Differentiable Optimization of Similarity Scores Between Models and Brains (2407.07059v2)

Abstract: How do we know if two systems - biological or artificial - process information in a similar way? Similarity measures such as linear regression, Centered Kernel Alignment (CKA), Normalized Bures Similarity (NBS), and angular Procrustes distance, are often used to quantify this similarity. However, it is currently unclear what drives high similarity scores and even what constitutes a "good" score. Here, we introduce a novel tool to investigate these questions by differentiating through similarity measures to directly maximize the score. Surprisingly, we find that high similarity scores do not guarantee encoding task-relevant information in a manner consistent with neural data; and this is particularly acute for CKA and even some variations of cross-validated and regularized linear regression. We find no consistent threshold for a good similarity score - it depends on both the measure and the dataset. In addition, synthetic datasets optimized to maximize similarity scores initially learn the highest variance principal component of the target dataset, but some methods like angular Procrustes capture lower variance dimensions much earlier than methods like CKA. To shed light on this, we mathematically derive the sensitivity of CKA, angular Procrustes, and NBS to the variance of principal component dimensions, and explain the emphasis CKA places on high variance components. Finally, by jointly optimizing multiple similarity measures, we characterize their allowable ranges and reveal that some similarity measures are more constraining than others. While current measures offer a seemingly straightforward way to quantify the similarity between neural systems, our work underscores the need for careful interpretation. We hope the tools we developed will be used by practitioners to better understand current and future similarity measures.

• The paper demonstrates that differentiating through similarity measures uncovers biases, revealing that high scores do not always reflect task-relevant encoding.
• It shows that methods like CKA and linear regression prioritize high-variance features while angular Procrustes captures lower variance dimensions, affecting interpretation.
• The study introduces a Python package to standardize similarity measures, offering a practical tool for consistent neural and model representation comparisons.

Differentiable Optimization of Similarity Scores Between Models and Brains

In the field of neural and artificial system comparisons, accurately measuring the representational similarity is pivotal. The paper "Differentiable Optimization of Similarity Scores Between Models and Brains" explores assessing common similarity measures such as linear regression, Centered Kernel Alignment (CKA), angular Procrustes distance, and Normalized Bures Similarity (NBS). It outlines the necessity of discerning what drives high similarity scores and proposes a method to address this through differentiation.

Key Findings

1. Variability in Good Scores: The paper demonstrates that a "good" similarity score isn't universal across datasets or measures. For instance, a high score in CKA or linear regression doesn't inherently signify encoding of task-relevant information consistent with neural data.
2. Principal Component Emphasis: CKA and some linear regression variants emphasize high-variance principal components. Angular Procrustes, in contrast, captures lower variance dimensions earlier during optimization. This discovery is crucial, as it indicates that similarity scores can be misleading if they discount dimensions holding significant task-related information.
3. Differentiation Analysis: By differentiating through similarity measures, it becomes apparent that different measures prioritize various aspects of data. This differentiation helps in understanding the specific emphasis each measure places on data features.
4. Similarity Measure Interdependencies: The paper reveals that some similarity measures are not independent. A high score in angular Procrustes often implies a high CKA score, but not vice versa. This finding highlights the intricacies of metric interactions.
5. Practical Tools: The researchers have developed a Python package aimed at standardizing similarity measures, which is a significant step for the research community. This package could potentially harmonize the interpretation and application of such metrics across studies.

Theoretical Insights

Through meticulous theoretical derivations, the paper illustrates the sensitivity of CKA and NBS to variances in data dimensions. CKA's quadratic dependence on high variance components and NBS's linear sensitivity are explored, elucidating why certain measures might overlook critical lower variance features.

Implications and Future Directions

The findings underscore the importance of carefully selecting and interpreting similarity measures when applying them to neural data. The research suggests cautious reliance on measures like CKA and linear regression without considering their limitations concerning principal component emphasis.

The paper's methodological contributions can guide future developments in similarity measures, particularly in formulating metrics that provide more comprehensive assessments of model-neural data alignments.

Furthermore, the exploration of joint optimization of similarity scores to determine independence or coupling between metrics can inform future studies in determining the robustness and reliability of these measures.

Conclusion

This work lays a foundational understanding of how similarity scores function across various dimensions of data. It offers a comprehensive examination of the pitfalls and potentials in using these scores as indicators of representational similarity. Moving forward, this research can inspire further scrutiny into the development of metrics that encapsulate the multifaceted nature of data, fostering advancements in both theoretical and practical domains within the AI research community.