Latent Functional Maps: A Powerful Machine Learning Framework for Analyzing Neural Network Representations

Neural networks (NNs) excel at transforming high-dimensional data into compact, lower-dimensional latent spaces. While research has traditionally focused on model outputs like classification or generation, understanding the internal geometry of these representations has become a key area of study. These internal representations provide valuable insights into how neural networks function, allowing researchers to repurpose learned features for various tasks and compare the structural properties of different models. Exploring these representations deepens our understanding of how neural networks process and encode information, uncovering patterns that go beyond individual model architectures.

Comparing representations learned by neural models is essential in various research fields, such as representation analysis and latent space alignment. Several methods have been developed to measure the similarity between different spaces, from functional performance matching to representational space comparisons. Techniques like Canonical Correlation Analysis (CCA) and its variants, including Singular Vector Canonical Correlation Analysis (SVCCA) and Projection-Weighted Canonical Correlation Analysis (PWCCA), have become widely used for this purpose. Another approach, Centered Kernel Alignment (CKA), measures latent space similarities, though recent studies have pointed out its sensitivity to local shifts, highlighting the need for more robust analytical methods.

Researchers from IST Austria and Sapienza University of Rome have introduced a new approach for understanding neural network representations by shifting focus from sample-level relationships to modeling mappings between function spaces. Their method, Latent Functional Map (LFM), uses spectral geometry principles to offer a comprehensive framework for representational alignment. By applying functional map techniques originally designed for 3D geometry and graph processing, LFM provides a flexible tool for comparing and finding correspondences between different representational spaces. This innovative approach allows unsupervised and weakly supervised methods to transfer information between neural network representations, marking a significant advancement in understanding the intrinsic structure of learned latent spaces.

LFM consists of three key steps: creating a graph representation of the latent space, encoding preserved quantities with descriptor functions, and optimizing the functional map between different representational spaces. The method begins by constructing a symmetric k-nearest neighbor graph to capture the underlying manifold geometry, enabling a detailed analysis of neural network representations. This approach can accommodate latent spaces of any dimension, making it a versatile tool for comparing and transferring information across various neural network models.

The LFM similarity measure shows remarkable robustness compared to the commonly used CKA method. While CKA is sensitive to local transformations that preserve linear separability, LFM remains stable under various perturbations. Experimental results demonstrate that LFM similarity remains consistently high, even when input spaces undergo significant changes, whereas CKA’s performance degrades. Visualization techniques, such as t-SNE projections, emphasize LFM’s ability to localize distortions and preserve semantic integrity, especially in challenging classification tasks with complex data representations.

The research introduces Latent Functional Maps as a novel method for understanding and analyzing neural network representations. By applying spectral geometry principles, this approach offers a comprehensive framework for comparing and aligning latent spaces across different models. It shows significant potential in tackling key challenges in representation learning, providing a robust methodology for finding correspondences and transferring information with minimal anchor points. Extending the functional map framework to high-dimensional spaces, this technique serves as a versatile tool for exploring the intrinsic structures and relationships between neural network representations.

The Importance of Representation in Neural Networks

Internal representations in neural networks are central to understanding their success in a wide range of tasks. These representations refer to the abstract features or concepts that a network learns through its hidden layers during training, which are crucial for its ability to solve complex problems. By encoding input data into internal representations, neural networks can learn intricate patterns and hierarchies without needing manual feature engineering. For instance, in convolutional neural networks (CNNs) used for image recognition, early layers might learn to detect edges, while deeper layers combine those edges into complex object parts or entire scenes​.

The significance of these representations has gained more attention as AI research has sought to understand how models generalize from their training data to new, unseen examples. Models that learn high-quality representations are more likely to perform well on tasks in real-world applications. This focus on learning efficient internal representations has helped explain the impressive performance of deep learning models, especially in complex areas like natural language processing and computer vision.

Moreover, the exploration of these representations has led to the idea of "repurposing" learned features. Researchers are now comparing representations across different models and tasks, allowing insights into how transferable or useful these features are in new contexts. This could lead to breakthroughs in the efficiency of machine learning systems and their ability to solve new problems with fewer adjustments. It also helps in model interpretability, where techniques like dimensionality reduction or visualization of hidden layer activations allow researchers to probe the learned representations and understand how decisions are made​.

