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Maximum Mean Discrepancy (MMD) — what is it?

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Answer

  • Distance between two distributions in RKHS: MMD(P, Q) = ||μP  −  μQ{\mu}_{P}\; - \;{\mu}_{Q}||_H where μP{\mu}_{P} is the mean embedding.
  • Estimated from samples using kernel k (Gaussian typical): MMD2  =  E[k(x,  x)]  +  E[k(y,  y)]  −  2\mathrm{MMD}^{2}\; = \;E[k(x, \;x)]\; + \;E[k(y, \;y)]\; - \;2 E[k(x, y)].
  • Uses: two-sample tests (do P and Q differ?), drift detection, generative model evaluation.
  • More powerful than univariate tests in multivariate settings.
Check yourself — multiple choice
  • Random
  • RKHS distance between two distributions via mean embeddings; MMD2  =  E[k(x,x)]  +  E[k(y,y)]  −  2\mathrm{MMD}^{2}\; = \;E[k(x, x)]\; + \;E[k(y, y)]\; - \;2E[k(x,y)]; two-sample test / drift detection
  • Same as KL
  • Not real

MMD: kernel two-sample distance; drift / two-sample tests.

#anomaly-detection#theory

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