Abstract
We consider the signal detection problem in the Gaussian design trace regression model with low rank alternative hypotheses. We derive the precise (Ingster-type) detection boundary for the Frobenius and the nuclear norm. We then apply these results to show that honest confidence sets for the unknown matrix parameter that adapt to all low rank sub-models in nuclear norm do not exist. This shows that recently obtained positive results in [5] for confidence sets in low rank recovery problems are essentially optimal.
Highlights
Consider the Gaussian design trace regression modelYi = tr(Xiθ) + i, i = 1, . . . , n, (1)where ∼ N (0, In) is an i.i.d. vector of Gaussian noise
We are interested in the case where the model dimension d2 is possibly large compared to sample size n, but where θ has low rank k, in which case we write θ ∈ R(k), 1 ≤ k ≤ d. This setting serves as a prototype for various matrix inference problems such as those occurring in compressed sensing [4] or in quantum tomography [7]
The first problem we study in this paper is the signal detection problem with low-rank alternatives: We want to test the hypothesis
Summary
Where ∼ N (0, In) is an i.i.d. vector of Gaussian noise. Here the matrices Xi are d × d square matrices with i.i.d. entries Xmi k ∼ N (0, 1), and θ is the unknown. [5] constructed another confidence set whose diameter adapts to low rank sub-models in the stronger nuclear norm distance, and that is honest for all θ’s that are non-negative definite and have trace equal to one, that is, whenever θ is the density matrix of a quantum state Such a constraint on θ is natural in a quantum physics context considered in [5], but not in general. In the present paper we show that the existence results of [5] are specific to the geometry induced by the Frobenius norm or to the quantum state constraint, and that nuclear-norm adaptive and honest confidence sets over general low rank parameter spaces do not exist in the model (1). Generalisations of our results to the matrix completion problem are currently under investigation
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