Super-Affine Transformations

• Super-affine transformations are a generalization of affine transformations that use enriched polynomial bases to capture complex, non-affine symmetries in data.
• The framework employs null-space learning with manifold constraints to efficiently recover symmetry generators and compute invariant features.
• Empirical evaluations show that super-affine methods outperform traditional affine approaches and models like LieGAN, particularly in non-affine tasks and scalability.

Super-affine transformations generalize classical affine transformations in Euclidean spaces by permitting higher-order polynomial or more complex basis expansions in the infinitesimal generator coefficients of transformation groups. This extension allows the discovery and exploitation of continuous symmetry structures in data that fall strictly outside the affine regime, interpolating methodically between affine symmetry and the full diffeomorphism group of ℝⁿ. The super-affine framework encompasses a versatile spectrum of polynomial and other function-based transformations, enabling symmetry-based analyses, model invariance, and geometric reasoning in machine learning beyond traditional affine groups (Shaw et al., 5 Jun 2024).

1. Mathematical Structure of Super-Affine Transformation Groups

The real affine group in n dimensions, Aff(n), has the standard action $S_A(x) = A x + b$ with $A \in \mathrm{GL}(n)$ and $b \in \mathbb{R}^n$, associated infinitesimally with vector fields $X_\mathrm{aff} = (B x + c)^i \partial_{x^i}$ for $B \in \mathbb{R}^{n \times n}$ and $c \in \mathbb{R}^n$. The super-affine construction replaces the affine constraint of first-degree polynomial coefficients by an ansatz involving richer feature bases (e.g., monomials up to degree d or even non-polynomial bases), leading to vector fields

$$X = \sum_{i=1}^n \sum_\alpha w_{i\alpha} \, \phi_\alpha(x) \, \partial_{x^i}$$

where $\{\phi_\alpha(x)\}$ spans the chosen feature space. The resulting transformation group, denoted here as $G_\mathrm{super}$, is a (potentially infinite-dimensional) Lie group strictly containing $\mathrm{Aff}(n)$ as a subgroup and lying within the diffeomorphism group of $\mathbb{R}^n$. By systematically enriching the basis $\{\phi_\alpha\}$, the framework interpolates from affine actions to higher-order polynomial or other functional symmetries.

2. Optimization and Null-Space Learning Objectives

Super-affine symmetry discovery relies on the principle that a vector field $X$ generates a symmetry of a function $f : \mathbb{R}^n \to \mathbb{R}$ if and only if $X(f) \equiv 0$ everywhere (or at all sampled points). By parameterizing $X$ with unknown weights $W = [w_{i\alpha}]$ spanning the chosen basis, and representing features as $B(x)_\alpha = \phi_\alpha(x)$, this symmetry condition is transcribed for $r$ data samples into a linear system $M W \approx 0$, where $M$ accumulates $\partial f/\partial x^i \cdot B(x)$ over all samples and coordinate directions.

The concrete optimization problem is

$$\min_{W} \|M W\|_F^2 \quad \text{subject to} \quad W^\top W = I$$

which seeks an orthonormal basis for the null-space of $M$, with orthonormality imposed on the Stiefel manifold. Manifold-aware optimizers, as implemented in McTorch, enforce these constraints efficiently. Symmetry-invariant features $h_j(x)$ are subsequently constructed by forming a similar null-space problem over the Jacobians of the basis functions under the identified generators, again with Stiefel or spherical manifold optimization.

3. Algorithmic Framework

The super-affine discovery algorithm proceeds via the following steps:

1. Basis Selection: Choose scalar feature functions $\{\phi_\alpha(x)\}$ (e.g., all monomials up to degree $d$).
2. Function Estimation: Fit $f(x) \approx \hat{y}(x)$ (e.g., regression, density, or level-set models).
3. Gradient Evaluation: Compute $\nabla \hat{y}(x_i)$ for all data points $x_i$.
4. Extended Matrix Construction: For each sample and each coordinate, set $$M_{(i,i'),(i',\alpha)} = \partial \hat{y}/\partial x^{i'}(x_i) \cdot \phi_\alpha(x_i)$$, yielding an extended feature matrix $$M \in \mathbb{R}^{(n \cdot r) \times (n \cdot \#\mathrm{features})}$$.
5. Infinitesimal Generator Recovery: Solve for $W$ as in the optimization above to obtain $k$ symmetry generators.
6. Invariant Computation (Optional): Construct a secondary matrix $M_2$ from the Jacobians under each $X_j$; solve a constrained null-space problem to obtain scalar invariants $h_j(x)$.

