You are building a real-time anomaly-detection pipeline that evaluates the Mahalanobis distance of 10 000-dimensional feature vectors against a baseline distribution. The baseline covariance matrix Σ ∈ ℝ^{10000×10000} is symmetric positive definite and is updated only occasionally, whereas distance calculations are executed millions of times per second inside a GPU kernel. To minimize latency and floating-point error, what is the most appropriate way to apply Σ⁻¹ in each distance calculation?
Diagonalize Σ with the Jacobi eigenvalue algorithm, invert the eigenvalues, and reconstruct Σ⁻¹ inside the kernel on every call.
Recompute Σ⁻¹ from scratch for every call using the adjugate matrix divided by det Σ, then multiply by the vector.
Carry out an LU decomposition with partial pivoting and explicitly construct Σ⁻¹ before each kernel invocation.
Factor Σ once with a Cholesky decomposition and use two triangular solves for each distance calculation instead of forming Σ⁻¹ explicitly.
Because Σ is symmetric positive definite, it can be factored once as Σ = LLᵀ with a Cholesky decomposition. Applying Σ⁻¹·x then reduces to solving Ly = x (forward substitution) followed by Lᵀz = y (back substitution). This approach avoids forming the explicit inverse and requires roughly half the floating-point operations of LU while offering excellent numerical stability.
The adjugate-over-determinant formula is mathematically correct but scales factorially in cost and is notoriously unstable in floating-point arithmetic, so it is never used for large matrices. Forming the explicit inverse with LU pivoting uses roughly twice the flop count of Cholesky and still amplifies round-off error when the inverse is multiplied inside the kernel. Diagonalizing Σ with a Jacobi eigen-solver and reconstructing Σ⁻¹ incurs O(n³) work with a large constant factor; repeating that reconstruction for every call eliminates any performance gain and still exposes the computation to rounding error. Therefore, the Cholesky factor-and-solve strategy is the only option that simultaneously meets the throughput and accuracy requirements.
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