Weight Matrix Compression via Kronecker Decomposition
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Solution Overview
Problem
In machine learning, the large size of initial weight matrices necessitates excessive memory space and lacks efficient methods for reusing hardware across different layers, leading to inefficient hardware usage and high computational costs.
Innovation Solution
The method involves generating a first and second weight matrix as a Kronecker product of the transposed second matrix and the first matrix, optimizing these matrices, and using them to perform calculations that reduce memory requirements by vectorizing data, allowing for reduced hardware usage and improved computation speed.
Engineering Contradictions & Design Principles
Engineering Contradiction Analysis
1Reliability
If an initial weight matrix is used for weighted calculations in machine learning, then the calculation can be performed, but the memory space required becomes excessively large
Solution Approach 1:
The initial weight matrix is segmented into two separate weight matrices (first weight matrix and second weight matrix). The Kronecker product structure allows the original matrix to be represented as the outer product of these two smaller matrices, significantly reducing the memory space required to store the weights while maintaining the same computational functionality.
2Adaptability or versatility
If a large initial weight matrix is used, then the model can capture complex patterns, but hardware resources are excessively consumed
Solution Approach 1:
The large weight matrix is divided into two smaller matrices, reducing the hardware resources needed to store and process the weights. The segmented structure maintains the model's capability to capture complex patterns through the Kronecker product relationship.
Solution Approach 2:
The weight matrix representation is changed from a single large matrix to a product of two smaller matrices. This parameter change in the mathematical representation reduces the total number of parameters that need to be stored and transmitted, thereby reducing hardware resource requirements.
3Ease of operation
If the initial weight matrix is used directly, then the calculation is straightforward, but computation efficiency is reduced due to large data size
Solution Approach 1:
The computation is segmented into multiple steps: first processing with the first weight matrix, then processing the intermediate result with the second weight matrix. This segmentation reduces the computational burden at each step compared to using the full large matrix, improving overall computation efficiency.
Solution Approach 2:
The data is pre-processed by the first weight matrix before being input to the second weight matrix. This preliminary action breaks down the computation into manageable stages, improving efficiency by avoiding the need to process all data through a single large matrix operation.
Data Source
AI summary
A method for compressing an initial weight matrix includes generating a first weight matrix and a second weight matrix according to the initial weight matrix where the initial weight matrix is a Kronecker product of a transposed matrix of the second weight matrix and the first weight matrix; optimizing the first and second weight matrixes to generate an optimized first weight matrix and an optimized second weight matrix; generating a processed data matrix according to an initial data matrix where the initial data matrix is vectorization of the processed data matrix; multiplying the processed data matrix by the optimized first weight matrix to generate a first product; multiplying the optimized second weight matrix by the first product to generate a second product; and vectorizing the second product. The initial weight matrix requires a larger memory space than a combined memory space of the first and second weight matrixes.