Concept of Equality and Similarity in Complex Data

When something, some object or concept, is comprised of tens or hundreds of dimensions or factors, then how do you compare two or more of such objects without loosing significant information?

Multidimensional scaling?
Dimensional Folding?
Cluster Analysis?
Factors Analysis?
Etc, Etc....

I have been intrigued by this question for many years.
To me this issue seems to share a connection with the issue of hash key generation because both can involve representing one thing with many parts with another thing that has one or a small fixed set of parts.

Finding a series of mathematical or other operations that yield unique values such that when a difference is computed between those unique values, that difference itself is also unique to the inputs.

NIST Next Gen Cryptographic Hash Algorithm Competition

NIST announced it is seeking a new cryptographic hash algorithm to replace or extend the existing SHA-2 family of algorithms.
The existing algorithm is described by the FIPS PUB 180-1 standard.
I am exploring whether or not such a new algorithm might be devised that uses, at least to some extent, new approaches to fulfill the requirements for NIST cryptographic hash algorithms rather than simply extending SHA-2 by modifying the number of rounds executed, the bit length of the final digest result, or the order/mix of bitwise compound operation performed per round.
I am sure that I am not the first to wonder about new approaches to this problem so maybe the current approach is the best that can be found.
I'll keep exploring the concepts just for the hell of it....

Feedback in Decision Support Systems

We monitor cell phone communications coming from certain parts of the world with the hope of detecting terrorist plans or intentions. They learn of this so they switch to other forms of communications. The question this raises is an old one.
How does or can the act of observing affect the behavior of the things being observed?
This question can be re-framed with regard to decision support system or decision analysis in which case the question becomes, in a much more indirect sense, how are the predictions inherent in generating data presented to users to aid in decision support affected by the ultimate decisions that are made based on them as those same predictions and decisions are made repeatedly over time?
In simpler terms, when does, if it does, the tail begin to wag the dog?
If it does, then how can such decision support systems compensate or dynamically adapt such that predictions remain valid and in fact become more accurate over time?