What is the maximum possible value of the cosine similarity between two vectors?

Understanding Cosine Similarity

Cosine similarity is a metric used to measure how similar two vectors are, which is calculated by the cosine of the angle between them. It ranges from -1 to 1.

Maximum Value Explanation

1. Vector Definitions: In the context of cosine similarity, let $\mathbf{A}$ and $\mathbf{B}$ be two non-zero vectors in a multi-dimensional space.

2. Formula for Cosine Similarity: The cosine similarity $\text{cosine\_similarity}$ between the two vectors can be expressed as: $\text{cosine\_similarity}(\mathbf{A}, \mathbf{B}) = \frac{\mathbf{A} \cdot \mathbf{B}}{\|\mathbf{A}\| \|\mathbf{B}\|}$ where:

   • $\mathbf{A} \cdot \mathbf{B}$ is the dot product of the vectors,
   • $\|\mathbf{A}\|$ and $\|\mathbf{B}\|$ are the magnitudes (norms) of the vectors.

3. Understanding the Dot Product: The dot product $\mathbf{A} \cdot \mathbf{B}$ reaches its maximum value when both vectors are in the same direction. This occurs when the cosine of the angle between them ($\theta$) is 1 (which means the angle is 0 degrees).

4. Conclusion: Therefore, the maximum cosine similarity occurs when: $\theta = 0 \Rightarrow \text{cosine\_similarity} = 1$

Final Answer

The maximum possible value of the cosine similarity between two vectors is 1.