Combining simple elements from the literature, we define a linear model that is geared toward sparse data, in particular implicit feedback data for recommender systems. We show that its training objective has a closed-form solution, and discuss the resulting conceptual insights. Surprisingly, this simple model achieves better ranking accuracy than various state-of-the-art collaborative-filtering approaches, including deep non-linear models, on most of the publicly available data-sets used in our experiments.
Compared to @cite_15 , we dropped the constraint of non-negative weights, which we found to greatly improve ranking accuracy in our experiments (see Table and Figure ). Moreover, we did not use L1-norm regularization for computational efficiency. We also did not find sparsity to noticeably improve ranking accuracy (see Section ). The learned weight matrix @math of is dense.
@cite_15
2949962875 2214177029 2170293972 2083945855
In sparse principal component analysis we are given noisy observations of a low-rank matrix of dimension @math and seek to reconstruct it under additional sparsity assumptions. In particular, we assume here each of the principal components @math has at most @math non-zero entries. We are particularly interested in the high dimensional regime wherein @math is comparable to, or even much larger than @math . In an influential paper, johnstone2004sparse introduced a simple algorithm that estimates the support of the principal vectors @math by the largest entries in the diagonal of the empirical covariance. This method can be shown to identify the correct support with high probability if @math , and to fail with high probability if @math for two constants @math . Despite a considerable amount of work over the last ten years, no practical algorithm exists with provably better support recovery guarantees. Here we analyze a covariance thresholding algorithm that was recently proposed by KrauthgamerSPCA . On the basis of numerical simulations (for the rank-one case), these authors conjectured that covariance thresholding correctly recover the support with high probability for @math (assuming @math of the same order as @math ). We prove this conjecture, and in fact establish a more general guarantee including higher-rank as well as @math much smaller than @math . Recent lower bounds berthet2013computational, ma2015sum suggest that no polynomial time algorithm can do significantly better. The key technical component of our analysis develops new bounds on the norm of kernel random matrices, in regimes that were not considered before. Motivated by the recent developments of nonconvex penalties in sparsity modeling, we propose a nonconvex optimization model for handing the low-rank matrix recovery problem. Different from the famous robust principal component analysis (RPCA), we suggest recovering low-rank and sparse matrices via a nonconvex loss function and a nonconvex penalty. The advantage of the nonconvex approach lies in its stronger robustness. To solve the model, we devise a majorization-minimization augmented Lagrange multiplier (MM-ALM) algorithm which finds the local optimal solutions of the proposed nonconvex model. We also provide an efficient strategy to speedup MM-ALM, which makes the running time comparable with the state-of-the-art algorithm of solving RPCA. Finally, empirical results demonstrate the superiority of our nonconvex approach over RPCA in terms of matrix recovery accuracy. Non-negative matrix factorization (NMF), i.e. V ap WH where both V, W and H are non-negative has become a widely used blind source separation technique due to its part based representation. The NMF decomposition is not in general unique and a part based representation not guaranteed. However, imposing sparseness both improves the uniqueness of the decomposition and favors part based representation. Sparseness in the form of attaining as many zero elements in the solution as possible is appealing from a conceptional point of view and corresponds to minimizing reconstruction error with an L0 norm constraint. In general, solving for a given L0 norm is an NP hard problem thus convex relaxation to regularization by the L1 norm is often considered, i.e., minimizing (1 2||V - WH||F 2 + lambda||H||1).An open problem is to control the degree of sparsity lambda imposed. We here demonstrate that a full regularization path for the L1 norm regularized least squares NMF for fixed W can be calculated at the cost of an ordinary least squares solution based on a modification of the least angle regression and selection (LARS) algorithm forming a non-negativity constrained LARS (NLARS). With the full regularization path, the L1 regularization strength lambda that best approximates a given L0 can be directly accessed and in effect used to control the sparsity of H. The MATLAB code for the NLARS algorithm is available for download. Abstract We investigate the recovery of almost s -sparse vectors x ∈ C N from undersampled and inaccurate data y = A x + e ∈ C m by means of minimizing ‖ z ‖ 1 subject to the equality constraints A z = y . If m ≍ s ln ( N s ) and if Gaussian random matrices A ∈ R m × N are used, this equality-constrained l 1 -minimization is known to be stable with respect to sparsity defects and robust with respect to measurement errors. If m ≍ s ln ( N s ) and if Weibull random matrices are used, we prove here that the equality-constrained l 1 -minimization remains stable and robust. The arguments are based on two key ingredients, namely the robust null space property and the quotient property. The robust null space property relies on a variant of the classical restricted isometry property where the inner norm is replaced by the l 1 -norm and the outer norm is replaced by a norm comparable to the l 2 -norm. For the l 1 -minimization subject to inequality constraints, this yields stability and robustness results that are also valid when considering sparsity relative to a redundant dictionary. As for the quotient property, it relies on lower estimates for the tail probability of sums of independent Weibull random variables.