Manifold

• Data manifolds are an abstraction
• Only geometric insights are important
• Locally around some point c where the PDF is large → It will stay large only for a small fraction of directions
  • Those directions span a low dimensional hyperplane around c
  • “low dimensional sheets”
  • curved path
• In an n dimensional real vector space ${\mathbb{R}}^n$ . Embedding space
  • $m\le n$ is a positive integer
  • An m dim manifold $\mathcal{M}$ is a subset of the vector space where one can smoothly map a neighborhood of that point to a neighborhood of the origin in m dim Euclidean space
    • Locally represents Euclidean space
• Only surface and not interior
• No sharp edges or spikes
• Can be exploited by Adversarial Learning
• Examples
  • 1 dim → Lines in some high dim figure : B
  • 2 dim → Surfaces : A
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Refs

• tds
• way more stuff : bjlkengtodo