On the well-spread property and its relation to linear regression
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We consider the robust linear regression model y = Xβ^* + η, where an adversary oblivious to the design X ∈ℝ^n × d may choose η to corrupt all but a (possibly vanishing) fraction of the observations y in an arbitrary way. Recent work [dLN+21, dNS21] has introduced efficient algorithms for consistent recovery of the parameter vector. These algorithms crucially rely on the design matrix being well-spread (a matrix is well-spread if its column span is far from any sparse vector). In this paper, we show that there exists a family of design matrices lacking well-spreadness such that consistent recovery of the parameter vector in the above robust linear regression model is information-theoretically impossible. We further investigate the average-case time complexity of certifying well-spreadness of random matrices. We show that it is possible to efficiently certify whether a given n-by-d Gaussian matrix is well-spread if the number of observations is quadratic in the ambient dimension. We complement this result by showing rigorous evidence – in the form of a lower bound against low-degree polynomials – of the computational hardness of this same certification problem when the number of observations is o(d^2).
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