Matrices are exceptionally useful in various fields of study as they provide a convenient framework to organize and manipulate data in a structured manner.

Matrices are exceptionally useful in various fields of study as they provide a convenient framework to organize and manipulate data in a structured manner.

Neural Information Processing Systems http://nips.cc/

In an attempt to reduce their description size, we study low rank approximation of such matrices.

Matrices are exceptionally useful in various fields of study as they provide a convenient framework to organize and manipulate data in a structured manner.

Large Language Models (LLMs) are reshaping the research landscape in artificial intelligence, particularly as model parameters scale up significantly, unlocking remarkable capabilities across various domains. Nevertheless, the scalability of model parameters faces constraints due to limitations in GPU memory and computational speed. To address these constraints, various weight compression methods have emerged, such as Pruning and Quantization. Given the low-rank nature of weight matrices in language models, the reduction of weights through matrix decomposition undoubtedly holds significant potential and promise. In this paper, drawing upon the intrinsic structure of LLMs, we propose a novel approach termed Data-free Joint Rank-k Approximation for compressing the parameter matrices. Significantly, our method is characterized by without necessitating additional involvement of any corpus, while simultaneously preserving orthogonality in conjunction with pruning and quantization methods. We achieve a model pruning of 80% parameters while retaining 93.43% of the original performance without any calibration data. Additionally, we explore the fundamental properties of the weight matrix of LLMs undergone Rank-k Approximation and conduct comprehensive experiments to elucidate our hypothesis.

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.

We consider the problem of approximating a $d \times d$ covariance matrix $M$ with a rank-$k$ matrix under $(\varepsilon,\delta)$-differential privacy. We present and analyze a complex variant of the Gaussian mechanism and show that the Frobenius norm of the difference between the matrix output by this mechanism and the best rank-$k$ approximation to $M$ is bounded by roughly $\tilde{O}(\sqrt{kd})$, whenever there is an appropriately large gap between the $k$'th and the $k+1$'th eigenvalues of $M$. This improves on previous work that requires that the gap between every pair of top-$k$ eigenvalues of $M$ is at least $\sqrt{d}$ for a similar bound. Our analysis leverages the fact that the eigenvalues of complex matrix Brownian motion repel more than in the real case, and uses Dyson's stochastic differential equations governing the evolution of its eigenvalues to show that the eigenvalues of the matrix $M$ perturbed by complex Gaussian noise have large gaps with high probability. Our results contribute to the analysis of low-rank approximations under average-case perturbations and to an understanding of eigenvalue gaps for random matrices, which may be of independent interest.