Related papers: Efficiently updating a covariance matrix and its L…
In many practical situations we would like to estimate the covariance matrix of a set of variables from an insufficient amount of data. More specifically, if we have a set of $N$ independent, identically distributed measurements of an $M$…
We present a simple formula to update the pseudoinverse of a full-rank rectangular matrix that undergoes a low-rank modification, and demonstrate its utility for solving least squares problems. The resulting algorithm can be dramatically…
This paper introduces the novel piecewise stationary covariate-assisted ranking estimation (PS-CARE) model for analyzing time-evolving pairwise comparison data, enhancing item ranking accuracy through the integration of covariate…
We address the estimation of seismic wavefields by means of Multidimensional Deconvolution (MDD) for various redatuming applications. While offering more accuracy than conventional correlation-based redatuming methods, MDD faces challenges…
We use available measurements to estimate the unknown parameters (variance, smoothness parameter, and covariance length) of a covariance function by maximizing the joint Gaussian log-likelihood function. To overcome cubic complexity in the…
Low-rank matrix approximations, such as the truncated singular value decomposition and the rank-revealing QR decomposition, play a central role in data analysis and scientific computing. This work surveys and extends recent research which…
In this paper, we study the problem of high-dimensional approximately low-rank covariance matrix estimation with missing observations. We propose a simple procedure computationally tractable in high-dimension and that does not require…
Large Language Models (LLMs) have transformed both everyday life and scientific research. However, adapting LLMs from general-purpose models to specialized tasks remains challenging, particularly in resource-constrained environments.…
The extension of ab initio quantum many-body theory to higher accuracy and larger systems is intrinsically limited by the handling of large data objects in form of wave-function expansions and/or many-body operators. In this work we present…
Large language models (LLMs) have made fundamental changes in human life. The attention scheme is one of the key components over all the LLMs, such as BERT, GPT-1, Transformers, GPT-2, 3, 3.5 and 4. Inspired by previous theoretical study of…
Kendall transformation is a conversion of an ordered feature into a vector of pairwise order relations between individual values. This way, it preserves ranking of observations and represents it in a categorical form. Such transformation…
A typical system of k difference (or differential) equations can be compressed, or folded into a difference (or ordinary differential) equation of order k. Such foldings appear in control theory as the canonical forms of the controllability…
A classical problem in matrix computations is the efficient and reliable approximation of a given matrix by a matrix of lower rank. The truncated singular value decomposition (SVD) is known to provide the best such approximation for any…
Matrix reordering is a task to permute the rows and columns of a given observed matrix such that the resulting reordered matrix shows meaningful or interpretable structural patterns. Most existing matrix reordering techniques share the…
Recent SVD-free matrix factorization formulations have enabled rank minimization for systems with millions of rows and columns, paving the way for matrix completion in extremely large-scale applications, such as seismic data interpolation.…
Low-Rank Adaptation (LoRA) has gained popularity for fine-tuning large foundation models, leveraging low-rank matrices $\mathbf{A}$ and $\mathbf{B}$ to represent weight changes (i.e., $\Delta \mathbf{W} = \mathbf{B} \mathbf{A}$). This…
This paper surveys randomized algorithms in numerical linear algebra for low-rank decompositions of matrices and tensors. The survey begins with a review of classical matrix algorithms that can be accelerated by randomized dimensionality…
In many applications, e.g., recommender systems and traffic monitoring, the data comes in the form of a matrix that is only partially observed and low rank. A fundamental data-analysis task for these datasets is matrix completion, where the…
Matrix reduction is the standard procedure for computing the persistent homology of a filtered simplicial complex with $m$ simplices. Its output is a particular decomposition of the total boundary matrix, from which the persistence diagrams…
Dynamic mode decomposition (DMD) is a popular technique for modal decomposition, flow analysis, and reduced-order modeling. In situations where a system is time varying, one would like to update the system's description online as time…