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Article overview
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Compressed Super-Resolution I: Maximal Rank Sum-of-Squares | | Augustin Cosse
; | | Date: |
1 Jan 2020 | | Abstract: | Let $mu(t) = sum_{ auin S} alpha_ au delta(t- au)$ denote an
$|S|$-atomic measure defined on $[0,1]$, satisfying $min_{ au
eq au’}| au
- au’|geq |S|cdot n^{-1}$. Let $eta( heta) = sum_{ auin S} a_ au
D_n( heta - au) + b_ au D’_n( heta - au)$, denote the polynomial
obtained from the Dirichlet kernel $D_n( heta) = frac{1}{n+1}sum_{|k|leq n}
e^{2pi i k heta}$ and its derivative by solving the system
$left{eta( au) = 1, eta’( au) = 0,; forall au in S
ight}$. We
provide evidence that for sufficiently large $n$, $Delta> |S|^2 n^{-1}$, the
non negative polynomial $1 - |eta( heta)|^2$ which vanishes at the atoms
$ au in S$, and is bounded by $1$ everywhere else on the $[0,1]$ interval,
can be written as a sum-of-squares with associated Gram matrix of rank $n-|S|$.
Unlike previous work, our approach does not rely on the Fejer-Riesz Theorem,
which prevents developing intuition on the Gram matrix, but requires instead a
lower bound on the singular values of a (truncated) large ($O(1e10)$) matrix.
Despite the memory requirements which currently prevent dealing with such a
matrix efficiently, we show how such lower bounds can be derived through Power
iterations and convolutions with special functions for sizes up to $O(1e7)$. We
also provide numerical simulations suggesting that the spectrum remains
approximately constant with the truncation size as soon as this size is larger
than $100$. | | Source: | arXiv, 2001.1644 | | Services: | Forum | Review | PDF | Favorites | |
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