EquiVLA: Integrating SO(2) Equivariance into Pretrained Vision–Language–Action Models

Statement. Suppose $\tau:G\to\mathrm{Perm}(N)$ is a group homomorphism, i.e., $\tau(gh)=\tau(g)\circ\tau(h)$ for all $g,h\in G$. Then

$$z^{\mathrm{eq}}(x)=\frac{1}{|G|}\sum_{h\in G}A(h^{-1})f_\theta(h\!\cdot\!x)$$

is exactly $G$-equivariant, $z^{\mathrm{eq}}(g\!\cdot\!x)=A(g)z^{\mathrm{eq}}(x)$, for every $g\in G$ and for any choice of $f_\theta$.

Proof

$$\begin{aligned} z^{\mathrm{eq}}(g\!\cdot\!x) &= \frac{1}{|G|}\sum_{h\in G}A(h^{-1})f_\theta(hg\!\cdot\!x) \\ &\overset{h'=hg}{=}\frac{1}{|G|}\sum_{h'}A\!\left(g(h')^{-1}\right)f_\theta(h'\!\cdot\!x) \\ &=A(g)\frac{1}{|G|}\sum_{h'}A\!\left((h')^{-1}\right)f_\theta(h'\!\cdot\!x) \\ &=A(g)z^{\mathrm{eq}}(x). \end{aligned}$$

The third equality uses $A(g(h')^{-1})=A(g)A((h')^{-1})$. This holds because $\tau$ is a homomorphism and $\rho_{\mathrm{reg}}$ is a group representation, hence $A=\tau\otimes\rho_{\mathrm{reg}}$ is also a homomorphism. $\square$

Invariant Stream

$$\begin{aligned} z^{\mathrm{inv}}(g\!\cdot\!x) &= \frac{1}{|G|}\sum_h f_\theta(hg\!\cdot\!x) \\ &=\frac{1}{|G|}\sum_{h'}f_\theta(h'\!\cdot\!x)=z^{\mathrm{inv}}(x). \end{aligned}$$

Thus $z^{\mathrm{inv}}(x)=|G|^{-1}\sum_h f_\theta(h\!\cdot\!x)$ is exactly invariant for any $f_\theta$ by the same change of variable $h'=hg$.

Definition — Representation Defect

$$D(g,h)\coloneqq\widetilde{A}(gh)-\widetilde{A}(g)\widetilde{A}(h), \qquad \Delta\coloneqq\max_{g,h\in G}\left\|D(g,h)\right\|_{\mathrm{op}}.$$

Because $\rho_{\mathrm{reg}}$ is an exact unitary representation, the defect reduces to the spatial component:

$$\left\|D(g,h)\right\|_{\mathrm{op}}=\left\|\widetilde{\tau}(gh)-\widetilde{\tau}(g)\circ\widetilde{\tau}(h)\right\|_{\mathrm{op}}.$$

Statement. Let $\widetilde{z}^{\mathrm{eq}}$ be token-level Frame Averaging with approximate spatial reassignment $\widetilde{\tau}$. For all $g\in G$,

$$\widetilde{z}^{\mathrm{eq}}(g\!\cdot\!x)=\widetilde{A}(g)\widetilde{z}^{\mathrm{eq}}(x)+\varepsilon(g,x),$$ $$\left\|\varepsilon(g,x)\right\|\leq\frac{1}{|G|}\sum_{h\in G}\left\|D(g,h^{-1})\right\|_{\mathrm{op}}\left\|f_\theta(h\!\cdot\!x)\right\|\leq\Delta B(x),$$

where $B(x)=\max_{h\in G}\|f_\theta(h\!\cdot\!x)\|$. In particular, $\Delta=0$ implies exact equivariance.

