Stimulus symmetries can confound representational similarity analyses

Technical Summary: Stimulus Symmetries Can Confound Representational Similarity Analyses

Problem Statement
Representational Similarity Analysis (RSA) and related metrics, such as the Representational Similarity Matrix (RSM) and Centered Kernel Alignment (CKA), are widely used to compare neural codes across biological and artificial networks. A fundamental assumption in these comparisons is that functionally equivalent representations—those that perform the same task or encode the same information—should yield similar summary statistics. While recent work has addressed intrinsic symmetries (e.g., global rotations of weights within a network layer that can be compensated by subsequent layers), there is a lack of systematic understanding regarding how stimulus symmetries (external data symmetries like rotation or translation) affect these analyses.

The core problem identified is that stimulus symmetries can generate multiple, functionally equivalent neural representations that are not related by simple orthogonal transformations (rotations) in the representation space. Consequently, these distinct configurations can produce qualitatively different RSMs. This creates a "gauge dependence" where the RSM varies based on an arbitrary, functionally irrelevant parameter (the gauge), potentially leading researchers to mistakenly conclude that two functionally equivalent codes are distinct.

Methodology
The authors employ a combination of theoretical formalization, analytical toy models, and empirical simulations on both synthetic and real-world data to demonstrate this phenomenon.

1. Theoretical Formalization: The paper formalizes the concept of data symmetries acting on a latent space $Z$ via a compact group $G$. It defines "gauge transformations" as actions of $G$ on the encoding function that leave decoding accuracy invariant. The authors prove that for an RSM to be gauge-invariant, the encoding must be an orthogonal linear representation of the symmetry group (i.e., the gauge transformation must act as a global rotation on the representation space). They demonstrate that for general nonlinear codes, this condition does not hold.
2. Toy Model Analysis: A simplified model is constructed where neurons tile a one-dimensional ring ($S^1$) with localized receptive fields (RFs). The global orientation of this tiling is defined by a gauge variable $\phi$. The authors analytically derive the RSM for this setup, showing that off-diagonal elements depend on $\phi$, leading to a variable RSM despite perfect reconstruction of the stimulus.
3. Parameter Sensitivity: The study investigates how factors such as the number of neurons, RF tuning width, and amplitude noise affect the magnitude of RSM variability. They derive closed-form expressions for the variability $\Delta$ as a function of neuron count $n$ and tuning width, showing that variability decays as $O(n^{-2})$ only in the limit of dense tiling.
4. Learning Dynamics: Using Stochastic Gradient Descent (SGD) on autoencoders with ReLU activations, the authors simulate learning scenarios. They demonstrate that SGD can lead to "collapsing" solutions where many neurons align, effectively reducing the network to a sparse tiling (e.g., 4 neurons for a circle), thereby maintaining high RSM variability. They also show that energetic regularization (L1 penalties) can induce sparsity without fixing the gauge.
5. Higher-Dimensional and Real Data: The framework is extended to higher-dimensional spherical manifolds ($S^{d-1}$), predicting structured clusters of RSM variability corresponding to the dimensionality of the gauge group. Finally, the authors train autoencoders on rotated images from the Kuzushiji-MNIST dataset and analyze pretrained vision models (ResNet-18, ConvNeXt, DINOv2) to show that latent rotational symmetries in image data induce systematic RSM drift and gauge dependence in realistic settings.

Key Contributions and Results

• Gauge Dependence of RSMs: The paper establishes that for nonlinear neural codes, functionally equivalent representations related by stimulus symmetries are generally not related by orthogonal transformations. Therefore, their RSMs differ, violating the invariance assumed by standard similarity metrics.
• Drifting Representations: The authors show that even with stable task performance (e.g., low reconstruction loss), representations can "drift" over time or across different training runs due to the arbitrary selection of a gauge. This drift manifests as a systematic change in the RSM, which correlates with the inferred gauge angle but not with reconstruction error.
• Sparsity and Variability: Contrary to the intuition that increasing the number of neurons suppresses variability, the authors find that learning-induced sparsity (via SGD noise or L1 regularization) can maintain high RSM variability even in large networks.
• Structured Variability in High Dimensions: For higher-dimensional symmetric manifolds, the RSM variability is not random noise but follows a structured pattern where off-diagonal elements cluster into groups that vary together, reflecting the underlying continuous symmetry group.
• Empirical Validation: In experiments with rotated image data, the inferred gauge angle of trained networks strongly predicts the off-diagonal elements of the RSM (Pearson's $r = 0.76$) and the similarity between different runs (CKA similarity drops as gauge difference increases, $r = -0.91$), while reconstruction loss remains uncorrelated with the gauge.

Significance and Claims
The paper claims that these findings highlight a critical limitation in comparing nonlinear neural codes using RSM-based metrics when stimulus symmetries are present. Specifically:

• Misinterpretation Risk: Researchers may erroneously identify functionally equivalent representations as distinct simply because they occupy different "gauges" in the solution space.
• Drift Mechanism: The work provides a mechanism for "representational drift" (observed in both brains and artificial networks) that is distinct from previous explanations based on internal symmetries; here, drift is driven by extrinsic stimulus symmetries and the lack of a unique preferred basis.
• Limitations of Current Metrics: Standard metrics like CKA and RSM are not invariant to data symmetries unless the network is explicitly constrained to be equivariant (which most standard networks are not).
• Future Directions: The authors suggest that to reliably compare representations in the presence of symmetries, new metrics invariant to the specific stimulus symmetries are required. They note that such methods would likely require knowledge of the stimulus distribution or the use of pullback metrics on the intrinsic data manifold, but acknowledge that without strong priors, accounting for gauge invariance remains an open challenge.

The paper concludes modestly, stating that while manifold-tiling codes served as a tractable test case, the generalization of this phenomenon to other nonlinear codes is expected, as most representations are not constrained to be orthogonally equivariant.