  The prohibitive sizes of Large Language Models (LLMs) today make it difficult
to deploy them on memory-constrained edge devices. This work introduces $\rm
CALDERA$ -- a new post-training LLM compression algorithm that harnesses the
inherent low-rank structure of a weight matrix $\mathbf{W}$ by approximating it
via a low-rank, low-precision decomposition as $\mathbf{W} \approx \mathbf{Q} +
\mathbf{L}\mathbf{R}$. Here, $\mathbf{L}$ and $\mathbf{R}$ are low rank
factors, and the entries of $\mathbf{Q}$, $\mathbf{L}$ and $\mathbf{R}$ are
quantized. The model is compressed by substituting each layer with its
$\mathbf{Q} + \mathbf{L}\mathbf{R}$ decomposition, and the zero-shot
performance of the compressed model is evaluated. Additionally, $\mathbf{L}$
and $\mathbf{R}$ are readily amenable to low-rank adaptation, consequently
enhancing the zero-shot performance. $\rm CALDERA$ obtains this decomposition
by formulating it as an optimization problem
$\min_{\mathbf{Q},\mathbf{L},\mathbf{R}}\lVert(\mathbf{Q} +
\mathbf{L}\mathbf{R} - \mathbf{W})\mathbf{X}^\top\rVert_{\rm F}^2$, where
$\mathbf{X}$ is the calibration data, and $\mathbf{Q}, \mathbf{L}, \mathbf{R}$
are constrained to be representable using low-precision formats. Theoretical
upper bounds on the approximation error of $\rm CALDERA$ are established using
a rank-constrained regression framework, and the tradeoff between compression
ratio and model performance is studied by analyzing the impact of target rank
and quantization bit budget. Results illustrate that compressing LlaMa-$2$
$7$B/$70$B and LlaMa-$3$ $8$B models obtained using $\rm CALDERA$ outperforms
existing post-training LLM compression techniques in the regime of less than
$2.5$ bits per parameter. The implementation is available at:
\href{https://github.com/pilancilab/caldera}{https://github.com/pilancilab/caldera}.
