Online Group-Structured Dictionary Learning Supplementary Material

Lemma 1. If ρ = 0, then Mt = M0 +M′t (∀t ≥ 1). When ρ > 0, then Mt = M ′ t (∀t ≥ 1). Proof. 1. Case ρ = 0: Since γt = 1 (∀t ≥ 1), thus Mt = M0 + ∑t i=1 Ni. We also have that ( i t 0 = 1 (∀i ≥ 1), and therefore M′t = ∑t i=1 Ni, which completes the proof. 2. Case ρ > 0: The proof proceeds by induction. • t = 1: In this case γ1 = 0, M1 = 0 × M0 + N1 = N1 and M′1 = N1, which proves that M1 = M ′ 1. • t > 1: Using the definitions of Mt and M ′ t, and exploiting the fact that Mt−1 = M ′ t−1 by induction, after some calculation we have that: