SlowGuess/NeurIPS_2023_OCR huggingface dataset:
$$
\begin{array}{l} \tau_ {\Theta} \left(\left(\theta_ {1}, x _ {1}\right), \left(\theta_ {2}, x _ {2}\right)\right) = \int_ {\left(\theta , x\right) \in S E (2)} \Theta (\theta , x) \Theta \left(\left(\theta , x\right) \cdot \left(\theta_ {1}, x _ {1}\right)\right) \Theta \left(\left(\theta , x\right) \cdot \left(\theta_ {2}, x _ {2}\right)\right) d \theta d x \\ = \int_ {(\theta , x) \in S E (2)} \Theta (\theta , x) \Theta (\theta + \theta_ {1}, R _ {\theta}. x _ {1} + x) \Theta (\theta + \theta_ {2}, R _ {\theta}. x _ {2} + x) d \theta d x, \\ \end{array}
$$
for any  $\theta_1, \theta_2 \in SO(2)$  and  $x_1, x_2 \in \mathbb{R}^2$ .
D Invariance of the  $G$ -Triple Correlation
Consider a real signal  $\Theta$  defined over a group  $G$ . The  $G$ -Triple Correlation is invariant to group actions on the domain of the signal  $\Theta$  as shown in [28].
Proposition 4. Consider two real signals  $\Theta_1, \Theta_2$  defined over a group  $G$ . If there exists  $h \in G$  such that one signal is obtained from the other by a group action, i.e.,  $\Theta_2 = L_h[\Theta_1]$ , then  $\tau_{\Theta_1} = \tau_{\Theta_2}$ .
We recall the proof of [28] below.
Proof. Consider two real signals  $\Theta_1, \Theta_2$  defined over a group  $G$ , such that  $\Theta_2 = L_h[\Theta_1]$  for a group action  $L_h$  of group element  $h$ . We show that this implies that  $\tau_{\Theta_1} = \tau_{\Theta_2}$ .
Taking  $g_{1}, g_{2} \in G$ , we have:
$$
\begin{array}{l} \tau_ {\Theta_ {2}} (g _ {1}, g _ {2}) = \int_ {g \in G} \Theta_ {2} (g) \Theta_ {2} (g g _ {1}) \Theta_ {2} (g g _ {2}) d g \\ = \int_ {g \in G} L _ {h} [ \Theta_ {1} ] (g) L _ {h} [ \Theta_ {1} ] (g g _ {1}) L _ {h} [ \Theta_ {1} ] (g g _ {2}) d g \\ = \int_ {g \in G} \Theta_ {1} (h g) \Theta_ {1} (h g g _ {1}) \Theta_ {1} (h g g _ {2}) d g \\ = \int_ {g \in G} \Theta_ {1} (g) \Theta_ {1} \left(g g _ {1}\right) \Theta_ {1} \left(g g _ {2}\right) d g \\ = \tau_ {\Theta_ {1}} (g _ {1}, g _ {2}). \\ \end{array}
$$
where we use the change of variable  $hg \to g$ .
This proves the invariance of the  $G$ -TC with respect to group actions on the signals.
E Symmetries of the  $G$ -Triple Correlation
The  $G$ -Triple Correlation ( $G$ -TC) enjoys some symmetries that we can leverage to avoid computing it for each pair of group elements (which would represent  $|G|^2$  computations), hence making the feedforward pass more efficient.
These symmetries are given in the main text. We recall them here for completeness.
Proposition 5. Consider two transformations  $g_1, g_2 \in G$ . The  $G$ -Triple Correlation of a signal  $\Theta$  has the following symmetry:
$$
\left(s 1\right) \quad T _ {\Theta} \left(g _ {1}, g _ {2}\right) = T _ {\Theta} \left(g _ {2}, g _ {1}\right).
$$
$$
\left(s 2\right) \quad T _ {\Theta} \left(g _ {1}, g _ {2}\right) = T _ {\Theta} \left(g _ {1} ^ {- 1}, g _ {2} g _ {1} ^ {- 1}\right) = T _ {\Theta} \left(g _ {2} g _ {1} ^ {- 1}, g _ {1} ^ {- 1}\right) = T _ {\Theta} \left(g _ {1} g _ {2} ^ {- 1}, g _ {2} ^ {- 1}\right) = T _ {\Theta} \left(g _ {2} ^ {- 1}, g _ {1} g _ {2} ^ {- 1}\right).
$$
Our proof extends the proof given in [39] for the group of translations.
