Testing Algebraic Complete Intersections

Key points

• We design an explicit and effective learning procedure which either certifies the nonexistence of such a manifold, up to a controlled relaxation of the approximation threshold, or returns a candidate regression manifold with controlled geometric complexity.1
• Equivalently, the procedure tests the manifold hypothesis within this hypothesis class.1
• The proposed procedure relies on quantitative geometric estimates for regular polynomial systems, which lead to a tractable auxiliary optimization problem.1
• We then develop a data-driven algorithm to solve this auxiliary optimization problem, establishing explicit bounds on its sample and arithmetic complexity.1

Sources (1)

• [1]Testing Algebraic Complete Intersections
  arXiv (AI, ML, NLP, CV, robotics, multi-agent) · Oct 8, 04:44 PM

  Given independent and identically distributed samples samples from a probability distribution in a potentially high-dimensional real space, we study the problem of testing whether the distribution is concentrated near a real algebraic complete intersection of prescribed dimension, bounded degree, and bounded condition number.

  We design an explicit and effective learning procedure which either certifies the nonexistence of such a manifold, up to a controlled relaxation of the approximation threshold, or returns a candidate regression manifold with controlled geometric complexity.

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