Cosine similarity is a Similarity Function that is often used in Information Retrieval

- it measures the angle between two vectors, and in case of IR - the angle between two documents

- recall the definition of the Dot Product: $\mathbf v \cdot \mathbf w = \| \mathbf v \| \cdot \| \mathbf w \| \cdot \cos \theta$
- or, by rearranging get $\cos \theta = \cfrac{\mathbf v \cdot \mathbf w}{\| \mathbf v \| \cdot \| \mathbf w \|}$
- so, let's define the cosine similarity function as $\text{cosine}(\mathbf d_1, \mathbf d_2) = \cfrac{\mathbf d_1^T \mathbf d_2}{\| \mathbf d_1 \| \cdot \| \mathbf d_2 \|}$
- cosine is usually $[-1, 1]$, but document vectors (see Vector Space Model) are usually non-negative, so the angle between two documents can never be greater than 90 degrees, and for document vectors $\text{cosine}(\mathbf d_1, \mathbf d_2) \in [0, 1]$
- min cosine is 0 (max angle: the documents are orthogonal)
- max cosine is 1 (min angle: the documents are the same)

If documents have unit length, then cosine similarity is the same as Dot Product

- $\text{cosine}(\mathbf d_1, \mathbf d_2) = \cfrac{\mathbf d_1^T \mathbf d_2}{\| \mathbf d_1 \| \cdot \| \mathbf d_2 \|} = \mathbf d_1^T \mathbf d_2$
- thus we can "unit-normalize" document vectors $\mathbf d' = \cfrac{\mathbf d}{\| \mathbf d \|}$ and then compute dot product on them and get cosine
- this "unit-length normalization" is often called "cosine normalization" in IR

- for documents $\text{cosine}(\mathbf d_1, \mathbf d_2) \in [0, 1]$
- it is max when two documents are the same
- how to define a distance? distance function should become larger as elements become less similar
- since maximal value of cosine is 1, we can define
**cosine distance**as - $d_c(\mathbf d_1, \mathbf d_2) = 1 - \text{cosine}(\mathbf d_1, \mathbf d_2) = 1 - \cfrac{\mathbf d_1^T \mathbf d_2}{\| \mathbf d_1 \| \cdot \| \mathbf d_2 \|}$

Let's check if cosine distance is a proper metric, i.e. it satisfies all the requirements

- Let $D$ be the document space and $\mathbf d_1, \mathbf d_2 \in D$
- $d_c(\mathbf d_1, \mathbf d_2) \geqslant 0$: checks - 0 is minimum
- $d_c(\mathbf d_1, \mathbf d_1) = 0$ checks - $1 - \cos 0 = 0$
- $d_c(\mathbf d_1, \mathbf d_2) = d_c(\mathbf d_2, \mathbf d_1)$: checks - angle is the same

What about the triangle inequality?

- under certain conditions is doesn't hold (Korenius2007) - so it's not a proper metric

Euclidean distance $\| \mathbf d_1 - \mathbf d_2 \| = \sqrt{(\mathbf d_1 - \mathbf d_2)^T (\mathbf d_1 - \mathbf d_2)}$

- ED is a proper metric

There's a connection between Cosine Distance end Euclidean Distance

- consider two unit-normalized vectors $\mathbf x_1 = \mathbf d_1 / \| \mathbf d_1 \|$ and $\mathbf x_2 = \mathbf d_1 / \| \mathbf d_1 \|$
- $\| \mathbf x_1 - \mathbf x_2 \|^2 = (\mathbf x_1 - \mathbf x_2)^T (\mathbf x_1 - \mathbf x_2) = \mathbf x_1^T \mathbf x_1 - 2 \, \mathbf x_1^T \mathbf x_2 + \mathbf x_2^T \mathbf x_2 = \| \mathbf x_1 \|^2 - 2 \, \mathbf x_1^T \mathbf x_2 + \| \mathbf x_2 \|^2 = 2 - 2 \, \mathbf x_1^T \mathbf x_2$
- if vectors are unit-normalized, cosine = dot product, so we have
- $\| \mathbf x_1 - \mathbf x_2 \|^2 = 2 \, (1 - \mathbf x_1^T \mathbf x_2) = 2 \, \big(1 - \text{cosine}(x_1, x_2)\big) = 2 \, d_c(x_1, x_2)$

It can also make some sense visually:

- recall the Cosine Theorem: $a^2 = b^2 + c^2 - 2 bc \cos \theta$
- $b = c = 1$, so we have $a^2 = 2 \, (1 - \cos \theta)$

Thus we can use Euclidean distance and interpret it as Cosine distance

For PCA we usually do mean-correction

- but mean-correction affects the angle between documents
- can show that visually: the angle between two vectors are not translation-resistant
- i.e. if we have two vectors $\mathbf x_1$ and $\mathbf x_2$ and with $\theta$ being the angle between them, when we translate the space by $m$, we get a new angle $\theta'$ s.t. $\theta \ne \theta'$
- also note that when we translate, some elements may become negative!

- Korenius, Tuomo, Jorma Laurikkala, and Martti Juhola. "On principal component analysis, cosine and Euclidean measures in information retrieval." 2007. [1]