论文标题
压缩超分辨率I：最大排名总和
Compressed Super-Resolution I: Maximal Rank Sum-of-Squares
论文作者
论文摘要
令$μ（t）= \ sum_ {τ\ in s}α_τδ（t-τ）$表示A $ | s | $ - $ - $ -ATOMIC度量定义在$ [0,1] $上，满足$ \ min_ {τ\ \ \ \ neq neq谢}} |令$η（θ）= \ sum_ {τ\ in s}a_τd_n（θ-τ） +b_τd'_n（θ-τ）$，表示从dirichlet内核$ d_n（θ）= \ frac = \ frac {n + 1} {n + 1} {n + 1} {n + sum__ { kθ} $及其导数通过求解系统$ \ left \ {η（τ）= 1，η'（τ）= 0，\; \ forallτ\在s \ right \} $中。 We provide evidence that for sufficiently large $n$, $Δ> |S|^2 n^{-1}$, the non negative polynomial $1 - |η(θ)|^2$ which vanishes at the atoms $τ\in S$, 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 | $。与以前的工作不同，我们的方法不依赖于Fejer-Riesz定理，该定理可以防止在革兰氏矩阵上发展直觉，而需要对（截断）大（$ o（$ o（1e10）$）矩阵的单数值进行下限。尽管记忆要求目前阻止有效地处理此类矩阵，但我们还展示了如何通过功率迭代和具有特殊功能的特殊功能来得出这样的低界限，最高为$ O（1E7）$。我们还提供数值模拟，表明频谱在此尺寸大于$ 100 $的情况下，截断尺寸与截断尺寸保持近乎恒定。
Let $μ(t) = \sum_{τ\in S} α_τδ(t-τ)$ denote an $|S|$-atomic measure defined on $[0,1]$, satisfying $\min_{τ\neq τ'}|τ- τ'|\geq |S|\cdot n^{-1}$. Let $η(θ) = \sum_{τ\in S} a_τD_n(θ- τ) + b_τD'_n(θ- τ)$, denote the polynomial obtained from the Dirichlet kernel $D_n(θ) = \frac{1}{n+1}\sum_{|k|\leq n} e^{2πi k θ}$ and its derivative by solving the system $\left\{η(τ) = 1, η'(τ) = 0,\; \forall τ\in S\right\}$. We provide evidence that for sufficiently large $n$, $Δ> |S|^2 n^{-1}$, the non negative polynomial $1 - |η(θ)|^2$ which vanishes at the atoms $τ\in S$, 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 $100$.
