Certifying Euclidean Sections and Finding Planted Sparse Vectors Beyond the √(n) Dimension Threshold

We consider the task of certifying that a random d-dimensional subspace X in ℝ^n is well-spread - every vector x ∈ X satisfies c√(n)x_2 ≤x_1 ≤√(n)x_2. In a seminal work, Barak et. al. showed a polynomial-time certification algorithm when d ≤ O(√(n)). On the other hand, when d ≫√(n), the certification task is information-theoretically possible but there is evidence that it is computationally hard [MW21,Cd22], a phenomenon known as the information-computation gap. In this paper, we give subexponential-time certification algorithms in the d ≫√(n) regime. Our algorithm runs in time exp(O(n^ε)) when d ≤O(n^(1+ε)/2), establishing a smooth trade-off between runtime and the dimension. Our techniques naturally extend to the related planted problem, where the task is to recover a sparse vector planted in a random subspace. Our algorithm achieves the same runtime and dimension trade-off for this task.