Gene

Big data

Function (biology)

Philosophy

Data Matrix

Low-rank approximation

Pure mathematics

Biology

Statistics

Chromatography

Rank (graph theory)

Latent variable

Variable (mathematics)

Biochemistry

Data mining

Computer science

Simple (philosophy)

Bounded function

Evolutionary biology

Matrix (chemical analysis)

Tensor (intrinsic definition)

Phylogenetic tree

Mathematics

Clade

Epistemology

Mathematical analysis

Combinatorics

Chemistry

Piecewise

SIAM Journal on Mathematics of Data Science

Matrices of (approximate) low rank are pervasive in data science, appearing in movie preferences, text documents, survey data, medical records, and genomics. While there is a vast literature on how to exploit low rank structure in these datasets, there is less attention paid to explaining why the low rank structure appears in the first place. Here, we explain the effectiveness of low rank models in data science by considering a simple generative model for these matrices: we suppose that each row or column is associated to a (possibly high dimensional) bounded latent variable, and entries of the matrix are generated by applying a piecewise analytic function to these latent variables. These matrices are in general full rank. However, we show that we can approximate every entry of an $m\times n$ matrix drawn from this model to within a fixed absolute error by a low rank matrix whose rank grows as $\mathcal{O}(\log(m+n))$. Hence any sufficiently large matrix from such a latent variable model can be approximated, up to a small entrywise error, by a low rank matrix.

Why Are Big Data Matrices Approximately Low Rank?