Proof

$$\widetilde{z}^{\mathrm{eq}}(g\!\cdot\!x)=\frac{1}{|G|}\sum_{h'}\widetilde{A}\!\left(g(h')^{-1}\right)f_\theta(h'\!\cdot\!x).$$

Use the defect decomposition $\widetilde{A}(g(h')^{-1})=\widetilde{A}(g)\widetilde{A}((h')^{-1})+D(g,(h')^{-1})$:

$$\begin{aligned} \widetilde{z}^{\mathrm{eq}}(g\!\cdot\!x) &= \widetilde{A}(g)\underbrace{\frac{1}{|G|}\sum_{h'}\widetilde{A}\!\left((h')^{-1}\right)f_\theta(h'\!\cdot\!x)}_{=\,\widetilde{z}^{\mathrm{eq}}(x)} \\ &\quad+\underbrace{\frac{1}{|G|}\sum_{h'}D\!\left(g,(h')^{-1}\right)f_\theta(h'\!\cdot\!x)}_{=\,\varepsilon(g,x)}. \end{aligned}$$

The first inequality follows from the triangle inequality and the definition of the operator norm. The second uses $\|D(g,h^{-1})\|_{\mathrm{op}}\leq\Delta$ and $\|f_\theta(h\!\cdot\!x)\|\leq B(x)$. $\square$

Definition — Maximum Spatial Displacement

$$\delta_h(i)=\left\|p_{\widetilde{\tau}(h)(i)}-h\!\cdot\!p_i\right\|_2, \qquad \delta_{\max}=\max_{h\in G,\,i\in[N]}\delta_h(i).$$

Here $p_i$ is a patch center on an $n\times n$ grid with spacing $p$. For interior patches whose rotated positions remain inside the grid's convex hull, $\delta_h(i)\leq p\sqrt{2}/2$; boundary patches can exceed this value.

Proposition

• (a) Exact case. If $\delta_h(i)=0$ for every $h$ and $i$, then $\Delta=0$. On a square grid this holds when $u$ divides $4$, because $90^\circ$ rotations map patch centers exactly to patch centers.
• **(b) $C_4$ subgroup.** In $C_8$, the subgroup $\{0^\circ,90^\circ,180^\circ,270^\circ\}$ has zero displacement. The defect comes only from the coset $\{45^\circ,135^\circ,225^\circ,315^\circ\}$.
• (c) Coset structure. For $g\in C_4\leq C_8$, $D(g,h)=0$ whenever $h\in C_4$. Thus at most four terms in the approximate bound are non-zero.
• (d) Permutation-norm bound. Universally, $\|D(g,h)\|_{\mathrm{op}}\leq2$ and $\|D(g,h)\|_F=\sqrt{2m(g,h)}$, where $m(g,h)$ counts positions at which composed nearest-neighbor reassignments disagree with reassignment under the composed rotation.
• (e) Displacement–defect connection. A disagreement can occur only when the accumulated displacement is at least one grid spacing.

$$\left\|\varepsilon(g,x)\right\|\leq\frac{|G\setminus C_4|}{|G|}\Delta B(x)=\frac{1}{2}\Delta B(x),\qquad g\in C_4.$$ $$m(g,h)=\left|\left\{i:[\widetilde{\tau}(g)\circ\widetilde{\tau}(h)](i)\neq\widetilde{\tau}(gh)(i)\right\}\right|.$$ $$\delta_h(i)+\delta_g\!\left(\widetilde{\tau}(h)(i)\right)+\delta_{gh}(i)\geq p.$$

Consequently, $m(g,h)$ is at most the number of positions satisfying the displacement inequality above.

Proof

(a) If every displacement vanishes, each rotation maps every patch center exactly to another. The unique tracking permutation therefore satisfies $\widetilde{\tau}(gh)=\widetilde{\tau}(g)\circ\widetilde{\tau}(h)$, so $\Delta=0$. On an integer square grid, $(x,y)\mapsto(-y,x)$ is a bijection, hence all multiples of $90^\circ$ are exact. A $45^\circ$ rotation maps $(x,y)$ to $((x-y)/\sqrt2,(x+y)/\sqrt2)$ and generally leaves the integer lattice.

(b) Apply part (a) to the $C_4$ subgroup.

(c) If $g,h\in C_4$, both spatial actions are exact and $gh\in C_4$ by closure. Hence $D(g,h)=0$. Restricting the residual sum to $h\notin C_4$ leaves at most $|G\setminus C_4|=4$ terms and gives the factor-$1/2$ bound.

(d) Both $\widetilde{\tau}(gh)$ and $\widetilde{\tau}(g)\circ\widetilde{\tau}(h)$ are $N\times N$ permutation matrices. Each disagreeing position contributes exactly two to the squared Frobenius norm, so $\|D\|_F=\sqrt{2m}$. The operator-norm bound follows from the triangle inequality and the unit operator norm of permutation matrices.

