Neural networks (NNs) remarkably transform high-dimensional data into compact, lower-dimensional latent spaces. While researchers traditionally focus on model outputs like classification or generation, understanding the internal representation geometry has emerged as a critical area of investigation. These internal representations offer profound insights into neural network functionality, enabling researchers to repurpose learned features for downstream tasks and compare different models’ structural properties. The exploration of these representations provides a deeper understanding of how neural networks process and encode information, revealing underlying patterns that transcend individual model architectures.
Comparing representations learned by neural models is crucial across various research domains, from representation analysis to latent space alignment. Researchers have developed multiple methodologies to measure similarity between different spaces, ranging from functional performance matching to representational space comparisons. Canonical Correlation Analysis (CCA) and its adaptations, such as Singular Vector Canonical Correlation Analysis (SVCCA) and Projection-Weighted Canonical Correlation Analysis (PWCCA), have emerged as classical statistical methods for this purpose. Centered Kernel Alignment (CKA) offers another approach to measure latent space similarities, though recent studies have highlighted its sensitivity to local shifts, indicating the need for more robust analytical techniques.
Researchers from IST Austria and Sapienza, University of Rome, have pioneered a robust approach to understanding neural network representations by shifting from sample-level relationships to modeling mappings between function spaces. The proposed method, Latent Functional Map (LFM), utilizes spectral geometry principles to provide a comprehensive framework for representational alignment. By applying functional map techniques originally developed for 3D geometry processing and graph applications, LFM offers a flexible tool for comparing and finding correspondences across distinct representational spaces. This innovative approach enables unsupervised and weakly supervised methods to transfer information between different neural network representations, presenting a significant advancement in understanding the intrinsic structures of learned latent spaces.
LFM involves three critical steps: constructing a graph representation of the latent space, encoding preserved quantities through descriptor functions, and optimizing the functional map between different representational spaces. By building a symmetric k-nearest neighbor graph, the method captures the underlying manifold geometry, allowing for a nuanced exploration of neural network representations. The technique can handle latent spaces of arbitrary dimensions and provides a flexible tool for comparing and transferring information across different neural network models.
LFM similarity measure demonstrates remarkable robustness compared to the widely used CKA method. While CKA is sensitive to local transformations that preserve linear separability, the LFM approach maintains stability across various perturbations. Experimental results reveal that the LFM similarity remains consistently high even as input spaces undergo significant changes, in contrast to CKA’s performance degradation. Visualization techniques, including t-SNE projections, highlight the method’s ability to localize distortions and maintain semantic integrity, particularly in challenging classification tasks involving complex data representations.
The research introduces Latent Functional Maps as an innovative approach to understanding and analyzing neural network representations. The method provides a comprehensive framework for comparing and aligning latent spaces across different models by applying spectral geometry principles. The approach demonstrates significant potential in addressing critical challenges in representation learning, offering a robust methodology for finding correspondences and transferring information with minimal anchor points. This innovative technique extends the functional map framework to high-dimensional spaces, presenting a versatile tool for exploring the intrinsic structures and relationships between neural network representations.
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