Cool. The plots are fun. The way gaussian spots project into spinning
chains is very educational about entropy.
For full Random Projection, a lame random number generator
(java.lang.Random) will generate a higher standard deviation than a
highquality one like MurmurHash.
On Fri, Jul 1, 2011 at 5:25 PM, Ted Dunning <ted.dunning@gmail.com> wrote:
> Here is R code that demonstrates what I mean by stunning (aka 15 significant
> figures). Note that I only check dot products here. From the fact that the
> final transform is orthonormal we know that all distances are preserved.
>
> # make a big random matrix with rank 20
>> a = matrix(rnorm(20000), ncol=20) %*% matrix(rnorm(20000), nrow=20);
>> dim(a)
> [1] 1000 1000
> # random projection
>> y = a %*% matrix(rnorm(30000), ncol=30);
> # get basis for y
>> q = qr.Q(qr(y))
>> dim(q)
> [1] 1000 30
> # reproject b, do svd on result
>> b = t(q) %*% a
>> v = svd(b)$v
>> d = svd(b)$d
> # note how singular values drop like a stone at index 21
>> plot(d)
>> dim(v)
> [1] 1000 30
> # finish svd just for fun
>> u = q %*% svd(b)$u
>> dim(u)
> [1] 1000 30
> # u is orthogonal, right?
>> diag(t(u)%*% u)
> [1] 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1
> # and u diag(d) v' reconstructs a very precisely, right?
>> max(abs(au %*% diag(d) %*% t(v)))
> [1] 1.293188e12
>
> # now project a into the reduced dimensional space
>> aa = a%*%v
>> dim(aa)
> [1] 1000 30
> # check a few dot products
>> sum(aa[1,] %*% aa[2,])
> [1] 6835.152
>> sum(a[1,] %*% a[2,])
> [1] 6835.152
>> sum(a[1,] %*% a[3,])
> [1] 3337.248
>> sum(aa[1,] %*% aa[3,])
> [1] 3337.248
>
> # wow, that's close. let's try a hundred dot products
>> dot1 = rep(0,100);dot2 = rep(0,100)
>> for (i in 1:100) {dot1[i] = sum(a[1,] * a[i,]); dot2[i] = sum(aa[1,]*
> aa[i,])}
>
> # how close to the same are those?
>> max(abs(dot1dot2))
> # VERY
> [1] 3.45608e11
>
>
>
> On Fri, Jul 1, 2011 at 4:54 PM, Ted Dunning <ted.dunning@gmail.com> wrote:
>
>> Yes. Been there. Done that.
>>
>> The correlation is stunningly good.
>>
>>
>> On Fri, Jul 1, 2011 at 4:22 PM, Lance Norskog <goksron@gmail.com> wrote:
>>
>>> Thanks!
>>>
>>> Is this true?  "Preserving pairwise distances" means the relative
>>> distances. So the ratios of new distance:old distance should be
>>> similar. The standard deviation of the ratios gives a rough&ready
>>> measure of the fidelity of the reduction. The standard deviation of
>>> simple RP should be highest, then this RP + orthogonalization, then
>>> MDS.
>>>
>>> On Fri, Jul 1, 2011 at 11:03 AM, Ted Dunning <ted.dunning@gmail.com>
>>> wrote:
>>> > Lance,
>>> >
>>> > You would get better results from the random projection if you did the
>>> first
>>> > part of the stochastic SVD. Basically, you do the random projection:
>>> >
>>> > Y = A \Omega
>>> >
>>> > where A is your original data, R is the random matrix and Y is the
>>> result.
>>> > Y will be tall and skinny.
>>> >
>>> > Then, find an orthogonal basis Q of Y:
>>> >
>>> > Q R = Y
>>> >
>>> > This orthogonal basis will be very close to the orthogonal basis of A.
>>> In
>>> > fact, there are strong probabilistic guarantees on how good Q is as a
>>> basis
>>> > of A. Next, you project A using the transpose of Q:
>>> >
>>> > B = Q' A
>>> >
>>> > This gives you a short fat matrix that is the projection of A into a
>>> lower
>>> > dimensional space. Since this is a left projection, it isn't quite what
>>> you
>>> > want in your work, but it is the standard way to phrase things. The
>>> exact
>>> > same thing can be done with left random projection:
>>> >
>>> > Y = \Omega A
>>> > L Q = Y
>>> > B = A Q'
>>> >
>>> > In this form, B is tall and skinny as you would like and Q' is
>>> essentially
>>> > an orthogonal reformulation of of the random projection. This
>>> projection is
>>> > about as close as you are likely to get to something that exactly
>>> preserves
>>> > distances. As such, you should be able to use MDS on B to get exactly
>>> the
>>> > same results as you want.
>>> >
>>> > Additionally, if you start with the original form and do an SVD of B
>>> (which
>>> > is fast), you will get a very good approximation of the prominent right
>>> > singular vectors of A. IF you do that, the first few of these should be
>>> > about as good as MDS for visualization purposes.
>>> >
>>> > On Fri, Jul 1, 2011 at 2:44 AM, Lance Norskog <goksron@gmail.com>
>>> wrote:
>>> >
>>> >> I did some testing and make a lot of pretty charts:
>>> >>
>>> >> http://ultrawhizbang.blogspot.com/
>>> >>
>>> >> If you want to get quick visualizations of your clusters, this is a
>>> >> great place to start.
>>> >>
>>> >> 
>>> >> Lance Norskog
>>> >> goksron@gmail.com
>>> >>
>>> >
>>>
>>>
>>>
>>> 
>>> Lance Norskog
>>> goksron@gmail.com
>>>
>>
>>
>

Lance Norskog
goksron@gmail.com
