torch.linalg.slogdet

function slogdet<S extends Shape, D extends DType, Dev extends DeviceType>(A: Tensor<S, D, Dev>): { sign: Tensor<DynamicShape, D, Dev>; logabsdet: Tensor<DynamicShape, D, Dev> }

Computes the sign and log absolute value of the determinant.

Numerical stability: Computing determinants without overflow/underflow
Log-likelihood: Gaussian distribution probabilities in log-space
Matrix properties: Checking invertibility (det ≠ 0) and orientation (sign)
Probabilistic models: Computing normalizing constants in Bayesian inference
Manifold learning: Jacobian determinants in change-of-variables formulas
Matrix computations: Separating magnitude and sign information

Uses LU decomposition: det(A) = sign(permutation) × ∏diag(U). The log operation is applied to absolute values to avoid numerical issues with very large or very small determinants.

\begin{aligned} \det(A) = \text{sign}(A) \cdot e^{\text{logabsdet}(A)} \\ \text{sign}(A) = \begin{cases} 0 & \text{if singular} \\ 1 & \text{if } \det(A) > 0 \\ -1 & \text{if } \det(A) < 0 \end{cases} \\ \text{logabsdet}(A) = \ln(|\det(A)|) = \sum_{i=1}^{n} \ln(|U_{ii}|) + \ln(|\text{perm sign}|) \end{aligned}

Sign interpretation: sign = 0 indicates singular matrix (det = 0)
Numerical safety: log(|det|) is safe for |det| ∈ [1e-30, 1e30] (no over/underflow)
Reconstruction: det(A) = sign × exp(logabsdet) avoids numerical issues
GPU efficient: Computed via LU factorization on GPU (no CPU readback)
Gradient flow: Both sign and logabsdet are differentiable (logabsdet gradient is continuous except at det=0)

Singular matrices: sign = 0 for singular matrices; logabsdet = -∞
Sign discontinuity: Gradient of sign is undefined at det = 0
Batches: All matrices must be square; shapes must be (..., n, n)

Parameters

ATensor<S, D, Dev>: Square matrix of shape (n, n) or batches of shape (..., n, n)

Returns

{ sign: Tensor<DynamicShape, D, Dev>; logabsdet: Tensor<DynamicShape, D, Dev> }– Object with two tensors: - sign: Tensor containing -1, 0, or 1 (0 for singular matrices) - logabsdet: Natural log of absolute determinant (log(|det(A)|))

Examples

// Simple 2x2 matrix
const A = torch.tensor([[3, 1], [1, 2]]);
const { sign, logabsdet } = torch.linalg.slogdet(A);
// sign: 1, logabsdet: ln(5) ≈ 1.609
// Reconstruct: det = 1 * exp(1.609) = 5

// Singular matrix (rank deficient)
const singular = torch.tensor([[1, 2], [2, 4]]);
const { sign: s, logabsdet: lad } = torch.linalg.slogdet(singular);
// sign: 0, logabsdet: -∞

// Gaussian log-likelihood computation
const cov = torch.tensor([[2, 0.5], [0.5, 1]]);
const x = torch.tensor([1, -1]);
const mean = torch.tensor([0, 0]);

const { logabsdet } = torch.linalg.slogdet(cov);
const diff = x.sub(mean);
const inv_cov = torch.linalg.inv(cov);
const quad_form = diff.matmul(inv_cov).matmul(diff);
const log_prob = -0.5 * (logabsdet + quad_form + 2 * Math.log(2 * Math.PI));

// Batched computation
const batch = torch.randn(32, 5, 5);  // 32 matrices of size 5x5
const { sign: signs, logabsdet: logdets } = torch.linalg.slogdet(batch);
// signs shape: [32], logdets shape: [32]

torch.linalg.slogdet

Parameters

Returns

Examples

See Also

torch.linalg.slogdet

Parameters

Returns

Examples

See Also