Matrix Compression via Randomized Low Rank and Low Precision
  Factorization
Rajarshi Saha Varun Srivastava Mert Pilanci
Matrices are exceptionally useful in various fields of study as they provide
a convenient framework to organize and manipulate data in a structured manner.
However, modern matrices can involve billions of elements, making their storage
and processing quite demanding in terms of computational resources and memory
usage. Although prohibitively large, such matrices are often approximately low
rank. We propose an algorithm that exploits this structure to obtain a low rank
decomposition of any matrix $\mathbf{A}$ as $\mathbf{A} \approx
\mathbf{L}\mathbf{R}$, where $\mathbf{L}$ and $\mathbf{R}$ are the low rank
factors. The total number of elements in $\mathbf{L}$ and $\mathbf{R}$ can be
significantly less than that in $\mathbf{A}$. Furthermore, the entries of
$\mathbf{L}$ and $\mathbf{R}$ are quantized to low precision formats $--$
compressing $\mathbf{A}$ by giving us a low rank and low precision
factorization. Our algorithm first computes an approximate basis of the range
space of $\mathbf{A}$ by randomly sketching its columns, followed by a
quantization of the vectors constituting this basis. It then computes
approximate projections of the columns of $\mathbf{A}$ onto this quantized
basis. We derive upper bounds on the approximation error of our algorithm, and
analyze the impact of target rank and quantization bit-budget. The tradeoff
between compression ratio and approximation accuracy allows for flexibility in
choosing these parameters based on specific application requirements. We
empirically demonstrate the efficacy of our algorithm in image compression,
nearest neighbor classification of image and text embeddings, and compressing
the layers of LlaMa-$7$b. Our results illustrate that we can achieve
compression ratios as aggressive as one bit per matrix coordinate, all while
surpassing or maintaining the performance of traditional compression
techniques.