Datasets:

yandex
/

yambda

@@ -1,21 +1,30 @@
-## ItemKNN
-We employ cosine similarity to measure the similarity between vectors. The item vectors are num_users-dimensional vectors derived from the user-item interaction matrix. In the simplest case, we generate top-k recommendations by retrieving the nearest vectors to the user’s most recent item (which corresponds to a decay parameter $\tau \rightarrow 0$). The more general formulation is as follows:
-$score(user_i, item_j) = cos(V[i, :], U[:,j])$,
-where $U$ is num_user x num_items interaction matrix,
-$V$ is num_users x num_users matrix, $V[i, :] = \sum_{(t, k) \in A_i} \tau^{max_t(i) - t} \cdot U[:, k]^T$, where $A_i$ is set of $i$-th user's (interaction_timestamp, item_index) pairs, $max_t(i)$ is last $i$-th user's interaction timestamp, $\tau$ is the decay coefficient (per second).
-The hyperparameter “hour” defines the time period (in hours) associated with a decay factor of 0.9.
-For 5b datasets, the matrix multiplications [num_users × num_item] and [num_item × num_items] to obtain [num_user x num_user]  matrix of user embeddings exceed memory constraints. We also use cosine similarity to the mean embedding instead of pairwise similarities to avoid new dimension (x basket_size).

+# ItemKNN
+We employ cosine similarity to measure similarity between vectors. Item vectors are represented as num_users-dimensional vectors derived from the user-item interaction matrix.
+Top-k recommendations are generated by retrieving vectors closest to the user's temporal interaction pattern (with decay parameter \\(\tau \rightarrow 0\\)) controlling historical influence).
+The formulation is: \\(score(user_i, item_j) = \cos(V[i, :], U[:,j])\\),
+where:
+- \\(U\\) [num user \\(\times\\) num items]: original user-item interaction matrix
+- \\(V\\) [num users \\(\times\\) num users]: user embedding matrix, where each row is:
+  \\(V[i, :] = \sum_{(t, k) \in A_i} \tau^{\max_t(i) - t} U[:, k]^T\\)
+    - \\(A_i\\): set of \\(i\\)-th user's (interaction timestamp, item index) pairs
+    - \\(\max_t(i)\\): last \\(i\\)-th user's interaction timestamp
+    - \\(\tau\\): time decay coefficient (per second)
+The hyperparameter `hour` defines the time period (in hours) associated with a decay factor of 0.9.
+## Memory Optimization
+For 5b datasets, the matrix multiplications between [num users \\(\times\\) num item] and [num item \\(\times\\) num items] exceeds memory constraints.
+To solve this, we leverage cosine similarity to the mean embedding instead of pairwise similarities to avoid new dimension (x basket_size).