arXiv subjects

Peyman Morteza

Publications and source records attributed to Peyman Morteza.

4 recordsLinked to original sources

arXiv2026

Parameter-Efficient Generative Modeling with Controlled Vector Fields

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.

cs.LG↗

arXiv2025

Representer Theorems for Metric and Preference Learning: Geometric Insights and Algorithms

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

cs.LG↗

arXiv2021

Provable Guarantees for Understanding Out-of-distribution Detection

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.

cs.LG↗

arXiv2019

The Calabi metric and desingularization of Einstein orbifolds

Consider an Einstein orbifold $(M_0,g_0)$ of real dimension $2n$ having a singularity with orbifold group the cyclic group of order $n$ in ${\rm{SU}}(n)$ which is generated by an $n$th root of unity times the identity. Existence of a Ricci-flat Kähler ALE metric with this group at infinity was shown by Calabi. There is a natural "approximate" Einstein metric on the desingularization of $M_0$ obtained by replacing a small neighborhood of the singular point of the orbifold with a scaled and truncated Calabi metric. In this paper, we identify the first obstruction to perturbing this approximate Einstein metric to an actual Einstein metric. If $M_0$ is compact, we can use this to produce examples of Einstein orbifolds which do not admit any Einstein metric in a neighborhood of the natural approximate Einstein metric on the desingularization. In the case that $(M_0,g_0)$ is asymptotically hyperbolic Einstein and non-degenerate, we show that if the first obstruction vanishes, then there does in fact exist an asymptotically hyperbolic Einstein metric on the desingularization. We also obtain a non-existence result in the asymptotically hyperbolic Einstein case, provided that the obstruction does not vanish. This work extends a construction of Biquard in the case $n =2$, in which case the Calabi metric is also known as the Eguchi-Hanson metric, but there are some key points for which the higher-dimensional case differs.

math.DG↗