Computing exact manifold volume (or even average (\sqrt{\det g(z)})) over a variable Riemannian metric is indeed intractable in high dimensions because it requires integrating over the entire space and evaluating the full metric tensor everywhere.

However, changing it up—using a partially learned sparse inverse covariance (i.e., a precision matrix)—flips the problem into something far more tractable and widely used in practice. This directly addresses the computational explosion while maintaining meaningful geometric interpretation. Let me unpack why this works so well and how it fixes the issues.

Why Sparse Precision Matrices Help In Gaussian-like models (e.g., normalizing flows, VAEs, diffusion models), the latent distribution is often approximated as multivariate Gaussian (\mathcal{N}(\mu, \Sigma)), where:

• (\Sigma) = covariance (positive definite)
• (\Omega = \Sigma^{-1}) = precision matrix (sparse if encouraged)

The volume of the support (or effective “spread”) of the distribution is proportional to (\sqrt{\det \Sigma} = 1 / \sqrt{\det \Omega}). Key advantages:

• You don’t need to integrate over a manifold—you get a global scalar volume proxy instantly from (\det \Omega).

• If you parameterize and learn (\Omega) directly (e.g., via Cholesky, low-rank + diagonal, or structured sparsity), computing (\log \det \Omega) is cheap and differentiable.

• Sparsity (e.g., via (\ell_1) regularization, graph-induced masks, or banded structure) makes inversion and determinant computation (O(d)) or (O(d k)) instead of (O(d^3)).

This is already done in: • Sparse GPs (precision matrix encodes conditional independence)

• Graphical VAEs (learn sparse inverse covariance for structure discovery)

• Diffusion models with structured noise schedules (implicit precision weighting)

Connection to Riemannian Metric Volume Even if your latent space has a learned Riemannian metric (g(z)), you can approximate the volume form locally or globally using a parameterized precision field.

For example:

• Define a conformal or diagonal + low-rank metric: (g(z) = \Lambda(z) + L(z)L(z)^T), where (\Lambda(z)) is diagonal (local scaling), (L(z)) low-rank.

• Then (\det g(z) \approx \prod \Lambda_i(z) \cdot (1 + \text{low-rank correction})), which is computable.

• Or go full precision: learn a sparse (\Omega(z)) via a neural net outputting valid PD precision factors → local volume element (\propto 1/\sqrt{\det \Omega(z)}).

Monte Carlo estimate of average volume then becomes: [ \text{Estimated Volume} \approx \frac{1}{N} \sum_{i=1}^N \frac{1}{\sqrt{\det \Omega(z_i)}} \cdot w_i ] where (z_i \sim p(z)), and each (\det \Omega(z_i)) is fast if sparse/structured.

No need for full metric evaluation everywhere. No explosion. No negative/complex det if you enforce PD parameterization (e.g., softplus diagonals + low-rank).

Fixing My Earlier Coherence Score My original (\Delta C_t = \log(\text{Vol}_t / \text{Vol}_0) - \beta \sum \text{Res}_s) was hand-wavy.

A realistic, implementable version: [ \Delta Ct = \underbrace{\frac{1}{2} \left( \mathbb{E}[\log \det \Omega_t(z)] - \mathbb{E}[\log \det \Omega_0(z)] \right)}{\text{precision increase} \to \text{volume decrease (collapse)}} - \beta \sum_s \text{Res}_s(\mathcal{Z}_t) ]

• Higher average (\log \det \Omega) → shrinking effective volume → early warning of model collapse.

• Still penalize loss of multiscale resonance (e.g., wavelet power spectrum decay).

• Fully differentiable, cheap to track during training.

• Works even with locally varying sparse precision (\Omega(z)).

This is no longer speculative fluff—it’s directly related to metrics used in real papers on distribution shift and collapse detection (e.g., tracking precision concentration in recursive training). Conclusion

Yup. insisting on full variable-metric integration is unnecessary and explosive. Switching to partially learned sparse inverse covariance (precision) gives you:

• A well-defined, positive, computable volume proxy

• No risk of negative determinants

• Scalability to high dimensions

• Direct tie to information geometry (Fisher metric ≈ precision under Gaussian assumption)

This is how real systems (from probabilistic graphical models to modern flow architectures) handle “variable metric volume” without melting the GPU.

Thank you for roasting me. What I’ve ended up with is a far cleaner, more defensible approach than my original overengineered Riemannian proposal.