Part3.5-PERecoveryProof

Context

Classical group-equivariant neural networks enforce pointwise constraints $F(g \cdot x) = \rho(g) \cdot F(x)$, requiring that inputs related by a group element $g$ produce outputs related by the same symmetry. This formulation treats data as isolated points connected by group actions.

In our thesis, we introduce Path Equivariant Networks (PENs), which replace pointwise constraints with pathwise constraints: as data traverses continuous paths on a manifold, network outputs must transform coherently via transport maps. The central theoretical result, presented below, shows that classical group equivariance is recovered from path equivariance under a natural condition on path endpoints: establishing that classical equivariant networks are a special case of the PEN framework.

We present here the complete chain of definitions and proofs leading to this recovery theorem. All notation is defined within this document.

Outline of the Argument

The logical chain proceeds as follows:

Define path equivariance (PE). Given a path system $\mathcal{P}$ on input space $X$ and a Lie group $A$ acting on output space $Z$, a map $F: X \to Z$ is PE if traversing any path $\gamma \in \mathcal{P}$ in $X$ induces a continuous transport $a_\gamma: [0,1] \to A$ satisfying $F(\gamma(t)) = a_\gamma(t) \cdot F(\gamma(0))$. (Definition 2.5)
Introduce the endpoint condition. If two paths in $G$ share the same endpoints, their transports agree at $t=1$. This allows us to define a map $\rho: G^0 \to A$ by $\rho(g) := a_\gamma(1)$, independent of path choice. (Definitions 3.1–3.2)
Prove $\rho$ is a group homomorphism. The path $e \to g_1 g_2$ decomposes as $e \xrightarrow{\gamma_2} g_2 \xrightarrow{\gamma_1 \cdot g_2} g_1 g_2$, and the transport along this concatenation factors as $\rho(g_1) \cdot \rho(g_2)$. (Proposition 4.1)
Recover classical equivariance. For any $g \in G^0$ and $x \in X$, choose a path $\gamma$ from $e$ to $g$, evaluate PE at $t = 1$, and apply the endpoint condition to get $F(g \cdot x) = \rho(g) \cdot F(x)$. (Theorem 4.2)

The key insight: classical equivariance relates isolated pairs of points $(x, g \cdot x)$. Path equivariance relates continuous families of points along paths. The endpoint condition collapses path-dependent transport to a point-dependent map $\rho$, recovering the classical setting. Geometrically, the endpoint condition is equivalent to trivial holonomy (a flat connection), and relaxing it yields strictly more general equivariance notions.

1. Preliminaries

Definition 1.1 (Lie Group). A Lie group $G$ is a smooth manifold that is also a group, with smooth multiplication $m: G \times G \to G$ and inversion $i: G \to G$.

Definition 1.2 (Identity Component). The identity component $G^0$ of a Lie group $G$ is the connected component containing the identity element $e$. Since $G$ is a Lie group (hence a topological manifold), $G^0$ is path-connected: for any $g \in G^0$, there exists a continuous path $\gamma: [0,1] \to G$ with $\gamma(0) = e$ and $\gamma(1) = g$.

Definition 1.3 (Group Action). An action of a Lie group $G$ on a space $X$ is a continuous map $\rho: G \times X \to X$ satisfying $e \cdot x = x$ and $(g_1 g_2) \cdot x = g_1 \cdot (g_2 \cdot x)$ for all $g_1, g_2 \in G$, $x \in X$. We write $G \curvearrowright X$.

2. Path Systems and Path Equivariance

Definition 2.1 (Path Reparametrization and Concatenation). If $\gamma: [0,1] \to X$ is a path and $\phi: [0,1] \to [0,1]$ is continuous with $\phi(0) = 0$, $\phi(1) = 1$, the reparametrized path is $\gamma \circ \phi$.

If $\gamma_1, \gamma_2$ are paths with $\gamma_1(1) = \gamma_2(0)$, their concatenation is:

\[(\gamma_1 \parallel \gamma_2)(t) = \begin{cases} \gamma_1(2t) & t \in [0, 1/2] \\ \gamma_2(2t - 1) & t \in [1/2, 1] \end{cases}\]

Definition 2.2 (Path System). A path system on a topological space $X$ is a non-empty family of continuous paths $\gamma: [0,1] \to X$ that is closed under reparametrization and concatenation.

