Related papers: Randomized Compression of Rank-Structured Matrices…
Low-rank approximation of a matrix by means of random sampling has been consistently efficient in its empirical studies by many scientists who applied it with various sparse and structured multipliers, but adequate formal support for this…
Scalability of statistical estimators is of increasing importance in modern applications and dimension reduction is often used to extract relevant information from data. A variety of popular dimension reduction approaches can be framed as…
We study distributed low rank approximation in which the matrix to be approximated is only implicitly represented across the different servers. For example, each of $s$ servers may have an $n \times d$ matrix $A^t$, and we may be interested…
Low-rank matrix approximation is extremely useful in the analysis of data that arises in scientific computing, engineering applications, and data science. However, as data sizes grow, traditional low-rank matrix approximation methods, such…
Dimensionality reduction is an essential technique for multi-way large-scale data, i.e., tensor. Tensor ring (TR) decomposition has become popular due to its high representation ability and flexibility. However, the traditional TR…
The singular value decomposition is widely used to approximate data matrices with lower rank matrices. Feng and He [Ann. Appl. Stat. 3 (2009) 1634-1654] developed tests on dimensionality of the mean structure of a data matrix based on the…
Retrieval-Augmented Generation (RAG) has significantly enhanced Large Language Models' ability to access external knowledge, yet current graph-based RAG approaches face two critical limitations in managing hierarchical information: they…
The combination of the sparse sampling and the low-rank structured matrix reconstruction has shown promising performance, enabling a significant reduction of the magnetic resonance imaging data acquisition time. However, the low-rank…
A new approach to solving eigenvalue optimization problems for large structured matrices is proposed and studied. The class of optimization problems considered is related to computing structured pseudospectra and their extremal points, and…
In a plethora of applications dealing with inverse problems, e.g. in image processing, social networks, compressive sensing, biological data processing etc., the signal of interest is known to be structured in several ways at the same time.…
We propose a novel class of kernels to alleviate the high computational cost of large-scale nonparametric learning with kernel methods. The proposed kernel is defined based on a hierarchical partitioning of the underlying data domain, where…
The power method is one of the most fundamental tools for extracting top principal components from data through low-rank matrix approximation. Yet, when the target rank is large, the cost of matrix multiplication associated with this…
This manuscript describes a technique for computing partial rank-revealing factorizations, such as, e.g, a partial QR factorization or a partial singular value decomposition. The method takes as input a tolerance $\varepsilon$ and an…
Current deep learning architectures are growing larger in order to learn from complex datasets. These architectures require giant matrix multiplication operations to train millions of parameters. Conversely, there is another growing trend…
We present memory-efficient and scalable algorithms for kernel methods used in machine learning. Using hierarchical matrix approximations for the kernel matrix the memory requirements, the number of floating point operations, and the…
We study the Sparse Plus Low-Rank decomposition problem (SLR), which is the problem of decomposing a corrupted data matrix into a sparse matrix of perturbations plus a low-rank matrix containing the ground truth. SLR is a fundamental…
We first propose a concise singular value decomposition of dual matrices. Then, the randomized version of the decomposition is presented. It can significantly reduce the computational cost while maintaining the similar accuracy. We analyze…
We consider the problem of estimating the factors of a rank-$1$ matrix with i.i.d. Gaussian, rank-$1$ measurements that are nonlinearly transformed and corrupted by noise. Considering two prototypical choices for the nonlinearity, we study…
A fast algorithm for the approximation of a low rank LU decomposition is presented. In order to achieve a low complexity, the algorithm uses sparse random projections combined with FFT-based random projections. The asymptotic approximation…
We consider whether algorithmic choices in over-parameterized linear matrix factorization introduce implicit regularization. We focus on noiseless matrix sensing over rank-$r$ positive semi-definite (PSD) matrices in $\mathbb{R}^{n \times…