A Longer Gibberish Essay: Ipsum Flows on Lorem Manifolds
Published:
Lorem ipsum dolor sit amet, consectetur adipiscing elit. This fictional essay pretends to introduce ipsum flows on lorem manifolds — a nonexistent object — so that the longer-form technical-writing look and feel of the blog can be evaluated: headings, paragraphs, display math, and several collapsible derivation panels.
1. Setting and notation
Let \((\mathcal{M}, g)\) be a (fake) Riemannian manifold with metric tensor \(g\), and let \(\pi \in \mathcal{P}(\mathcal{M})\) be a probability measure on it. Denote by \(\nabla^g\) the Levi-Civita connection and by \(\mathrm{div}_g\) the metric divergence. A lorem flow is a family \((\rho_t)_{t \ge 0} \subset \mathcal{P}(\mathcal{M})\) satisfying the continuity-style equation
\[\partial_t \rho_t + \mathrm{div}_g\!\bigl(\rho_t\, v_t\bigr) \;=\; 0, \qquad v_t := -\nabla^g \frac{\delta \mathcal{F}}{\delta \rho}(\rho_t), \label{eq:lorem-flow}\]for some (fake) energy functional \(\mathcal{F}: \mathcal{P}(\mathcal{M}) \to \mathbb{R}\). If \(\mathcal{F}\) were the KL divergence to a fixed target, this would be the gradient flow of relative entropy — but here it is nothing, because the whole setup is gibberish.
2. A “key identity”
The following identity drives most of what follows. Its one-line statement:
\[\frac{\mathrm{d}}{\mathrm{d}t} \mathcal{F}(\rho_t) \;=\; -\int_{\mathcal{M}} \bigl\lVert \nabla^g \tfrac{\delta \mathcal{F}}{\delta \rho}(\rho_t) \bigr\rVert_g^2\, \rho_t\, \mathrm{d}\mathrm{vol}_g. \label{eq:dissipation}\]Readers who want only the headline can skip the derivation.
Derivation of the "energy dissipation" identity
Differentiating \(\mathcal{F}(\rho_t)\) and applying the chain rule in \(\mathcal{P}(\mathcal{M})\):
\[\begin{aligned} \frac{\mathrm{d}}{\mathrm{d}t} \mathcal{F}(\rho_t) &= \int_{\mathcal{M}} \tfrac{\delta \mathcal{F}}{\delta \rho}(\rho_t)\, \partial_t \rho_t\, \mathrm{d}\mathrm{vol}_g \\ &\stackrel{\eqref{eq:lorem-flow}}{=} -\int_{\mathcal{M}} \tfrac{\delta \mathcal{F}}{\delta \rho}(\rho_t)\, \mathrm{div}_g\!\bigl(\rho_t\, v_t\bigr)\, \mathrm{d}\mathrm{vol}_g \\ &\stackrel{\text{(IBP)}}{=} \int_{\mathcal{M}} \bigl\langle \nabla^g \tfrac{\delta \mathcal{F}}{\delta \rho}(\rho_t),\, v_t \bigr\rangle_g\, \rho_t\, \mathrm{d}\mathrm{vol}_g \\ &\stackrel{(v_t = -\nabla^g \tfrac{\delta \mathcal{F}}{\delta \rho})}{=} -\int_{\mathcal{M}} \bigl\lVert \nabla^g \tfrac{\delta \mathcal{F}}{\delta \rho}(\rho_t) \bigr\rVert_g^2\, \rho_t\, \mathrm{d}\mathrm{vol}_g. \end{aligned}\]Integration by parts requires the obvious (fake) boundary / decay conditions. $\blacksquare$
The sign of \eqref{eq:dissipation} is the only substantive content: \(\mathcal{F}\) is non-increasing along the flow. Donec placerat, nibh ac aliquam mattis, ligula nisi porta justo.
3. Discretization
A plausible-looking numerical scheme for \eqref{eq:lorem-flow} is the (totally invented) implicit ipsum step:
\[\rho_{k+1} \;\in\; \arg\min_{\rho \in \mathcal{P}(\mathcal{M})}\; \Bigl\{\, \mathcal{F}(\rho) \;+\; \tfrac{1}{2\tau}\, W_2^2(\rho, \rho_k) \,\Bigr\},\]where \(W_2\) is the 2-Wasserstein distance and \(\tau > 0\) is a step size. For Euclidean \(\mathcal{M}\) and quadratic \(\mathcal{F}\) this has a closed-form update; in general it does not.
Show the closed-form update for quadratic $$\mathcal{F}$$
Take \(\mathcal{M} = \mathbb{R}^d\) and \(\mathcal{F}(\rho) = \tfrac{1}{2}\,\mathbb{E}_{X \sim \rho}\bigl[\lVert X - \mu \rVert^2\bigr]\) for some fixed \(\mu \in \mathbb{R}^d\). Using the fact that \(W_2^2\) factorizes on means + covariances for Gaussian-like measures (we assume \(\rho_k\) is Gaussian without loss of generality — this is also fake),
\[\begin{aligned} \mu_{k+1} &= \arg\min_m\; \tfrac{1}{2}\,\lVert m - \mu \rVert^2 + \tfrac{1}{2\tau}\,\lVert m - \mu_k \rVert^2 \\ &= \frac{\tau\, \mu + \mu_k}{\tau + 1} \;=\; \mu_k + \frac{\tau}{1 + \tau}\,(\mu - \mu_k). \end{aligned}\]i.e., an implicit gradient step with rate \(\tau / (1 + \tau)\) — which, for small \(\tau\), recovers the explicit flow as expected.
4. Consequences (none)
Neque porro quisquam est, qui dolorem ipsum quia dolor sit amet. The two identities above are, in the real literature, genuinely true statements about gradient flows in Wasserstein space — see Jordan–Kinderlehrer–Otto (1998), Ambrosio–Gigli–Savaré (2008), and Otto (2001) for actual references. Nothing specific to this post is real.
4.1 A list of fake consequences
- Ipsum entropy is monotone, therefore everything is fine.
- Lorem manifolds are geodesically convex (not checked).
- Consectetur \(\ge 0\) for integrable elits (undefined).
- The scheme converges at rate \(\mathcal{O}(\tau)\), obviously.
4.2 A pair of unordered nonsense
- Adipiscing incididunt, except when it does.
- Ut labore sed do eiusmod tempor.
- Excepteur sint occaecat cupidatat non proident.
5. Where to go from here
Aside: connection to ipsum-regularized transport
If one adds an entropic penalty \(\varepsilon H(\rho)\) to \(\mathcal{F}\), the flow becomes a noisy SDE of Fokker–Planck type with diffusion \(\varepsilon\). None of that matters here; this aside exists only to show the lighter “aside” variant of the toggle component.
Curabitur pretium tincidunt lacus. Nulla gravida orci a odio. End of placeholder.
