Learning through function-space geometry.
My research connects functional analysis with statistical learning theory. I use reproducing kernel Hilbert spaces (RKHSs), variation spaces, and operator methods to study representation, approximation, generalization, and the dynamics of optimization.
Recursive RKHSs & variation spaces
Brownian Kernel Ladders encode depth through recursive Brownian geometry. I study their regularity, approximation properties, variational structure, and learning guarantees. Variation Brownian Kernel Ladders separate nonlinear recursion from outermost signed-measure superposition, supporting compactness, attainment, and constructive finite approximation.
Adaptive representations & covariance geometry
What is the statistical cost of choosing a representation? I study activation-mass characterizations for Brownian heads, same-sample selection through threshold traces, and common-covariance envelopes connecting Gaussian complexity, simultaneous approximation, and semidefinite certification.
Representation & optimization
I investigate how different finite realizations of the same function space affect optimizer trajectories. Current work examines Brownian-metric preconditioning, basis dependence, observable quotient geometry, and relaxed multilayer dynamics, combining mathematical analysis with controlled computational studies.
Operator methods for multitask learning
I study shared representations and task relationships through functions and operators. This includes Koopman- and composition-operator generalization bounds for deep multi-output models, vector-valued RKHSs, and trace-norm geometry.
Mathematical tools
Functional analysis · Operator theory · Empirical processes · Approximation theory · Convex and variational analysis · Matrix and operator geometry
Computational work
Python · PyTorch · NumPy · SciPy · scikit-learn · CVXPY · Finite element and spectral methods · Numerical certification · Reproducible experiments