Causal inference on time series data is a challenging problem, especially in
the presence of unobserved confounders. This work focuses on estimating the
causal effect between two time series, which are confounded by a third,
unobserved time series. Assuming spectral sparsity of the confounder, we show
how in the frequency domain this problem can be framed as an adversarial
outlier problem. We introduce Deconfounding by Robust regression (DecoR), a
novel approach that estimates the causal effect using robust linear regression
in the frequency domain. Considering two different robust regression
techniques, we first improve existing bounds on the estimation error for such
techniques. Crucially, our results do not require distributional assumptions on
the covariates. We can therefore use them in time series settings. Applying
these results to DecoR, we prove, under suitable assumptions, upper bounds for
the estimation error of DecoR that imply consistency. We show DecoR's
effectiveness through experiments on synthetic data. Our experiments
furthermore suggest that our method is robust with respect to model
misspecification.

通过在频域内使用鲁棒线性回归，估计由第三个未观测时间序列混淆的两个时间序列之间的因果效应。

DecoR：使用鲁棒回归解除时间序列中的混杂

DecoR: Deconfounding Time Series with Robust Regression

The paper studies the problem of recovering a spectrally sparse object from a
small number of time domain samples. Specifically, the object of interest with
ambient dimension $n$ is assumed to be a mixture of $r$ complex
multi-dimensional sinusoids, while the underlying frequencies can assume any
value in the unit disk. Conventional compressed sensing paradigms suffer from
the {\em basis mismatch} issue when imposing a discrete dictionary on the
Fourier representation. To address this problem, we develop a novel
nonparametric algorithm, called enhanced matrix completion (EMaC), based on
structured matrix completion. The algorithm starts by arranging the data into a
low-rank enhanced form with multi-fold Hankel structure, then attempts recovery
via nuclear norm minimization. Under mild incoherence conditions, EMaC allows
perfect recovery as soon as the number of samples exceeds the order of
$\mathcal{O}(r\log^{2} n)$. We also show that, in many instances, accurate
completion of a low-rank multi-fold Hankel matrix is possible when the number
of observed entries is proportional to the information theoretical limits
(except for a logarithmic gap). The robustness of EMaC against bounded noise
and its applicability to super resolution are further demonstrated by numerical
experiments.

首次提出了一种新的增强矩阵完成（EMaC）算法，它使用多倍 Hankel 结构将数据排列成低秩增强形式，能够恢复具有谱稀疏性的对象，其底层频率可以在单位圆中具有任何值，该算法对有限噪声和超分辨率具有去噪和应用优势。