🌴 Brian

Experiment Outline

Question: How does reducing the precision of vector components to various extents (half, third, quarter, fifth) using different methods (toFixed, Math.round) affect the cosine similarity between two vectors?

Hypothesis: Reducing the precision of vectors will alter the cosine similarity, with more significant reductions leading to larger differences. The method of precision reduction might not significantly impact the cosine similarity.

Experiment Design:

Data Collection: Implement JavaScript code to calculate cosine similarities in each case and run multiple iterations to average the results.

Analysis: Evaluate how different levels and methods of precision reduction impact the cosine similarity value.

Code Specification

Functions for Cosine Similarity and Vector Generation: Functions to compute the dot product, magnitude, cosine similarity, and generate random vectors with specific bit-depth and dimensions.

Functions for Precision Reduction:

Implementation Considerations:

Function for Averaging Differences: A function to calculate the average difference in cosine similarity over multiple iterations for each precision reduction level and method.

Execution of Experiment: Run the experiment with 1000 iterations for each combination of vector type, dimension, and precision reduction method.

Interpretation of Results

The results of this experiment will help understand the extent to which precision reduction affects the similarity of high-dimensional vectors. This is particularly relevant in applications like data compression or optimization in machine learning, where a balance between precision and computational efficiency is often sought. The findings indicate that while precision reduction does impact cosine similarity, the effects are relatively minor, even with significant reductions. This suggests potential flexibility in the precision of vector representations in certain applications, without substantially compromising their comparative similarity.

Checkout the full code on Github.

node embeddings/precision-reduction-impact-on-cosine-similarity.js

Results

Average differences for 16-bit vectors (384-dim): {
  precision_half_to_fixed: '0.0000007919%',
  precision_half_math_round: '0.0000007919%',
  precision_third_to_fixed: '0.0001907019%',
  precision_third_math_round: '0.0001907019%',
  precision_quarter_to_fixed: '0.0019869799%',
  precision_quarter_math_round: '0.0019869799%',
  precision_fifth_to_fixed: '0.0182798241%',
  precision_fifth_math_round: '0.0182798241%'
}
Average differences for 16-bit vectors (1536-dim): {
  precision_half_to_fixed: '0.0000009126%',
  precision_half_math_round: '0.0000009126%',
  precision_third_to_fixed: '0.0001162906%',
  precision_third_math_round: '0.0001162906%',
  precision_quarter_to_fixed: '0.0010907664%',
  precision_quarter_math_round: '0.0010907664%',
  precision_fifth_to_fixed: '0.0099245784%',
  precision_fifth_math_round: '0.0099245784%'
}
Average differences for 8-bit vectors (384-dim): {
  precision_half_to_fixed: '0.0000009423%',
  precision_half_math_round: '0.0000009423%',
  precision_third_to_fixed: '0.0003219478%',
  precision_third_math_round: '0.0003219478%',
  precision_quarter_to_fixed: '0.0038977933%',
  precision_quarter_math_round: '0.0038977933%',
  precision_fifth_to_fixed: '0.0331642817%',
  precision_fifth_math_round: '0.0331642817%'
}
Average differences for 8-bit vectors (1536-dim): {
  precision_half_to_fixed: '0.0000012704%',
  precision_half_math_round: '0.0000012704%',
  precision_third_to_fixed: '0.0001971148%',
  precision_third_math_round: '0.0001971148%',
  precision_quarter_to_fixed: '0.0021234765%',
  precision_quarter_math_round: '0.0021234765%',
  precision_fifth_to_fixed: '0.0219500987%',
  precision_fifth_math_round: '0.0219500987%'
}

The experiment results show the average difference in cosine similarity between the original and precision-reduced vectors, for different methods of precision reduction (toFixed and Math.round) and for varying degrees of precision reduction (half, third, quarter, fifth). The experiment was conducted on two types of vectors: 16-bit and 8-bit, with two different dimensions (384 and 1536).

Key Observations

Impact of Precision Reduction:

Comparison of Reduction Methods:

Effect of Vector Dimensionality:

16-bit vs. 8-bit Vectors:

Interpretation

The experiment's results suggest that reducing the precision of vectors has a measurable but minor impact on their cosine similarity. This impact becomes slightly more pronounced as the degree of precision reduction increases, but even the most significant changes are relatively small. The method of precision reduction (rounding vs. truncating) does not appear to significantly affect the results.

These findings could have practical implications in applications that utilize vector embeddings, where high-dimensional vectors are used to represent complex data. The results suggest that it's possible to reduce the precision of these vectors (for instance, for storage or computation efficiency) with only a minimal impact on their comparative similarity. However, the degree to which precision can be reduced without significantly affecting the results will depend on the specific requirements and tolerances of the application.