The Traditional Approaches to Representation Analysis

Several methods are widely used to measure the similarity between neural network representations, with three notable techniques being Canonical Correlation Analysis (CCA), Singular Vector Canonical Correlation Analysis (SVCCA), and Centered Kernel Alignment (CKA).

• Canonical Correlation Analysis (CCA): This method focuses on maximizing the correlation between two sets of variables by finding linear combinations of the original features that are most highly correlated. CCA is robust and widely used to compare the representations of neural networks. Its key strength lies in its invariance to affine transformations, making it useful for comparing representations even when models differ in architecture​.

  
  

• Singular Vector Canonical Correlation Analysis (SVCCA): SVCCA enhances CCA by incorporating Singular Value Decomposition (SVD). This method allows for a more nuanced analysis of the relationships between neural network layers. By applying SVD, SVCCA identifies a subspace that captures the most relevant features of the network’s output. This is especially valuable when working with large networks, where certain dimensions contribute disproportionately to the network’s performance. SVCCA has been shown to offer insights into model compression and interpretability, as it reduces dimensionality without significant loss of accuracy​.

  
  

• Centered Kernel Alignment (CKA): CKA is a kernel-based method that aligns feature representations using a kernel function. The method measures the alignment between two feature sets through their inner products, normalized to ensure invariance to isotropic scaling. CKA is particularly useful for comparing deep network representations, offering flexibility in the choice of kernel (e.g., linear or RBF). CKA has been shown to be highly effective in evaluating similarity, as it is invariant to isotropic scaling and offers an improved alignment measure for more complex neural networks​.

  
  

Each of these methods has its strengths, with CCA being simpler but less powerful in capturing complex relationships, SVCCA providing detailed insights into the dynamics of neural networks, and CKA offering flexibility with kernel choices for more diverse network comparisons.

When discussing the limitations of traditional methods like Canonical Correlation Analysis (CCA) and Centered Kernel Alignment (CKA) for comparing neural network representations, several issues arise, especially regarding their sensitivity to certain transformations and the robustness of the metrics used.

One key limitation of CKA, for instance, is its sensitivity to local transformations. While it excels in distinguishing between representations from different architectures or initialization, it struggles in cases where there are subtle differences due to factors like training seeds, layer depth, or slight changes in data distribution. This makes CKA less reliable in certain contexts where robustness across different model variations is critical.

Additionally, while CKA is praised for its specificity (i.e., it performs well when comparing similar network layers), it has been shown to falter in sensitivity tests, particularly when comparing representations with significant differences in performance on out-of-distribution data or tasks. This indicates that CKA might miss crucial distinctions that more advanced or specialized metrics can identify, which is an important consideration when evaluating the functional behavior of models across various tasks.

In contrast, while methods like CCA may handle these transformations better in some scenarios, they are also not without flaws. CCA assumes that transformations between representations are linear, which might not always hold true in complex neural networks.

Ultimately, these limitations suggest that a more comprehensive, multi-metric approach is needed to truly capture the differences and similarities between neural network representations, beyond the constraints of any single metric like CKA or CCA.

Introducing Latent Functional Maps (LFM)

Latent Functional Maps (LFM) represent a groundbreaking approach in analyzing and aligning the internal representations of neural networks by leveraging principles from spectral geometry. This framework provides a robust methodology for comparing and transferring information between different latent spaces of neural networks, making it highly valuable in representation learning.

LFM operates by mapping the latent spaces of neural networks to functional domains, facilitating a more interpretable and mathematically grounded alignment of these spaces. This approach significantly mitigates the complexity traditionally associated with aligning high-dimensional spaces. In essence, LFM makes it possible to compare the geometric structure of different representations, find correspondences between them, and even transfer learned features from one network to another in an efficient and robust manner.

The core of the LFM method involves constructing a graph representation of each latent space, where nodes represent data points in the latent space, and edges are weighted based on the similarity between these points. This graph is modeled using the k-nearest neighbor (k-NN) approach, which captures the underlying manifold geometry of the data. By using the graph Laplacian, LFM computes eigenvectors that serve as a functional basis for these latent spaces. These eigenvectors help define the relationship between the different latent spaces, providing a framework for mapping functions between them.

One of the standout features of LFM is its robustness to perturbations, such as those caused by changes in the input space. Traditional methods like Canonical Correlation Analysis (CCA) are often sensitive to such transformations, leading to less stable results. In contrast, LFM remains stable even when there are significant changes in the spaces, making it a more reliable tool for analyzing complex neural network behaviors.