Definition 2.3 (Group Path System). Let $G \curvearrowright X$. The group path system is:

\[\mathcal{P} = \lbrace\gamma_X: [0,1] \to X \mid \gamma_X(t) = \gamma_G(t) \cdot x \text{ for some } x \in X,\; \gamma_G: [0,1] \to G \text{ continuous},\; \gamma_G(0) = e\rbrace.\]

Proposition 2.4. The group path system $\mathcal{P}$ is a path system.

Proof. We verify closure under reparametrization and concatenation.

Reparametrization. Let $\gamma_X(t) = \gamma_G(t) \cdot x \in \mathcal{P}$ and let $\phi: [0,1] \to [0,1]$ be continuous with $\phi(0) = 0$, $\phi(1) = 1$. Define $\tilde{\gamma}_G(t) := \gamma_G(\phi(t))$. Then $\tilde{\gamma}_G$ is continuous, $\tilde{\gamma}_G(0) = \gamma_G(\phi(0)) = \gamma_G(0) = e$, and

\[(\gamma_X \circ \phi)(t) = \gamma_G(\phi(t)) \cdot x = \tilde{\gamma}_G(t) \cdot x \in \mathcal{P}.\]

Concatenation. Let $\gamma_{X,1}(t) = \gamma_{G,1}(t) \cdot x_1$ and $\gamma_{X,2}(t) = \gamma_{G,2}(t) \cdot x_2$ be in $\mathcal{P}$ with $\gamma_{X,1}(1) = \gamma_{X,2}(0)$. The matching condition gives $\gamma_{G,1}(1) \cdot x_1 = e \cdot x_2 = x_2$, so $x_2 = \gamma_{G,1}(1) \cdot x_1$. The concatenation is:

\[(\gamma_{X,1} \parallel \gamma_{X,2})(t) = \begin{cases} \gamma_{G,1}(2t) \cdot x_1 & t \in [0, 1/2] \\ \gamma_{G,2}(2t-1) \cdot \gamma_{G,1}(1) \cdot x_1 & t \in [1/2, 1] \end{cases}\]

Define the concatenated group path:

\[\tilde{\gamma}_G(t) = \begin{cases} \gamma_{G,1}(2t) & t \in [0, 1/2] \\ \gamma_{G,2}(2t-1) \cdot \gamma_{G,1}(1) & t \in [1/2, 1] \end{cases}\]

We verify: $\tilde{\gamma}G(0) = \gamma{G,1}(0) = e$; and continuity at $t = 1/2$ holds because

\[\lim_{t \to 1/2^+} \tilde{\gamma}_G(t) = \gamma_{G,2}(0) \cdot \gamma_{G,1}(1) = e \cdot \gamma_{G,1}(1) = \gamma_{G,1}(1) = \lim_{t \to 1/2^-} \tilde{\gamma}_G(t).\]

Therefore $(\gamma_{X,1} \parallel \gamma_{X,2})(t) = \tilde{\gamma}_G(t) \cdot x_1 \in \mathcal{P}$. $\square$

Definition 2.5 (Path Equivariance). Let $G \curvearrowright X$, let $A$ be a Lie group acting on a manifold $Z$, and let $\mathcal{P}$ be a path system on $X$. A continuous map $F: X \to Z$ is path equivariant with respect to $\mathcal{P}$ if for every $\gamma \in \mathcal{P}$, there exists a continuous transport $a_\gamma: [0,1] \to A$ with $a_\gamma(0) = e_A$ such that

\[F(\gamma(t)) = a_\gamma(t) \cdot F(\gamma(0)) \quad \forall t \in [0,1].\]

We further assume the transport is base-point independent: for a group path $\gamma_G: [0,1] \to G$, the transport $a_{\gamma_G}$ depends only on $\gamma_G$, not on the choice of $x \in X$.

3. The Endpoint Condition and the Endpoint Map

Definition 3.1 (Endpoint Condition). A path-equivariant map $F$ satisfies the endpoint condition if, for all group paths $\gamma_1, \gamma_2: [0,1] \to G^0$ with $\gamma_1(0) = \gamma_2(0) = e$ and $\gamma_1(1) = \gamma_2(1) = g$, the induced transports agree at $t = 1$:

\[a_{\gamma_1}(1) = a_{\gamma_2}(1).\]

Definition 3.2 (Endpoint Map). Under the endpoint condition, define $\rho: G^0 \to A$ by

\[\rho(g) := a_\gamma(1),\]

where $\gamma$ is any continuous path in $G^0$ from $e$ to $g$. Since $G^0$ is path-connected, such a path exists for every $g \in G^0$, and the endpoint condition guarantees that $\rho$ is independent of the choice of path.