Proof. Consider two transformations  $g_1, g_2 \in G$ .
Symmetry (s1) relies on the fact that  $\Theta(gg_2)$  and  $\Theta(gg_1)$  are scalar values that commute:
$$
T _ {\Theta} (g _ {2}, g _ {1}) = \frac {1}{| G |} \sum_ {g \in G} \Theta (g) \Theta (g g _ {2}) \Theta (g g _ {1}) = \frac {1}{| G |} \sum_ {g \in G} \Theta (g) \Theta (g g _ {1}) \Theta (g g _ {2}) = T _ {\Theta} (g _ {2}, g _ {1}).
$$
For symmetry (s2), we assume that  $G$  is commutative. We prove the first equality:
$$
\begin{array}{l} T _ {\Theta} (g _ {1} ^ {- 1}, g _ {2} g _ {1} ^ {- 1}) = \frac {1}{| G |} \sum_ {g \in G} \Theta (g) \Theta (g g _ {1} ^ {- 1}) \Theta (g g _ {2} g _ {1} ^ {- 1}) \\ = \frac {1}{| G |} \sum_ {g ^ {\prime} \in G} \Theta \left(g ^ {\prime} g _ {1}\right) \Theta \left(g ^ {\prime}\right) \Theta \left(g ^ {\prime} g _ {1} g _ {2} g _ {1} ^ {- 1}\right) \quad (\text {w i t h} g ^ {\prime} = g g _ {1} ^ {- 1}, \text {i . e . ,} g = g ^ {\prime} g _ {1}) \\ = \frac {1}{| G |} \sum_ {g ^ {\prime} \in G} \Theta \left(g ^ {\prime} g _ {1}\right) \Theta \left(g ^ {\prime}\right) \Theta \left(g ^ {\prime} g _ {2}\right) (G \text {c o m m u t a t i v e :} g _ {2} g _ {1} ^ {- 1} = g _ {1} ^ {- 1} g _ {2}) \\ = \frac {1}{| G |} \sum_ {g \in G} \Theta (g g _ {1}) \Theta (g) \Theta (g g _ {2}) \\ = \frac {1}{| G |} \sum_ {g \in G} \Theta (g) \Theta (g g _ {1}) \Theta (g g _ {2}) (\Theta \text {t a k e s o n r e a l v a l u e s t h a t c o m m u t e}) \\ = T _ {\Theta} \left(g _ {2}, g _ {1}\right) \\ \end{array}
$$
The second equality of symmetry (s2) follows using (s1). The third and fourth equality of symmetry (s2) have the same proof.
This result and its proof are also valid for the extension of the  $G$ -TC that we propose in Appendix B. They naturally emerge by replacing the regular representation by the induced representation in the proof above.
Specifically, consider a signal  $\Theta_{2} = \pi (g_{0})[\Theta_{1}]$  obtained from the action of  $g_{0}$  on a signal  $\Theta_{1}$ . We want to show that  $\tau_{\Theta_1} = \tau_{\Theta_2}$ . The key ingredient of the proof is the change of variable within the integral  $\int_{G^{\prime}}$ , which follows the semi-direct product structure of  $\mathrm{G}$ :
$$
h ^ {\prime} = h h _ {0} \text {a n d} t ^ {\prime} = \phi (h) t _ {0} + t
$$
where  $\phi(h)$  is a matrix representing  $h$  that acts on  $\mathbb{Z}_2$  via matrix multiplication: e.g., a rotation matrix  $R = \phi(r)$  in the case of  $SE(n)$ . This concludes the adaptation of the proof for the steerable case.
