tag:blogger.com,1999:blog-13710336.post812342920624561680..comments2022-09-18T11:14:36.876-04:00Comments on Neurochannels: Visualizing the SVDEric Thomsonhttp://www.blogger.com/profile/06847717704454032165noreply@blogger.comBlogger9125tag:blogger.com,1999:blog-13710336.post-83087731343994374682022-09-18T11:14:36.876-04:002022-09-18T11:14:36.876-04:00Pawan, sorry I missed this question: the singular ...Pawan, sorry I missed this question: the singular value matrix (S) indeed scaled the elements according to the singular values, but as stated in the post, only the first r (rank of the matrix A) have nonnegative values. The remaining values are zero, so they scale the values to zero along those dimensions, flattening data clusters along those axes (in this case the third axis). In this example, AEric Thomsonhttps://www.blogger.com/profile/06847717704454032165noreply@blogger.comtag:blogger.com,1999:blog-13710336.post-77860915324142591472015-06-29T17:49:54.304-04:002015-06-29T17:49:54.304-04:00Hi Eric
Thanks for the beautiful explanation of S...Hi Eric<br /><br />Thanks for the beautiful explanation of SVD. I am confused about the flattening of the curves in the U-basis. As I understand, when S applied to the V-basis, it would effectively stretch the cylinder in V-basis right ? How does this translate to flattening in U-basis ?<br />Further clarification would be very appreciated.Pawanhttps://www.blogger.com/profile/14764841355863198041noreply@blogger.comtag:blogger.com,1999:blog-13710336.post-84096684369186999362014-10-22T12:29:36.163-04:002014-10-22T12:29:36.163-04:00Note also, Unknown, something that the equal matri...Note also, Unknown, something that the equal matrix sizes doesn't make clear. Note that we multiply by V' (transpose) to bring from standard into V-basis. We multiply by U (not transposed) to bring from U-basis back into standard basis. I believe we could multiply by U' to bring from standard into U-basis. This is something I had actually forgotten since writing this. :)Eric Thomsonhttps://www.blogger.com/profile/06847717704454032165noreply@blogger.comtag:blogger.com,1999:blog-13710336.post-29091717818518734442014-10-15T11:13:06.503-04:002014-10-15T11:13:06.503-04:00Unknown: Good question: I need to think about this...Unknown: Good question: I need to think about this more, as I think I was not clear in my description. Indeed, there are a few places where I could improve it. <br /><br />While V projects the data in an n dimensional space, S projects it into an m-dimensional space with a different basis set (given by the columns of U). And when you multiply by U, you rotate back into the standard basis. I thinkEric Thomsonhttps://www.blogger.com/profile/06847717704454032165noreply@blogger.comtag:blogger.com,1999:blog-13710336.post-7454380429683412142014-10-15T02:31:51.306-04:002014-10-15T02:31:51.306-04:00Thank you very much for this intuitive illustratio...Thank you very much for this intuitive illustration of SVD. However I find a statement confusing, i.e., the U transformation brings the standard basis back into alignment with the main axes of the plot. As A = U \Sigma V, this is equivalent to say after A transformation, the standard basis still aligns with the standard basis. I guess this probably doesn't hold. Or maybe I misunderstood your Unknownhttps://www.blogger.com/profile/15797413064239466971noreply@blogger.comtag:blogger.com,1999:blog-13710336.post-82933746482751677112011-01-17T20:22:12.107-05:002011-01-17T20:22:12.107-05:00Awesome! thanksAwesome! thanksAnonymousnoreply@blogger.comtag:blogger.com,1999:blog-13710336.post-69838807617907281432010-12-29T09:59:34.090-05:002010-12-29T09:59:34.090-05:00Very good stuff to recap SVD stuffVery good stuff to recap SVD stuffAnonymousnoreply@blogger.comtag:blogger.com,1999:blog-13710336.post-6003330572085170742010-01-14T06:38:33.775-05:002010-01-14T06:38:33.775-05:00Thanks for this, pretty good explanation!Thanks for this, pretty good explanation!ngiannhttps://www.blogger.com/profile/10312031533945961871noreply@blogger.comtag:blogger.com,1999:blog-13710336.post-20343259190762491342008-07-27T19:21:00.000-04:002008-07-27T19:21:00.000-04:00Very good visualization.Very good visualization.Anonymousnoreply@blogger.com