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  9. LowRankMultivariateNormal

torch.distributions.LowRankMultivariateNormal

class LowRankMultivariateNormal extends Distribution
new LowRankMultivariateNormal(loc: number[] | Tensor, cov_factor: number[][] | Tensor, cov_diag: number[] | Tensor, options?: DistributionOptions)
readonlyloc(Tensor)
– Mean of the distribution.
readonlycov_factor(Tensor)
– Low-rank factor W such that covariance = W @ W.T + D.
readonlycov_diag(Tensor)
– Diagonal of D in the covariance decomposition.
readonlyarg_constraints(unknown)
readonlysupport(unknown)
readonlyhas_rsample(unknown)
readonlycovariance_matrix(Tensor)
– Full covariance matrix: W @ W.T + D.
readonlyprecision_matrix(Tensor)
– Precision matrix (inverse of covariance). Uses Woodbury identity for efficient computation.
readonlymean(Tensor)
readonlymode(Tensor)
readonlyvariance(Tensor)

Low Rank Multivariate Normal distribution: multivariate normal with efficient low-rank covariance.

Essential for:

  • High-dimensional Gaussian processes (rank-reduced approximations)
  • Scalable Bayesian models with covariance structure
  • Matrix-normal variational inference
  • Structured covariance modeling

Covariance matrix of low-rank plus diagonal form: Σ = W @ W.T + D where W is a (d x r) low-rank factor and D is a diagonal matrix.

This parametrization is useful for high-dimensional distributions where a full covariance would be too expensive to store/compute.

Covariance: Σ=WWT+DW∈Rd×r,D=diag(d)Cholesky: Σ=LLT computed efficiently via rank updatesPDF: f(x)=(2π)−d/2∣Σ∣−1/2exp⁡(−12(x−μ)TΣ−1(x−μ))Determinant: ∣Σ∣=∏idi⋅∣I+WTD−1W∣(Weinstein-Aronszajn)\begin{aligned} \text{Covariance: } \Sigma = WW^T + D \quad W \in \mathbb{R}^{d \times r}, D = \text{diag}(\mathbf{d}) \\ \text{Cholesky: } \Sigma = LL^T \text{ computed efficiently via rank updates} \\ \text{PDF: } f(x) = (2\pi)^{-d/2} |\Sigma|^{-1/2} \exp\left(-\frac{1}{2}(x-\mu)^T\Sigma^{-1}(x-\mu)\right) \\ \text{Determinant: } |\Sigma| = \prod_i d_i \cdot |I + W^T D^{-1} W| \quad \text{(Weinstein-Aronszajn)} \end{aligned}Covariance: Σ=WWT+DW∈Rd×r,D=diag(d)Cholesky: Σ=LLT computed efficiently via rank updatesPDF: f(x)=(2π)−d/2∣Σ∣−1/2exp(−21​(x−μ)TΣ−1(x−μ))Determinant: ∣Σ∣=i∏​di​⋅∣I+WTD−1W∣(Weinstein-Aronszajn)​

Examples

const m = new LowRankMultivariateNormal(
  torch.tensor([0.0, 0.0, 0.0]),       // mean
  torch.tensor([[1.0], [0.5], [0.2]]), // cov_factor (d x r)
  torch.tensor([0.1, 0.1, 0.1])        // cov_diag (d,)
);
m.sample();  // sample from the distribution
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