bernoulli_restricted_sampling_injective_under_general_sample_bound

Role. It belongs to the tangent-space injectivity branch, where concentration of the sampled tangent operator rules out nonzero feasible tangent perturbations.

Problem and notation. Exact matrix completion asks when an unknown low-rank real matrix can be recovered from a random subset of its entries. Here $M\in\mathbb R^{n_1\times n_2}$ has rank $r$, $m$ entries are observed, and $n=\max(n_1,n_2)$. Recovery means nuclear-norm minimization: minimize $\|X\|_*$ among matrices $X$ agreeing with $M$ on the observed entries. Probability notation. $\operatorname{successProb}(m,M)$ is the fixed-cardinality success probability: $\Omega$ is chosen uniformly among all subsets of $n_1n_2$ entries with $|\Omega|=m$, and the event is that the convex program uniquely returns $M$. In Bernoulli nodes, $\mathbb P_p(E)$ or $\operatorname{bernoulliEventProb}(p,E)$ means each entry is sampled independently with probability $p$, usually $p=m/(n_1n_2)$. Coherence notation. The object $S$ records SVD/singular-vector data for $M$. The hypotheses $A0(S,\mu_0)$ and $A1(S,\mu_1)$ are the Candes-Recht incoherence assumptions: $\mu_0$ measures how spread out the singular vector spaces are, and $\mu_1$ measures the largest entry of the sign matrix $UV^\top$. The parameter $\beta>2$ controls polynomial failure probabilities such as $n^{-\beta}$. For tangent-space nodes, the main event is that the sampled tangent operator is well conditioned; this prevents a nonzero tangent perturbation from agreeing with M on the sampled entries.

Claim. Under the general Candes-Recht sample lower bound, Bernoulli sampling is injective on the tangent space with high probability. This packages the operator concentration/invertibility part of Section 4.2 in the same sample-complexity regime used by Theorem 1.3.

Lecture-note formulation:

The constants in this node are universal existential constants; the theorem asserts that some positive constants with these roles exist.

Decomposition status. A corresponding proof sketch reduces this node to smaller mathematical subclaims. The checked reduction uses 3 subclaims: Bernoulli tangent sampling concentration under general sample bound; Bernoulli restricted sampling injectivity from tangent concentration; sample ratio between zero and one.