Yes; I'm pretty sure that it is exactly the repeated eigenvalues that are
the issue. The matrices I am using are all nonsingular, and the various
algorithms have no problem computing the eigenvalues correctly (up to
numerical errors that I can bound and thus account for on tests by rounding
appropriately). But an eigenvalue of multiplicity M has an M-dimensional
eigenspace with no preferred basis. So, any M-dimensional (unitary) change
of basis is permitted. That's what give rise to the lack of reproducibility
across architectures. The choice of basis appears to use different
heuristics on 32-bit windows than on 64-bit Windows or Linux machines. As a
result, I can't include the tests I'd like as part of a CRAN submission.
On Thu, May 17, 2018, 2:29 PM William Dunlap <wdun...@tibco.com> wrote:
> Your explanation needs to be a bit more general in the case of identical
> eigenvalues - each distinct eigenvalue has an associated subspace, whose
> dimension is the number repeats of that eigenvalue and the eigenvectors for
> that eigenvalue are an orthonormal basis for that subspace. (With no
> repeated eigenvalues this gives your 'unique up to sign'.)
>
> E.g., for the following 5x5 matrix with two eigenvalues of 1 and two of 0
>
> > x <- tcrossprod( cbind(c(1,0,0,0,1),c(0,1,0,0,1),c(0,0,1,0,1)) )
> > x
> [,1] [,2] [,3] [,4] [,5]
> [1,] 1 0 0 0 1
> [2,] 0 1 0 0 1
> [3,] 0 0 1 0 1
> [4,] 0 0 0 0 0
> [5,] 1 1 1 0 3
> the following give valid but different (by more than sign) eigen vectors
>
> e1 <- structure(list(values = c(4, 1, 0.999999999999999, 0,
> -2.22044607159862e-16
> ), vectors = structure(c(-0.288675134594813, -0.288675134594813,
> -0.288675134594813, 0, -0.866025403784439, 0, 0.707106781186547,
> -0.707106781186547, 0, 0, 0.816496580927726, -0.408248290463863,
> -0.408248290463863, 0, -6.10622663543836e-16, 0, 0, 0, -1, 0,
> -0.5, -0.5, -0.5, 0, 0.5), .Dim = c(5L, 5L))), .Names = c("values",
> "vectors"), class = "eigen")
> e2 <- structure(list(values = c(4, 1, 1, 0, -2.29037708937563e-16),
> vectors = structure(c(0.288675134594813, 0.288675134594813,
> 0.288675134594813, 0, 0.866025403784438, -0.784437556312061,
> 0.588415847923579, 0.196021708388481, 0, 4.46410900710223e-17,
> 0.22654886208902, 0.566068420404321, -0.79261728249334, 0,
> -1.11244069540181e-16, 0, 0, 0, -1, 0, -0.5, -0.5, -0.5,
> 0, 0.5), .Dim = c(5L, 5L))), .Names = c("values", "vectors"
> ), class = "eigen")
>
> I.e.,
> > all.equal(crossprod(e1$vectors), diag(5), tol=0)
> [1] "Mean relative difference: 1.407255e-15"
> > all.equal(crossprod(e2$vectors), diag(5), tol=0)
> [1] "Mean relative difference: 3.856478e-15"
> > all.equal(e1$vectors %*% diag(e1$values) %*% t(e1$vectors), x, tol=0)
> [1] "Mean relative difference: 1.110223e-15"
> > all.equal(e2$vectors %*% diag(e2$values) %*% t(e2$vectors), x, tol=0)
> [1] "Mean relative difference: 9.069735e-16"
>
> > e1$vectors
> [,1] [,2] [,3] [,4] [,5]
> [1,] -0.2886751 0.0000000 8.164966e-01 0 -0.5
> [2,] -0.2886751 0.7071068 -4.082483e-01 0 -0.5
> [3,] -0.2886751 -0.7071068 -4.082483e-01 0 -0.5
> [4,] 0.0000000 0.0000000 0.000000e+00 -1 0.0
> [5,] -0.8660254 0.0000000 -6.106227e-16 0 0.5
> > e2$vectors
> [,1] [,2] [,3] [,4] [,5]
> [1,] 0.2886751 -7.844376e-01 2.265489e-01 0 -0.5
> [2,] 0.2886751 5.884158e-01 5.660684e-01 0 -0.5
> [3,] 0.2886751 1.960217e-01 -7.926173e-01 0 -0.5
> [4,] 0.0000000 0.000000e+00 0.000000e+00 -1 0.0
> [5,] 0.8660254 4.464109e-17 -1.112441e-16 0 0.5
>
>
>
>
>
> Bill Dunlap
> TIBCO Software
> wdunlap tibco.com
>
> On Thu, May 17, 2018 at 10:14 AM, Martin Maechler <
> maech...@stat.math.ethz.ch> wrote:
>
>> >>>>> Duncan Murdoch ....
