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Is there a way around the numerical problem of calculating the denominator of the Multivariate Gaussian distribution of high dimensional vectors?
Given the following formula $$f_{\mathbf X}(x_1,\ldots,x_k) = \frac{\exp\left(-\frac 1 2 ({\mathbf x}-{\boldsymbol\mu})^\mathrm{T}{\boldsymbol\Sigma}^{-1}({\mathbf x}-{\boldsymbol\mu})\right)}{\sqrt{(2\pi)^k|\boldsymbol\Sigma|}}$$ With a feature vector of size, $>128$ the denominator cannot be computed with normal precision(64 bit) frameworks. My feature vectors are $1024$ long, and I haven't found a way to come across the problem.

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It's better to operate on log domain in such cases. There are implementations (employing factorizations like Cholesky, LU) available in common languages:

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