Both generator and invariant learning steps utilize manifold optimization to preserve the orthonormality (for $W$) or norm (for $v$) constraints.

4. Specialization to the Affine Case

Super-affine methods recover the classical affine framework as a special case. If the basis $\{\phi_\alpha\}$ contains only constants and coordinate functions, i.e., $\phi_0(x)=1$, $\phi_i(x)=x^i$, then the ansatz for $X$ reduces to

$X = \sum_i (c_i + B_{ij} x^j) \partial_{x^i}$

which forms precisely the Lie algebra $\mathrm{aff}(n)$. In this reduction, the extended matrix $M$ encodes only first-degree features, and the solution space has dimension $n+n^2$, corresponding exactly to existing "Lie-algebra learning" methods such as LieGAN but within a vector field null-space framework.

5. Empirical Evaluation and Benchmarks

Comparative experiments demonstrate super-affine symmetry detection on both affine and genuinely non-affine tasks, benchmarked primarily against LieGAN. The following table summarizes salient results:

Task	LieGAN Similarity	Super-affine Similarity	LieGAN Time (s)	Super-affine Time (s)
Affine (N=200)	0.50 ± 0.28	0.99 ± 0.03	1.94 ± 0.06	1.28 ± 0.08
Affine (N=2,000)	0.998 ± 0.002	0.97 ± 0.08	15.4 ± 1.0	1.69 ± 0.11
Affine (N=20,000)	0.9999 ± 5e-5	0.97 ± 0.07	177 ± 7	3.90 ± 0.09
Super-affine (poly 2)	0.19 ± 0.11	0.992 ± 0.011	-	-

For finite $N$, the super-affine method more reliably locates the generator, achieving substantially higher similarity for small sample sizes and is significantly faster at large $N$. On super-affine (e.g., quadratic) symmetry targets, LieGAN is incapable ($0.19 \pm 0.11$) while super-affine recovers near the ground truth ($0.992 \pm 0.011$).

Additional applications include:

• Supervised Classification Invariants: On the Palmer Penguins dataset, the scalar invariant

$h(x,y) = 0.4335\, x - 0.9011\, y$

computed via super-affine symmetry aligns with model probability contours for the "Adelie" class.

• Symmetry-based Regression: For functions with $7$-fold discrete rotational symmetry, symmetry-adapted features increase $R^2$ from $0.5965$ (test) in the original coordinates to $0.9965$ (test) after symmetry-based transformation.
• Infinitesimal Isometries in Pulled-back Metrics: Recovery of the 6-D Killing algebra of a nontrivial Riemannian metric on ℝ³ is achieved to cosine similarity $\approx 1.000$.

6. Scope, Flexibility, and Limitations

The super-affine transformation framework extends symmetry discovery capabilities:

• The inclusion of polynomial or otherwise enriched feature bases allows continuous interpolation from classical affine symmetry to broader polynomial and isometric symmetries.
• Null-space learning with manifold constraints produces explicit generators and invariants.
• Reduction to affine forms recovers existing algorithms exactly as a special case.
• Capability to detect high-order ("beyond-affine") continuous symmetries empirically distinguishes it.
• Super-affine methods match or outperform LieGAN for both sample efficiency and computational cost in all tested domains, particularly excelling at capturing non-affine symmetries.

A plausible implication is that this framework defines a new algorithmic regime for learning and exploiting symmetry in data manifolds, with potential applicability across regression, density modeling, geometric learning, and invariant representation extraction (Shaw et al., 5 Jun 2024).