Delete Duplicates by a distance test

Question

Consider the following list which is of length 245:

list = {{1.7821782709267973`, 1.782178215568537`, 
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    1.09853200574019`}, {1.7821782241937663`, 1.7821782217911688`, 
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    1.098329810431893`}, {1.7821782657916128`, 1.782178182563235`, 
    1.7821781351169887`}, {4.631752709518959`, 1.0974384186951538`, 
    1.0953328414826287`}, {4.629100018056158`, 1.109735335556899`, 
    1.0776020488399833`}, {1.7821782406783102`, 1.7821782197467901`, 
    1.7821781576231295`}, {1.7821782142229652`, 1.7821782134084538`, 
    1.7821782131318817`}, {4.6347259226908095`, 1.099551036963113`, 
    1.0987973177176564`}};

I wish to eliminate duplicates here, however I have difficulty of finding the proper criterion

after applying DeleteDuplicates[Floor[list, 0.001]] I reduce the number of elements in the list to 91

{{1.782, 1.782, 1.782}, {4.632, 1.106, 1.087}, {4.627, 1.106, 
  1.077}, {4.631, 1.096, 1.096}, {4.632, 1.097, 1.096}, {4.627, 1.092,
   1.092}, {4.629, 1.094, 1.094}, {4.63, 1.095, 1.095}, {4.634, 1.099,
   1.098}, {4.634, 1.101, 1.096}, {4.633, 1.099, 1.097}, {4.631, 
  1.095, 1.095}, {4.632, 1.095, 1.095}, {4.632, 1.097, 1.097}, {4.628,
   1.093, 1.093}, {4.629, 1.093, 1.093}, {4.631, 1.098, 
  1.094}, {4.634, 1.102, 1.094}, {4.63, 1.097, 1.093}, {4.633, 1.098, 
  1.098}, {4.626, 1.091, 1.091}, {4.633, 1.099, 1.095}, {4.634, 1.099,
   1.099}, {4.631, 1.099, 1.096}, {4.633, 1.097, 1.097}, {4.61, 1.072,
   1.065}, {4.619, 1.089, 1.077}, {4.633, 1.099, 1.094}, {4.634, 1.1, 
  1.097}, {4.63, 1.095, 1.094}, {4.63, 1.094, 1.094}, {4.633, 1.104, 
  1.091}, {4.628, 1.109, 1.077}, {4.62, 1.108, 1.06}, {4.633, 1.1, 
  1.096}, {4.61, 1.075, 1.066}, {4.633, 1.102, 1.094}, {4.634, 1.098, 
  1.098}, {4.629, 1.095, 1.091}, {4.631, 1.096, 1.095}, {4.627, 1.099,
   1.099}, {4.632, 1.101, 1.091}, {4.636, 1.1, 1.095}, {4.632, 1.107, 
  1.086}, {4.625, 1.09, 1.09}, {4.632, 1.104, 1.088}, {4.629, 1.096, 
  1.092}, {4.632, 1.107, 1.087}, {4.633, 1.102, 1.091}, {4.626, 1.11, 
  1.069}, {4.626, 1.09, 1.09}, {4.625, 1.09, 1.089}, {4.631, 1.097, 
  1.095}, {4.633, 1.103, 1.094}, {4.627, 1.092, 1.091}, {4.634, 1.1, 
  1.096}, {4.634, 1.1, 1.095}, {4.625, 1.093, 1.086}, {4.62, 1.113, 
  1.048}, {4.639, 1.099, 1.096}, {4.622, 1.107, 1.066}, {4.633, 1.102,
   1.093}, {4.632, 1.1, 1.095}, {4.63, 1.103, 1.087}, {4.617, 1.083, 
  1.08}, {4.619, 1.089, 1.078}, {4.624, 1.099, 1.078}, {4.633, 1.101, 
  1.093}, {4.631, 1.099, 1.099}, {4.631, 1.105, 1.087}, {4.617, 1.097,
   1.066}, {4.632, 1.098, 1.095}, {4.633, 1.098, 1.095}, {4.639, 1.1, 
  1.096}, {4.618, 1.091, 1.062}, {4.628, 1.103, 1.082}, {4.633, 1.099,
   1.096}, {4.634, 1.103, 1.094}, {4.634, 1.099, 1.097}, {4.628, 
  1.093, 1.092}, {4.631, 1.097, 1.097}, {4.632, 1.098, 1.096}, {4.633,
   1.1, 1.095}, {4.634, 1.1, 1.098}, {4.635, 1.099, 1.098}, {4.631, 
  1.104, 1.084}, {4.627, 1.093, 1.09}, {4.629, 1.095, 1.092}, {4.632, 
  1.103, 1.091}, {4.632, 1.098, 1.097}, {4.629, 1.109, 1.077}}

however, many of the elements in the list are still considered (by me) as duplicates, for example:

the elements {4.631, 1.099, 1.099}, {4.632, 1.098, 1.097},{4.634, 1.1, 1.098} are duplicates. at total I expect the length of the list after removing all the duplicates to be no longer than 10.

What criterion should I apply to eliminate these duplicates? A metric distance for example can be useful, how can it be applied?

Answer 1

The problem of clustering is much more difficult than what DeleteDuplicates can deal with. Graph clustering methods are required here, lest you fall into edge cases when the clusters aren't well separated.

FindClusters with default settings only finds two major clusters. We can compute their means and covariances with

{Mean[#], Covariance[#]} & /@ FindClusters[list]
(*    {{{1.78218, 1.78218, 1.78218},
        {{3.43653*10^-15, 2.67606*10^-15, 2.26368*10^-15},
         {2.67606*10^-15, 3.42421*10^-15, 3.58434*10^-15},
         {2.26368*10^-15, 3.58434*10^-15, 4.77886*10^-15}}},
       {{4.63073, 1.09729, 1.09322},
        {{0.000016663, 0.0000109568, 0.0000239168},
         {0.0000109568, 0.0000240003, -2.33322*10^-7},
         {0.0000239168, -2.33322*10^-7, 0.0000554542}}}}        *)

We can also find three clusters, depending on taste:

Mean /@ FindClusters[list, 10]
(*    {{1.78218, 1.78218, 1.78218},
       {4.63136, 1.09757, 1.0944},
       {4.61709, 1.09121, 1.06747}}    *)

If you want even more fine-grained clusters, you can do several iterations of FindClusters, starting with FindClusters /@ FindClusters[list] etc.