Recently, I worked on recommendation system in SSENSE. I have employed the collaborative filtering algorithm based on implicit feedback, which is the number of hits per item.

The methodology part is based on:

Hu, Yifan, Yehuda Koren, and Chris Volinsky. “Collaborative filtering for implicit feedback datasets.” Data Mining, 2008. ICDM’08. Eighth IEEE International Conference on. Ieee, 2008.

While the implicit feedback is the number of hits on the product/item page.

## Methodology

### Setting

u - user, v another user
i - item, j another item

$r_{ui}$ - observations, e.g. hits, time spent, $r_{ui}=0$ means missing observation

Preference:

$$P_{ui} = \begin{cases} 1, & r_{ui} > 0 \\ 0, & r_{ui} = 0 \end{cases}$$

Confidence:

$$C_{ui} = \begin{cases} 1 + \alpha \times r_{ui}, & r_{ui} > 0 \\ 1, & r_{ui} = 0 \end{cases}$$

### Goal

Find a vector $x_u \in \mathbb{R}^f$ for each user u, and a vector $y_i \in \mathbb{R}^f$ for each item i, thus $p_{ui} = x_u^T y_i$

### Math to English

The vectors strive to map users and items into a common latent factor space where they can be directly compared.

### Cost Function

$$\min_{x_*, y_*} \sum_{u,i} c_{ui}(p_{ui}-x^T_uy_i)^2 + \lambda(\sum_{u} | x_u | ^ 2 + \sum_{i} | y_i | ^2)$$

### Algorithm: alternating-least-squares (als)

alternate between recomputing user-factors and item-factors.

Step 1. recomputing all user factors $X$

$$x_u = (Y^T C^u Y + \lambda I)^{-1} Y^T C^u p(u) \\ where\ Y^T C^u Y = Y^T Y + Y^T (C^u - I)Y$$

Step 2. recomputing all item factors $Y$

$$y_i = (X^T C^i X + \lambda I)^{-1} X^T C^i p(i)$$

Step 3. Iterate over step 1 and step 2, till stablize

### Explaining Recommendation

As the preference for item i given user u is

$$\hat{p_{ui}} = y^T_i x_u = y_i^T (Y^T C^u Y + \lambda I)^{-1} Y^T C^u p(u).$$

Now, we denote $W^u = (Y^T C^u Y + \lambda I)^{-1}$ as weighting matrix associated with user u. Accordingly, the weighted similarity between items i and j from u‘s viewpoint is denoted by $s^u_{ij} = y_i^T W^u y_j$.

Hence, the predicted user i‘s preference on item u is:

$$\hat{p_{ui}} = \sum_{j:r_{uj}>0} s^u_{ij}c_{uj}$$

This means each past actions ( $r_{uj}>0$ ) receives a separate term in forming the predicted $\hat{p_{ui}}$, and thus we can isolate its unique contribution.