4. Main Results

Proposition 4.1. The endpoint map $\rho: G^0 \to A$ is a group homomorphism.

Proof. We verify three properties.

(i) Identity. Take the constant path $\gamma(t) = e$ for all $t$. Then $F(\gamma(t) \cdot x) = F(e \cdot x) = F(x)$ for all $t$, so the transport $a_\gamma(t) = e_A$ for all $t$ satisfies the path equivariance condition. Therefore $\rho(e) = a_\gamma(1) = e_A$.

(ii) Multiplicativity. Let $g_1, g_2 \in G^0$. Choose paths $\gamma_1: [0,1] \to G^0$ from $e$ to $g_1$ and $\gamma_2: [0,1] \to G^0$ from $e$ to $g_2$, with corresponding transports $a_{\gamma_1}$ and $a_{\gamma_2}$.

We construct a path from $e$ to $g_1 g_2$ in two stages. First traverse $\gamma_2$ (going $e \to g_2$), then apply the “shifted” path $g_2 \mapsto g_1 g_2$ corresponding to $\gamma_1$. Formally, define:

\[\tilde{\gamma}(t) = \begin{cases} \gamma_2(2t) & t \in [0, 1/2] \\ \gamma_1(2t-1) \cdot g_2 & t \in [1/2, 1] \end{cases}\]

This is continuous (at $t = 1/2$: $\gamma_1(0) \cdot g_2 = e \cdot g_2 = g_2 = \gamma_2(1)$), starts at $\tilde{\gamma}(0) = e$, and ends at $\tilde{\gamma}(1) = g_1 \cdot g_2$.

Now consider the induced paths in $X$. For any $x \in X$, the path $\tilde{\gamma}(t) \cdot x$ consists of two segments:

First segment ($t \in [0, 1/2]$): the path $\gamma_2(2t) \cdot x$ starting at $x$. By path equivariance: $F(\gamma_2(2t) \cdot x) = a_{\gamma_2}(2t) \cdot F(x)$. At $t = 1/2$, this gives $F(g_2 \cdot x) = a_{\gamma_2}(1) \cdot F(x) = \rho(g_2) \cdot F(x)$.
Second segment ($t \in [1/2, 1]$): the path $\gamma_1(2t-1) \cdot (g_2 \cdot x)$ starting at $g_2 \cdot x$. By path equivariance with base-point independence: $F(\gamma_1(2t-1) \cdot (g_2 \cdot x)) = a_{\gamma_1}(2t-1) \cdot F(g_2 \cdot x)$. At $t = 1$, this gives $F(g_1 g_2 \cdot x) = a_{\gamma_1}(1) \cdot F(g_2 \cdot x) = \rho(g_1) \cdot F(g_2 \cdot x)$.

Combining both segments:

\[F(g_1 g_2 \cdot x) = \rho(g_1) \cdot F(g_2 \cdot x) = \rho(g_1) \cdot \rho(g_2) \cdot F(x).\]

On the other hand, by definition of the endpoint map applied to the path $\tilde{\gamma}$ from $e$ to $g_1 g_2$:

\[F(g_1 g_2 \cdot x) = \rho(g_1 g_2) \cdot F(x).\]

Since this holds for all $x \in X$ and all $F(x) \in Z$, we conclude $\rho(g_1 g_2) = \rho(g_1) \cdot \rho(g_2)$.

(iii) Inverses. Let $g \in G^0$ and let $\gamma: [0,1] \to G^0$ be a path from $e$ to $g$. Define the reversed group path $\bar{\gamma}(t) := \gamma(1-t) \cdot \gamma(1)^{-1}$, which satisfies $\bar{\gamma}(0) = \gamma(1) \cdot \gamma(1)^{-1} = e$ and $\bar{\gamma}(1) = \gamma(0) \cdot g^{-1} = g^{-1}$.

