We study robust regression under a contamination model in which covariates are clean while the responses may be corrupted in an adaptive manner. Unlike the classical Huber's contamination model, where both covariates and responses may be contaminated and consistent estimation is impossible when the contamination proportion is a non-vanishing constant, it turns out that the clean-covariate setting admits strictly improved statistical guarantees. Specifically, we show that the additional information in the clean covariates can be carefully exploited to construct an estimator that achieves a better estimation rate than that attainable under Huber contamination. In contrast to the Huber model, this improved rate implies consistency even when the contamination is a constant. A matching minimax lower bound is established using Fano's inequality together with the construction of contamination processes that match $m> 2$ distributions simultaneously, extending the previous two-point lower bound argument in Huber's setting. Despite the improvement over the Huber model from an information-theoretic perspective, we provide formal evidence -- in the form of Statistical Query and Low-Degree Polynomial lower bounds -- that the problem exhibits strong information-computation gaps. Our results strongly suggest that the information-theoretic improvements cannot be achieved by polynomial-time algorithms, revealing a fundamental gap between information-theoretic and computational limits in robust regression with clean covariates.

This paper studies the statistical and computational limits of high-order clustering with planted structures. We focus on two clustering models, constant high-order clustering (CHC) and rank-one higher-order clustering (ROHC), and study the methods and theory for testing whether a cluster exists (detection) and identifying the support of cluster (recovery). Specifically, we identify the sharp boundaries of signal-to-noise ratio for which CHC and ROHC detection/recovery are statistically possible. We also develop the tight computational thresholds: when the signal-to-noise ratio is below these thresholds, we prove that polynomial-time algorithms cannot solve these problems under the computational hardness conjectures of hypergraphic planted clique (HPC) detection and hypergraphic planted dense subgraph (HPDS) recovery. We also propose polynomial-time tensor algorithms that achieve reliable detection and recovery when the signal-to-noise ratio is above these thresholds. Both sparsity and tensor structures yield the computational barriers in high-order tensor clustering. The interplay between them results in significant differences between high-order tensor clustering and matrix clustering in literature in aspects of statistical and computational phase transition diagrams, algorithmic approaches, hardness conjecture, and proof techniques. To our best knowledge, we are the first to give a thorough characterization of the statistical and computational trade-off for such a double computational-barrier problem. Finally, we provide evidence for the computational hardness conjectures of HPC detection and HPDS recovery.