BBP Phase Transition for a Doubly Sparse Deformed Model

We prove the equivalent of the Baik, Ben Arous, Péché (2004) phenomenon for a novel, doubly sparse model where both the Wigner noise matrix and signal vector(s) are sparse. Specifically, we consider a deformed sub-Gaussian sparse Wigner ensemble with a fixed number of sub-Gaussian spike vectors of the same-order sparsity added. We show that spike vectors with signals greater than one are correlated with the top eigenvectors of the deformed ensemble and that each spike vector of signal greater than one induces an outlier eigenvalue. Notably, our results hold in the supercritical sparsity regime for the Wigner matrix ($q \gg \frac{\log n}{n}$) and for any sparse spike vector with an unbounded number of entries ($np\to \infty$). No further relationship between the sparsities of the noise matrix ($q$) and spike vectors ($p$) is necessary. This generalizes the work of Benaych-Georges and Nadakuditi (2010) and Péché (2005).

💡 Research Summary

The paper establishes a Baik‑Ben‑Arous‑Péché (BBP) phase transition for a “doubly sparse” spiked random matrix model in which both the noise matrix and the signal vectors are sparse. The authors consider a deformed Wigner ensemble: a symmetric matrix W whose off‑diagonal entries are independent sub‑Gaussian random variables multiplied by independent Bernoulli(q) masks, with q≫(log n)/n (the super‑critical sparsity regime). The signal consists of a fixed number r of spike vectors v^{(ℓ)} (ℓ=1,…,r). Each spike is generated by multiplying independent sub‑Gaussian entries with independent Bernoulli(p) masks, where the expected number of non‑zero entries np diverges (np→∞) but p itself may be as small as any function satisfying this condition. No orthogonal invariance is assumed for either the noise or the spikes, and no relationship between q and p is required.

The main results are twofold. First, a distinguishability theorem: if the signal strength θ_ℓ≤1 for all spikes, the top eigenvalue λ₁ of the deformed matrix X = ∑_{ℓ}θ_ℓ v^{(ℓ)}(v^{(ℓ)})ᵀ + W converges almost surely to the edge of the semicircle law (2), and no outlier eigenvalues appear. If any θ_ℓ>1, then λ₁ converges almost surely to θ_ℓ + 1/θ_ℓ > 2, producing a detectable outlier. Second, a recovery theorem: for each spike with θ_ℓ>1, the corresponding top eigenvector u₁ aligns with the spike, in the sense that the squared inner product ⟨u₁, v^{(ℓ)}⟩² converges almost surely to 1 − 1/θ_ℓ² > 0. Thus, PCA not only detects the presence of a spike but also yields a non‑trivial estimator of its direction.

To prove these statements, the authors combine several recent advances in random matrix theory. They invoke a result (AB26) that bounds the operator norm of a sparse sub‑Gaussian Wigner matrix in the regime q≫log n/n, showing ‖W‖=O(√(q n)) with high probability. They also adapt a local law for sparse Wigner matrices (HWZ26) to the same sparsity regime, establishing that the empirical spectral distribution follows the semicircle law down to scales of order 1/(q n). From these tools they derive a “no‑outlier” lemma for the pure noise case, guaranteeing that all eigenvalues lie within