Last year I had the opportunity to attend this great summer school in the beautiful and lovely city of Lisbon. It was a great week together with a lot of interesting and intelligent people, all of them interested in the amazing and exciting area of machine learning and NLP. I liked it so much last year that I decided to come back this year to volunteer as an assistant in the summer school. Today was the -1 day, where we organized some of the registration stuff, welcomed some student and had some beers. Looks like it will be, again, a great time here in Lisbon
On February I visited Cambridge to attend WSDM Doctoral Consortium. It happened during the first day of the conference, in parallel to some tutorials. It was a great time, we had excellent discussions about our projects with senior researchers and fellow Ph.D. candidates. Here is a photo for the posterity.
And the program: http://www.wsdm-conference.org/2017/doctoral-consortium/
Lately, I have been working with Poisson Matrix Factorization models and
at some point a needed to work a lower bound for . After seeing some people using this lower bound without a good explanation, I decided to write this blog post. Also, this is included as an appendix to my ECML-PKDD 2017 paper about poisson factorizatiom model for recommendation.
The function is a concave function, which means that:
By induction this property can be generalized to any convex combination of ( with and ):
Now with the a random variable we can create a similar convex combination by multiplying and dividing each random variable by and apply the sum of of expectation property:
If we want a tight lower bound we should use Lagrange multipliers to choose the set of that maximize the lower-bound given that they should sum to 1.
The final formula for is exactly the same that we can find for the parameters of the the Multinomial distribution of the auxiliary variables in a Poisson model with rate parameter as sum of Gamma distributed latent variables. Also using this optimal we can show a tight bound without the auxiliary variables.
This results in:
We have a paper accepted at ECML-PKDD 2017: “Content-Based Social Recommendation with Poisson Matrix Factorization” (Eliezer de Souza da Silva, Helge Langseth and Heri Ramampiaro). This is our first full paper resulting from our research on Poisson factorization and integration of multiple sources of information in a single recommendation model. If you have interest on the paper please email me and I will be happy to discuss.
Also, I am uploading the supplement of the paper here (you can find it also on my publications page)