Hi Sujit ,
Firstly, I must show my deep appreciation and respect towards your kind help
and excellent knowledge.It would be the best if you and me are in the same
place then I shall specially go to express my thanks and respect to you.
I will try your way by spark mllib SVD .
For Linear Regression, Ax = b, in fact I want to view their variables and
coefficient conversely, just as (1): x1 * a1 + x2 * a2 + ... + xn * an = b ,
there is only with one linear formula for it.There are also training data set
with n number of point tuple [a11, a21, ..., an1, b1] just from [A, b]
(variables ), then the coefficient x = [x1, x2, ..., xn]T may be got by mllib
linear regression.
However, I tested spark mllib LR, while the point tuple dimension is more than
6, it would need more than 100 000 number of iterations to get enough accurate
solution about its coefficient, the time complexity is too much, the time cost
would be very tremendous while the dimension is hundreds of.
In effect, I am working on algorithm optimization with specific model not in
MLlib, that is object quadratic functionf(x1, x2, ..., xn) with lots of linear
constraint conditions, then I use Lagrange way to convert the question as
linear system of equations.My last problem is that, whether spark is properly
used to algorithm optimization , or just directly use
org.apache.spark.mllib.optimization, or by some other way, or it is not much
convenient for this application...
Thank you very much~~Zhiliang
On Saturday, October 24, 2015 12:41 AM, Sujit Pal <sujitatgt...@gmail.com>
wrote:
Hi Zhiliang,
For a system of equations AX = y, Linear Regression will give you a best-fit
estimate for A (coefficient vector) for a matrix of feature variables X and
corresponding target variable y for a subset of your data. OTOH, what you are
looking for here is to solve for x a system of equations Ax = b, where A and b
are known and you want the vector x.
This Math Stackexchange page [2] explains the math in more detail, but
basically...
A * x = b can be rewritten as x = A.I * b. You can get the pseudo-inverse of A
using SVD (Spark MLLib supports SVD [1]). So the SVD decomposition would make A
a product of three other matrices.
A = U * S * V.T
and the pseudo-inverse can be written as:
A.I = V * S * U.T
Then x can be found by multiplying A.I with b.
-sujit
[1] https://spark.apache.org/docs/1.2.0/mllib-dimensionality-reduction.html[2]
http://math.stackexchange.com/questions/458404/how-can-we-compute-pseudoinverse-for-any-matrix
On Fri, Oct 23, 2015 at 2:19 AM, Zhiliang Zhu <zchl.j...@yahoo.com> wrote:
Hi Sujit, and All,
Currently I lost in large difficulty, I am eager to get some help from you.
There is some big linear system of equations as:Ax = b, A with N number of row
and N number of column, N is very large, b = [0, 0, ..., 0, 1]TThen, I will
sovle it to get x = [x1, x2, ..., xn]T.
The simple solution would be to get inverse(A), and then x = (inverse(A)) * b
.A would be some JavaRDD<Interable<double>> , however, for RDD/matrix there is
add/multply/transpose APIs, no inverse API for it!
Then, how would it conveniently get inverse(A), or just solve the linear system
of equations by some other way...In Spark MLlib, there was linear regression,
the training process might be to solve the coefficients to get some specific
linear model, just is,
Ax = y, just train by (x, y) to get A , this might be used to solve the linear
system of equations. It is like that? I could not decide.
I must show my deep appreciation torwards your all help.
Thank you very much!Zhiliang
