A Collaborative Kalman Filter for Time-Evolving Dyadic Processes

TL;DR

The paper introduces the collaborative Kalman filter (CKF), a dynamic matrix factorization model that captures time-evolving latent factors in large-scale collaborative filtering tasks using a Kalman filter framework.

Contribution

It proposes a novel CKF model that incorporates Brownian motion for latent factors, enabling dynamic updates in collaborative filtering with a variational inference approach.

Findings

Achieved state-of-the-art results on large-scale datasets like Movielens and Netflix.

Demonstrated effective modeling of stock returns over decades.

Validated the model's ability to handle time-evolving user preferences.

Abstract

We present the collaborative Kalman filter (CKF), a dynamic model for collaborative filtering and related factorization models. Using the matrix factorization approach to collaborative filtering, the CKF accounts for time evolution by modeling each low-dimensional latent embedding as a multidimensional Brownian motion. Each observation is a random variable whose distribution is parameterized by the dot product of the relevant Brownian motions at that moment in time. This is naturally interpreted as a Kalman filter with multiple interacting state space vectors. We also present a method for learning a dynamically evolving drift parameter for each location by modeling it as a geometric Brownian motion. We handle posterior intractability via a mean-field variational approximation, which also preserves tractability for downstream calculations in a manner similar to the Kalman filter. We evaluate the model on several large datasets, providing quantitative evaluation on the 10 million Movielens and 100 million Netflix datasets and qualitative evaluation on a set of 39 million stock returns divided across roughly 6,500 companies from the years 1962-2014.