[HN Gopher] The geometry of data: the missing metric tensor and ...
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       The geometry of data: the missing metric tensor and the Stein score
       [Part II]
        
       Author : perone
       Score  : 36 points
       Date   : 2024-11-14 11:49 UTC (11 hours ago)
        
 (HTM) web link (blog.christianperone.com)
 (TXT) w3m dump (blog.christianperone.com)
        
       | openrisk wrote:
       | Establishing linkages between ML and Differential Geometry is
       | intriguing (to say the least). But I have this nagging sense that
       | "data manifolds" are too rigidly tied to numerical
       | representations for this program to flourish. Differential
       | geometry is all about invariance. Geometric objects have a life
       | of their own so to speak, irrespective of any particular
       | representation. In the broader data science world such an
       | internal structure is not accessible in general. The systems
       | modeled are too complex and their capture in data too superficial
       | to be a reflection of the "true state". In a sense this is
       | analogous to the "blind men touching a elephant in different
       | parts and disagreeing about what it is".
        
         | perone wrote:
         | I'm not sure I agree about the data manifolds being too rigid.
         | When we look at the quality score-based generative models and
         | diffusion we can see a clear evidence of how flexible these
         | representations are. We could say the same about statistical
         | manifolds, but the fact that the Fisher is the fundamental
         | metric tensor for the statistical manifold is a fundamental
         | piece of many 1st and 2nd order optimizers today.
        
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