Given a dataset D
a_1 a_2
x_1 0.55, 0.99
x_2 0.41, 1
x_3 0.76, 0.83
x_4 0.22, 0.65
where each row represents an object x_i and the column refers to the features describing the objects. In general, computing the Euclidean distance L(x_i,x_j) between the two objects x_i and x_j is a straightforward strategy. However, determining the similarity degree depends on a threshold level. For example x_i is similar to x_j if L(x_i,x_j)
I have dealt with this problem extensively in several very different applications, for example: gas turbine performance, turbulent boundary layer, and groundwater well pumping response. A simple difference in values often mischaracterizes the real difference you can see when you look at the data all at once because humans are very good at recognizing patterns. Another way to look at the differences is to calculate the area between two curves (different data sets, ideal response, theoretical or analytical solution, etc.). You might be able to calculate the area directly or numerically integrate it. You might even integrate between two cubic splines. See the examples at this site http://dudleybenton.altervista.org/projects/WellTest/WellTest.html
I think rater than the bounds in distance based non parametric methods you are interested in general methods without such bounds or thresholds. The alternative methods are Bayesian parametric methods defined on certain data distribution models or density functions, which do not need predefined threshold, boundedeness, error defining or initial guess. However, the difficulties here arising, is to calculate normalizing constants that are naturally appearing in Bayesian parametric probability density functions. Here, recognition is based on pixel densities estimations (for the cases of images or videos data), that is for qualitative data classification. If your data is quantitative or continuous, you may than need regression models which do not need the calculation of normalizing constants. Here, however., you may need some smallest step sizes and bounds etc again as it is non parametric regression.
See e.g., Parametric modelling : https://www.researchgate.net/project/Statistical-Learning-on-manifolds-with-its-applications-in-computer-vision?_sg=p3auPX6bo1l07aj8WDFVg3ducF8zgMrm0nT1Kyj7VVO3rmSwQoo4lX_VyvnasHaxWjLtyrZVhqehn0PT58sdIrhO7epu_bMctBS9
See e.g., Parametric modelling : https://www.researchgate.net/project/Statistical-Learning-on-manifolds-with-its-applications-in-computer-vision?_sg=p3auPX6bo1l07aj8WDFVg3ducF8zgMrm0nT1Kyj7VVO3rmSwQoo4lX_VyvnasHaxWjLtyrZVhqehn0PT58sdIrhO7epu_bMctBS9
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