F Algorithmic Reduction
In this section, we show that we can reduce the complexity of the  $G$ -Triple Correlation of a real signal. This computational reduction requires that we consider, instead, the spectral dual of the  $G$ -TC, the bispectrum [31]. In what follows, we consider a signal  $\Theta$  defined over a finite group  $G$ . The signal can be real or complex valued.
F.1 Reduction for Commutative Groups
Consider a commutative group  $G$ . The bispectrum for a signal  $\Theta$  is defined over a pair of irreducible representations  $\rho_1, \rho_2$  of the group  $G$  as:
$$
\beta (\Theta) _ {\rho_ {1}, \rho_ {2}} = \mathcal {F} (\Theta) _ {\rho_ {1}} ^ {\dagger} \mathcal {F} (\Theta) _ {\rho_ {2}} ^ {\dagger} \mathcal {F} (\Theta) _ {\rho_ {1} \rho_ {2}} \quad \in \mathbb {C}, \tag {13}
$$
where  $\mathcal{F}(\Theta)$  is the Fourier transform of  $\Theta$  that generalizes the classical Fourier transformation to signals defined over a group. We note that, in group theory, the irreducible representations (irreps) of commutative groups map to scalar values. Hence, the bispectrum is a complex scalar in this case.
For a discrete commutative group, the number of irreps is equal to the size of the group. [KondorThesis] proved that, for cyclic groups, it is enough to compute  $|G| + 1$  bispectral coefficients to fully describe the signal  $\Theta$  up to group action, instead of the  $|G|^2$  that would otherwise be required from its definition.
F.2 Reduction for Non-Commutative Groups
Consider a non-commutative group  $G$ . The bispectrum of a signal  $\Theta$  is defined over a pair of irreducible representations  $\rho_1, \rho_2$  of  $G$  as:
$$
\beta (\Theta) _ {\rho_ {1}, \rho_ {2}} = [ \mathcal {F} (\Theta) _ {\rho_ {1}} \otimes \mathcal {F} (\Theta) _ {\rho_ {2}} ] ^ {\dagger} C _ {\rho_ {1}, \rho_ {2}} \Big [ \bigoplus_ {\rho \in \rho_ {1} \otimes \rho_ {2}} \mathcal {F} (\Theta) _ {\rho} \Big ] C _ {\rho_ {1}, \rho_ {2}} ^ {\dagger} \qquad \in \mathbb {C} ^ {D _ {1} D _ {2} \times D _ {1} D _ {2}},
$$
where  $\otimes$  is the tensor product, and  $\oplus$  is a direct sum over irreps. The unitary Clebsch-Gordan matrix  $C_{\rho_1,\rho_2}$  is analytically defined for each pair of representations  $\rho_1,\rho_2$  as:
$$
\left(\rho_ {1} \otimes \rho_ {2}\right) (g) = C _ {\rho_ {1}, \rho_ {2}} ^ {\dagger} \left[ \bigoplus_ {\rho \in \rho_ {1} \otimes \rho_ {2}} \rho (g) \right] C _ {\rho_ {1}, \rho_ {2}}. \tag {14}
$$
We note that the irreps of non-commutative groups map to matrices, hence the bispectrum is a complex matrix in this case.
We provide an algorithmic approach reducing the computational complexity of the bispectrum for non-commutative finite groups. We show that we can recover the signal  $\Theta$  from a small subset of its bispectral coefficients. That is, we can recover  $\Theta$  from coefficients  $\beta (\Theta)_{\rho_1,\rho_2}$  computed for a few, well-chosen irreps  $\rho_{1},\rho_{2}$ . In practice, we only need to compute a few bispectral coefficients to have a complete invariant of the signal  $\Theta$  — hence reducing the computational complexity of the layer.
Generalizing [28], we will show that a subset of bispectral coefficients allows us to recover the Fourier transform of the signal  $\Theta$  for every irreducible representation  $\rho$  of the group. This will show that we can recover the signal  $\Theta$  itself by applying the inverse Fourier transform.
