Supplement to ’ Sparse recovery by thresholded non - negative least squares ’

We here provide additional proofs, definitions, lemmas and derivations omitted in the paper. Note that material contained in the latter are referred to by the captions used there (e.g. Theorem 1), whereas auxiliary statements contained exclusively in this supplement are preceded by a capital Roman letter (e.g. Theorem A.1). A Sub-Gaussian random variables and concentration inequalities A random variable Z is called sub-Gaussian if there exists a positive constant K such that E[|Z|] ≤ K√q. The smallest suchK is called the sub-Gaussian norm ‖Z‖ψ2 ofZ. If E[Z] = 0, which shall be assumed for the remainder of this paragraph, then the moment-generating function of Z satisfies E[exp(tZ)] ≤ exp(−t/(2σ)) for a parameter σ > 0 which is related to ‖Z‖ψ2 by a multiplicative constant, cf. [1]. It follows that if Z1, . . . , Zn are i.i.d. copies of Z and v ∈ R, then ∑n i=1 viZi is sub-Gaussian with parameter ‖v‖ 2 2 σ . We have the well-known tail bound P(|Z| > z) ≤ 2 exp ( − z 2 2σ2 ) , z ≥ 0. (A.1) Combining the previous two facts and using a union bound, with Z = (Z1, . . . , Zn), it follows that for any collection of vectors vj ∈ R, j = 1, . . . , p, P ( max 1≤j≤p |v> j Z| > σ max 1≤j≤p ‖vj‖2 √ 2 log p+ σz ) ≤ 2 exp ( − 2 z ) , z ≥ 0. (A.2) A.1 Bernstein-type inequality for squared sub-Gaussian random variables The following exponential inequality combines Lemma 14, Proposition 16 and Remark 18 in [1]. Lemma A. 1. Let Z1, . . . , Zn be i.i.d. centered sub-Gaussian random variables with sub-Gaussian norm K. Then for every a = (a1, . . . , an) ∈ R and every z ≥ 0, one has