Large language models (LLMs) have shown impressive capabilities across various tasks. However, training LLMs from scratch requires significant computational power and extensive memory capacity. Recent studies have explored low-rank structures on weights for efficient fine-tuning in terms of parameters and memory, either through low-rank adaptation or factorization. While effective for fine-tuning, low-rank structures are generally less suitable for pretraining because they restrict parameters to a low-dimensional subspace. In this work, we propose to parameterize the weights as a sum of low-rank and sparse matrices for pretraining, which we call SLTrain. The low-rank component is learned via matrix factorization, while for the sparse component, we employ a simple strategy of uniformly selecting the sparsity support at random and learning only the non-zero entries with the fixed support. While being simple, the random fixed-support sparse learning strategy significantly enhances pretraining when combined with low-rank learning. Our results show that SLTrain adds minimal extra parameters and memory costs compared to pretraining with low-rank parameterization, yet achieves substantially better performance, which is comparable to full-rank training. Remarkably, when combined with quantization and per-layer updates, SLTrain can reduce memory requirements by up to 73% when pretraining the LLaMA 7B model.

Abstract: he problem of approximating a dense matrix by a product of sparse factors is a fundamental problem for many signal processing and machine learning tasks. It can be decomposed into two subproblems: finding the position of the non-zero coefficients in the sparse factors, and determining their values. While the first step is usually seen as the most challenging one due to its combinatorial nature, this paper focuses on the second step, referred to as sparse matrix approximation with fixed support. First, we show its NP-hardness, while also presenting a nontrivial family of supports making the problem practically tractable with a dedicated algorithm. Then, we investigate the landscape of its natural optimization formulation, proving the absence of spurious local valleys and spurious local minima, whose presence could prevent local optimization methods to achieve global optimality.

K-means -- and the celebrated Lloyd algorithm -- is more than the clustering method it was originally designed to be. It has indeed proven pivotal to help increase the speed of many machine learning and data analysis techniques such as indexing, nearest-neighbor search and prediction, data compression; its beneficial use has been shown to carry over to the acceleration of kernel machines (when using the Nystr\"om method). Here, we propose a fast extension of K-means, dubbed QuicK-means, that rests on the idea of expressing the matrix of the $K$ centroids as a product of sparse matrices, a feat made possible by recent results devoted to find approximations of matrices as a product of sparse factors. Using such a decomposition squashes the complexity of the matrix-vector product between the factorized $K \times D$ centroid matrix $\mathbf{U}$ and any vector from $\mathcal{O}(K D)$ to $\mathcal{O}(A \log A+B)$, with $A=\min (K, D)$ and $B=\max (K, D)$, where $D$ is the dimension of the training data. This drastic computational saving has a direct impact in the assignment process of a point to a cluster, meaning that it is not only tangible at prediction time, but also at training time, provided the factorization procedure is performed during Lloyd's algorithm. We precisely show that resorting to a factorization step at each iteration does not impair the convergence of the optimization scheme and that, depending on the context, it may entail a reduction of the training time. Finally, we provide discussions and numerical simulations that show the versatility of our computationally-efficient QuicK-means algorithm.

This paper examines fundamental error characteristics for a general class of matrix completion problems, where the matrix of interest is a product of two a priori unknown matrices, one of which is sparse, and the observations are noisy. Our main contributions come in the form of minimax lower bounds for the expected per-element squared error for this problem under under several common noise models. Specifically, we analyze scenarios where the corruptions are characterized by additive Gaussian noise or additive heavier-tailed (Laplace) noise, Poisson-distributed observations, and highly-quantized (e.g., one-bit) observations, as instances of our general result. Our results establish that the error bounds derived in (Soni et al., 2016) for complexity-regularized maximum likelihood estimators achieve, up to multiplicative constants and logarithmic factors, the minimax error rates in each of these noise scenarios, provided that the nominal number of observations is large enough, and the sparse factor has (on an average) at least one non-zero per column.

In this article, we derive a Bayesian model to learning the sparse and low rank PARAFAC decomposition for the observed tensor with missing values via the elastic net, with property to find the true rank and sparse factor matrix which is robust to the noise. We formulate efficient block coordinate descent algorithm and admax stochastic block coordinate descent algorithm to solve it, which can be used to solve the large scale problem. To choose the appropriate rank and sparsity in PARAFAC decomposition, we will give a solution path by gradually increasing the regularization to increase the sparsity and decrease the rank. When we find the sparse structure of the factor matrix, we can fixed the sparse structure, using a small to regularization to decreasing the recovery error, and one can choose the proper decomposition from the solution path with sufficient sparse factor matrix with low recovery error. We test the power of our algorithm on the simulation data and real data, which show it is powerful.

Latent factor models are the canonical statistical tool for exploratory analyses of low-dimensional linear structure for an observation matrix with p features across n samples. We develop a structured Bayesian group factor analysis model that extends the factor model to multiple coupled observation matrices; in the case of two observations, this reduces to a Bayesian model of canonical correlation analysis. The main contribution of this work is to carefully define a structured Bayesian prior that encourages both element-wise and column-wise shrinkage and leads to desirable behavior on high-dimensional data. In particular, our model puts a structured prior on the joint factor loading matrix, regularizing at three levels, which enables element-wise sparsity and unsupervised recovery of latent factors corresponding to structured variance across arbitrary subsets of the observations. In addition, our structured prior allows for both dense and sparse latent factors so that covariation among either all features or only a subset of features can both be recovered. We use fast parameter-expanded expectation-maximization for parameter estimation in this model. We validate our method on both simulated data with substantial structure and real data, comparing against a number of state-of-the-art approaches. These results illustrate useful properties of our model, including i) recovering sparse signal in the presence of dense effects; ii) the ability to scale naturally to large numbers of observations; iii) flexible observation- and factor-specific regularization to recover factors with a wide variety of sparsity levels and percentage of variance explained; and iv) tractable inference that scales to modern genomic and document data sizes.

This paper focuses on applications of sparse principal component analysis to clustering and feature selection problems, with a particular focus on gene expression data analysis. Sparse methods have had a significant impact in many areas of statistics, in particular regression and classification (see [CT05], [DT05] and [Vap95] among others). As in these areas, our motivation for developing sparse multivariate visualization tools is the potential of these methods for yielding statistical results that are both more interpretable and more robust than classical analyses, while giving up little statistical efficiency. Principal component analysis (PCA) is a classic tool for analyzing large scale multivariate data. It seeks linear combinations of the data variables (often called factors or principal components) that capture a maximum amount of variance.