(e) The composed path sends $i$ to the grid point nearest $g\!\cdot\!p_{\widetilde{\tau}(h)(i)}$, whereas the direct path sends it to the grid point nearest $(gh)\!\cdot\!p_i$. Since $g$ is an isometry,

$$\left\|g\!\cdot\!p_{\widetilde{\tau}(h)(i)}-(gh)\!\cdot\!p_i\right\|=\left\|p_{\widetilde{\tau}(h)(i)}-h\!\cdot\!p_i\right\|=\delta_h(i).$$

By the triangle inequality, the distance between the two selected target grid points is bounded by $\delta_h(i)+\delta_g(\widetilde{\tau}(h)(i))+\delta_{gh}(i)$. Distinct grid points are separated by at least $p$, which gives the stated condition by contraposition. $\square$

Computed $C_8$ Defect Values

Unit grid spacing $p=1$. Every $C_4\times C_4$ pair has $m=0$; the $45^\circ$-offset coset is the only source of non-zero defect.

Statement. Let $\widetilde{z}^{\mathrm{eq}}$ be $G$-equivariant in the regular representation; let $s^{\mathrm{inv}},s^{\mathrm{lang}},s^{\mathrm{vis}}$ be invariant; and let $W_g$ and $W_s$ be $G$-equivariant linear maps. Then

$$z^{\mathrm{out}}_{\mathrm{eq}}=\alpha^{\mathrm{reg}}\odot W_s(s^{\mathrm{inv}}\otimes\mathbf{1}_{|G|})+(\mathbf{1}-\alpha^{\mathrm{reg}})\odot W_g(\widetilde{z}^{\mathrm{eq}})$$

is equivariant: $z^{\mathrm{out}}_{\mathrm{eq}}(g\!\cdot\!x)=\rho_{\mathrm{reg}}(g)z^{\mathrm{out}}_{\mathrm{eq}}(x)$.

Proof

The gate $\alpha=\sigma(W_{\mathrm{gate}}[s^{\mathrm{lang}};s^{\mathrm{vis}};\bar z^{\mathrm{eq}}])$ depends only on invariant inputs, so it is invariant. Its tiling $\alpha^{\mathrm{reg}}=\alpha\otimes\mathbf{1}_{|G|}$ remains invariant because $\rho_{\mathrm{reg}}(g)$ only cyclically permutes group coordinates and fixes $\mathbf{1}_{|G|}$.

Semantic branch. The tiled semantic feature $s^{\mathrm{inv}}\otimes\mathbf{1}_{|G|}$ is invariant in regular-representation space. Equivariance of $W_s$ therefore makes $W_s(s^{\mathrm{inv}}\otimes\mathbf{1}_{|G|})$ invariant.

Equivariant branch. By assumption, $W_g(\rho_{\mathrm{reg}}(g)\widetilde z^{\mathrm{eq}})=\rho_{\mathrm{reg}}(g)W_g(\widetilde z^{\mathrm{eq}})$.

Combination. Since $\alpha^{\mathrm{reg}}$ is invariant and $\rho_{\mathrm{reg}}(g)$ is a coordinate permutation, Hadamard products commute with the action:

$$\rho_{\mathrm{reg}}(g)[\alpha^{\mathrm{reg}}\odot v]=\alpha^{\mathrm{reg}}\odot\rho_{\mathrm{reg}}(g)v.$$

Applying this identity to both branches, using invariance of the semantic branch and equivariance of $W_g$, gives exactly the adapter output evaluated at $g\!\cdot\!x$. $\square$

Statement. Let $\rho_a$ be the $SO(2)$ representation on action space, and suppose EquiActor implements a $G$-equivariant velocity field:

$$v_\theta\!\left(\rho_a(g)a^k,k,\widetilde A(g)z,\rho(g)z^s\right)=\rho_a(g)v_\theta(a^k,k,z,z^s),\qquad\forall g\in G.$$

• (i) The flow-matching loss is $G$-invariant in distribution: $\mathcal L(g\!\cdot\!o,\rho_a(g)a_t)=\mathcal L(o,a_t)$ in expectation over $\epsilon$ and $k$.
• (ii) ODE integration is equivariant: $\widehat a_t(g\!\cdot\!o,\rho(g)z^s,\rho_a(g)\epsilon)=\rho_a(g)\widehat a_t(o,z^s,\epsilon)$.

