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In Google Earth Engine I have a FeatureCollection of 14,002 points. For each of the points I want to find the next nearest point's distance away. I don't need to record the nearest point ID, only the distance to it and save it in the feature. I haven't been able to find anything in the Docs that would do this.

Essentially I am trying to implement ArcMap's NEAR tool (found here: http://desktop.arcgis.com/en/arcmap/10.3/tools/analysis-toolbox/near.htm into something similar in GEE but haven't found a way yet.

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It's a multi-step process, but can be done with a spatial join (see this doc):

var points = ee.FeatureCollection.randomPoints(geometry, 14000, 0, 10);
Map.addLayer(points, {}, 'points')

// Max search distance is 5 km.
var spatialFilter = ee.Filter.withinDistance({
  distance: 5000,
  leftField: '.geo',
  rightField: '.geo',
  maxError: 10
})

// Join the points to themselves.
var joined = ee.Join.saveAll({
  matchesKey: 'neighbors', 
  measureKey: 'distance',
  ordering: 'distance'
}).apply({
  primary: points, 
  secondary: points, 
  condition: spatialFilter
});

// Get rid of points w/o neighbors.
// Increase distance in the spatialFilter if
// you want them all to have neighbors.
var hasNearest = joined.map(function(f) {
  var neighsSize = ee.List(f.get('neighbors')).size();
  return f.set('neighsSize', neighsSize);
}).filter(ee.Filter.gt('neighsSize', 1));
Map.addLayer(hasNearest, {color: 'red'}, 'hasNearest');

// Get distance to nearest point.
var withNearestDist = hasNearest.map(function(f) {
  var nearestDist = ee.Feature(ee.List(f.get('neighbors')).get(1))
      .get('distance');
  return f.set('nearestDist', nearestDist);
});
Map.addLayer(withNearestDist, {color: 'blue'}, 'withNearestDist');

Note that you get all the neighbors, whether you want them or not.

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