# Hardy-Basis Equivariant Mixer

- ID: 2271
- Canonical URL: https://synthcore.org/idea/2271/hardy-basis-equivariant-mixer
- API JSON: https://synthcore.org/api/idea/2271.json
- API Markdown: https://synthcore.org/api/idea/2271.md
- Verification status: unverified
- Source: [arXiv:2608.13965](https://arxiv.org/abs/2608.13965)
- Category: architecture
- Solves: accuracy, stability
- ML areas: graph-nn, embedding, training-dynamics
- Math tags: representation-theory, lie-groups, geometry, harmonic-analysis
- Ratings: usefulness 5/10; difficulty 5/10; novelty 8/10

## Idea description

Insert a fixed or partially learnable equivariant change-of-basis module into a spherical or SO(3)-equivariant network. At each angular frequency \(\ell\), the module maps the line selected by the line-bundle quantization to the line selected by the Grauert-tube quantization, allowing the network to represent both holomorphic/base-local and geodesic-flow-adapted features without breaking rotation equivariance.

## Mathematical statement

Theorem 1.1 represents the two Hardy summands in the \(\ell\)-th irreducible representation \(V_\ell\) by multiplicity-space vectors \(u_\ell=z_0^{2\ell}\) and \(w_\ell(\tau)=(z_0^2+z_1^2+2\coth(\tau)z_0z_1)^\ell\), where \(z_0,z_1\) are homogeneous coordinates on \(\mathbb{C}^2\), \(\tau>0\) is the Grauert-tube radius, and \(\ell\in\mathbb{N}_0\). The normalized line overlap is \(c_\ell(\tau)=|\langle\widehat w_\ell(\tau),\widehat u_\ell\rangle|\), with \(\widehat u_\ell=u_\ell/\|u_\ell\|\) and \(\widehat w_\ell=w_\ell/\|w_\ell\|\). The explicit formula uses the Legendre polynomial \(P_\ell\): \(c_\ell(\tau)=2^{-\ell}\sqrt{\binom{2\ell}{\ell}}\sinh^\ell(\tau)/\sqrt{P_\ell(\cosh(2\tau))}\). Since the transformation acts only in the multiplicity factor and leaves every \(V_\ell\) block intact, it commutes with the left SO(3) action. In an ML layer, use a unitary block \(R_\ell(\tau)\) satisfying \(R_\ell\widehat u_\ell=\widehat w_\ell(\tau)\); a Householder realization is \(R_\ell=I-2vv^*/(v^*v)\), with \(v=\widehat u_\ell-\widehat w_\ell\).

## Key formulas

- $$u_{\ell}=z_{0}^{2\ell},\qquad w_{\ell}(\tau)=\bigl(z_{0}^{2}+z_{1}^{2}+2\operatorname{coth}\tau\,z_{0}z_{1}\bigr)^{\ell}.$$
- $$\bigl|\langle\widehat{w}_{\ell}(\tau),\widehat{u}_{\ell}\rangle\bigr|=c_\ell(\tau)=\frac{\sqrt{\binom{2\ell}{\ell}}}{2^{\ell}}\frac{\sinh^{\ell}\!\tau}{\sqrt{P_{\ell}(\cosh 2\tau)}}.$$
- $$R_\ell(\tau)=I-2\frac{v_\ell v_\ell^{*}}{v_\ell^{*}v_\ell},\qquad v_\ell=\widehat u_\ell-\widehat w_\ell(\tau),\qquad R_\ell(\tau)\widehat u_\ell=\widehat w_\ell(\tau).$$
- $$\mathcal{E}_{\mathrm{mix}}(x)=\bigoplus_{\ell=0}^{L}R_\ell(\tau)x_\ell.$$

## Implementation notes

1. Integration point: use a Fourier-domain feature tensor from an SO(3)-equivariant or spherical network. Decompose each feature map into angular-frequency blocks \(x_\ell\); each block has an irrep index carrying the left SO(3) action and a multiplicity index carrying channels, orientations, or basis states. Apply the mixer only to the multiplicity index, immediately before an equivariant tensor product or after a spectral convolution. It cannot alter the transformation law of the irrep index.

2. Pseudocode:
```text
for ell in 0..L:
    u = normalized(coefficients(z0^(2*ell)))
    w = normalized(coefficients((z0^2 + z1^2 + 2*coth(tau)*z0*z1)^ell))
    v = u - w
    if norm(v) > eps:
        R = I - 2 * outer(v, conjugate(v)) / dot(conjugate(v), v)
    else:
        R = I
    x_ell = einsum("ab,...bq->...aq", R, x_ell)
concatenate(x_ell)
```
Here `q` indexes the SO(3) irrep component and `a,b` index multiplicity channels. Compute polynomial coefficients with stable recurrence or log-space normalization. Keep \(\tau\) fixed initially; a learnable version can use \(\tau=\operatorname{softplus}(\rho)+10^{-4}\), recomputing the basis periodically. Compute the exact overlap \(c_\ell(\tau)\) with stable log-binomial and Legendre recurrences, and optionally initialize a scalar residual gate from this value.

3. Computed from the paper: \(u_\ell\), \(w_\ell(\tau)\), the unitary Householder map, and \(c_\ell(\tau)\). Estimated empirically: which frequencies benefit, whether the mixer improves gradient conditioning, and whether a learned residual is useful.

4. First experiment: train a small SO(3)-equivariant classifier or spherical CNN on rotated ModelNet40 projections, spherical MNIST, or a synthetic task combining local orientation and long-range geodesic structure. Compare a baseline spectral equivariant network, the fixed mixer, and a learnable mixer initialized at the analytic map, using \(L=8\) or \(16\), equal parameter count, and equal FLOPs. Sweep \(\tau\in\{0.2,0.5,1.0,2.0\}\). Success means higher rotated-test accuracy or faster loss descent at equal compute, while equivariance error remains at baseline numerical tolerance. Plot per-frequency gradient norms and accuracy versus \(\ell\) to determine whether the basis improves conditioning rather than merely adding capacity.

## Disclaimer

AI-generated research hypothesis, automatically tested. Not peer-reviewed.