Moreover, LFM enables various applications, such as unsupervised or weakly supervised learning tasks. For instance, it can be used for transfer learning, where representations from one neural model are transferred to another, or for finding correspondences between different models. This versatility has made LFM an essential tool in research fields focused on understanding the inner workings of neural networks and advancing representation learning techniques.

The Latent Functional Map (LFM) method is a powerful framework used to establish mappings between two different latent spaces, such as in neural networks or 3D shape matching. The method follows three key steps:

1. Graph Representation:

The first step involves constructing a k-nearest neighbor (k-NN) graph that captures the geometry of the latent spaces. This graph allows us to understand the underlying structure of the data by connecting nearby points. It helps to model the relationships between data points as edges in an undirected graph. Each node represents a sample, and the edges are weighted based on some distance metric, such as L2 or angular distance. The graph's Laplace-Beltrami operator approximates the geometry of the underlying manifold, enabling the identification of critical features, like eigenvalues and eigenvectors, which serve as the foundation for the next steps​.

2. Descriptor Functions

In the second step, the method uses descriptor functions to encode essential information. These functions are designed to preserve key characteristics of the latent spaces. They provide a means to define a functional map that relates points between the spaces. For instance, geodesic distance functions can be used to characterize the spatial relationships between points in the graph. By solving a set of optimization equations, the descriptor functions are refined to ensure that they map between the spaces accurately, minimizing distortion and maintaining important structural properties​.

3. Optimization

The final step involves the optimization of the functional map. Given the descriptors, an optimization problem is set up to find the best possible correspondence between the latent spaces. The optimization aims to minimize discrepancies in the mapping while ensuring that the relations between data points are preserved. This step typically includes regularization techniques to prevent overfitting and to preserve the integrity of the map. The result is a refined functional map that captures the transformation between the two spaces, allowing for accurate transfer of information​.

These three steps together enable Latent Functional Maps to be an efficient tool for applications like shape matching, neural network analysis, and transferring information between different representational spaces.

Advantages of LFM Over Traditional Methods

Latent Functional Maps (LFM) outperform the Centered Kernel Alignment (CKA) in terms of robustness to perturbations because they maintain stable similarity measures even when the input spaces are perturbed, as long as the linear separability is preserved. This is an important characteristic in tasks where preserving the high-level structure of data, rather than being sensitive to small changes, is crucial.

In experiments, LFM consistently shows greater stability when subjected to perturbations, such as those that preserve linear separability (e.g., translations or distortions in the latent space). The CKA metric, on the other hand, tends to degrade under these conditions, reflecting its sensitivity to such transformations. This makes LFM a more reliable tool for assessing the similarity between latent representations in deep learning, where small perturbations in the data should not drastically affect the learned feature space or similarity assessments.

Additionally, LFM’s ability to localize distortion to specific areas in the perturbed space makes it interpretable, which is an advantage when trying to understand how different regions of the space respond to changes. This feature of LFM can be particularly useful for identifying how perturbations influence particular classes or features in data.

In the context of Latent Functional Maps (LFM), visualization techniques like t-SNE (t-Distributed Stochastic Neighbor Embedding) can be valuable for understanding the preservation of semantic integrity and the localization of distortions in data. These techniques help visualize the high-dimensional data in a lower-dimensional space, making it easier to observe patterns and relationships between data points.

LFM models work by mapping one latent space to another, with an optimization process that takes into account various constraints like preserving local area and minimizing distortion. t-SNE is used in this context to help visualize how well the functional map preserves the semantic meaning of the data, even when the shapes or manifolds being mapped are not perfectly isometric. By plotting the outputs of LFM in a 2D or 3D space, t-SNE can highlight areas where the map has maintained the underlying structure of the data, as well as areas where distortions may be more prominent.

Moreover, t-SNE projections can show the advantages of LFMs in managing complex transformations between datasets. The inherent structure captured by LFMs allows for better localization of distortions, meaning that even when data is transformed, the semantic meaning remains largely intact. This is particularly important in applications like neural network latent spaces, where LFM helps maintain the integrity of learned representations across transformations.

In essence, t-SNE offers a direct way to visualize the effectiveness of LFMs by showcasing how well the data's high-dimensional relationships are preserved in a simplified, interpretable format.

Applications and Potential of LFM

Latent Functional Maps (LFM) offer a versatile tool for transferring information between neural networks in unsupervised and weakly supervised settings. These maps work by establishing correspondences between different latent spaces, facilitating the transfer of knowledge even when labeled data is sparse or absent.