We first show relationships between the bispectral coefficients and the Fourier coefficients of the signal  $\Theta$  in the following Lemma. We denote  $\rho_0$  the trivial representation of the group  $G$ , which is the representation that sends every group element to the scalar 1.
Lemma 1. Consider  $\rho_0$  the trivial representation of the group  $G$ . Consider  $\rho$  any other irreducible representation of dimension  $D$ . The bispectral coefficients write:
$$
\beta_ {\rho_ {0}, \rho_ {0}} = | \mathcal {F} (\Theta) _ {\rho_ {0}} | ^ {2} \mathcal {F} (\Theta) _ {\rho_ {0}} \in \mathbb {C}
$$
$$
\beta_ {\rho , \rho_ {0}} = \mathcal {F} (\Theta) _ {\rho_ {0}} ^ {\dagger} \mathcal {F} (\Theta) _ {\rho} ^ {\dagger} \mathcal {F} (\Theta) _ {\rho} \in \mathbb {C} ^ {D \times D}.
$$
Here, and in what follows, we denote  $\beta$  the bispectrum of  $\Theta$ , i.e., we omit the argument  $\Theta$  for clarity of notations.
Proof. For  $\rho_0$  the trivial representation, and  $\rho$  an irreps, the Clebsh-Jordan (CJ) matrix  $C_{\rho \rho_0}$  is the identity matrix, and the matrix  $C_{\rho \rho_0}$  is the scalar 1.
We compute the bispectral coefficient  $\beta_{\rho_0,\rho_0}$ :
$$
\begin{array}{l} \beta_ {\rho_ {0}, \rho_ {0}} = (\mathcal {F} (\Theta) _ {\rho_ {0}} \otimes \mathcal {F} (\Theta) _ {\rho_ {0}}) ^ {\dagger} C _ {\rho_ {0} \rho_ {0}} \left[ \bigoplus_ {\rho \in \rho_ {0} \otimes \rho_ {0}} \mathcal {F} (\Theta) _ {\rho} \right] C _ {\rho_ {0} \rho_ {0}} ^ {\dagger} \\ = \left(\mathcal {F} (\Theta) _ {\rho_ {0}} \otimes \mathcal {F} (\Theta) _ {\rho_ {0}}\right) ^ {\dagger} C _ {\rho_ {0} \rho_ {0}} \mathcal {F} (\Theta) _ {\rho_ {0}} C _ {\rho_ {0} \rho_ {0}} ^ {\dagger} \quad \left(\rho_ {0} \otimes \rho_ {0} = \rho_ {0} \text {w h i c h i s i r r e d u c i b l e}\right) \\ = \left(\mathcal {F} (\Theta) _ {\rho_ {0}} ^ {2}\right) ^ {\dagger} C _ {\rho_ {0} \rho_ {0}} \mathcal {F} (\Theta) _ {\rho_ {0}} C _ {\rho_ {0} \rho_ {0}} ^ {\dagger} \quad \left(\mathcal {F} (\Theta) _ {\rho_ {0}} \text {i s a s c a l a r f o r w h i c h t e n s o r p r o d u c t i s m u l t i p l i c a t i o n}\right) \\ = \left| \mathcal {F} (\Theta) _ {\rho_ {0}} \right| ^ {2} \mathcal {F} (\Theta) _ {\rho_ {0}} \quad (\text {C J m a t r i c e s a r e e q u a l t o 1 .}) \\ \end{array}
$$
Take any irreducible representation  $\rho$  of dimension  $D$ , we have:
$$
\begin{array}{l} \beta_ {\rho , \rho_ {0}} = (\mathcal {F} (\Theta) _ {\rho} \otimes \mathcal {F} (\Theta) _ {\rho_ {0}}) ^ {\dagger} C _ {\rho \rho_ {0}} \left[ \bigoplus_ {\rho \in \rho \otimes \rho_ {0}} \mathcal {F} (\Theta) _ {\rho} \right] C _ {\rho \rho_ {0}} ^ {\dagger} \\ = \mathcal {F} (\Theta) _ {\rho_ {0}} ^ {\dagger} \mathcal {F} (\Theta) _ {\rho} ^ {\dagger} C _ {\rho \rho_ {0}} \mathcal {F} (\Theta) _ {\rho} C _ {\rho \rho_ {0}} ^ {\dagger} \quad (\mathcal {F} (\Theta) _ {\rho_ {0}} \text {i s a s c a l a r a n d} \rho \otimes \rho_ {0} = \rho) \\ = \mathcal {F} (\Theta) _ {\rho_ {0}} ^ {\dagger} \mathcal {F} (\Theta) _ {\rho} ^ {\dagger} \mathcal {F} (\Theta) _ {\rho} \qquad (\text {C J m a t r i c e s a r e i d e n t i t y m a t r i c e s}). \\ \end{array}
$$
Next, we summarize our main result.