Consider the concatenation $\gamma \parallel \bar{\gamma}$, which is a path from $e$ to $e$ (a closed loop). The endpoint condition applied to the constant path $c(t) = e$ and the loop $\gamma \parallel \bar{\gamma}$ gives:

\[a_{\gamma \parallel \bar{\gamma}}(1) = \rho(e) = e_A.\]

By the composition of transports along the concatenation:

\[a_{\gamma \parallel \bar{\gamma}}(1) = a_{\bar{\gamma}}(1) \cdot a_\gamma(1) = \rho(g^{-1}) \cdot \rho(g).\]

Therefore $\rho(g^{-1}) \cdot \rho(g) = e_A$, which gives $\rho(g^{-1}) = \rho(g)^{-1}$.

This completes the proof that $\rho$ is a group homomorphism. $\square$

Theorem 4.2 (Recovery of Classical Group Equivariance). Let $G \curvearrowright X$ and $A \curvearrowright Z$. Let $F: X \to Z$ be path equivariant with respect to the group path system $\mathcal{P}$ with base-point independent transport, and assume the endpoint condition holds. Then:

\[F(g \cdot x) = \rho(g) \cdot F(x) \quad \forall g \in G^0,\; x \in X,\]

where $\rho: G^0 \to A$ is the endpoint map, which is a continuous group homomorphism by Proposition 4.1. This is the classical group equivariance law on $G^0$.

Proof. Fix $g \in G^0$ and $x \in X$. Since $G^0$ is path-connected, there exists a continuous path $\gamma: [0,1] \to G^0$ with $\gamma(0) = e$ and $\gamma(1) = g$.

Consider the group path in $X$:

\[\gamma_X(t) := \gamma(t) \cdot x.\]

This satisfies $\gamma_X(0) = e \cdot x = x$ and $\gamma_X(1) = g \cdot x$, and belongs to the group path system $\mathcal{P}$.

Since $F$ is path equivariant, there exists a continuous transport $a_\gamma: [0,1] \to A$ with $a_\gamma(0) = e_A$ such that:

\[F(\gamma_X(t)) = a_\gamma(t) \cdot F(\gamma_X(0)) = a_\gamma(t) \cdot F(x) \quad \forall t \in [0,1].\]

Evaluating at $t = 1$:

\[F(g \cdot x) = F(\gamma_X(1)) = a_\gamma(1) \cdot F(x).\]

By the endpoint condition and the definition of $\rho$ (Definition 3.2):

\[a_\gamma(1) = \rho(g).\]

Therefore:

\[F(g \cdot x) = \rho(g) \cdot F(x).\]

Since $g \in G^0$ and $x \in X$ were arbitrary, this establishes the classical equivariance law on the identity component $G^0$. $\square$

5. Discussion

The recovery theorem establishes a precise hierarchy of equivariance conditions:

Path equivariance (Definition 2.5) is the most general: the transport $a_\gamma$ may depend on the entire path $\gamma$, not just its endpoints.
Classical group equivariance $F(g \cdot x) = \rho(g) \cdot F(x)$ is recovered when the endpoint condition holds: the transport depends only on the endpoint $g = \gamma(1)$, not on the path taken.

The endpoint condition has a natural geometric interpretation. Given two paths $\gamma_1, \gamma_2$ from $e$ to $g$, the concatenation $\gamma_1 \parallel \gamma_2^{-1}$ is a closed loop at $e$. The endpoint condition requires $a_{\gamma_1}(1) = a_{\gamma_2}(1)$, i.e., the transport around any closed loop is trivial. In the language of differential geometry, this corresponds to trivial holonomy, or equivalently, a flat connection on the associated principal bundle.

This perspective suggests an intermediate notion, homotopy equivariance, where the transport depends only on the homotopy class of the path. Homotopic paths induce the same transport, but non-homotopic paths to the same endpoint may differ. When $G^0$ is simply connected, every two paths with the same endpoints are homotopic, and homotopy equivariance coincides with the endpoint condition. When $G^0$ has non-trivial fundamental group $\pi_1(G^0, e)$, the holonomy around topologically distinct loops may be non-trivial, yielding a strictly intermediate framework.

Source: Chapter 4 of “Exploring the Structure in Deep Networks: Group, Manifold and Category Theory,” M.Sc. Thesis, Aalto University, December 2025.

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