Ziv Goldfeld

Goldfeld's research interests include statistical machine learning, optimal transport theory, information theory, high-dimensional statistic, applied probability and interacting particle systems. He seeks to understand and design engineering systems by formulating and solving mathematical models. A main focus is working towards a comprehensive statistical learning theory to obtain better understanding and strong performance guarantees for modern ML methods that operate on real-world high-dimensional data.

In a recent project, Goldfeld aims to create a global, high-dimensional inference framework that prompts a scalable (in dimension) generalization and sample complexity theory for ML. To that end, a new class of discrepancy measures between probability distributions adapted to high-dimensional spaces is developed. Termed smooth statistical distances (SSDs), these distances level out irregularities in the considered distributions (via convolution with a chosen smoothing kernel) in a way that preserves inference capability, but alleviates the curse of dimensionality when estimating them from data. Measuring or optimizing distances between distributions is central to basic inference setups and advanced ML tasks. Therefore, research questions include: (i) SSD fundamentals, encompassing geometric, topological and functional properties; (ii) high-dimensional statistical questions, such as empirical approximation, limit distributions, testing, goodness-of-fit, etc.; and (iii) learning-theoretic applications, including generalization theory for generative models, efficient barycenter computation/estimation, anomaly detection, etc.

Another focus is developing tools, rooted in information theory, for measuring the flow of information through deep neural networks (DDNs). The goal here is to explain the process by which DNNs progressively build representations of data—from crude and over-redundant representations in shallow layers to highly-clustered and interpretable ones in deeper layers—and to give the designer more control over that process. To that end, the project develops efficient estimators of information measures over the network (possibly via built-in dimensionality reduction techniques). Such estimators also lead to new visualization, optimization, and pruning methods of DDNs. New instance-dependent generalization bounds based on information measures are also of interest.

Additional research trajectories include causal machine learning and relation to the directed information functional, information-theoretic security, high-dimensional nonparametric estimation, and interacting particle systems.