Peyman Morteza

Peyman Morteza

We introduce a continuous-time generative modeling framework, motivated by the Chow-Rashevskii theorem, that builds expressive flows from a small set of fixed vector fields and learned scalar controls. Instead of learning an unconstrained high-dimensional vector field, our framework constructs the velocity by modulating fixed vector fields with learned scalar control functions. When the fixed fields are bracket-generating, their Lie algebra spans the ambient space, providing a mechanism for expressive transport with only a small number of learned control channels and offering a parameter-efficient geometric alternative to standard vector-field parameterizations. This decoupled formulation yields a structured and interpretable generative model in which the number of learned scalar output channels can be chosen independently of the ambient dimension. We formulate an expressivity principle showing that, under suitable controllability and well-posedness assumptions, such controlled flows can transport a source distribution to a target distribution. We train the resulting model using a continuous-normalizing-flow likelihood objective and present proof-of-concept experiments on synthetic distributions.

Peyman Morteza

We develop a mathematical framework to address a broad class of metric and preference learning problems within a Hilbert space. We obtain a novel representer theorem for the simultaneous task of metric and preference learning. Our key observation is that the representer theorem for this task can be derived by regularizing the problem with respect to the norm inherent in the task structure. For the general task of metric learning, our framework leads to a simple and self-contained representer theorem and offers new geometric insights into the derivation of representer theorems for this task. In the case of Reproducing Kernel Hilbert Spaces (RKHSs), we illustrate how our representer theorem can be used to express the solution of the learning problems in terms of finite kernel terms similar to classical representer theorems. Lastly, our representer theorem leads to a novel nonlinear algorithm for metric and preference learning. We compare our algorithm against challenging baseline methods on real-world rank inference benchmarks, where it achieves competitive performance. Notably, our approach significantly outperforms vanilla ideal point methods and surpasses strong baselines across multiple datasets. Code available at: https://github.com/PeymanMorteza/Metric-Preference-Learning-RKHS

Peyman Morteza, Yixuan Li

Out-of-distribution (OOD) detection is important for deploying machine learning models in the real world, where test data from shifted distributions can naturally arise. While a plethora of algorithmic approaches have recently emerged for OOD detection, a critical gap remains in theoretical understanding. In this work, we develop an analytical framework that characterizes and unifies the theoretical understanding for OOD detection. Our analytical framework motivates a novel OOD detection method for neural networks, GEM, which demonstrates both theoretical and empirical superiority. In particular, on CIFAR-100 as in-distribution data, our method outperforms a competitive baseline by 16.57% (FPR95). Lastly, we formally provide provable guarantees and comprehensive analysis of our method, underpinning how various properties of data distribution affect the performance of OOD detection.