| Summary: | This paper proposes an effective extended reweighted <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msub><mo>ℓ</mo><mn>1</mn></msub></mrow></semantics></math></inline-formula> minimization algorithm (ERMA) to solve the basis pursuit problem <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><munder><mrow><mi>min</mi></mrow><mrow><mi>u</mi><mo>∈</mo><msup><mi>R</mi><mi>n</mi></msup></mrow></munder><mrow><mo stretchy="false">{</mo><mrow><msub><mrow><mo stretchy="false">|</mo><mo stretchy="false">|</mo><mi>u</mi><mo stretchy="false">|</mo><mo stretchy="false">|</mo></mrow><mn>1</mn></msub><mo>:</mo><mrow><mrow><mi>A</mi><mi>u</mi><mo>=</mo><mi>f</mi></mrow><mo stretchy="false">}</mo></mrow></mrow></mrow></mrow></semantics></math></inline-formula> in compressed sensing, where <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>A</mi><mo>∈</mo><msup><mi>R</mi><mrow><mi>m</mi><mo>×</mo><mi>n</mi></mrow></msup></mrow></semantics></math></inline-formula>, <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>m</mi><mo>≪</mo><mi>n</mi></mrow></semantics></math></inline-formula>. The fast algorithm is based on linearized Bregman iteration with soft thresholding operator and generalized inverse iteration. At the same time, it also combines the iterative reweighted strategy that is used to solve <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><munder><mrow><mi>min</mi></mrow><mrow><mi>u</mi><mo>∈</mo><msup><mi>R</mi><mi>n</mi></msup></mrow></munder><mrow><mo stretchy="false">{</mo><mrow><msubsup><mrow><mo stretchy="false">|</mo><mo stretchy="false">|</mo><mi>u</mi><mo stretchy="false">|</mo><mo stretchy="false">|</mo></mrow><mi>p</mi><mi>p</mi></msubsup><mo>:</mo><mrow><mrow><mi>A</mi><mi>u</mi><mo>=</mo><mi>f</mi></mrow><mo stretchy="false">}</mo></mrow></mrow></mrow></mrow></semantics></math></inline-formula> problem, with the weight <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msub><mi>ω</mi><mi>i</mi></msub><mrow><mo stretchy="false">(</mo><mrow><mi>u</mi><mo>,</mo><mi>p</mi></mrow><mo stretchy="false">)</mo></mrow><mo>=</mo><msup><mrow><mrow><mo stretchy="false">(</mo><mrow><mi>ε</mi><mo>+</mo><msup><mrow><mrow><mo stretchy="false">|</mo><mrow><msub><mi>u</mi><mi>i</mi></msub></mrow><mo stretchy="false">|</mo></mrow></mrow><mn>2</mn></msup></mrow><mo stretchy="false">)</mo></mrow></mrow><mrow><mi>p</mi><mo>/</mo><mn>2</mn><mo>−</mo><mn>1</mn></mrow></msup></mrow></semantics></math></inline-formula>. Numerical experiments show that this <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msub><mo>ℓ</mo><mn>1</mn></msub></mrow></semantics></math></inline-formula> minimization persistently performs better than other methods. Especially when <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>p</mi><mo>=</mo><mn>0</mn></mrow></semantics></math></inline-formula>, the restored signal by the algorithm has the highest signal to noise ratio. Additionally, this approach has no effect on workload or calculation time when matrix A is ill-conditioned.
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