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A restricted two-operator class with a positive stability modulus

Example

Let the truth space be $U = [1, 2] \cup [4, 5]$ with its absolute-value metric, the operator labels be $A = {1, 2}$ , and the data space be $R$ with its absolute-value metric. For each $a \in A$ , define $F_{a} : U \to R$ by $F_{a} (x) = a x$ . Let the truth-target map be $q_{U} : U \to Q : = U$ , $q_{U} (x) = x$ . Restrict the admissible truth/operator pairs to $C = {(x, 1) : 1 \leq x \leq 2} \cup {(x, 2) : 4 \leq x \leq 5} \subseteq U \times A,$ with the restriction to $C$ of the metric $d_{1} ((x, a), (x^{'}, a^{'})) = ∣ x - x^{'} ∣ + ∣ a - a^{'} ∣$ on $R^{2}$ . The first-coordinate projection $π_{U} : C \to U$ is bijective, since every $x \in U$ occurs in exactly one admissible pair. Thus the configuration target $q_{C} : = q_{U} \circ π_{U} : C \to Q$ , given by $q_{C} (x, a) = x$ , is also bijective and has equality as its recovery equivalence. For the stability-modulus definition, take $C$ as the parameter class, with configuration forward map $G : C \to R$ given by $G (x, a) = F_{a} (x) = a x$ and target $q_{C}$ . Then $ω (t) = t$ is a uniform stability modulus: for every $c, c^{'} \in C$ , $∣ q_{C} (c) - q_{C} (c^{'}) ∣ \leq ω (∣ G (c) - G (c^{'}) ∣)$ .

Facts & Assumptions

Given: The truth space and target map $q_{U}$ , operator labels and maps $F_{a}$ , restricted admissible-pair set $C$ , configuration target $q_{C}$ , and configuration forward map $G$ in the Example.

[F1]

Operator, noise-law, and prior-support mismatch: admissible truth/operator pairs may form a specified subset of the truth-space/operator-label product; each operator label gives a forward map on the truth space.

[F2]

Identifiability and stability moduli: for a parameter class with a target map and a forward map defined on that class, a uniform stability modulus is nondecreasing, vanishes at $0$ , tends to $0$ as its argument decreases to $0$ , and bounds target distance by data distance for every pair in the class.

[F3]

$R^{n}$ as the set of functions $n \to R$ , and $d_{1}$ , $d_{2}$ , $d_{\infty}$ are metrics on it The absolute value makes $R$ a metric space: $d (x, y) = ∣ x - y ∣$ is a metric, its open balls are the intervals $(x - r, x + r)$ , and it is unbounded Metric space: $d (x, y) = 0$ iff $x = y$ , symmetry, and the triangle inequality; pseudometric and ultrametric: $d_{1}$ is a metric on $R^{2}$ , and absolute value is a metric on $R$ ; the three metric axioms remain true when restricted to a subset.

Verification

1.1givenF1F3algebra

By [F1], $C$ is the specified admissible set of operator-labelled configurations. By [F3], $d_{1}$ is a metric on $R^{2}$ ; restricting its three metric axioms to $C$ makes the restriction a metric. Each $x \in U$ belongs to exactly one of $[1, 2]$ and $[4, 5]$ , so it occurs in exactly one admissible pair. Thus $π_{U}$ and $q_{C} = q_{U} \circ π_{U}$ are bijections onto $U = Q$ , and equality is the stated recovery equivalence.

1.2givenF1F3algebra

If $c = (x, 1)$ and $c^{'} = (x^{'}, 1)$ , then $∣ q_{C} (c) - q_{C} (c^{'}) ∣ = ∣ G (c) - G (c^{'}) ∣$ . If $c = (x, 2)$ and $c^{'} = (x^{'}, 2)$ , then $∣ q_{C} (c) - q_{C} (c^{'}) ∣ = \frac{1}{2} ∣ G (c) - G (c^{'}) ∣ \leq ∣ G (c) - G (c^{'}) ∣$ .

1.3givenF3algebra

For $c = (x, 1)$ with $1 \leq x \leq 2$ and $c^{'} = (x^{'}, 2)$ with $4 \leq x^{'} \leq 5$ , we have $∣ x^{'} - x ∣ \leq 4$ and $∣ 2 x^{'} - x ∣ \geq 2 \cdot 4 - 2 = 6$ , so $∣ q_{C} (c) - q_{C} (c^{'}) ∣ \leq ∣ G (c) - G (c^{'}) ∣$ . The reversed ordering follows by symmetry of absolute value.

2.1givenF2step 1.2step 1.3algebra∎

Steps 1.2 and 1.3 cover every pair in $C$ , so $∣ q_{C} (c) - q_{C} (c^{'}) ∣ \leq ∣ G (c) - G (c^{'}) ∣$ for all $c, c^{'}$ . The function $ω (t) = t$ is nondecreasing, satisfies $ω (0) = 0$ , and has $\lim_{t ↓ 0} ω (t) = 0$ ; by [F2] it is the stated uniform stability modulus.

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