The Gaussian Entropy Pair is Noise-Closed, and Its Penalty Bounds the Squared Noise Wasserstein Distance to the Reference Measure

In the setting of A Diagonal Gaussian Reference Measure on the Noise Wasserstein Space, Rescaled Heads and Gaussian Tails: Standing Notation, so that the reference measure is $\rho=\gamma_{c}$, let $\beta$ and $\kappa$ be positive real numbers with $c_{k}\le\kappa\,a_{k}$ for every $k\in\mathbb{N}$, and let $(\mathcal{D},\mathcal{D}_{\Sigma},\mathcal{E},\Sigma)$ be the Gaussian entropy pair with temperature $\beta$, whose hypothesis holds with this $\kappa$; it is a noise penalty pair on $\mathcal{P}^{a}_{\rho}$ by The Gaussian Entropy Pair is a Noise Penalty Pair, with Nonnegative Penalty and Dense Score Domain §pair. Noise-closedness is read in the metric space $(\mathcal{P}^{a}_{\rho},W_{a})$ of The Noise Wasserstein Distance is a Metric on the Measures Noise-Connected to the Reference Measure: Existence of Noise-Optimal Couplings, Comparison with the Quadratic Wasserstein Distance and Lower Semicontinuity §metric; the real number quantified as $c$ in Noise-Closed Noise Penalty Pairs §noise-closed is a bound variable there, distinct from the variance sequence $c$. For $\mu\in\mathcal{D}$ the number $W_{a}(\mu,\rho)$ is defined by The Noise Wasserstein Distance is a Metric on the Measures Noise-Connected to the Reference Measure: Existence of Noise-Optimal Couplings, Comparison with the Quadratic Wasserstein Distance and Lower Semicontinuity §connected, since $\mathcal{D}\subseteq\mathcal{P}^{a}_{\rho}$ by The Gaussian Entropy Pair on the Noise Wasserstein Space: Relative Entropy and the Noise Score Field §penalty-domain and $\rho\in\mathcal{P}^{a}_{\rho}$ by The Noise Wasserstein Distance is a Metric on the Measures Noise-Connected to the Reference Measure: Existence of Noise-Optimal Couplings, Comparison with the Quadratic Wasserstein Distance and Lower Semicontinuity §reference, and $|s|$ is the absolute value of $s\in\mathbb{R}$.

1. (Noise-closed)¶ The pair is noise-closed.

2. (Growth bound)¶ For every $\mu\in\mathcal{D}$,

$$W_{a}(\mu,\rho)^{2}\le\frac{2\kappa}{\beta}\,\mathcal{E}(\mu)\le\frac{2\kappa}{\beta}\,\bigl(1+|\mathcal{E}(\mu)|\bigr).$$