Lower bounds for integration and recovery in L2

On lower bounds for the L2-discrepancy

The L2-discrepancy measures the irregularity of the distribution of a finite point set. In this note, we prove lower bounds for the L2-discrepancy of arbitrary N-point sets. Our main focus is on the two-dimensional case. Asymptotic upper and lower estimates of the L2-discrepancy in dimension 2 are well known, and are of the sharp order √ logN . Nevertheless, the gap in the constants between the...

Algorithms and Lower Bounds for Sparse Recovery

We consider the following k-sparse recovery problem: design a distribution of m× n matrix A, such that for any signal x, given Ax with high probability we can efficiently recover x̂ satisfying ‖x− x̂‖1 ≤ C mink-sparse x′ ‖x− x‖1. It is known that there exist such distributions with m = O(k log(n/k)) rows; in this thesis, we show that this bound is tight. We also introduce the set query algorithm,...

Some Lower Bounds for Estrada Index

Upper and lower bounds for numerical radii of block shifts

For an n-by-n complex matrix A in a block form with the (possibly) nonzero blocks only on the diagonal above the main one, we consider two other matrices whose nonzero entries are along the diagonal above the main one and consist of the norms or minimum moduli of the diagonal blocks of A. In this paper, we obtain two inequalities relating the numeical radii of these matrices and also determine ...

Lower Bounds for Adaptive Sparse Recovery

We give lower bounds for the problem of stable sparse recovery from adaptive linear measurements. In this problem, one would like to estimate a vector x ∈ R from m linear measurements A1x, . . . , Amx. One may choose each vector Ai based on A1x, . . . , Ai−1x, and must output x̂ satisfying ‖x̂− x‖p ≤ (1 + ) min k-sparse x′ ‖x− x‖p with probability at least 1−δ > 2/3, for some p ∈ {1, 2}. For p = ...