Hubs in high-dimensional point clouds (distribution of “being within nearest neighbours of how_many other...












2














The hubness problem is a phenomenon mentioned in this paper, for instance. It claims that for random data point clouds in high-dimensional spaces, there will be "many" hubs -- that is, points that are within k nearest neighbors of many other points.



We define, for each point $x$, the number $N_k(x)$ to be the number of other points $y$ such that $x$ belongs to $k$ nearest members of $y$. Radonovic claims that $N_k(x)$ has approximately normal distribution for random point clouds in low dimensional manifolds, but a highly skewed distribution with long right-tail in high dimensions. Radonovic illustrates this with uniformly distributed points in $[0,1]^N$.



I tried to reproduce some of that in Python and came to a surprising observation. While the $N_k(x)$ distribution is indeed skewed in high dimensions when generating uniform point cloud in the cube and using the Euclidean distance function, I didn't observe this when generating uniform point clouds in the sphere.



Experiment with a cube (uniformly generated 2000 points in $[0,1]^{500}$)
enter image description hereenter image description here



The same experiment with a sphere (uniformly generated 2000 points in $S^{500}$)
enter image description hereenter image description here



My intuition is that there should be no difference between hub-histograms in an $n$-cube or an $n$-sphere, because this hubeness seems to be a local phenomenon. Unless I have some stupid mistake in the code, my intuition seems to be wrong.



Generating more points, I think I could see some skewness in the sphere as well, but only very slightly, if at all.



So here are my questions:





  1. How is the cube different from a sphere? Could the difference be only explained by boundaries, corners or corners of corners?

  2. Any insight into this hubeness-phenomena?






statistics euclidean-geometry experimental-mathematics







share|cite|improve this question





























    2














    The hubness problem is a phenomenon mentioned in this paper, for instance. It claims that for random data point clouds in high-dimensional spaces, there will be "many" hubs -- that is, points that are within k nearest neighbors of many other points.



    We define, for each point $x$, the number $N_k(x)$ to be the number of other points $y$ such that $x$ belongs to $k$ nearest members of $y$. Radonovic claims that $N_k(x)$ has approximately normal distribution for random point clouds in low dimensional manifolds, but a highly skewed distribution with long right-tail in high dimensions. Radonovic illustrates this with uniformly distributed points in $[0,1]^N$.



    I tried to reproduce some of that in Python and came to a surprising observation. While the $N_k(x)$ distribution is indeed skewed in high dimensions when generating uniform point cloud in the cube and using the Euclidean distance function, I didn't observe this when generating uniform point clouds in the sphere.



    Experiment with a cube (uniformly generated 2000 points in $[0,1]^{500}$)
    enter image description hereenter image description here



    The same experiment with a sphere (uniformly generated 2000 points in $S^{500}$)
    enter image description hereenter image description here



    My intuition is that there should be no difference between hub-histograms in an $n$-cube or an $n$-sphere, because this hubeness seems to be a local phenomenon. Unless I have some stupid mistake in the code, my intuition seems to be wrong.



    Generating more points, I think I could see some skewness in the sphere as well, but only very slightly, if at all.



    So here are my questions:





    1. How is the cube different from a sphere? Could the difference be only explained by boundaries, corners or corners of corners?

    2. Any insight into this hubeness-phenomena?






    statistics euclidean-geometry experimental-mathematics







    share|cite|improve this question



























      2












      2








      2


      1





      The hubness problem is a phenomenon mentioned in this paper, for instance. It claims that for random data point clouds in high-dimensional spaces, there will be "many" hubs -- that is, points that are within k nearest neighbors of many other points.



      We define, for each point $x$, the number $N_k(x)$ to be the number of other points $y$ such that $x$ belongs to $k$ nearest members of $y$. Radonovic claims that $N_k(x)$ has approximately normal distribution for random point clouds in low dimensional manifolds, but a highly skewed distribution with long right-tail in high dimensions. Radonovic illustrates this with uniformly distributed points in $[0,1]^N$.



      I tried to reproduce some of that in Python and came to a surprising observation. While the $N_k(x)$ distribution is indeed skewed in high dimensions when generating uniform point cloud in the cube and using the Euclidean distance function, I didn't observe this when generating uniform point clouds in the sphere.



      Experiment with a cube (uniformly generated 2000 points in $[0,1]^{500}$)
      enter image description hereenter image description here



      The same experiment with a sphere (uniformly generated 2000 points in $S^{500}$)
      enter image description hereenter image description here



      My intuition is that there should be no difference between hub-histograms in an $n$-cube or an $n$-sphere, because this hubeness seems to be a local phenomenon. Unless I have some stupid mistake in the code, my intuition seems to be wrong.



