Eugene Livshitz

Eugene Livshitz, Vladimir Temlyakov

We study sparse approximation by greedy algorithms. Our contribution is two-fold. First, we prove exact recovery with high probability of random $K$-sparse signals within $\lceil K(1+\e)\rceil$ iterations of the Orthogonal Matching Pursuit (OMP). This result shows that in a probabilistic sense the OMP is almost optimal for exact recovery. Second, we prove the Lebesgue-type inequalities for the Weak Chebyshev Greedy Algorithm, a generalization of the Weak Orthogonal Matching Pursuit to the case of a Banach space. The main novelty of these results is a Banach space setting instead of a Hilbert space setting. However, even in the case of a Hilbert space our results add some new elements to known results on the Lebesque-type inequalities for the RIP dictionaries. Our technique is a development of the recent technique created by Zhang.

Eugene Livshitz

We show that if a matrix $Φ$ satisfies the RIP of order $[CK^{1.2}]$ with isometry constant $\dt = c K^{-0.2}$ and has coherence less than $1/(20 K^{0.8})$, then Orthogonal Matching Pursuit (OMP) will recover $K$-sparse signal $x$ from $y=Φx$ in at most $[CK^{1.2}]$ iterations. This result implies that $K$-sparse signal can be recovered via OMP by $M=O(K^{1.6}\log N)$ measurements.

Eugene Livshitz

We show that Orthogonal Greedy Algorithms (Orthogonal Matching Pursuit) provides almost optimal approximation on the first [1/(20M)] steps for M-coherent dictionaries

Eugene Livshitz

We discuss the upper and lower estimates for the rate of convergence of Pure and Orthogonal Greedy Algorithms for dictionary with bounded cumulative coherence.