>> >>>>> on Thu, 17 May 2018 12:13:01 -0400 writes:
>>
>> > On 17/05/2018 11:53 AM, Martin Maechler wrote:
>> >>>>>>> Kevin Coombes ... on Thu, 17
>> >>>>>>> May 2018 11:21:23 -0400 writes:
>>
>> >> [..................]
>>
>> >> > [3] Should the documentation (man page) for "eigen" or
>> >> > "mvrnorm" include a warning that the results can change
>> >> > from machine to machine (or between things like 32-bit and
>> >> > 64-bit R on the same machine) because of difference in
>> >> > linear algebra modules? (Possibly including the statement
>> >> > that "set.seed" won't save you.)
>>
>> >> The problem is that most (young?) people do not read help
>> >> pages anymore.
>> >>
>> >> help(eigen) has contained the following text for years,
>> >> and in spite of your good analysis of the problem you
>> >> seem to not have noticed the last semi-paragraph:
>> >>
>> >>> Value:
>> >>>
>> >>> The spectral decomposition of ‘x’ is returned as a list
>> >>> with components
>> >>>
>> >>> values: a vector containing the p eigenvalues of ‘x’,
>> >>> sorted in _decreasing_ order, according to ‘Mod(values)’
>> >>> in the asymmetric case when they might be complex (even
>> >>> for real matrices). For real asymmetric matrices the
>> >>> vector will be complex only if complex conjugate pairs
>> >>> of eigenvalues are detected.
>> >>>
>> >>> vectors: either a p * p matrix whose columns contain the
>> >>> eigenvectors of ‘x’, or ‘NULL’ if ‘only.values’ is
>> >>> ‘TRUE’. The vectors are normalized to unit length.
>> >>>
>> >>> Recall that the eigenvectors are only defined up to a
>> >>> constant: even when the length is specified they are
>> >>> still only defined up to a scalar of modulus one (the
>> >>> sign for real matrices).
>> >>
>> >> It's not a warning but a "recall that" .. maybe because
>> >> the author already assumed that only thorough users would
>> >> read that and for them it would be a recall of something
>> >> they'd have learned *and* not entirely forgotten since
>> >> ;-)
>> >>
>>
>> > I don't think you're really being fair here: the text in
>> > ?eigen doesn't make clear that eigenvector values are not
>> > reproducible even within the same version of R, and
>> > there's nothing in ?mvrnorm to suggest it doesn't give
>> > reproducible results.
>>
>> Ok, I'm sorry ... I definitely did not want to be unfair.
>>
>> I've always thought the remark in eigen was sufficient, but I'm
>> probably wrong and we should add text explaining that it
>> practically means that eigenvectors are only defined up to sign
>> switches (in the real case) and hence results depend on the
>> underlying {Lapack + BLAS} libraries and therefore are platform
>> dependent.
>>
>> Even further, we could consider (optionally, by default FALSE)
>> using defining a deterministic scheme for postprocessing the current
>> output of eigen such that at least for the good cases where all
>> eigenspaces are 1-dimensional, the postprocessing would result
>> in reproducible signs, by e.g., ensuring the first non-zero
>> entry of each eigenvector to be positive.
>>
>> MASS::mvrnorm() and mvtnorm::rmvnorm() both use "eigen",
>> whereas mvtnorm::rmvnorm() *does* have method = "chol" which
>> AFAIK does not suffer from such problems.
>>
>> OTOH, the help page of MASS::mvrnorm() mentions the Cholesky
>> alternative but prefers eigen for better stability (without
>> saying more).
>>
>> In spite of that, my personal recommendation would be to use
>>
>> mvtnorm::rmvnorm(.., method = "chol")
>>
>> { or the 2-3 lines of R code to the same thing without an extra package,
>> just using rnorm(), chol() and simple matrix operations }
>>
>> because in simulations I'd expect the var-cov matrix Sigma to
>> be far enough away from singular for chol() to be stable.
>>
>> Martin
>>
>> ______________________________________________
>> R-package-devel@r-project.org mailing list
>> https://stat.ethz.ch/mailman/listinfo/r-package-devel
>>
>
>
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