      Generating more points, I think I could see some skewness in the sphere as well, but only very slightly, if at all.



      So here are my questions:





      1. How is the cube different from a sphere? Could the difference be only explained by boundaries, corners or corners of corners?

      2. Any insight into this hubeness-phenomena?






      statistics euclidean-geometry experimental-mathematics







      share|cite|improve this question















      The hubness problem is a phenomenon mentioned in this paper, for instance. It claims that for random data point clouds in high-dimensional spaces, there will be "many" hubs -- that is, points that are within k nearest neighbors of many other points.



      We define, for each point $x$, the number $N_k(x)$ to be the number of other points $y$ such that $x$ belongs to $k$ nearest members of $y$. Radonovic claims that $N_k(x)$ has approximately normal distribution for random point clouds in low dimensional manifolds, but a highly skewed distribution with long right-tail in high dimensions. Radonovic illustrates this with uniformly distributed points in $[0,1]^N$.



      I tried to reproduce some of that in Python and came to a surprising observation. While the $N_k(x)$ distribution is indeed skewed in high dimensions when generating uniform point cloud in the cube and using the Euclidean distance function, I didn't observe this when generating uniform point clouds in the sphere.



      Experiment with a cube (uniformly generated 2000 points in $[0,1]^{500}$)
      enter image description hereenter image description here



      The same experiment with a sphere (uniformly generated 2000 points in $S^{500}$)
      enter image description hereenter image description here



      My intuition is that there should be no difference between hub-histograms in an $n$-cube or an $n$-sphere, because this hubeness seems to be a local phenomenon. Unless I have some stupid mistake in the code, my intuition seems to be wrong.



      Generating more points, I think I could see some skewness in the sphere as well, but only very slightly, if at all.



      So here are my questions:





      1. How is the cube different from a sphere? Could the difference be only explained by boundaries, corners or corners of corners?

      2. Any insight into this hubeness-phenomena?






      statistics euclidean-geometry experimental-mathematics




      statistics euclidean-geometry experimental-mathematics






      share|cite|improve this question















      share|cite|improve this question













      share|cite|improve this question




      share|cite|improve this question








      edited Nov 21 '18 at 12:43

























      asked Nov 21 '18 at 10:21









      Peter Franek

      7,21631935




      7,21631935






















          0






          active

          oldest

          votes











          Your Answer





          StackExchange.ifUsing("editor", function () {
          return StackExchange.using("mathjaxEditing", function () {
          StackExchange.MarkdownEditor.creationCallbacks.add(function (editor, postfix) {
          StackExchange.mathjaxEditing.prepareWmdForMathJax(editor, postfix, [["$", "$"], ["\\(","\\)"]]);
          });
          });
          }, "mathjax-editing");

          StackExchange.ready(function() {
          var channelOptions = {
          tags: "".split(" "),
          id: "69"
          };
          initTagRenderer("".split(" "), "".split(" "), channelOptions);

          StackExchange.using("externalEditor", function() {
          // Have to fire editor after snippets, if snippets enabled
          if (StackExchange.settings.snippets.snippetsEnabled) {
          StackExchange.using("snippets", function() {
          createEditor();
          });
          }
          else {
          createEditor();
          }
          });

          function createEditor() {
          StackExchange.prepareEditor({
          heartbeatType: 'answer',
          autoActivateHeartbeat: false,
          convertImagesToLinks: true,
          noModals: true,
          showLowRepImageUploadWarning: true,
          reputationToPostImages: 10,
          bindNavPrevention: true,
          postfix: "",
          imageUploader: {
          brandingHtml: "Powered by u003ca class="icon-imgur-white" href="https://imgur.com/"u003eu003c/au003e",
          contentPolicyHtml: "User contributions licensed under u003ca href="https://creativecommons.org/licenses/by-sa/3.0/"u003ecc by-sa 3.0 with attribution requiredu003c/au003e u003ca href="https://stackoverflow.com/legal/content-policy"u003e(content policy)u003c/au003e",
          allowUrls: true
          },
          noCode: true, onDemand: true,
          discardSelector: ".discard-answer"
          ,immediatelyShowMarkdownHelp:true
          });


          }
          });














          draft saved

          draft discarded


















          StackExchange.ready(
          function () {
          StackExchange.openid.initPostLogin('.new-post-login', 'https%3a%2f%2fmath.stackexchange.com%2fquestions%2f3007531%2fhubs-in-high-dimensional-point-clouds-distribution-of-being-within-nearest-nei%23new-answer', 'question_page');
          }
          );

          Post as a guest















          Required, but never shown

























          0






          active

          oldest

          votes








          0






          active

          oldest

          votes









          active

          oldest

          votes






          active

          oldest

          votes
















          draft saved

          draft discarded




















































          Thanks for contributing an answer to Mathematics Stack Exchange!


