An Efficient Sparse Identification Algorithm For Stochastic Systems With General Observation Sequences

The paper’s Algorithm 1 selects thresholds {α_n} with sqrt(log R_n / λ_min^n) = o(α_n) (Step 0), runs recursive LS (Step 1), then hard-thresholds to produce β_{n+1} (Step 2) . Under Assumptions 1–2, it proves (i) β_{n+1}(l) → θ(l) a.s. (Theorem 1) and (ii) finite-time correct identification of the zero set H_{N+1} = H* for all N≥N0(ω) (Theorem 2), relying on the standard LS error bound ∥θ_{n+1}−θ∥^2 ≤ C0·(log R_n)/λ_min^n (Lemma 1) and the chosen α_n scaling . The candidate solution assumes the same LS bound and non-excited condition, chooses α_n with ρ_n := sqrt(log R_n/λ_min^n) = o(α_n), and then proves (a) parameter convergence and (b) finite-time support recovery by showing zero coordinates are eventually thresholded to 0 and nonzeros stay above α_n. This mirrors the paper’s logic and uses the same key ingredients; the argument is slightly more explicit (e.g., introducing c_min = min_{i≤d}|θ(i)|>0) but imposes no extra substantive assumptions. Hence both are correct, and the proofs are substantially the same.