In an unsupervised learning context, LFMs allow different models, even with distinct structures, to align their representations by mapping their latent spaces onto a common framework. This is particularly useful in tasks like zero-shot learning or domain adaptation, where models trained on different datasets or tasks need to share knowledge without direct supervision. The ability of LFMs to handle such cross-model analysis is invaluable for improving the generalization and performance of models in real-world scenarios where labeled data is limited​.

Weakly supervised learning benefits from LFMs by enabling models to learn from partial or noisy supervision. For example, if only a few corresponding examples are labeled across different models or datasets, LFMs can interpolate and extrapolate this sparse information, creating robust mappings between the spaces. This makes them particularly powerful for tasks like image retrieval or cross-modal learning, where weak annotations are often more feasible than full annotations​.

The practical application of LFMs in these settings is evident in their robustness. Unlike traditional similarity measures like CKA (Centered Kernel Alignment), LFMs maintain stability under perturbations that preserve the linear separability of latent spaces. This characteristic is particularly valuable in unsupervised contexts where maintaining the integrity of learned representations is crucial​.

By leveraging LFMs, models can share knowledge efficiently without requiring full supervision, unlocking potential for advanced applications in domains such as multi-task learning, transfer learning, and cross-domain analysis. This makes LFMs a powerful tool for enhancing the flexibility and scalability of machine learning models, especially in scenarios with limited labeled data.

Conclusion

Latent Functional Maps (LFMs) offer several significant benefits for enhancing our understanding of neural network representations and transferring learned features across different models. One key advantage is their ability to bridge the gap between diverse latent spaces, providing a flexible and robust framework for aligning representations from different neural networks. This capability facilitates the reuse of learned features for downstream tasks, reducing the need for retraining models and thus improving efficiency, especially in fields like medical imaging, where multimodal data (such as images and text) need to be aligned for enhanced model performance.

LFMs work by modeling the internal geometry of latent spaces through spectral geometry principles, which allows them to capture the structural properties of different neural network representations. This makes them highly effective in comparing the latent spaces of different models, even when the models have varying architectures. By doing so, LFMs provide a way to understand how neural networks process information, enabling researchers to detect patterns and relationships that go beyond the output layer of the model.

Additionally, LFMs have proven to be more stable than some traditional methods, like Centered Kernel Alignment (CKA), particularly when dealing with local transformations that preserve linear separability. This stability is crucial when exploring complex and high-dimensional latent spaces where other methods might fail to capture important nuances. As a result, LFMs offer a more reliable tool for transferring information between models, even when the input spaces undergo significant changes.

In summary, LFMs enhance our ability to understand and transfer neural network representations across different models. They provide a deeper understanding of how networks encode and process information, allowing for more effective use of learned features and improving model transferability. By extending traditional functional map techniques to the high-dimensional spaces of neural networks, LFMs open new avenues for research and application in areas like representation learning and cross-model feature transfer.

The future of Latent Functional Maps (LFM) in the context of neural representation analysis holds exciting potential, particularly with the continuing advancements in deep learning and manifold learning. Researchers are focusing on improving the robustness and flexibility of LFMs for comparing representational spaces across different models. Current studies have shown that LFMs offer significant advantages over traditional similarity measures, like CKA (Centered Kernel Alignment), especially in handling perturbations that preserve the linear separability of spaces. This indicates that LFMs are more stable and can provide more interpretable insights into neural representations.

Looking ahead, one of the most promising areas of exploration is the application of LFMs to more complex neural architectures, such as transformers and multi-modal systems. The ability of LFMs to bridge representational spaces between different domains (e.g., vision and language) can significantly enhance model generalization, particularly in zero-shot tasks and cross-domain transfer learning. This is particularly important in applications where deep neural networks need to integrate diverse types of data or adapt to new tasks with minimal additional training.

Another key direction for future research is the optimization of the descriptor functions that align different latent spaces. By refining these descriptors, researchers can improve the precision with which LFMs map between domains, making them even more effective in neural network analysis. This could lead to more accurate representations of underlying data structures, enabling better model interpretability and performance in tasks such as anomaly detection, clustering, and transfer learning.

In conclusion, as the theory and application of LFMs continue to evolve, we can expect significant breakthroughs in neural representation analysis, particularly in enhancing cross-domain learning, improving model stability, and advancing our understanding of how neural networks represent and process information.