Variable Group Calculation Using Mollifier Function
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Solution Overview
Problem
In sparse learning, existing methods for calculating undetermined variable groups face challenges with calculation speed and identifying main factors due to limitations in regularization terms, particularly with L0 norm, L1 norm, and L2 norm, which result in insufficient speed and difficulty in identifying main factors.
Innovation Solution
A variable group calculation apparatus and method that converts the regularization term to a convolution value for an L1 norm using a mollifier function, enabling faster calculation and easier identification of undetermined variable groups by smoothing the L1 regularization term, allowing for the application of Newton's method or its variations.
Engineering Contradictions & Design Principles
Engineering Contradiction Analysis
1Measurement precision
If L0 norm regularization term is used, then sparsity is achieved, but calculation speed becomes insufficient
Solution Approach 1:
The patent introduces a mollifier function as an intermediary to transform the L0 norm regularization term into a smooth approximation. This mediator function preserves the sparsity-inducing property while enabling efficient gradient-based optimization, thus resolving the contradiction between achieving sparsity and maintaining calculation speed
Solution Approach 2:
The patent changes the parameter form of the regularization term by replacing the non-differentiable L0 norm with a smooth approximated version using the mollifier function. This parameter transformation maintains the essential sparsity characteristic while making the optimization problem computationally tractable with faster convergence
2Device complexity
If L1 norm regularization term is used, then calculation is simplified, but many non-consecutive parts cannot be differentiated
Solution Approach 1:
The patent transforms the L1 norm parameter into a smooth approximated form using the mollifier function, which maintains the simplicity of the regularization approach while resolving the differentiability issue. The smooth approximation allows continuous differentiation throughout the optimization process
3Productivity
If L2 norm regularization term is used, then calculation speed improves, but identification of main factors becomes difficult
Solution Approach 1:
The patent changes the regularization parameter from L2 norm to a smooth approximated L0 norm using the mollifier function. This parameter transformation restores the sparsity-inducing capability that enables accurate identification of main factors, while maintaining the calculation speed advantage through the smooth differentiable form
Data Source
AI summary
The present disclosure is applied to a variable group calculation apparatus for calculating an undetermined variable group that simultaneously minimizes a difference value and a data value. The difference value is a difference between an added composite value, which is obtained by adding and combining the undetermined variable group and a dictionary data group, and an observation data group. The data value includes the difference value and a regularization term of the undetermined variable group. The variable group calculation apparatus of the present disclosure includes a convolution unit configured to convert the regularization term to a convolution value for an L1 norm using the undetermined variable group and a mollifier function, and a calculation unit configured to perform the calculation using the regularization term, which is converted to the convolution value by the convolution unit.