In many practical applications, we need to recover block sparse signals. In this paper, we encounter the system model where joint sparse signals exhibit block structure. To reconstruct this category of signals, we propose a new algorithm called block signal subspace matching pursuit (BSSMP) for the block joint sparse recovery problem in compressed sensing, which simultaneously reconstructs the support of block jointly sparse signals from a common sensing matrix. To begin with, we consider the case where block joint sparse matrix X has full column rank and any r nonzero row-blocks are linearly independent. Based on these assumptions, our theoretical analysis indicates that the BSSMP algorithm could reconstruct the support of X through at most <InlineEquation ID="IEq1"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="10114_2025_3171_Article_IEq1.gif" Format="GIF" Height="23" Rendition="HTML" Resolution="72" Type="Linedraw" Width="87" /> </InlineMediaObject> <EquationSource Format="TEX">$k - r + \left\lceil {{r \over L}} \right\rceil$</EquationSource> <EquationSource Format="MATHML"><math display="block"> <mi>k</mi> <mo>−</mo> <mi>r</mi> <mo>+</mo> <mrow> <mo>⌈</mo> <mrow> <mrow> <mfrac> <mi>r</mi> <mi>L</mi> </mfrac> </mrow> </mrow> <mo>⌉</mo> </mrow> </math></EquationSource> </InlineEquation> iterations if sensing matrix A satisfies the block restricted isometry property of order L(K − r) + r + 1 with <InlineEquation ID="IEq2"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="10114_2025_3171_Article_IEq2.gif" Format="GIF" Height="34" Rendition="HTML" Resolution="72" Type="Linedraw" Width="292" /> </InlineMediaObject> <EquationSource Format="TEX">${\delta _{{B_{L( {K - r}) + r + 1}}}}< \max \{{{{\sqrt r} \over {\sqrt {K + {r \over 4}} + \sqrt {{r \over 4}} }},{{\sqrt L} \over {\sqrt {Kd} + \sqrt L}}}\}$</EquationSource> <EquationSource Format="MATHML"><math display="block"> <mrow> <msub> <mi>δ</mi> <mrow> <mrow> <msub> <mi>B</mi> <mrow> <mi>L</mi> <mo stretchy="false">(</mo> <mrow> <mi>K</mi> <mo>−</mo> <mi>r</mi> </mrow> <mo stretchy="false">)</mo> <mo>+</mo> <mi>r</mi> <mo>+</mo> <mn>1</mn> </mrow> </msub> </mrow> </mrow> </msub> </mrow> <mo><</mo> <mo form="prefix" movablelimits="true">max</mo> <mo fence="false" stretchy="false">{</mo> <mrow> <mrow> <mfrac> <mrow> <msqrt> <mi>r</mi> </msqrt> </mrow> <mrow> <msqrt> <mi>K</mi> <mo>+</mo> <mrow> <mfrac> <mi>r</mi> <mn>4</mn> </mfrac> </mrow> </msqrt> <mo>+</mo> <msqrt> <mrow> <mfrac> <mi>r</mi> <mn>4</mn> </mfrac> </mrow> </msqrt> </mrow> </mfrac> </mrow> <mo>,</mo> <mrow> <mfrac> <mrow> <msqrt> <mi>L</mi> </msqrt> </mrow> <mrow> <msqrt> <mi>K</mi> <mi>d</mi> </msqrt> <mo>+</mo> <msqrt> <mi>L</mi> </msqrt> </mrow> </mfrac> </mrow> </mrow> <mo fence="false" stretchy="false">}</mo> </math></EquationSource> </InlineEquation>. This condition improves the existing result.

The Analysis of Block Joint Sparse Recovery Using Block Signal Space Matching Pursuit

期刊论文
发表日期：

2025年6月15日

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Haifeng Li,
Hao Ying,
Jinming Wen

摘要

In many practical applications, we need to recover block sparse signals. In this paper, we encounter the system model where joint sparse signals exhibit block structure. To reconstruct this category of signals, we propose a new algorithm called block signal subspace matching pursuit (BSSMP) for the block joint sparse recovery problem in compressed sensing, which simultaneously reconstructs the support of block jointly sparse signals from a common sensing matrix. To begin with, we consider the case where block joint sparse matrix X has full column rank and any r nonzero row-blocks are linearly independent. Based on these assumptions, our theoretical analysis indicates that the BSSMP algorithm could reconstruct the support of X through at most $k - r + \left\lceil {{r \over L}} \right\rceil$ $k - r + ⌈ \frac{r}{L} ⌉$ iterations if sensing matrix A satisfies the block restricted isometry property of order L(K − r) + r + 1 with ${\delta _{{B_{L( {K - r}) + r + 1}}}}< \max \{{{{\sqrt r} \over {\sqrt {K + {r \over 4}} + \sqrt {{r \over 4}} }},{{\sqrt L} \over {\sqrt {Kd} + \sqrt L}}}\}$ $δ_{B_{L (K - r) + r + 1}} < max {\frac{\sqrt{r}}{\sqrt{K + \frac{r}{4}} + \sqrt{\frac{r}{4}}}, \frac{\sqrt{L}}{\sqrt{K d} + \sqrt{L}}}$ . This condition improves the existing result.