Technical Summary: "Too Symmetric to See: Diagnosing Over-Invariance in Finite-Symmetry Geometric Learning"

1. Problem Statement: Silent Over-Invariance

The paper addresses a specific failure mode in geometric deep learning termed over-invariance. While incorporating known symmetries into neural networks is standard practice to reduce sample complexity and ensure physical consistency, the author argues that representations can be too symmetric.

Specifically, when a target function possesses only a finite phase symmetry (e.g., invariance under a discrete root-of-unity group), but the model representation imposes a continuous phase symmetry (e.g., invariance under the full circle group $U(1)$ by using only magnitude features), the model silently discards target-relevant information.

• The Mechanism: Replacing complex coordinates with their magnitudes (or squared magnitudes) creates a representation that is invariant under a continuous torus. If the true target is only invariant under a finite subgroup of that torus, the representation collapses fibers containing distinct target values.
• The Consequence: This results in an "irreducible error" (information loss) that cannot be fixed by better optimization or more data. Crucially, because the model remains exactly invariant under the declared group actions, this failure does not trigger standard symmetry-error warnings, making it "silent."

2. Methodology and Diagnostic Framework

2.1 The Diagnostic: Irreducible Error as Conditional Variance

The author formalizes over-invariance as a diagnostic problem. For a target $T$ and representation $R$, the irreducible squared loss of any predictor restricted to $R$ is the conditional variance:
$$\inf_f \mathbb{E}[(T - f \circ R)^2] = \mathbb{E}[\text{Var}(T | R)]$$
If $\text{Var}(T | R) > 0$, the representation $R$ has discarded information relevant to $T$. The paper provides a controlled synthetic benchmark where this irreducible error can be computed in closed form, allowing for a direct comparison between the theoretical lower bound of magnitude-only models and trained models.

2.2 The Repair: Projective Character Pooling (PCP)

The paper proposes Projective Character Pooling (PCP) as a scoped, orbit-free repair for a specific class of finite-symmetry actions. PCP constructs a strictly invariant residual that retains phase information without requiring expensive orbit averaging.

• Construction: It operates on aligned blocks of coordinates. It normalizes blocks to remove projective scaling, then constructs "phasors" by raising normalized coordinates to specific powers ($d_k$) that trivialize the declared root-phase rotations.
• Mechanism: These phasors are contracted with shared slot encoders. The construction includes "charge-cancelling" global products to ensure invariance under the specific finite subgroup while preserving relative phase distinctions that magnitudes discard.
• Guarantee: The construction is exactly invariant under the declared diagonal root-phase actions and simultaneous slot permutations, and it descends to a smooth function on the product projective space.

2.3 Evaluation Protocol

The study employs a multi-layered evaluation strategy to distinguish representation-level information loss from optimization artifacts:

1. Synthetic Benchmark: A tunable target with a known analytic "moduli-only floor" to verify that magnitude-only models hit a performance ceiling determined by information loss, not optimization.
2. Molecular Control ($H_3^+$): A local electronic-structure surface with $D_3$ symmetry to test if the phenomenon extends beyond complex projective geometry.
3. Geometric Benchmarks:
   • Dwork Quintic: A one-parameter family of Calabi–Yau manifolds where the symmetry group changes with the modulus, allowing the "over-invariance penalty" to be tuned.
   • Symmetric Bicubic: A product-projective hypersurface ($CP^2 \times CP^2$) to test transferability to different ambient spaces and group structures.
4. Controls: The study includes matched-capacity baselines, "wrong-charge" controls, generic-phase controls, and exact orbit averaging (where computationally feasible) to isolate the specific contribution of the phase-aware construction.

3. Key Results

3.1 Synthetic and Molecular Controls

• Synthetic: In the controlled benchmark, magnitude-only models consistently hit the analytic irreducible error floor as the phase-content fraction ($\gamma$) increased. PCP models remained significantly below this floor, reducing MSE by up to 95% at high phase content.
• Molecular ($H_3^+$): A radial (magnitude-only) model failed to improve beyond a certain error threshold regardless of training budget (up to 9,000 updates), confirming the loss of the $D_3$-allowed angular component. In contrast, the character-aware model reached low error quickly, and exact orbit averaging only caught up at very large budgets.

3.2 Geometric Results (Dwork Quintic)

• Determinant-Ratio Flatness: On the Dwork quintic, character-aware residuals (PCP) achieved significantly lower validation flatness ($\sigma(\eta)$) than magnitude-only baselines across all tested moduli ($\psi = 0.3, 0.5, 0.8$).
• Controls: Widening the magnitude-only model, using generic projective MLPs, or training for longer did not close the gap. "Wrong-charge" controls (miscalibrated phases) failed to improve, confirming the effect relies on correct phase calibration.
• Canonicalization Failure: A canonicalization-based approach (piecewise defined) showed low local error but failed to glue smoothly across boundaries, highlighting that pointwise invariance does not guarantee global geometric validity.

3.3 Geometric Results (Bicubic Threefold)

• Transferability: PCP significantly outperformed magnitude-only and generic projective controls on the bicubic threefold.
• Budget Trade-off: A comparison with exact orbit averaging (972 elements) revealed a finite-budget trade-off: PCP achieved lower error early in training (30 updates), while exact orbit averaging surpassed it at longer budgets (200 and 1,000 updates). This suggests PCP is an efficient coordinate system for early convergence, while orbit averaging is asymptotically superior given sufficient compute.

4. Contributions and Significance

The paper makes the following specific contributions:

1. Diagnosis of Over-Invariance: It identifies and formalizes "over-invariance" as a silent failure mode where continuous-invariant representations (like magnitudes) discard information required by finite-symmetry targets.
2. Analytic Lower Bound: It provides a controlled synthetic setting with a closed-form irreducible error floor, separating representational limits from optimization limits.
3. Projective Character Pooling (PCP): It introduces a concrete, orbit-free neural construction that is exactly invariant for a declared class of finite root-phase actions while retaining necessary phase distinctions.
4. Empirical Validation: It demonstrates that this failure mode and the proposed repair are relevant in non-trivial geometric settings (Calabi–Yau metric learning), where they improve the quality of learned Kähler potentials compared to standard magnitude-only approaches.

5. Claims and Limitations (Modest Scope)

The author is explicit about the boundaries of their claims:

• Not a Universal Compiler: PCP is a "scoped construction" for diagonal root-phase actions and simultaneous permutations. It does not cover arbitrary unitary finite groups, continuous groups, or independent block permutations (e.g., the bicubic block-exchange symmetry required a separate hand-written symmetrization).
• No Universal Approximation: The paper does not claim that PCP generates the full algebra of invariants or provides a universal approximation theorem.
• Finite-Budget Results: The comparison with orbit averaging is a finite-budget observation (PCP wins early, orbit averaging wins late), not a claim of asymptotic superiority for PCP.
• Diagnostic Focus: The primary goal is to make over-invariance measurable and falsifiable. The paper does not claim to solve all geometric learning problems or replace existing numerical solvers, but rather to provide a tool to detect when a representation is "too symmetric to see" the target.

In summary, the paper argues that when a target has finite phase symmetry, using continuous-invariant magnitude features imposes an avoidable information bottleneck. PCP offers a targeted repair for this specific mismatch, validated through analytic bounds and geometric benchmarks.