How can I find a projection to preserve the relative value of inner product?

Question

I want to do dimension reduction with a 100-dimension vector v, then get a 10-dimension vector v'.

And the property below must be preserved:

For arbitrary vector w1, w2(100-dimension)
if v * w1 > v * w2(* rep inner product)
After reduction....
v' * w1' > v' * w2'

I learn that random projection is a method( http://scikit-learn.org/stable/modules/random_projection.html ), but it preserve the value of distance and inner product. But I only want to keep the relative > or < property in stead of absolute distance/inner-product value.

The other problem in random projection is that it suits for large dimension reduction(10000-3000).

from sklearn.random_projection import johnson_lindenstrauss_min_dim
johnson_lindenstrauss_min_dim gives us a bound.

Below is my Python-Pseudo-Code to explain what I need:

import sys
import math
import numpy as np
def compare(a, b_lst):
    d_lst = []
    indx = 0
    for b in b_lst:
        d_lst.append((index, np.dot(a, b)))
        indx += 1
   return sorted(d_lst, key = lambda v : v[1])

x = np.random.rand(1, 100)
y = np.random.rand(5, 100)
result1 = compare(x, y)

# do projection
transformer = projection_method(object_dimension = 10)
x1 = transformer.transform(x)
y1 = transformer.transform(y)
result2 = compare(x1, y1)

for i in xrange(len(result1)):
    if result1[i][0] != result2[i][0]: # compare sorted index
        print 'failed'
        sys.exit(-1)
print 'passed'

Answer 1

There are no such ready made transforms. Even if there are that I am not aware of, no transformation is going to preserve such a property exactly. By reducing the dimension you are intrinsically losing information.