Majorization-Minimization by IST and DG-IST.

Abstract

<p>(A) the <i>J</i>(<i>x</i>) function to be minimized. (B) The <i>G</i>(<i>x</i>) (yellow surface) function where <i>G</i>(<i>x</i>) ≥ <i>J</i>(<i>x</i>) ∀ <i>x</i> and <i>G</i>(<i>x<sub>k</sub></i>) = <i>J</i>(<i>x<sub>k</sub></i>), IST algorithm. (C) <i>G</i>(<i>x</i>) ≥ <i>J</i>(<i>x</i>) ∀ <i>x<sub>2</sub></i> with <i>x</i><sub>1</sub> = <i>const</i> and <i>G</i>(<i>x<sub>k</sub></i>) = <i>J</i>(<i>x<sub>k</sub></i>), DG-IST algorithm. (D) <i>x<sub>k</sub></i> is the initialization point of vector <i>x</i> (black arrow) and blue arrow indicates the <math><mrow><mi>x</mi><mo>′</mo><mo>=</mo><mrow><mo>(</mo><mrow><msub><mi>x</mi><mo>′</mo><mn>1</mn></msub><mo>,</mo><msub><mi>x</mi><mo>′</mo><mn>2</mn></msub></mrow><mo>)</mo></mrow></mrow></math> where <i>G</i>(<i>x</i>) is minimized, IST algorithm. (E) <i>x<sub>k</sub></i> is the initialization point of vector <i>x</i> (black arrow, same as in (D)) and blue arrow indicates the <math><mrow><mi>x</mi><mo>″</mo><mo>=</mo><mrow><mo>(</mo><mrow><msub><mi>x</mi><mo>″</mo><mn>1</mn></msub><mo>,</mo><msub><mi>x</mi><mo>″</mo><mn>2</mn></msub></mrow><mo>)</mo></mrow></mrow></math> where <i>G</i>(<i>x</i>) is minimized, DG-IST algorithm. Notice that, <math><mrow><msub><mi>x</mi><mo>″</mo><mn>2</mn></msub><mo><</mo><msub><mi>x</mi><mo>′</mo><mn>2</mn></msub></mrow></math>, thus closer to the point where <i>J</i>(<i>x</i>) is minimized (see (A)).</p

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