# Efficient calculation of Mahalanobis distance

**URL:** https://discourse.julialang.org/t/efficient-calculation-of-mahalanobis-distance/90121
**Category:** Performance
**Tags:** question, linearalgebra
**Created:** [November 11, 2022, 2:16pm UTC](https://discourse.julialang.org/t/efficient-calculation-of-mahalanobis-distance/90121 "2022-11-11T14:16:04Z")
**Posts on this page:** 1
**Showing post:** 4

<div class="post-metadata">

### Author: ![stevengj](https://sea2.discourse-cdn.com/julialang/user_avatar/discourse.julialang.org/stevengj/32/71_2.png) [@stevengj](https://discourse.julialang.org/u/stevengj)
#### Post date: [November 11, 2022, 5:55pm UTC](https://discourse.julialang.org/t/efficient-calculation-of-mahalanobis-distance/90121/4 "2022-11-11T17:55:31Z")

</div>

From your example, it seems that A \in \mathbb{R}^{n\times M} is a “wide” matrix, i.e. more columns than rows (M \ge n). In this case, maybe use the LQ factorization?

That is, `lq(A)` forms a factorization A=LQ, where L \mathbb{R}^{n\times n} and Q \in \mathbb{R}^{M\times n} has orthonormal rows (QQ^T = I), equivalent to the QR factorization of A^T. Given this, \Sigma = AA^T = LQQ^T L^T = LL^T, and \Sigma^{-1} = L^{-T} L^{-1}. Hence your distance is simply d(x^i) = \Vert L^{-1} (x^i - \mu) \Vert.

In Julia:

```julia
L = LowerTriangular(lq(A).L)
dmahal = norm(L \ (x - μ))

```

Or, using your example code, I get:

```julia
julia> norm.(eachcol(LowerTriangular(lq(AX).L) \ AX)).^2 ≈ dmahal
true

```

(Note that you can apply `L \` to every column of a matrix at once, which can be much more efficient than solving column-by-column, especially when the matrix is large.)

In general, the standard approaches to avoiding A^T A or AA^T computations (which square condition numbers) often involve QR or LQ factorization (respectively), or the SVD.

PS. I think the [Mahalanobis distance](https://en.wikipedia.org/wiki/Mahalanobis_distance) is technically just the `norm(...)`, not the norm squared as in your code.

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