Closed-Form Models of Accuracy Loss due to Subsampling in SVD Collaborative Filtering

We postulate and analyze a nonlinear subsampling accuracy loss(SSAL)model based on the root mean square error(RMSE)and two SSAL models based on the mean square error(MSE),suggested by extensive preliminary simulations.The SSAL models predict accuracy loss in terms of subsampling parameters like the fraction of users dropped(FUD)and the fraction of items dropped(FID).We seek to investigate whether the models depend on the characteristics of the dataset in a constant way across datasets when using the SVD collaborative filtering(CF)algorithm.The dataset characteristics considered include various densities of the rating matrix and the numbers of users and items.Extensive simulations and rigorous regression analysis led to empirical symmetrical SSAL models in terms of FID and FUD whose coefficients depend only on the data characteristics.The SSAL models came out to be multi-linear in terms of odds ratios of dropping a user(or an item)vs.not dropping it.Moreover,one MSE deterioration model turned out to be linear in the FID and FUD odds where their interaction term has a zero coefficient.Most importantly,the models are constant in the sense that they are written in closed-form using the considered data characteristics(densities and numbers of users and items).The models are validated through extensive simulations based on 850 synthetically generated primary(pre-subsampling)matrices derived from the 25M MovieLens dataset.Nearly 460000 subsampled rating matrices were then simulated and subjected to the singular value decomposition(SVD)CF algorithm.Further validation was conducted using the 1M MovieLens and the Yahoo!Music Rating datasets.The models were constant and significant across all 3 datasets.

Optimal Dependence of Performance and Efficiency of Collaborative Filtering on Random Stratified Subsampling

Dropping fractions of users or items judiciously can reduce the computational cost of Collaborative Filtering(CF)algorithms.The effect of this subsampling on the computing time and accuracy of CF is not fully understood,and clear guidelines for selecting optimal or even appropriate subsampling levels are not available.In this paper,we present a Density-based Random Stratified Subsampling using Clustering(DRSC)algorithm in which the desired Fraction of Users Dropped(FUD)and Fraction of Items Dropped(FID)are specified,and the overall density during subsampling is maintained.Subsequently,we develop simple models of the Training Time Improvement(TTI)and the Accuracy Loss(AL)as functions of FUD and FID,based on extensive simulations of seven standard CF algorithms as applied to various primary matrices from MovieLens,Yahoo Music Rating,and Amazon Automotive data.Simulations show that both TTI and a scaled AL are bi-linear in FID and FUD for all seven methods.The TTI linear regression of a CF method appears to be same for all datasets.Extensive simulations illustrate that TTI can be estimated reliably with FUD and FID only,but AL requires considering additional dataset characteristics.The derived models are then used to optimize the levels of subsampling addressing the tradeoff between TTI and AL.A simple sub-optimal approximation was found,in which the optimal AL is proportional to the optimal Training Time Reduction Factor(TTRF)for higher values of TTRF,and the optimal subsampling levels,like optimal FID/(1-FID),are proportional to the square root of TTRF.