Proposition 6. We can recover the Fourier coefficients of a signal  $\Theta$  from only  $L$  bispectral coefficients, where  $L$  is a number computed from the Kronecker product table of the group  $G$ .
In the proof, we propose an algorithmic method that iteratively computes bispectral coefficients until the Fourier coefficients of the signal are all recovered. We note that, for arbitrary groups and their representations, Clebsch-Gordan (CJ) matrices are not known in general, yet they can be computed numerically. Thus, the proof below assumes that the CJ matrices are given for the group  $G$  of interest.
Proof. Algorithmic Approach.
First, we show how we can recover the first Fourier coefficient (DC component), i.e., the Fourier transform of the signal at the trivial representation  $\rho_0$ , from a single bispectral coefficient.
$$
\mathcal {F} (\Theta) _ {\rho_ {0}} = \hat {f} _ {\rho_ {0}} = \int_ {G} \Theta (g) \rho_ {0} (g) d g = \int_ {G} \Theta (g) d g \in \mathbb {C}. \tag {15}
$$
Using Lemma 1, we can recover this Fourier component from the bispectral coefficient  $\beta_{\rho_0,\rho_0}$ , as:
$$
\left| \mathcal {F} (\Theta) _ {\rho_ {0}} \right| = \left(\left| \beta_ {\rho_ {0}, \rho_ {0}} \right|\right) ^ {1 / 3}, \quad \arg \left(\mathcal {F} (\Theta) _ {\rho_ {0}}\right) = \arg \left(\beta_ {\rho_ {0}, \rho_ {0}}\right). \tag {16}
$$
Next, consider an irreducible representation  $\rho_{1}$  of dimension  $D$ . We seek to recover the Fourier coefficient  $\mathcal{F}(\Theta)_{\rho_1}$ . This Fourier coefficient is a matrix in  $\mathbb{C}^{D\times D}$ . Using Lemma 1, we can recover it from a single bispectral coefficient:
$$
\mathcal {F} (\Theta) _ {\rho} ^ {\dagger} \mathcal {F} (\Theta) _ {\rho} = \frac {\beta_ {\rho , \rho_ {0}}}{\mathcal {F} (\Theta) _ {\rho_ {0}} ^ {\dagger}} \in \mathbb {C} ^ {D \times D}, \tag {17}
$$
since we have already recovered the Fourier coefficient  $\mathcal{F}(\Theta)_{\rho_0}$ . The matrix  $\mathcal{F}(\Theta)_{\rho}^{\dagger}\mathcal{F}(\Theta)_{\rho}$  is hermitian, and thus admits a square-root, that we denote  $\mathcal{F}(\Theta)_{\rho}^{\prime}$ :
$$
\mathcal {F} (\Theta) _ {\rho} ^ {\prime} = \left(\frac {\beta_ {\rho , \rho_ {0}}}{\mathcal {F} (\Theta) _ {\rho_ {0}} ^ {\dagger}}\right) ^ {1 / 2}. \tag {18}
$$
The square-root  $\mathcal{F}(\Theta)_{\rho}'$  only corresponds to  $\mathcal{F}(\Theta)_{\rho}$  up to a matrix factor. This is an unidentifiability similar to the commutative case [28]. Specifically, consider the singular value decomposition (SVD) of  $\mathcal{F}(\Theta)_{\rho}$ :
$$
