1. What are the contributions in "Lossless analog compression" ?
The authors establish the fundamental limits of lossless analog compression by considering the recovery of arbitrary random vectors x R m from the noiseless linear measurements y =. Specifically, their achievability result states that, for Lebesgue-almost all A, the random vector x can be recovered with zero error probability provided that n > K ( x ), where K ( x ) is given by the infimum of the lower modified Minkowski dimension over all support sets U of x ( i. e., sets U R m with P [ x U ] = 1 ).. The authors then particularize this achievability result to the class of s-rectifiable random vectors as introduced in Koliander et al. ( 2016 ) ; these are random vectors of absolutely continuous distributionwith respect to the s-dimensional Hausdorff measure-supported on countable unions of s-dimensional C 1 -submanifolds of Rm. Countable unions of C 1 -submanifolds include essentially all signal models used in the compressed sensing literature such as the standard union of subspaces model underlying much of compressed sensing theory and spectrum-blind sampling, smooth submanifolds, block-sparsity, and low-rank matrices as considered in the matrix completion problem.. Specifically, the authors prove that, for Lebesgue-almost all A, s-rectifiable random vectors x can be recovered with zero error probability from n > s linear measurements.. Motivated by this observation, the authors introduce the new class of s-analytic random vectors, which admit a strong converse in the sense of n s being necessary for recovery with probability of error smaller than one.. FA ] 1 7 Ju l 2 01 9 1 Lossless Analog Compression Giovanni Alberti, Helmut Bölcskei, Camillo De Lellis, Günther Koliander, and Erwin Riegler Abstract—We establish the fundamental limits of lossless analog compression by considering the recovery of arbitrary random vectors x ∈ R from the noiseless linear measurements y =. Specifically, their achievability result states that, for Lebesgue-almost all A, the random vector x can be recovered with zero error probability provided that n > K ( x ), where K ( x ) is given by the infimum of the lower modified Minkowski dimension over all support sets U of x ( i. e., sets U ⊆ R with P [ x ∈ U ] = 1 ).. The authors then particularize this achievability result to the class of s-rectifiable random vectors as introduced in Koliander et al. ( 2016 ) ; these are random vectors of absolutely continuous distribution—with respect to the sdimensional Hausdorff measure—supported on countable unions of s-dimensional C-submanifolds of R. Countable unions of C -submanifolds include essentially all signal models used in the compressed sensing literature such as the standard union of subspaces model underlying much of compressed sensing theory and spectrum-blind sampling, smooth submanifolds, block-sparsity, and low-rank matrices as considered in the matrix completion problem.. Specifically, the authors prove that, for Lebesgue-almost all A, s-rectifiable random vectors x can be recovered with zero error probability from n > s linear measurements.. Motivated by this observation, the authors introduce the new class of s-analytic random vectors, which admit a strong converse in the sense of n ≥ s being necessary for recovery with probability of error smaller than one.. The authors establish the fundamental limits of lossless analog compression by considering the recovery of arbitrary random vectors x ∈ R from the noiseless linear measurements y =. Specifically, their achievability result states that, for Lebesgue-almost all A, the random vector x can be recovered with zero error probability provided that n > K ( x ), where K ( x ) is given by the infimum of the lower modified Minkowski dimension over all support sets U of x ( i. e., sets U ⊆ R with P [ x ∈ U ] = 1 ).. The authors then particularize this achievability result to the class of s-rectifiable random vectors as introduced in Koliander et al. ( 2016 ) ; these are random vectors of absolutely continuous distribution—with respect to the sdimensional Hausdorff measure—supported on countable unions of s-dimensional C-submanifolds of R. Countable unions of C -submanifolds include essentially all signal models used in the compressed sensing literature such as the standard union of subspaces model underlying much of compressed sensing theory and spectrum-blind sampling, smooth submanifolds, block-sparsity, and low-rank matrices as considered in the matrix completion problem.. Specifically, the authors prove that, for Lebesgue-almost all A, s-rectifiable random vectors x can be recovered with zero error probability from n > s linear measurements.. Motivated by this observation, the authors introduce the new class of s-analytic random vectors, which admit a strong converse in the sense of n ≥ s being necessary for recovery with probability of error smaller than one.
read more
2. What is the case with s-rectifiable random vectors?
Since s-rectifiable random vectors cannot have positive probability measure on sets of Hausdorff dimension t < s, it is natural to ask whether taking n ≥ s linear measurements is necessary for zero error recovery of s-rectifiable random vectors.
read more
3. What is the sanalytic error probability of a linear measurement?
Random vectors that are both sanalytic and s-rectifiable can be recovered with zero error probability from n > s linear measurements, and n ≥ s linear measurements are necessary for recovery with error probabilitysmaller than one.
read more
4. What is the proof for x Rm s-analytic?
Rm s-analytic, n ≥ s is necessary for the existence of a measurement matrix A ∈ Rn×m and a Borel measurable mapping g : Rn×m × Rn →
read more
![Figure 3. ([38, Figure 3.3]) Graph of H d(U) as a function of d ∈ [0,m] for a set U ⊆ Rm.](/figures/figure-3-38-figure-3-3-graph-of-h-d-u-as-a-function-of-d-0-m-2hvde4ve.webp)
