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st: latent factor model with stochastic gradient method/alternating least squares

From   "Dimitriy V. Masterov" <[email protected]>
To   Statalist <[email protected]>
Subject   st: latent factor model with stochastic gradient method/alternating least squares
Date   Tue, 5 Oct 2010 17:22:45 -0400

Suppose I observe lots of consumer ratings (explicit or implicit) of
thousands of items, which may themselves be combinations of other
items. The rating matrix is pretty sparse because most consumers only
try/rate a few of the items. I would like to model how items ratings
are related to each other for the purpose of making recommendations of
new items to try. I am thinking of this as a missing rating problem.
The characteristics of the items are many and are not easily modelled
with fewer dimensions.

My approach to this problem is to map consumers and items into a joint
latent factor space of dimension f, so that consumer-item interactions
are modeled as inner products in that space. Each item i is associated
with a vector q_i in R^f, which measures the extent to which that item
possesses the latent factors. The vector p_u in R^f measures the
interest of the consumer in each of the latent factors.

I would like to model the rating for item i by consumer u as:

r_ui = mu + b_u + b_i + b_u*b_i + q_i'*p_u,

where mu is a constant which is the same for all products, b_u is a
user fixed effect, b_i is an item fixed effect, and the inner product
of q and p captures the consumer's overall interest in the item's
latent characteristics. The fixed effects are meant to capture the
idea that some items may be more popular and the fact that some users
may rate more harshly, and that these may interact. For example, a
popular item my be judged to be especially poor by a harsh critic.

For a given f, I would like to find b_i, b_u, mu, and the vectors q_i
and p_u to minimize the sum of squared residuals:

sum[(r_ui - mu - b_u - b_i - b_u*b_i - q_i'*p_i)^2] for all items and
users that are observed.

I would like to use these parameters to estimate the rankings of
products that have not been sampled by some consumers.

I believe it is possible to estimate these parameters with stochastic
gradient descent optimization or with alternating least squares. Does
anyone know if those methods are possible with Mata/Stata or if
there's a way to recast this problem in another way?

Dimitriy Masterov
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