• Machine learning (ML) is the study of designing and implementing statistical algorithms that can learn from data and generalize to unseen data, extract knowledge or hypothesis from data, make predictions on data, and perform tasks without explicit instructions.

• Stability analysis of multi-basin systems (MBS) with toolkits from differential/algebraic topology and dynamical systems can be applied to developing efficient and stable machine learning models.

Stable Clustering

• The clustering algorithm consists of three steps: (i) construct a support function f that estimates the support of a given data distribution; (ii) construct an MBS associated with f to decompose the data space into several disjoint groups, each represented by a stable equilibrium vector (SEV); and (iii) label each data into a cluster based on the similarity or connectivity of the topological graph of the constructed SEV.

• Equilibrium-based support vector clustering, Multi-basin support-based clustering, Gaussian processes clustering, Fast support-based clustering, Dynamic and hierarchical support-based clustering, Voronoi cell-based clustering using a kernel support, Multi-basin kernels for dynamic pattern denoising.

• [J Lee, D Lee, 2005], [J Lee, D Lee, 2006], [HC Kim, J Lee, 2007], [D Lee, J Lee, 2010], [KH Jung, D Lee, J Lee, 2010], [KH Jung, N Kim, J Lee, 2011], [K Kim, Y Son, J Lee, 2014], [Y Son, S Lee, S Park, J Lee, 2018]

Stable Manifold Learning

• Many practical high-dimensional real data such as images are often confined to a region of the space having lower effective dimension

• The algorithms aim to find effective and stable low-dimensional structures in high-dimensional data spaces: Sequential manifold learning, Semi-supervised nonlinear dimensionality reduction, Nonlinear dynamic projection for noise reduction of dispersed manifolds.

• [K Kim, J Lee, 2014], [S Park, W Lee, J Lee, 2019], [S Park, J Lee, K Kim*, 2019], [S Park, J Lee, 2020]

Dynamical Systems

• Study of the long-term behavior of evolutionary nonlinear systems.

• Deterministic Differential Equation

The laws of Nature are expressed by differential equations, so it is useful to solve differential equations - Isaac Newton

Stochastic Differential Equation 

The laws of artificial systems (financial systems & artificial intelligence systems) can be expressed by (stochastic) differential equations, so it is useful to apply (stochastic) dynamical systems approach to analyze and stabilize artificial systems – J. Lee.

Convergence Analysis for Nonlinear Optimization (1999-2007)

• Convergence analysis for nonlinear optimization := Stability analysis for nonlinear systems

• The construction of the multi-basin systems (MBS) ,i.e. completely stable dynamical systems on manifolds, associated with objective function and/or constraint functions can be applied to developing efficient numerical methods towards global optimization as well as to establishing theoretical foundations of them.
  [J Lee, HD Chiang, 2001b], [J Lee, HD Chiang, 2002], [J Lee, HD Chiang, 2004], [J Lee, 2005]

• Develop novel deterministic methods for systematically computing multiple optimal solutions of general nonconvex optimization problems.
  [J Lee, HD Chiang, 2001a], [J Lee, HD Chiang, 2004], [J Lee, 2007]

Stability Analysis for Nonlinear Systems (1997-2005)

• Study of the stability of solutions of differential equations and of trajectories of dynamical systems under small perturbations of initial conditions.