\begin{array}{l} \mathcal {F} (\Theta) _ {\rho} = U. \Sigma . V ^ {\dagger} \\ \Rightarrow \mathcal {F} (\Theta) _ {\rho} ^ {\dagger} \mathcal {F} (\Theta) _ {\rho} = (U. \Sigma . V ^ {\dagger}) ^ {\dagger}. U. \Sigma . V ^ {\dagger} = V \Sigma^ {2} V ^ {\dagger} \\ \Rightarrow \mathcal {F} (\Theta) _ {\rho} ^ {\prime} = V \Sigma V ^ {\dagger}. \\ \end{array}
$$
Thus, we have:  $\mathcal{F}(\Theta)_{\rho} = UV^{\dagger}.\mathcal{F}(\Theta)_{\rho}'$  where  $U,V$  are unitary matrices that come from the (unknown) SVD decomposition of the (unknown)  $\mathcal{F}(\Theta)_{\rho}$ . By recovering  $\mathcal{F}(\Theta)_{\rho}$  as  $\mathcal{F}(\Theta)_{\rho}'$ , we fix  $UV^{\dagger} = I$ . This is similar to the same way in which [28] fixed  $\phi = 0$  (rotation of angle 0, i.e., the identity) in the commutative case.
Next, we seek to find the remaining Fourier coefficients of the signal  $\Theta$ , from a limited subset of the bispectral coefficients. To this aim, we denote  $\mathcal{R}$  the set of irreducible representations of the group
$G$ . We recall that, for a finite group  $G$ , the set  $\mathcal{R}$  is also finite, with its size equal to the number of conjugacy classes of  $G$ .
We consider the following bispectral coefficient:
$$
\begin{array}{l} \beta_ {\rho_ {1}, \rho_ {1}} = (\mathcal {F} (\Theta) _ {\rho_ {1}} \otimes \mathcal {F} (\Theta) _ {\rho_ {1}}) ^ {\dagger} C _ {\rho_ {1} \rho_ {1}} \left[ \bigoplus_ {\rho \in \rho_ {1} \otimes \rho_ {1}} \mathcal {F} (\Theta) _ {\rho} \right] C _ {\rho_ {1} \rho_ {1}} ^ {\dagger} \\ = (\mathcal {F} (\Theta) _ {\rho_ {1}} \otimes \mathcal {F} (\Theta) _ {\rho_ {1}}) ^ {\dagger} C _ {\rho_ {1} \rho_ {1}} \left[ \bigoplus_ {\rho \in \mathcal {R}} \mathcal {F} (\Theta) _ {\rho} ^ {n _ {\rho , \rho_ {1}}} \right] C _ {\rho_ {1} \rho_ {1}} ^ {\dagger}, \\ \end{array}
$$
where  $n_{\rho, \rho_1}$  is the multiplicity of irreps  $\rho$  in the decomposition of  $\rho_1, \rho_1$ . This multiplicity is known as it only depends on the group, not on the signal  $\Theta$ .
We get an equation that allows us to recover additional Fourier coefficients of the signal  $\Theta$ :
$$
\bigoplus_ {\rho \in \mathcal {R}} \mathcal {F} (\Theta) _ {\rho} ^ {n _ {\rho , \rho_ {1}}} = C _ {\rho_ {1} \rho_ {1}} ^ {- 1} \left(\mathcal {F} (\Theta) _ {\rho_ {1}} \otimes \mathcal {F} (\Theta) _ {\rho_ {1}}\right) ^ {- \dagger} \beta_ {\rho_ {1}, \rho_ {1}} C _ {\rho_ {1} \rho_ {1}} ^ {- \dagger}, \tag {19}
$$
where everything on the right hand side is known. Therefore, every Fourier coefficient  $\mathcal{F}(\Theta)_{\rho}$  that appears in the decomposition of  $\rho_{1} \otimes \rho_{1}$  into irreducible irreps  $\rho$  can be computed, by reading off the elements of the block diagonal matrix defined by the direct sum. We recover the Fourier coefficients  $\mathcal{F}(\Theta)_{\rho}$  for which  $n_{\rho, \rho_1} \neq 0$ .
