Kokot et al. (2025)

It is commonly speculated that machine learning algorithms more effectively leverage structured data than their classical counterparts. The supervised noisy manifold hypothesis is designed to capture this, with covariates concentrated about a low-dimensional manifold, and labels invariant to orthogonal deviations.

This is an instance of our newly introduced setting of continuous-index learning. I study this problem via kernel smoothing in an adaptive Mahalanobis metric. The objective is to induce anisotropy, reducing estimator variance by elongating along the normal space and pooling additional low bias data points.

Kokot & Luedtke (2025)

We seek to construct a 'coreset' of m observations such that, relative to a specified divergence D, the coreset's deviation from the empirical is of the same order as the empirical's deviation from the truth. This presents a substantial generalization of typical clustering and coreset selection settings, as the loss D is arbitrary.

I developed the Coresets of Order 2 (CO2) algorithm, which reduces the problem to minimizing a quadratic form under affine and sparsity constraints. By verifying Hadamard differentiability of the entropic potentials in the Gaussian RKHS, we showed that only poly-log(n) samples are required to approximate the population distribution up to negligible error.

Targeted Sampling & Fine-Tuning

Beyond theoretical bounds, this framework allows for the selection of coresets to optimize arbitrary functionals, and this includes key settings such as empirical risk minimization. In ongoing research, these methods are applied to select data to accelerate model fitting, with applications ranging from identifying corpuses for fine-tuning to batch selection in SGD.

Ongoing Research

In my analysis of the Sinkhorn divergence, I verified Hadamard differentiability via an inverse function theorem applied to the Schrödinger equations, refining earlier approaches in the field. In current research, we are building on this methodology, deriving limits for self-EOT (entropic transport from a distribution to itself) as the regularization approaches zero.

Kokot, Murad, & Meila (2025)

We study Laplacian spectral embeddings for manifold data injected with high-dimensional noise. The traditional folklore is that low-dimensional spectral embeddings are insensitive to such contamination. To verify this rigorously, I developed a metric perturbation argument, comparing the induced tube geometry to the "flattened" Sasaki metric.

The Sasaki metric presents a natural splitting of manifold and noise information, decomposing the Neumann Laplacian into an intrinsic component and a high frequency perturbation. By leveraging perturbation theory for unbounded operators, I showed that in the continuum, low frequency eigenfunctions are nearly invariant to deviations away from the manifold.

Rather than a stability result, we take the dual point of view: Laplacian spectral embeddings detect fundamental data structure, with a dependence that is not strictly dimensional.