use the `get_coefficient` method:
```python
x = TensorSphericalHarmonics.normal(s=2, lmax=2, parity=1, key=jax.random.PRNGKey(0))
coefficient = x.get_coefficient(l=1, j=2, mj=0)
```
or to set the coefficient for a specific $(l, j, m_j)$:
```python
x = TensorSphericalHarmonics.zeros(s=2, lmax=2, parity=1)
x = x.set_coefficient(l=1, j=2, mj=0, value=1.0)
```
Similarly, you can get or set all $(2j + 1)$ coefficients for a given $(l, j)$:
```python
x = TensorSphericalHarmonics.normal(s=2, lmax=2, parity=1, key=jax.random.PRNGKey(0))
coefficients = x.get_coefficients(l=1, j=2)  # shape (2j + 1,) containing coefficients
x = x.set_coefficients(l=1, j=2, values=jnp.ones(2 * 2 + 1))  # set all coefficients for (l=1, j=2) to 1.0
```

## Irrep Signal Tensor Product (ISTP)

Further, we have implemented the corresponding *irrep signal tensor product* (ISTP) which interacts two irrep (tensor) fields on the sphere with a pointwise tensor product.
See our [preprint](https://arxiv.org/abs/2602.21466) to understand how the irrep signal tensor product enables faster Clebsch-Gordan tensor products.

```python
x = TensorSphericalHarmonics.normal(s=2, lmax=2, parity=1, key=jax.random.PRNGKey(0))
y = TensorSphericalHarmonics.normal(s=2, lmax=2, parity=1, key=jax.random.PRNGKey(1))

z = x.reduce_pointwise_tensor_product(y, s_out=3, res_beta=100, res_alpha=99, quadrature="gausslegendre")
```

## Plotting

To plot the tensor spherical harmonics as a 3D cone plot over $S^2$, you can do:
```python
x.plot(dims=(0, 1, 2), res_beta=100, res_alpha=99, quadrature="gausslegendre")
```
Since we can only plot 3D vector fields, you need to specify the `dims` argument to indicate which three dimensions of the $SO(3)$ irrep to plot. For example, `dims=(0, 1, 2)` plots the first three dimensions of the irrep, while `dims=(2, 3, 4)` plots the last three dimensions of the irrep for $s = 2$.
This requires the `plotly` package, which you can install via `uv pip install plotly 'nbformat>=4.2.0'`.

To visualize the intensity (squared l2-norm of all $(2s + 1)$ entries at each point) of the tensor spherical harmonic on the sphere, you can do:
```python
x.plot_intensity(res_beta=100, res_alpha=99, quadrature="gausslegendre")
```

## Vector Spherical Harmonics

Our implementation directly leads to the *vector spherical harmonics* by simply setting $s = 1$:
```python
from harmonyx import VectorSphericalHarmonics
x = VectorSphericalHarmonics.vector_spherical_harmonic(l=1, j=2, mj=0, parity=1)
```
as well as the corresponding *vector signal tensor product* (VSTP) which interacts two vector fields on the sphere:
```python
x = VectorSphericalHarmonics.normal(lmax=2, parity=1, key=jax.random.PRNGKey(0))
y = VectorSphericalHarmonics.normal(lmax=2, parity=1, key=jax.random.PRNGKey(1))

z = x.reduce_pointwise_cross_product(y, res_beta=100, res_alpha=99, quadrature="gausslegendre")
```
All other methods described above for the tensor spherical harmonics are also available for the vector spherical harmonics.

See our [example notebooks](https://github.com/atomicarchitects/tensor-spherical-harmonics/tree/main/examples)!

## Citation

Please cite our [preprint](https://arxiv.org/abs/2602.21466) if you use this repository:

```bibtex
@misc{xie2026asymptoticallyfastclebschgordantensor,
  title={Asymptotically Fast Clebsch-Gordan Tensor Products with Vector Spherical Harmonics}, 
  author={YuQing Xie and Ameya Daigavane and Mit Kotak and Tess Smidt},
  year={2026},
  eprint={2602.21466},
  archivePrefix={arXiv},
  primaryClass={cs.LG},
  url={https://arxiv.org/abs/2602.21466}, 
}
```

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