We assume that this procedure provides at least one other Fourier coefficient, for an irreps  $\rho_{2}$ , that we fix. We can then compute the following bispectral coefficient:
$$
\beta_ {\rho_ {1}, \rho_ {2}} = (\mathcal {F} (\Theta) _ {\rho_ {1}} \otimes \mathcal {F} (\Theta) _ {\rho_ {2}}) ^ {\dagger} C _ {\rho_ {1} \rho_ {2}} \left[ \bigoplus_ {\rho \in \rho_ {1} \otimes \rho_ {2}} \mathcal {F} (\Theta) _ {\rho} ^ {n _ {\rho , \rho_ {2}}} \right] C _ {\rho_ {1} \rho_ {2}} ^ {\dagger},
$$
to get a novel equation:
$$
\bigoplus_ {\rho \in \mathcal {R}} \mathcal {F} (\Theta) _ {\rho} ^ {n _ {\rho , \rho_ {2}}} = C _ {\rho_ {1} \rho_ {2}} ^ {- 1} \left(\mathcal {F} (\Theta) _ {\rho_ {1}} \otimes \mathcal {F} (\Theta) _ {\rho_ {2}}\right) ^ {- \dagger} \beta_ {\rho_ {1}, \rho_ {2}} C _ {\rho_ {1} \rho_ {2}} ^ {- \dagger}, \tag {20}
$$
where everything on the right-hand side is known. Thus, every Fourier coefficient  $\mathcal{F}(\Theta)_{\rho}$  that appears in the decomposition of  $\rho_{1} \otimes \rho_{1}$  into irreducible irreps can be recovered, by reading off the elements of the block diagonal matrix. We get the  $\mathcal{F}(\Theta)_{\rho}$  for which  $n_{\rho, \rho_2} \neq 0$ . We iterate this procedure to recover more Fourier coefficients.
Number of bispectral coefficients.
Now, we show that our procedure can indeed recover all of the Fourier coefficients of the signal  $\Theta$ . Additionally, we show that it only requires a limited number  $L$  of bispectral coefficients, where  $L$  depends on the group  $G$ . Specifically, it depends on the Kronecker product table of  $G$ , which is the  $|\mathcal{R}| \times |\mathcal{R}|$  table of the decomposition of the tensor product of two irreducible representations into a direct sum of elementary irreps. In this table, the element at row  $i$  and column  $j$  lists all of the multiplicity of the irreps that appear in the decomposition of  $\rho_i \otimes \rho_j$ .
The Kronecker table, i.e., any multiplicity  $m_{k} = n_{\rho_{k}}$  of  $\rho_{k}$  in the decomposition  $\tilde{\rho} = \rho_{i}\otimes \rho_{j}$  can be computed with a procedure inspired by [9] and described below.
