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Norm symmetrization at the actual matrix mean

Statement

Let $(Ω, w)$ be a finite probability space, and let $A, A^{'} : Ω \to M_{m \times k} (R)$ be matrix-valued functions, where $m, k$ are nonnegative integers. A matrix-valued random variable on this space means such a function. For any such function $B$ , its distribution assigns to each matrix value $M$ the probability $q_{B} (M) : = P (B = M) = \sum_{ω : B (ω) = M} w (ω)$ , with the empty sum equal to zero. Having the same distribution means $q_{A} (M) = q_{A^{'}} (M)$ for every $M$ ; independence of the two matrices means $P (A = M, A^{'} = N) = q_{A} (M) q_{A^{'}} (N)$ for every pair $M, N$ . Its matrix mean is $E B : = \sum_{ω \in Ω} w (ω) B (ω)$ , using entrywise matrix operations. Suppose they are independent and have the same distribution, and equip their matrix vector space with any norm $∥ \cdot ∥$ . Matrix expectation is taken entrywise. Then $E ∥ A - E A ∥ \leq E ∥ A - A^{'} ∥$ .

For the adjacency matrix $A$ of the simple graph $G (n, p)$ , with $n \geq 1$ and $0 \leq p \leq 1$ , let $J$ be the all-ones matrix and $I$ the identity matrix. Then $E A = p (J - I)$ and $A - p J = (A - E A) - p I$ . Consequently $E ∥ A - p J ∥ \leq E ∥ A - A^{'} ∥ + p ∥ I ∥$ for an independent copy $A^{'}$ under the same norm. The graph model has independent indicators for distinct unordered edges; independence of $A$ and $A^{'}$ concerns the two whole matrices.

Statement references

Pairwise and mutual independence of finite-valued random variables

Expectation of a real random variable on a finite probability space

A norm on a real vector space, the induced metric, and the dictionary with the metric axioms

The Erdős-Rényi finite random graph G(n,p)

The vector space M_m × n(F) := F^ m × n of m by n matrices over a field, with entrywise operations

Rectangular matrix multiplication and the identity matrix Iₙ, including zero-sized shapes

The adjacency matrix of a finite simple graph

Finite probability spaces, outcome weights, events, and event probabilities

Facts & Assumptions

[F1]

For finite-valued variables, independence means that each joint value probability is the product of its marginal value probabilities; finite expectation is the weighted sum over outcomes (Pairwise and mutual independence of finite-valued random variables, Expectation of a real random variable on a finite probability space).

[F2]

A norm is absolutely homogeneous and satisfies the triangle inequality (A norm on a real vector space, the induced metric, and the dictionary with the metric axioms). Expectation is linear for finite families of real random variables, without an independence hypothesis (Expectation is linear for every finite family of random variables, without any independence hypothesis).

[F3]

In $G (n, p)$ , the indicators of distinct unordered pairs of vertices are mutually independent Bernoulli variables with success probability $p$ ; the graph is simple (The Erdős-Rényi finite random graph G(n,p), Bernoulli random variables and binomial random variables as sums of independent Bernoulli trials).

[F4]

Real matrices form a vector space under entrywise addition and scalar multiplication. Equality can be checked entrywise, and the identity matrix has entries one on the diagonal and zero elsewhere (The vector space M_m × n(F) := F^ m × n of m by n matrices over a field, with entrywise operations, Rectangular matrix multiplication and the identity matrix Iₙ, including zero-sized shapes).

[F5]

In a chosen vertex order, the adjacency matrix of a finite simple graph has entry one exactly for an edge, zero otherwise; it is symmetric and has zero diagonal (The adjacency matrix of a finite simple graph).

[F6]

A finite probability space has nonnegative outcome weights summing to one; an event has probability equal to the sum of its outcome weights (Finite probability spaces, outcome weights, events, and event probabilities). The matrix-valued distribution in the Statement applies this event formula to the sets of outcomes with a fixed matrix value.

Proof

Given: Independent identically distributed finite-valued matrices $A, A^{'}$ and a norm on their real matrix space; for the graph application, the specified simple $G (n, p)$ model.

Goal: Prove the norm inequality and the stated graph centering identities and bound.

1.1F1F2F4F6algebra

List the possible matrix values as $M_{1}, \dots, M_{r}$ with probabilities $q_{1}, \dots, q_{r}$ , including all values of either copy. The sets ${ω : A (ω) = M_{j}}$ are pairwise disjoint and their union is $Ω$ , and $q_{j} = \sum_{ω : A (ω) = M_{j}} w (ω)$ . Thus the $q_{j}$ are nonnegative and sum to one; grouping the outcome sums entrywise gives $μ = E A = \sum_{j} q_{j} M_{j}$ . For each $i$ , $M_{i} - μ = \sum_{j} q_{j} (M_{i} - M_{j})$ ; repeated application of the triangle inequality and absolute homogeneity gives $∥ M_{i} - μ ∥ \leq \sum_{j} q_{j} ∥ M_{i} - M_{j} ∥$ .

2.1F1F6step 1.1

Multiply this inequality by $q_{i}$ and sum over $i$ . Independence gives $P (A = M_{i}, A^{'} = M_{j}) = q_{i} q_{j}$ , so the resulting inequality is $E ∥ A - E A ∥ \leq \sum_{i, j} q_{i} q_{j} ∥ M_{i} - M_{j} ∥ = E ∥ A - A^{'} ∥$ .

3.1F1F2F3F4F5algebra

For the simple graph, $A_{i i} = 0$ , while for $i \neq j$ the entry $A_{i j}$ is the indicator of the unordered edge ${i, j}$ and has expectation $p$ . Thus entrywise $E A = p (J - I)$ , and subtraction gives $A - p J = (A - E A) - p I$ . This uses the graph's edge-coordinate independence as part of its model, whereas step 2.1 uses independence of the whole matrices.

4.1F1F2step 2.1step 3.1∎

The triangle inequality and homogeneity yield $∥ A - p J ∥ \leq ∥ A - E A ∥ + p ∥ I ∥$ pointwise. Taking finite expectations and applying step 2.1 proves the asserted bound.

Depends on

Used by · 3

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