Proof

(i) The noisy action $a^k=(1-k)\epsilon+ka_t$ transforms linearly as $\rho_a(g)a^k=(1-k)\rho_a(g)\epsilon+k\rho_a(g)a_t$. Because $\rho_a(g)$ is orthogonal and $\epsilon\sim\mathcal N(0,I)$, rotational invariance gives $\rho_a(g)\epsilon\sim\mathcal N(0,I)$. The target velocity becomes $\rho_a(g)(a_t-\epsilon)$, and the predicted velocity transforms identically. Orthogonality preserves the squared norm, so the loss is unchanged under the simultaneous group action.

(ii) For an ODE step $a^{k+\Delta k}=a^k+\Delta k\,v_\theta(a^k,k,\ldots)$, apply $\rho_a(g)$ to both sides and use velocity-field equivariance:

$$\rho_a(g)a^{k+\Delta k}=\rho_a(g)a^k+\Delta k\,v_\theta\!\left(\rho_a(g)a^k,k,\ldots\right).$$

Induction over all integration steps propagates equivariance from the initial noise to the final action prediction. $\square$

Assumption — Lipschitz Continuity of EquiActor

For any visual contexts $z_1,z_2$ with shared proprioceptive state and noise, EquiActor $\mathcal A$ is $L$-Lipschitz:

$$\left\|\mathcal A(z_1)-\mathcal A(z_2)\right\|\leq L\left\|z_1-z_2\right\|.$$

With bounded weights, $L$ is governed by attention and MLP layer norms and may be controlled by spectral normalization or weight decay.

Statement. For the full policy $\pi(o)=\mathcal A(\mathcal P(o))$,

$$\left\|\pi(g\!\cdot\!o)-\rho_a(g)\pi(o)\right\|\leq L\Delta B(o),\qquad B(o)=\max_{h\in G}\left\|f_\theta(h\!\cdot\!x)\right\|.$$

Proof

By Theorem 1, write $\mathcal P(g\!\cdot\!o)=\widetilde A(g)\mathcal P(o)+\varepsilon$, with $\|\varepsilon\|\leq\Delta B(o)$. Proposition 3 preserves equivariance through the Adapter, while Proposition 4 gives exact equivariance of EquiActor. Therefore

$$\begin{aligned} \pi(g\!\cdot\!o)&=\mathcal A\!\left(\widetilde A(g)\mathcal P(o)+\varepsilon\right)\\ &=\underbrace{\mathcal A\!\left(\widetilde A(g)\mathcal P(o)\right)}_{=\,\rho_a(g)\pi(o)}\\ &\quad+\left[\mathcal A\!\left(\widetilde A(g)\mathcal P(o)+\varepsilon\right)-\mathcal A\!\left(\widetilde A(g)\mathcal P(o)\right)\right]. \end{aligned}$$

The first term is exactly $\rho_a(g)\pi(o)$. The Lipschitz assumption bounds the bracketed remainder by $L\|\varepsilon\|\leq L\Delta B(o)$, proving the result. $\square$

Let $\mathcal F$ be a policy function class and define its $G$-equivariant subclass by

$$\mathcal F_G=\left\{f\in\mathcal F:f(g\!\cdot\!o)=\rho_a(g)f(o)\ \forall g\in G\right\}.$$

The empirical Rademacher complexity satisfies

$$\widehat{\mathcal R}_n(\mathcal F_G)\leq\frac{1}{|G|}\widehat{\mathcal R}_n(\mathcal F).$$

By standard generalization bounds, the equivariant class therefore requires a factor of $|G|$ fewer samples to reach the same generalization error, giving up to an $8\times$ reduction for $C_8$. For an approximately equivariant pipeline satisfying $\|\varepsilon\|\leq\epsilon_0$, the appendix states the modulated gap

$$\frac{1}{|G|}\widehat{\mathcal R}_n(\mathcal F)+\mathcal O(\epsilon_0),$$

which interpolates between the fully equivariant benefit at $\epsilon_0=0$ and the unconstrained regime as the residual grows.