DeepAI

# Near-Optimal Entrywise Sampling for Data Matrices

We consider the problem of selecting non-zero entries of a matrix A in order to produce a sparse sketch of it, B, that minimizes A-B_2. For large m × n matrices, such that n ≫ m (for example, representing n observations over m attributes) we give sampling distributions that exhibit four important properties. First, they have closed forms computable from minimal information regarding A. Second, they allow sketching of matrices whose non-zeros are presented to the algorithm in arbitrary order as a stream, with O(1) computation per non-zero. Third, the resulting sketch matrices are not only sparse, but their non-zero entries are highly compressible. Lastly, and most importantly, under mild assumptions, our distributions are provably competitive with the optimal offline distribution. Note that the probabilities in the optimal offline distribution may be complex functions of all the entries in the matrix. Therefore, regardless of computational complexity, the optimal distribution might be impossible to compute in the streaming model.

• 8 publications
• 20 publications
• 9 publications
12/28/2021

### The full rank condition for sparse random matrices

We derive a sufficient condition for a sparse random matrix with given n...
11/03/2020

### Near-Optimal Entrywise Sampling of Numerically Sparse Matrices

Many real-world data sets are sparse or almost sparse. One method to mea...
10/29/2020

### Active Sampling Count Sketch (ASCS) for Online Sparse Estimation of a Trillion Scale Covariance Matrix

Estimating and storing the covariance (or correlation) matrix of high-di...
01/24/2023

### Logarithmically Sparse Symmetric Matrices

A positive definite matrix is called logarithmically sparse if its matri...
07/11/2019

### Schatten Norms in Matrix Streams: Hello Sparsity, Goodbye Dimension

The spectrum of a matrix contains important structural information about...
02/15/2022

### Bohemian Matrix Geometry

A Bohemian matrix family is a set of matrices all of whose entries are d...
12/19/2022

### Non-asymptotic bounds for inclusion probabilities in rejective sampling

We provide non-asymptotic bounds for first and higher order inclusion pr...