Interpreting Cosine Similarity via Normalized Independent Component Analysis-transformed Embeddings

แนวคิดหลัก

Cosine similarity can be interpreted as the sum of semantic similarities along the interpretable axes of normalized ICA-transformed embeddings.

บทคัดย่อ

The paper proposes a novel interpretation of cosine similarity by focusing on embeddings transformed by Independent Component Analysis (ICA). ICA aims to maximize the independence of the embedding components, resulting in interpretable axes that represent specific semantic meanings.

The key insights are:

1. Normalized ICA-transformed embeddings exhibit sparsity, enhancing the interpretability of each axis. ICA provides better interpretability than Principal Component Analysis (PCA), and normalization further improves this interpretability.

2. Cosine similarity can be decomposed into the sum of semantic similarities along the axes of the normalized ICA-transformed embeddings. The semantic similarity on each axis is defined as the component-wise product of the normalized embeddings.

3. By deriving the probability distributions governing the component values and their products, the authors propose a method to statistically select the most significant axes for interpreting the similarity between two words.

The experiments demonstrate the effectiveness of this approach through numerical examples and thorough quantitative evaluations. The authors show that ICA-transformed embeddings can represent cosine similarity with fewer dimensions compared to PCA-transformed embeddings, due to the sparsity of the component-wise products.

สถิติ

The cosine similarity between ultraviolet and light is 0.485.
The semantic similarity on the [spectrum] axis between ultraviolet and light is 0.296.
The p-value for the [spectrum] axis of ultraviolet is 9.97 × 10^-21, and the Bonferroni-corrected p-value is 2.99 × 10^-18.
The inverse of the observed variance of the component values is approximately equal to the embedding dimension d = 300.
The inverse of the observed variance of the component-wise products is approximately equal to the square of the embedding dimension d^2 = 90,000.

คำพูด

"The sum of the component-wise products forms the inner product, yielding an identical cosine similarity value of 0.485 for both transformations."
"The expression for the cosine similarity in (6) can be rewritten as in (1) with the definition (7). Thus, the cosine similarity can be interpreted as the sum of the semantic similarities over all axes."
"By deriving the probability distributions that govern each component and the product of components, we propose a method for selecting statistically significant axes."

ข้อมูลเชิงลึกที่สำคัญจาก

Revisiting Cosine Similarity via Normalized ICA-transformed Embeddings

by Hiroaki Yama... ที่ arxiv.org 09-19-2024

https://arxiv.org/pdf/2406.10984.pdf

สอบถามเพิ่มเติม

How can the proposed interpretation of cosine similarity be extended to other similarity measures beyond cosine, such as Euclidean distance or Jaccard similarity?

The proposed interpretation of cosine similarity as the sum of semantic similarities across axes can be extended to other similarity measures by adapting the underlying mathematical frameworks that define these measures. For instance, in the case of Euclidean distance, which measures the straight-line distance between two points in a multi-dimensional space, one could interpret the distance in terms of the contributions of individual axes. Specifically, the Euclidean distance between two normalized ICA-transformed embeddings can be expressed as:
[
d(w_i, w_j) = \sqrt{\sum_{\ell=1}^{d} (s_{\ell}(w_i) - s_{\ell}(w_j))^2}
]
This formulation allows for the decomposition of the distance into contributions from each axis, similar to how cosine similarity was decomposed. By analyzing the squared differences along each axis, one can identify which axes contribute most significantly to the overall distance, thereby providing an interpretable framework for understanding the semantic differences between embeddings.
For Jaccard similarity, which is defined as the size of the intersection divided by the size of the union of two sets, the interpretation can be approached through the lens of shared features. If we consider the embeddings as sets of features (or non-zero components), the Jaccard similarity can be expressed as:
[
J(w_i, w_j) = \frac{|A(w_i) \cap A(w_j)|}{|A(w_i) \cup A(w_j)|}
]
Here, (A(w_i)) and (A(w_j)) represent the sets of features (or axes) activated by the embeddings of words (w_i) and (w_j). By analyzing the features that contribute to the intersection and union, one can derive insights into the shared semantic content and the distinctiveness of each embedding. This approach allows for a similar axis-based interpretation as seen with cosine similarity, thus broadening the applicability of the proposed framework.

What are the potential limitations or drawbacks of the statistical approach used to select significant axes, and how could it be further improved?

The statistical approach employed to select significant axes in the context of normalized ICA-transformed embeddings has several potential limitations. One major drawback is the reliance on the Bonferroni correction for multiple hypothesis testing, which, while conservative, can lead to a high rate of false negatives. This means that potentially significant axes may be overlooked due to the stringent criteria imposed by the correction, resulting in a loss of interpretability.
Additionally, the assumption that the component values follow a normal distribution may not hold true in all cases, particularly in high-dimensional spaces where the distribution of embeddings can be complex and multi-modal. This could lead to inaccurate p-value calculations and, consequently, the selection of axes that do not truly represent significant semantic features.
To improve this statistical approach, alternative methods such as the False Discovery Rate (FDR) could be employed, which allows for a more balanced trade-off between false positives and false negatives. Furthermore, incorporating bootstrapping techniques could provide a more robust estimation of the significance of axes by generating empirical distributions based on resampling, thus enhancing the reliability of the selected axes.
Lastly, integrating cross-validation methods could help in assessing the stability of the selected axes across different subsets of data, ensuring that the identified axes are not artifacts of a particular sample but rather reflect consistent semantic features across the embedding space.

How might the insights from this work on interpreting word embeddings be applied to interpreting the representations learned by other types of neural networks, such as those used for computer vision or speech recognition tasks?

The insights gained from interpreting word embeddings through the lens of normalized ICA-transformed embeddings can be effectively applied to other types of neural networks, including those used in computer vision and speech recognition. In these domains, the representations learned by neural networks can also be viewed as high-dimensional embeddings, where understanding the underlying structure and semantics is crucial for interpretability.
For computer vision, similar techniques can be employed to analyze the feature maps produced by convolutional neural networks (CNNs). By applying ICA to these feature maps, one can extract independent components that represent distinct visual features, such as edges, textures, or shapes. The proposed interpretation framework can then be utilized to assess the contributions of these features to the overall classification or detection tasks, allowing for a clearer understanding of how specific visual elements influence model predictions.
In the realm of speech recognition, the embeddings generated from audio signals can be analyzed using the same ICA-based approach. By transforming the learned representations into a space where independent components can be identified, one can interpret how different phonetic or prosodic features contribute to the recognition of spoken words or phrases. This can lead to insights into the model's decision-making process, such as identifying which acoustic features are most relevant for distinguishing between similar-sounding words.
Overall, the application of these interpretative techniques across various domains can enhance the transparency of neural networks, facilitating better understanding and trust in their outputs. By decomposing complex representations into interpretable components, researchers and practitioners can gain valuable insights into the learned features, ultimately leading to improved model performance and user confidence.