Error bounds on the SCISSORS approximation method.

Area of Science:

• Computational chemistry
• Machine learning
• Data science

Background:

• The SCISSORS method demonstrates strong empirical performance in approximating chemical similarities.
• Theoretical bounds for SCISSORS' worst-case error performance are currently lacking.
• Understanding error bounds is crucial for reliable chemical data analysis.

Purpose of the Study:

• To establish theoretical error bounds for the SCISSORS method.
• To demonstrate the theoretical equivalence of SCISSORS to established kernel methods.
• To analyze the implications of these bounds for chemical similarity approximations.

Main Methods:

• Proving reductions to establish SCISSORS' equivalence with kernel principal components analysis and rank-k Nyström approximation.
• Applying generalization bounds from related kernel methods to SCISSORS.
• Analyzing error bounds in terms of expected pairwise inner product error, matrix norms, and root-mean-square deviation.

Main Results:

• SCISSORS is theoretically equivalent to kernel principal components analysis and rank-k Nyström approximation.
• Expected error bounds were established for SCISSORS approximations of molecular similarity kernels.
• Actual SCISSORS performance significantly exceeds the derived worst-case theoretical bounds.

Conclusions:

• The SCISSORS method possesses theoretically proven error bounds, validating its use in chemical similarity approximation.
• The findings suggest that chemical space is inherently well-structured for sampling algorithms.
• This work bridges the gap between empirical success and theoretical guarantees for SCISSORS.