Hadamard Product Full-Rank Parameterization for Neural Network Compression
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Solution Overview
Problem
Current data reduction techniques, such as low-rank parameterization, impose rank constraints that limit the expressiveness and performance of neural network models, particularly in reducing the number of parameters while maintaining model capacity.
Innovation Solution
The implementation of full-rank reduced parameterization using the Hadamard product of two low-rank inner matrices, which allows for the construction of a higher-rank matrix without rank constraints, enabling a more efficient reduction of parameters while maintaining or exceeding the rank of the original matrix, as demonstrated by Propositions 1 and 2.
Engineering Contradictions & Design Principles
Engineering Contradiction Analysis
1Quantity of substance
If low-rank parameterization is used to reduce parameters, then the number of parameters is reduced, but the rank constraint limits model expressiveness and performance
Solution Approach 1:
The parameter matrix W is segmented into two low-rank matrices W1 and W2, where W1 has rank r1 and W2 has rank r2. This segmentation allows each matrix to be independently optimized with lower rank constraints, reducing the total number of parameters while maintaining the ability to achieve full rank through their combination.
Solution Approach 2:
The patent combines two low-rank matrices W1 and W2 using the Hadamard product to create a composite parameter matrix W. This composite structure leverages the properties of both component matrices, achieving full rank capability while maintaining parameter efficiency. The composite approach allows the system to benefit from the low-rank properties of individual matrices while achieving the full expressiveness of a high-rank matrix through their interaction.
2Productivity
If traditional data reduction techniques are applied, then processing speed increases, but model performance deteriorates due to rank constraints
Solution Approach 1:
By segmenting the parameter matrix into two separately optimized low-rank matrices, the system achieves efficient computation (improving processing speed) while the flexible combination of these segments maintains model performance. The segmentation enables independent optimization of each component.
Solution Approach 2:
The composite structure of W = W1 ⊙ W2 enables the system to achieve both computational efficiency and high model performance. The Hadamard product combination allows the model to capture complex patterns while maintaining the computational benefits of low-rank representations.
3Adaptability or versatility
If full-rank reduced parameterization is used, then model capacity is maintained, but computational complexity increases
Solution Approach 1:
Segmenting the parameter matrix into two low-rank components simplifies the computational complexity compared to optimizing a single full-rank matrix. Each low-rank matrix can be optimized independently with fewer parameters, reducing the overall computational burden while maintaining full rank capability through their combination.
Solution Approach 2:
The composite approach using Hadamard product provides a computationally efficient way to achieve full rank. The complexity of computing the Hadamard product of two low-rank matrices is significantly lower than optimizing a single full-rank matrix, as it leverages the structured properties of low-rank decompositions.
Data Source
AI summary
Provided are a data reduction method and a data processing device. The data processing device includes a memory configured to store target data expressed as a vector matrix and instructions for performing control over data reduction and a processor configured to determine low-rank matrices W1 and W2 from which a target parameter matrix W is constructable. The target parameter matrix W is constructed as the Hadamard product between the low-rank matrices W1 and W2.