Motivation:
My plan is to get the overall euclidean distance matrix for all the vectors in N number of dataset. Each dataset is basically an array of n-dimensional points. For e.g: A dataset can be like [[1,2,3,...n],[1,2,3,..n],...[1,2,3,n]]. However, the datasets are not gonna be shared to a single entity for which I cannot compile them and thus will not know all the points to calculate pairwise euclidean distance for some similarity calculation and clustering analysis and modeling. However, I will be only notified what are the common vectors in the datasets and distance of the vectors in any dataset with respect to those common vectors without knowing or passing the point's coordinates. If the dataset could be shared in the same model, I wouldn't have to face this distributed calculation problem.
Details:
A and C are two n-dimensional vectors from two different dataset. They have a common vector B. I want to calculate euclidean distance between A and C without exposing A and C, rather using B. Lets say ED(A,B) is calculated in model_1 and ED(B,C) is calculated in model_2, I can easily use ED(A,B) i.e AB and ED(B,C) i.e BC to compute ED(A,C). However, if someone know B they can easily find out A and C. Even if B is a randomly created common point, if it available, A and C can also be found. Is it possible to use B such that A and C cannot be retrieve in anyway. I have looked into differential privacy, PATE, SMPC techniques but they have their limitations in preserving privacy.
Work done so far :
https://www.online-python.com/Pq7OhM8FJe
Question
Given the euclidean distance between vector A and vector B, and that between vector B and vector C, how to calculate the euclidean-distance between point A and C such that there is no way to retrieve A and C even if B is known?
Any computation technique I can follow?
Thanks in advance
I would consult Kauffman and Rousseau, Finding Groups in Data, for some ideas. the z-library has the book. Software is in R package, cluster. Best wishes, David Booth
https://math.stackexchange.com/questions/4217205/get-distance-between-point-a-and-c-when-their-distances-with-a-common-point-b-is
@Aravinda, this is similar question I post in math-exchange. My approach mentioned in the python scripted attached in the question has huge privacy issue.
I think the main privacy problem is in line 19 where we can get direction of the vectors from the difference:
https://www.online-python.com/puVbWm8yce
Let's say, instead of (vectorB - VectorAD1) and (vectorB - VectorAD2), we take euclidean distance of AD1 and B and that of B and AD2 which are scalar.
Are there any way to compute the aggregated euclidean distance of AD1 and AD2 using those scalar? If we can minimize the error as much as possible is also great.
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