The procedure relies on the character  $\chi_{\tilde{\rho}}(g) = \mathrm{Tr}(\tilde{\rho}(g))$  of the representation  $\tilde{\rho}$  to be decomposed. From group theory, we know that the characters of irreps  $\rho_i, \rho_j$  are orthogonal, in the following sense:
$$
\left\langle \chi_ {\rho_ {i}}, \chi_ {\rho_ {j}} \right\rangle \equiv \frac {1}{| G |} \sum_ {h \in G} \chi_ {\rho_ {i}} (h) \chi_ {\rho_ {j}} (h) = \delta_ {i j}. \tag {21}
$$
Thus, we can obtain the multiplicity of irrep  $\rho_{k}$  in  $\tilde{\rho}$  by computing the inner product with the  $k$ -th character:
$$
\left\langle \chi_ {\tilde {\rho}}, \chi_ {\rho_ {k}} \right\rangle = \left\langle \chi_ {\oplus l} m _ {l} \rho_ {l}, \chi_ {\rho_ {k}} \right\rangle = \left\langle \sum_ {l} m _ {l} \chi_ {\rho_ {l}}, \chi_ {\rho_ {k}} \right\rangle = \sum_ {l} m _ {l} \left\langle \chi_ {\rho_ {l}}, \chi_ {\rho_ {k}} \right\rangle = m _ {k},
$$
⊗	A1	A2	B1	B2	E
A1	(1,0,0,0,0)	(0,1,0,0,0)	(0,0,1,0,0)	(0,0,0,1,0)	(0,0,0,0,1)
A2	(0,1,0,0,0)	(1,0,0,0,0)	(0,0,0,1,0)	(0,0,1,0,0)	(0,0,0,0,1)
B1	(0,0,1,0,0)	(0,0,0,1,0)	(1,0,0,0,0)	(0,1,0,0,0)	(0,0,0,0,1)
B2	(0,0,0,1,0)	(0,0,1,0,0)	(0,1,0,0,0)	(1,0,0,0,0)	(0,0,0,0,1)
E	(0,0,0,0,1)	(0,0,0,0,1)	(0,0,0,0,1)	(0,0,0,0,1)	(1,1,1,1,0)
Table 2: Kronecker table for the dihedral group  ${D}_{4}$  which has 5 irreps called  ${A}_{1},{A}_{2},{B}_{1},{B}_{2}$  and  $E$  .
using the fact that the trace of a direct sum equals the sum of the traces (i.e.  $\chi_{\rho \oplus \rho^{\prime}} = \chi_{\rho} + \chi_{\rho^{\prime}}$ ). Thus, we can determine the Kronecker product table of interest. For example, the Kronecker product table for the dihedral group  $D_4$  is shown in Table 2.
The Kronecker product table shows us how many bispectral coefficients we need to complete our algorithmic procedure. Our procedure essentially uses a breadth-first search algorithm on the space of irreducible representations, starting with  $\rho_{1}$  and using the tensor product with  $\rho_{1}$  as the mechanism to explore the space. Whether this procedure succeeds in including all the irreps in  $\mathcal{R}$  might on the group and its Kronecker (tensor) product table. Specifically, consider all the irreps that appear in the row corresponding to  $\rho_{1}$  in the Kronecker product table. If these irreps do not cover the set  $\mathcal{R}$  of all possible irreps, then the approach will need more than the tensor products of the form  $\rho_{1} \otimes \rho_{j}$  to get all of the Fourier coefficients of the signal  $\Theta$ .
We observe in our experiments that this procedure does indeed succeed in computing all of the Fourier coefficients of the signal  $\Theta$  for most of the groups of interest. We provide detailed examples of these computations on our github repository, for the diedral groups  $D_4$ ,  $D_{16}$  and for the chiral octahedral group  $O$ . The procedure does not succeed in the case of the full octahedral group  $O_h$  does not succeed.
For the diedral group  $D_4$ , which has  $\mathcal{R} = 5$  irreps, our approach allows us to recover a signal on  $D_4$  from only 3 bispectral coefficients, instead of  $5^2 = 25$ . For the diedral group  $D_{16}$ , which has  $\mathcal{R} = 11$  irreps, we recover the signal from 9 bispectral coefficients instead of  $11^2 = 121$ . For the octahedral group, which has  $\mathcal{R} = 5$  irreps, we use 4 bispectral coefficients instead of  $5^2 = 25$ . This represents a substantial complexity reduction. More theoretical work is needed to prove that our approach applies to a wide range of discrete groups, or to further generalize it for groups such as  $O_h$ .
G Training and Implementation Details
The code to implement all models and experiments in this paper can be found at github.com/sophiaas/gtc-invariance.