          • Please be sure to answer the question. Provide details and share your research!

          But avoid …



          • Asking for help, clarification, or responding to other answers.

          • Making statements based on opinion; back them up with references or personal experience.


          Use MathJax to format equations. MathJax reference.


          To learn more, see our tips on writing great answers.





          Some of your past answers have not been well-received, and you're in danger of being blocked from answering.


          Please pay close attention to the following guidance:


          • Please be sure to answer the question. Provide details and share your research!

          But avoid …



          • Asking for help, clarification, or responding to other answers.

          • Making statements based on opinion; back them up with references or personal experience.


          To learn more, see our tips on writing great answers.




          draft saved


          draft discarded














          StackExchange.ready(
          function () {
          StackExchange.openid.initPostLogin('.new-post-login', 'https%3a%2f%2fmath.stackexchange.com%2fquestions%2f3007531%2fhubs-in-high-dimensional-point-clouds-distribution-of-being-within-nearest-nei%23new-answer', 'question_page');
          }
          );

          Post as a guest















          Required, but never shown





















































          Required, but never shown














          Required, but never shown












          Required, but never shown







          Required, but never shown

































          Required, but never shown














          Required, but never shown












          Required, but never shown







          Required, but never shown







          Popular posts from this blog

          Can a sorcerer learn a 5th-level spell early by creating spell slots using the Font of Magic feature?

          Image
          28 6 $begingroup$ Per the Font of Magic feature, sorcerer can use Flexible Casting to create 5th-level spell slots at level 7, even though they are not typically available until level 9. You can transform unexpended sorcery points into one spell slot as a bonus action on your turn. The Creating Spell Slots table shows the cost of creating a spell slot of [5th level is 7] If such a sorcerer levels up while still having this 5th-level spell slot, can they choose a 5th-level spell as the spell they gain upon levelling up? The Spellcasting feature states: Additionally, when you gain a level in this class, you can choose one of the sorcerer spells you know and replace it with another spell from the sorcerer spell list, which also must be of a level for which you have spell slots.
          Read more

          Does disintegrating a polymorphed enemy still kill it after the 2018 errata?

          Image
          up vote 13 down vote favorite 1 After the 2018 errata, the disintegrate spell description now reads: A creature targeted by this spell must make a Dexterity saving throw. On a failed save, the target takes 10d6 + 40 force damage. The target is disintegrated if this damage leaves it with 0 hit points. If you were to polymorph an enemy into a rat and then disintegrate it, would the enemy be disintegrated or would it just return to its original form? I know that for Druids, it’s not an instant kill anymore, but is this the case for polymorph as well? dnd-5e spells polymorph errata share | improve this question edited 18 hours ago
          Read more

          A Topological Invariant for $pi_3(U(n))$

          Image
          3 3 $begingroup$ Recently, I saw a construction of topological invariant for $pi_3(U(n))$ with $ngeq 2$ : $$ N=frac{1}{24pi^2}int_{S^3} d^3x epsilon^{ijk} Tr[(U^{-1}partial_{x_i}U)(U^{-1}partial_{x_j}U)(U^{-1}partial_{x_k}U)] , $$ where $Uin U(n)$ depends on $boldsymbol{x}=(x_1,x_2,x_3)in S^3$ , $epsilon^{ijk}$ is the Levi-Civita symbol, $i,j,k=1,2,3$ , and the duplicated indexes are summed over. It is claimed that $N$ is an integer, but why? Update 02/02/2019 I think I got an argument for $n=2$ . In this case, $U=e^{i varphi} q$ with $qin SU(2)$ . Due to the trace and the Levi-Civita symbol in $N$ , $varphi$ does not contribute to $N$ . As $Tr[q^{dagger}partial_i q]=0$ and $(q^{dagger}partial_i q)^{dagger}=-q^{dagger}partial_i q$ , $q^{dagger}partial_i q$ in geneeral has the form $q^{dagger}partia
          Read more