On Ambiguity in Linear Inverse Problems: Entrywise Bounds on Nearly Data-Consistent Solutions and Entrywise Condition...

Area of Science:

• Signal Processing
• Applied Mathematics
• Computational Imaging

Background:

• Ill-posed linear inverse problems are common in signal processing.
• Traditional measures of ill-posedness (e.g., condition number) are global and may lack entry-specific insights.
• Understanding solution ambiguity is crucial for reliable reconstruction.

Purpose of the Study:

• To develop novel theoretical bounds for quantifying entrywise ill-posedness in linear inverse problems.
• To provide a more nuanced characterization of solution ambiguity beyond global measures.
• To introduce an entrywise condition number for improved sensitivity analysis.

Main Methods:

• Derivation of novel theoretical lower and upper bounds for individual solution vector entries.
• Analysis of bounds for data-consistent solutions, independent of noise statistics and reconstruction methods.
• Introduction of an entrywise condition number based on derived bounds.

Main Results:

• New tight, entrywise bounds are established for ill-posed inverse problems.
• The derived bounds are agnostic to noise statistics and chosen solution methods.
• An entrywise condition number is introduced, offering nuanced characterization of solution sensitivity.

Conclusions:

• The novel bounds and entrywise condition number offer significant improvements in characterizing ill-posed inverse problems.
• These tools provide deeper insights into solution ambiguity and sensitivity for specific solution components.
• The findings have direct applications in areas like magnetic resonance imaging reconstruction.