Learning and 1-bit Compressed Sensing under Asymmetric Noise

We study the approximate recovery problem: Given corrupted 1-bit measurements of the form sign(w^*⋅x_i), recover a vector w that is a good approximation to w^* ∈ R^d. This problem has been studied by both the learning theory and signal processing communities. In learning theory, this is known as the problem of learning halfspaces with noise, and in signal processing, as 1-bit compressed sensing, in which there is an additional assumption that w^* is t-sparse. The challenge in both cases is to design computationally efficient algorithms that are tolerant to large amounts of noise under realistic noise models. Furthermore, in the case of 1-bit compressed sensing, we require the number of measurements x_i to scale polynomially in t and only polylogarithmically in d, the ambient dimension. In this work, we introduce algorithms with nearly optimal guarantees for both problems under two realistic noise models, bounded (Massart) noise and adversarial (agnostic) noise, when the measurements x_i’s are drawn from any isotropic log-concave distribution. In bounded (Massart) noise, an adversary can flip the measurement of each point x with probability η(x)≤η