We present Sketch 'n Solve, an open-source Python package that implements efficient randomized numerical linear algebra (RandNLA) techniques for solving large-scale least squares problems. While sketch-and-solve algorithms have demonstrated theoretical promise, their practical adoption has been limited by the lack of robust, user-friendly implementations. Our package addresses this gap by providing an optimized implementation built on NumPy and SciPy, featuring both dense and sparse sketching operators with a clean API. Through extensive benchmarking, we demonstrate that our implementation achieves up to 50x speedup over traditional LSQR while maintaining high accuracy, even for ill-conditioned matrices. The package shows particular promise for applications in machine learning optimization, signal processing, and scientific computing.

In statistics and machine learning, logistic regression is a widely-used supervised learning technique primarily employed for binary classification tasks. When the number of observations greatly exceeds the number of predictor variables, we present a simple, randomized sampling-based algorithm for logistic regression problem that guarantees high-quality approximations to both the estimated probabilities and the overall discrepancy of the model. Our analysis builds upon two simple structural conditions that boil down to randomized matrix multiplication, a fundamental and well-understood primitive of randomized numerical linear algebra. We analyze the properties of estimated probabilities of logistic regression when leverage scores are used to sample observations, and prove that accurate approximations can be achieved with a sample whose size is much smaller than the total number of observations. To further validate our theoretical findings, we conduct comprehensive empirical evaluations. Overall, our work sheds light on the potential of using randomized sampling approaches to efficiently approximate the estimated probabilities in logistic regression, offering a practical and computationally efficient solution for large-scale datasets.

Symmetric positive definite matrices (symmetric matrices with positive eigenvalues) are ubiquitous, not least because they arise in the solution of many minimization problems. However, a matrix that is supposed to be positive definite may fail to be so for a variety of reasons. Missing or inconsistent data in forming a covariance matrix or a correlation matrix can cause a loss of definiteness, and rounding errors can cause a tiny positive eigenvalue to go negative. The best way to check definiteness is to compute a Cholesky factorization, which is often needed anyway. The MATLAB function chol returns an error message if the factorization fails, and a second output argument can be requested, which is set to the number of the stage on which the factorization failed, or to zero if the factorization succeeded.

Numerical linear algebra is concerned with the practical implications of implementing and executing matrix operations in computers with real data. It is an area that requires some previous experience of linear algebra and is focused on both the performance and precision of the operations. In this post, you will discover the fast.ai Computational Linear Algebra for Coders Review Photo by Ruocaled, some rights reserved. The course "Computational Linear Algebra for Coders" is a free online course provided by fast.ai.