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- W4295847366 abstract "Abstract The paper considers the problem of unique recovery of sparse finite-valued integer signals using a single linear integer measurement. For l -sparse signals in $${mathbb {Z}}^n$$ <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML> <mml:msup> <mml:mrow> <mml:mi>Z</mml:mi> </mml:mrow> <mml:mi>n</mml:mi> </mml:msup> </mml:math> , $$2l<n$$ <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML> <mml:mrow> <mml:mn>2</mml:mn> <mml:mi>l</mml:mi> <mml:mo><</mml:mo> <mml:mi>n</mml:mi> </mml:mrow> </mml:math> , with absolute entries bounded by r , we construct an $$1times n$$ <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML> <mml:mrow> <mml:mn>1</mml:mn> <mml:mo>×</mml:mo> <mml:mi>n</mml:mi> </mml:mrow> </mml:math> measurement matrix with maximum absolute entry $$Delta =O(r^{2l-1})$$ <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML> <mml:mrow> <mml:mi>Δ</mml:mi> <mml:mo>=</mml:mo> <mml:mi>O</mml:mi> <mml:mo>(</mml:mo> <mml:msup> <mml:mi>r</mml:mi> <mml:mrow> <mml:mn>2</mml:mn> <mml:mi>l</mml:mi> <mml:mo>-</mml:mo> <mml:mn>1</mml:mn> </mml:mrow> </mml:msup> <mml:mo>)</mml:mo> </mml:mrow> </mml:math> . Here the implicit constant depends on l and n and the exponent $$2l-1$$ <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML> <mml:mrow> <mml:mn>2</mml:mn> <mml:mi>l</mml:mi> <mml:mo>-</mml:mo> <mml:mn>1</mml:mn> </mml:mrow> </mml:math> is optimal. Additionally, we show that, in the above setting, a single measurement can be replaced by several measurements with absolute entries sub-linear in $$Delta$$ <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML> <mml:mi>Δ</mml:mi> </mml:math> . The proofs make use of results on admissible $$(n-1)$$ <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML> <mml:mrow> <mml:mo>(</mml:mo> <mml:mi>n</mml:mi> <mml:mo>-</mml:mo> <mml:mn>1</mml:mn> <mml:mo>)</mml:mo> </mml:mrow> </mml:math> -dimensional integer lattices for m -sparse n -cubes that are of independent interest." @default.
- W4295847366 created "2022-09-15" @default.
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- W4295847366 date "2022-09-13" @default.
- W4295847366 modified "2023-10-01" @default.
- W4295847366 title "On unique recovery of finite-valued integer signals and admissible lattices of sparse hypercubes" @default.
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- W4295847366 doi "https://doi.org/10.1007/s11590-022-01927-0" @default.
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