Structured sparsity regularization

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2024 Jan 15 16:24

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Edited

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2025 Mar 24 23:57

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Compressed sensing

Model Regularization

Structured Sparse Model

\min_\alpha||x-D\alpha||^2 + \lambda||\alpha||_0

||\alpha||_0

represents counts of element which are not 0 valued.

Sparse
Total Variation

This regularization favors solutions whose coefficients are constant within contiguous regions but also promotes sparsity.

J(w) = \lambda(|\nabla w| + |w|)

Sparse Total
Laplacian Kernel

This regularization encourages both smooth variations within regions and sparsity.

J(w) = \frac{1}{2} \lambda(1 -\alpha) \sum_{i\sim j} (w_i - w_j)^2 + \lambda\alpha |w| \text{ where } \alpha \in [0,1]

Structured sparsity regularization

Structured sparsity regularization is a class of methods, and an area of research in statistical learning theory, that extend and generalize sparsity regularization learning methods.[1] Both sparsity and structured sparsity regularization methods seek to exploit the assumption that the output variable Y {\displaystyle Y} (i.e., response, or dependent variable) to be learned can be described by a reduced number of variables in the input space X {\displaystyle X} (i.e., the domain, space of features or explanatory variables). Sparsity regularization methods focus on selecting the input variables that best describe the output. Structured sparsity regularization methods generalize and extend sparsity regularization methods, by allowing for optimal selection over structures like groups or networks of input variables in X {\displaystyle X} .[2][3]

Structured sparsity regularization

https://en.wikipedia.org/wiki/Structured_sparsity_regularization

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