V* in SVD should be transposed

[nblumoe]

I am not 100% sure if vectorz does it wrong or other libs, I am continuing to investigate this. However, here is the issue, illustrated with an example:

user=> (clojure.core.matrix/set-current-implementation :vectorz)
:vectorz
(:V* (decomp-svd (covariance [[1 2 3 ] [ 4 5 6 ] [7 8 9] [10 11 12]])))
#<mikera.matrixx.Matrix@492d4c90
[[0.5773502691896257,0.5773502691896255,0.5773502691896257],
[0.816496580927726,-0.40824829046386285,-0.40824829046386296],
[0.0,-0.7071067811865475,0.7071067811865475]]>

user=> (clojure.core.matrix/set-current-implementation :clatrix)
:clatrix
user=> (:V* (decomp-svd (covariance [[1 2 3 ] [ 4 5 6 ] [7 8 9] [10 11 12]])))
((-0.5773502691896256 0.8164965809277263 0.0)
 (-0.5773502691896258 -0.40824829046386285 -0.7071067811865476)
 (-0.577350269189626 -0.40824829046386296 0.7071067811865475))

This is done via Incanter / core.matrix. As you can see the resulting V* are different as one is the transpose of the other. There are also differences in the signs, but they are not significant for SVD and we don't need to bother.

R returns the same data as the clatrix implementation:

> svd(cov(dat))
...
$v
           [,1]       [,2]       [,3]
[1,] -0.5773503  0.0000000  0.8164966
[2,] -0.5773503 -0.7071068 -0.4082483
[3,] -0.5773503  0.7071068 -0.4082483

Thus I think vectorz does it wrong. The other parts of the return value (U,S) are correct (aka the same) in all implementations and R.

[mikera]

I think Vectorz is correct - V* is the transpore of V, and this is the natural form to return given the decomposition:

M = U. Σ . V*

If you want V, you can just transpose it as you observe.

Here's my test which I believe demonstrates that Vectorz has the correct behaviour:

(let [m (array :vectorz [[1 2 3 ] [ 4 5 6 ] [7 8 9]])
        {:keys [U S V*]} (svd m)]
    (inner-product U (diagonal-matrix S) V*))
=> #<Matrix [[0.9999999999999991,1.9999999999999991,2.999999999999998],
[4.000000000000001,5.000000000000003,6.000000000000001],
[7.000000000000001,8.000000000000002,9.0]]>

i.e. you can reconstruct the matrix from the decomposition (modulo minor numerical inaccuracy)

[mikera]

I suspect this is Clatrix doing it wrong, and returning V instead of V*

[mikera]

Closing this because not a vectorz-clj issue

[nblumoe]

Yes, you are absolutely right.

In contrast to what I said earlier, this is also in line with R. The subtle difference is, that svd in R returns V, whereas La.svd in R returns V*. Somehow I confused those.

So, everything is good. vectorz is doing it right - just like you said.