bernoulli_energy_moment_from_count_moment_and_pointwise_domination

Role. It controls sampled row/column counts or energies, which feed the moment bounds for random sampled matrices.

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 sampled row/column nodes, $N_i(\Omega)$ counts observed entries in row $i$, $N^j(\Omega)$ counts observed entries in column $j$, and the corresponding energies sum $X_{ij}^2$ over sampled entries. These estimates feed the noncommutative Khintchine and spectral-norm concentration bounds.

Claim. Generic moment comparison for sampled energies. If a nonnegative energy statistic is pointwise dominated by $\lVert X\rVert_{\infty}^2$ times a nonnegative count statistic, then a moment bound for the count statistic implies the corresponding energy moment bound.

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. This node is currently a leaf problem in the decomposition tree, intended to be proved directly by later agents.