In 1-bit compressed sensing, the aim is to estimate a $k$-sparse unit vector $x\in S^{n-1}$ within an $\epsilon$ error (in $\ell_2$) from minimal number of linear measurements that are quantized to just their signs, i.e., from measurements of the form $y = \mathrm{Sign}(\langle a, x\rangle).$ In this paper, we study a noisy version where a fraction of the measurements can be flipped, potentially by an adversary. In particular, we analyze the Binary Iterative Hard Thresholding (BIHT) algorithm, a proximal gradient descent on a properly defined loss function used for 1-bit compressed sensing, in this noisy setting. It is known from recent results that, with $\tilde{O}(\frac{k}{\epsilon})$ noiseless measurements, BIHT provides an estimate within $\epsilon$ error. This result is optimal and universal, meaning one set of measurements work for all sparse vectors. In this paper, we show that BIHT also provides better results than all known methods for the noisy setting. We show that when up to $\tau$-fraction of the sign measurements are incorrect (adversarial error), with the same number of measurements as before, BIHT agnostically provides an estimate of $x$ within an $\tilde{O}(\epsilon+\tau)$ error, maintaining the universality of measurements. This establishes stability of iterative hard thresholding in the presence of measurement error. To obtain the result, we use the restricted approximate invertibility of Gaussian matrices, as well as a tight analysis of the high-dimensional geometry of the adversarially corrupted measurements.

翻译：在1比特压缩感知中，目标是从线性测量量的符号信息（即形如$y = \mathrm{Sign}(\langle a, x\rangle)$的测量值）中，以最小测量次数估计出一个$k$-稀疏单位向量$x\in S^{n-1}$，并使其误差（$\ell_2$范数）控制在$\epsilon$以内。本文研究了一种含噪版本，其中部分测量值可能被翻转（甚至可能由对手恶意操控）。具体而言，我们分析了二元迭代硬阈值算法（BIHT）——该算法是针对1比特压缩感知所定义的恰当损失函数的近端梯度下降法——在含噪情形下的性能。根据近期研究结果，当使用$\tilde{O}(\frac{k}{\epsilon})$次无噪声测量时，BIHT能够提供误差在$\epsilon$以内的估计。该结果具有最优性和普适性，即一组测量值适用于所有稀疏向量。本文进一步证明，在含噪情形下，BIHT的性能优于所有已知方法。当符号测量值中最多有$\tau$比例存在错误（对抗性误差）时，在保持相同测量次数的情况下，BIHT能够以不可知方式提供误差不超过$\tilde{O}(\epsilon+\tau)$的估计值$x$，并维持测量值的普适性。这证实了迭代硬阈值方法在测量误差存在下的稳定性。为得到该结果，我们利用了高斯矩阵的受限近似可逆性，并对对抗性破坏测量值的高维几何结构进行了严谨分析。