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What is an Eigenfunction?

An eigenfunction is a type of eigenvector that is also a function and used in multi-dimensional analysis, in particular spectral clustering and computer vision. Like eigenvectors, the function’s direction remains the same when a linear transformation is applied and instead it is only multiplied by a scaling factor (the eigenvalue).

If you imagine resizing a picture, eigenfunctions are the unmoving axes along which the linear transformation stretches, compresses or flips the data. In data analysis, using a function in place of a simple eigenvector allows you to model all the dimensions of any given space in one formula. Which drastically simplifies the deep learning process and improves accuracy.

Practical Uses of Eigenfunctions

  • Image Processing

    – When designing facial recognition programs, eigenfunctions provide a way to interpret faces across different angles/lightning levels for identification purposes.

  • Spectral Clustering – These functions are the mathematic underpinning of dimensional reduction and what makes multi-dimensional similarity matrixes possible.
  • Financial Markets Trading Strategies – While useful in all large-scale data analysis, eigenfunctions are quite useful in developing high-speed trading algorithms. These functions are a powerful tool to examine many different linear equations and find a unique solution for them all, or in market trading terms the least risky trade to make.