Surpassing Cosine Similarity for Multidimensional Comparisons: Dimension Insensitive Euclidean Metric (2407.08623v4)

Abstract: Advances in computational power and hardware efficiency have enabled tackling increasingly complex, high-dimensional problems. While AI achieves remarkable results, the interpretability of high-dimensional solutions remains challenging. A critical issue is the comparison of multidimensional quantities, essential in techniques like Principal Component Analysis. Metrics such as cosine similarity are often used, for example in the development of natural language processing algorithms or recommender systems. However, the interpretability of such metrics diminishes as dimensions increase. This paper analyzes the effects of dimensionality, revealing significant limitations of cosine similarity, particularly its dependency on the dimension of vectors, leading to biased and poorly interpretable outcomes. To address this, we introduce a Dimension Insensitive Euclidean Metric (DIEM) which demonstrates superior robustness and generalizability across dimensions. DIEM maintains consistent variability and eliminates the biases observed in traditional metrics, making it a reliable tool for high-dimensional comparisons. An example of the advantages of DIEM over cosine similarity is reported for a LLM application. This novel metric has the potential to replace cosine similarity, providing a more accurate and insightful method to analyze multidimensional data in fields ranging from neuromotor control to machine learning.

• The paper introduces DIEM to overcome cosine similarity’s bias in high-dimensional vector comparisons.
• It derives DIEM by normalizing Euclidean distance with expected value and variance, ensuring consistency across dimensions.
• Numerical simulations confirm DIEM’s robustness, offering improved accuracy for clustering and machine learning applications.

Dimension Insensitive Euclidean Metric (DIEM): An Analysis and Evaluation

Overview of the Paper

The paper entitled "Surpassing Cosine Similarity for Multidimensional Comparisons: Dimension Insensitive Euclidean Metric (DIEM)" by Federico Tessari and Neville Hogan proposes a novel metric for comparing high-dimensional vectors. Traditional metrics like cosine similarity and Euclidean and Manhattan distances show significant limitations when applied to multidimensional data. This paper introduces the Dimension Insensitive Euclidean Metric (DIEM) as a robust alternative.

Cosine Similarity and Its Limitations

Cosine similarity is a widely used metric for comparing multidimensional vectors due to its bounded range and intuitive interpretation in the context of angular measurements. However, the paper identifies crucial drawbacks of cosine similarity, particularly its sensitivity to the dimensionality of vectors. It tends to yield biased results that converge to a specific value as dimensions increase, reducing the interpretability of similarity measures. Numerical simulations demonstrated that randomly generated vectors rapidly converged to cosine similarity values that suggest a misleadingly high degree of similarity regardless of actual vector relationships.

Euclidean Distance Analysis

Euclidean distance has also been employed for multidimensional comparisons. The paper presents detailed analyses of the Euclidean distance, showing how it scales with the number of dimensions (n). Notably, the variability of Euclidean distance remains roughly constant as dimensionality increases, which is an improvement over cosine similarity. The theoretical underpinnings of Euclidean distance are explored, demonstrating how its distribution changes and why it maintains a consistent variance across dimensions.

Introduction of the Dimension Insensitive Euclidean Metric (DIEM)

To address the limitations identified in traditional metrics, the authors propose the DIEM, derived from the Euclidean distance but normalized in such a way that it remains insensitive to the dimensionality of the vectors. DIEM detrends the Euclidean distance by subtracting its expected value and normalizing by its variance, scaled to the range of values being analyzed. This innovative metric maintains consistent variability and eliminates the dimensionality-induced bias inherent to the other metrics.

1. Mathematical Properties: Several mathematical properties of the Euclidean distance are used to derive DIEM. The expected value and variances are computed considering vectors sampled from both uniform and Gaussian distributions. Analytical derivations show that DIEM's values are independent of the dimensions of the vectors, granting it robustness and generalizability.
2. Practical Implementation: The DIEM is demonstrated through numerical simulations to show how its distribution remains stable across various dimensions. Histograms of detrended Euclidean distances for different dimensionalities confirmed that DIEM exhibits steady behavior expected of a reliable comparison metric.

Implications and Speculations

The implications of adopting DIEM in multidimensional data comparison are profound. Traditional methods like PCA, k-means clustering, and singular value decomposition rely heavily on metrics that are often biased by the dimensionality of the data. By using DIEM, researchers can achieve more accurate and interpretable comparisons, leading to more reliable conclusions.

In practical fields such as neuromotor control or machine learning, DIEM can facilitate better modeling and understanding of complex, high-dimensional data. For example, analyzing muscle synergies through DIEM could provide more precise insights into motor control strategies by eliminating the bias introduced by higher dimensions.

Future Developments

Future research should further validate the advantages of DIEM across other types of multidimensional data and domains beyond those investigated in this paper. Potential extensions include evaluating the performance of DIEM in real-time applications and integrating it with existing machine learning frameworks to enhance large-scale data analysis tasks.

Conclusion

The Dimension Insensitive Euclidean Metric (DIEM) represents a sophisticated response to the challenges posed by traditional metrics in high-dimensional comparisons. By addressing the intrinsic biases of cosine similarity and Euclidean distance, DIEM provides a robust, dimension-independent tool for researchers to analyze multidimensional data more effectively. With its potential to enhance interpretability and reliability, DIEM could significantly impact many areas of data science and machine learning. Future research efforts should focus on expanding its applications and further scrutinizing its theoretical foundations to cement its status as a foundational tool for multidimensional comparisons.