Consider L groups of point sources or spike trains, with the lth group represented by \(x_l(t)\). For a function \(g:\mathbb {R}\rightarrow \mathbb {R}\), let \(g_l(t) = g(t/\mu _l)\) denote a point spread function with scale \(\mu _l > 0\), and with \(\mu _1< \cdots < \mu _L\). With \(y(t) = \sum _{l=1}^{L} (g_l \star x_l)(t)\), our goal is to recover the source parameters given samples of y, or given the Fourier samples of y. This problem is a generalization of the usual super-resolution setup wherein \(L = 1\); we call this the multi-kernel unmixing super-resolution problem. Assuming access to Fourier samples of y, we derive an algorithm for this problem for estimating the source parameters of each group, along with precise non-asymptotic guarantees. Our approach involves estimating the group parameters sequentially in the order of increasing scale parameters, i.e., from group 1 to L. In particular, the estimation process at stage \(1 \le l \le L\) involves (i) carefully sampling the tail of the Fourier transform of y, (ii) a deflation step wherein we subtract the contribution of the groups processed thus far from the obtained Fourier samples, and (iii) applying Moitra’s modified Matrix Pencil method on a deconvolved version of the samples in (ii).

中文翻译：

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改进的矩阵铅笔法实现多核分解和超分辨率

考虑L组点源或尖峰列，第l组由\（x_l（t）\）表示。对于函数\（g：\ mathbb {R} \ rightarrow \ mathbb {R} \），令\（g_l（t）= g（t / \ mu _l）\）表示标度为\（\ mu _l> 0 \）并带有\（\ mu _1 <\ cdots <\ mu _L \）。使用\（y（t）= \ sum _ {l = 1} ^ {L}（g_l \ star x_l）（t）\），我们的目标是在给定y样本或给定Fourier样本的情况下恢复源参数。的ÿ。此问题是通常的超分辨率设置的一般化，其中\（L = 1 \）; 我们称此为多核分解超分辨率问题。假设访问y的傅立叶样本，我们导出了针对此问题的算法，用于估计每个组的源参数以及精确的非渐近保证。我们的方法涉及在规模越来越大的参数，即顺序依次估计所述组参数，从组1到大号。特别地，阶段\（1 \ le l \ le L \）的估计过程涉及（i）仔细采样y的Fourier变换的尾部，（ii）放气 步骤，其中我们从获得的傅里叶样本中减去到目前为止已处理的组的贡献，并且（iii）在（ii）中对样本的反卷积版本应用Moitra的改进的Matrix Pencil方法。