In order to achieve state-of-the-art performance, modern machine learning techniques require careful data pre-processing and hyperparameter tuning.

Neural Information Processing Systems http://nips.cc/

This is a course project report with complete methodology, experiments, references and mathematical derivations. Matrix factorization [1] is a widely used technique in recommendation systems. Probabilistic Matrix Factorization (PMF) [2] extends traditional matrix factorization by incorporating probability distributions over latent factors, allowing for uncertainty quantification. However, computing the posterior distribution is intractable due to the high-dimensional integral. To address this, we employ two Bayesian inference methods: Markov Chain Monte Carlo (MCMC) [3, 4] and Variational Inference (VI) [5, 6] to approximate the posterior. We evaluate their performance on MovieLens dataset [7] and compare their convergence speed, predictive accuracy, and computational efficiency. Experimental results demonstrate that VI offers faster convergence, while MCMC provides more accurate posterior estimates.

In this paper, we exploit the observation that stochastic variational inference (SVI) is a form of annealing and present a modified SVI approach -- applicable to both large and small datasets -- that allows the amount of annealing done by SVI to be tuned. We are motivated by the fact that, in SVI, the larger the batch size the more approximately Gaussian is the intrinsic noise, but the smaller its variance. This low variance reduces the amount of annealing which is needed to escape bad local optimal solutions. We propose a simple method for achieving both goals of having larger variance noise to escape bad local optimal solutions and more data information to obtain more accurate gradient directions. The idea is to set an actual batch size, which may be the size of the data set, and a smaller effective batch size that matches the larger level of variance at this smaller batch size. The result is an approximation to the maximum entropy stochastic gradient at this variance level. We theoretically motivate our approach for the framework of conjugate exponential family models and illustrate the method empirically on the probabilistic matrix factorization collaborative filter, the Latent Dirichlet Allocation topic model, and the Gaussian mixture model.

Using data from professional bouldering competitions from 2008 to 2022, we train a logistic regression to predict climber results and measure climber skill. However, this approach is limited, as a single numeric coefficient per climber cannot adequately capture the intricacies of climbers' varying strengths and weaknesses in different boulder problems. For example, some climbers might prefer more static, technical routes while other climbers may specialize in powerful, dynamic problems. To this end, we apply Probabilistic Matrix Factorization (PMF), a framework commonly used in recommender systems, to represent the unique characteristics of climbers and problems with latent, multi-dimensional vectors. In this framework, a climber's performance on a given problem is predicted by taking the dot product of the corresponding climber vector and problem vectors. PMF effectively handles sparse datasets, such as our dataset where only a subset of climbers attempt each particular problem, by extrapolating patterns from similar climbers. We contrast the empirical performance of PMF to the logistic regression approach and investigate the multivariate representations produced by PMF to gain insights into climber characteristics. Our results show that the multivariate PMF representations improve predictive performance of professional bouldering competitions by capturing both the overall strength of climbers and their specialized skill sets. We provide our code open-source at https://github.com/baronet2/boulder2vec.

Define that fact as an assumption, parameter space should be formalized.

SUMMARY This paper studies the problem of low rank matrix completion which exists in many real-world applications such as collaborative filtering for recommender systems. A previous work (ref [4]) proposed a scalable algorithm called Soft-Impute for solving a convex optimization problem involving the nuclear norm as a regularizer. Like previous work such as probabilistic matrix factorization (PMF), this paper gives the problem a probabilistic interpretation by relating the (non-probabilistic) optimization problem to a MAP estimation problem. Different (concave) penalty functions of the nuclear norm are proposed and then an EM algorithm is proposed to solve the MAP estimation problem. The algorithms proposed in this paper are more general than the Soft-Impute algorithm proposed in [4] in that the latter comes as a particular case.

Many existing approaches to collaborative filtering can neither handle very large datasets nor easily deal with users who have very few ratings. In this paper we present the Probabilistic Matrix Factorization (PMF) model which scales linearly with the number of observations and, more importantly, performs well on the large, sparse, and very imbalanced Netflix dataset. We further extend the PMF model to include an adaptive prior on the model parameters and show how the model capacity can be controlled automatically. Finally, we introduce a con- strained version of the PMF model that is based on the assumption that users who have rated similar sets of movies are likely to have similar preferences. The result- ing model is able to generalize considerably better for users with very few ratings.

Matrix factorization exploits the idea that, in complex high-dimensional data, the actual signal typically lies in lower-dimensional structures. These lower dimensional objects provide useful insight, with interpretability favored by sparse structures. Sparsity, in addition, is beneficial in terms of regularization and, thus, to avoid over-fitting. By exploiting Bayesian shrinkage priors, we devise a computationally convenient approach for high-dimensional matrix factorization. The dependence between row and column entities is modeled by inducing flexible sparse patterns within factors. The availability of external information is accounted for in such a way that structures are allowed while not imposed. Inspired by boosting algorithms, we pair the the proposed approach with a numerical strategy relying on a sequential inclusion and estimation of low-rank contributions, with data-driven stopping rule. Practical advantages of the proposed approach are demonstrated by means of a simulation study and the analysis of soccer heatmaps obtained from new generation tracking data.

Co-administration of two or more drugs simultaneously can result in adverse drug reactions. Identifying drug-drug interactions (DDIs) is necessary, especially for drug development and for repurposing old drugs. DDI prediction can be viewed as a matrix completion task, for which matrix factorization (MF) appears as a suitable solution. This paper presents a novel Graph Regularized Probabilistic Matrix Factorization (GRPMF) method, which incorporates expert knowledge through a novel graph-based regularization strategy within an MF framework. An efficient and sounded optimization algorithm is proposed to solve the resulting non-convex problem in an alternating fashion. The performance of the proposed method is evaluated through the DrugBank dataset, and comparisons are provided against state-of-the-art techniques. The results demonstrate the superior performance of GRPMF when compared to its counterparts.