Brofos, James A; Brubaker, Marcus A; Lederman, Roy R Manifold Density Estimation via Generalized Dequantization Technical Report 2021, (arXiv: 2102.07143). Abstract | Links | BibTeX | Tags: Algorithms, Computer Science - Machine Learning, Density estimation, Manifolds, Statistics - Machine Learning @techreport{brofos_manifold_2021, title = {Manifold Density Estimation via Generalized Dequantization}, author = {James A Brofos and Marcus A Brubaker and Roy R Lederman}, url = {http://arxiv.org/abs/2102.07143}, year = {2021}, date = {2021-07-01}, urldate = {2021-07-14}, abstract = {Density estimation is an important technique for characterizing distributions given observations. Much existing research on density estimation has focused on cases wherein the data lies in a Euclidean space. However, some kinds of data are not well-modeled by supposing that their underlying geometry is Euclidean. Instead, it can be useful to model such data as lying on a textbackslashit manifold with some known structure. For instance, some kinds of data may be known to lie on the surface of a sphere. We study the problem of estimating densities on manifolds. We propose a method, inspired by the literature on "dequantization," which we interpret through the lens of a coordinate transformation of an ambient Euclidean space and a smooth manifold of interest. Using methods from normalizing flows, we apply this method to the dequantization of smooth manifold structures in order to model densities on the sphere, tori, and the orthogonal group.}, note = {arXiv: 2102.07143}, keywords = {Algorithms, Computer Science - Machine Learning, Density estimation, Manifolds, Statistics - Machine Learning}, pubstate = {published}, tppubtype = {techreport} } Density estimation is an important technique for characterizing distributions given observations. Much existing research on density estimation has focused on cases wherein the data lies in a Euclidean space. However, some kinds of data are not well-modeled by supposing that their underlying geometry is Euclidean. Instead, it can be useful to model such data as lying on a textbackslashit manifold with some known structure. For instance, some kinds of data may be known to lie on the surface of a sphere. We study the problem of estimating densities on manifolds. We propose a method, inspired by the literature on "dequantization," which we interpret through the lens of a coordinate transformation of an ambient Euclidean space and a smooth manifold of interest. Using methods from normalizing flows, we apply this method to the dequantization of smooth manifold structures in order to model densities on the sphere, tori, and the orthogonal group. |

Brofos, James; Lederman, Roy R Evaluating the Implicit Midpoint Integrator for Riemannian Hamiltonian Monte Carlo Inproceedings International Conference on Machine Learning, pp. 1072–1081, PMLR, 2021, (ISSN: 2640-3498). Links | BibTeX | Tags: HMC, Manifolds, MCMC, Numerical Analysis @inproceedings{brofos_evaluating_2021, title = {Evaluating the Implicit Midpoint Integrator for Riemannian Hamiltonian Monte Carlo}, author = {James Brofos and Roy R Lederman}, url = {http://proceedings.mlr.press/v139/brofos21a.html}, year = {2021}, date = {2021-07-01}, urldate = {2021-07-14}, booktitle = {International Conference on Machine Learning}, pages = {1072--1081}, publisher = {PMLR}, note = {ISSN: 2640-3498}, keywords = {HMC, Manifolds, MCMC, Numerical Analysis}, pubstate = {published}, tppubtype = {inproceedings} } |

Brofos, James A; Lederman, Roy R Magnetic Manifold Hamiltonian Monte Carlo Technical Report 2020, (arXiv: 2010.07753). Abstract | Links | BibTeX | Tags: Algorithms, Computer Science - Machine Learning, HMC, Manifolds, MCMC, Statistics - Machine Learning @techreport{brofos_magnetic_2020, title = {Magnetic Manifold Hamiltonian Monte Carlo}, author = {James A Brofos and Roy R Lederman}, url = {http://arxiv.org/abs/2010.07753}, year = {2020}, date = {2020-10-01}, urldate = {2020-11-25}, abstract = {Markov chain Monte Carlo (MCMC) algorithms offer various strategies for sampling; the Hamiltonian Monte Carlo (HMC) family of samplers are MCMC algorithms which often exhibit improved mixing properties. The recently introduced magnetic HMC, a generalization of HMC motivated by the physics of particles influenced by magnetic field forces, has been demonstrated to improve the performance of HMC. In many applications, one wishes to sample from a distribution restricted to a constrained set, often manifested as an embedded manifold (for example, the surface of a sphere). We introduce magnetic manifold HMC, an HMC algorithm on embedded manifolds motivated by the physics of particles constrained to a manifold and moving under magnetic field forces. We discuss the theoretical properties of magnetic Hamiltonian dynamics on manifolds, and introduce a reversible and symplectic integrator for the HMC updates. We demonstrate that magnetic manifold HMC produces favorable sampling behaviors relative to the canonical variant of manifold-constrained HMC.}, note = {arXiv: 2010.07753}, keywords = {Algorithms, Computer Science - Machine Learning, HMC, Manifolds, MCMC, Statistics - Machine Learning}, pubstate = {published}, tppubtype = {techreport} } Markov chain Monte Carlo (MCMC) algorithms offer various strategies for sampling; the Hamiltonian Monte Carlo (HMC) family of samplers are MCMC algorithms which often exhibit improved mixing properties. The recently introduced magnetic HMC, a generalization of HMC motivated by the physics of particles influenced by magnetic field forces, has been demonstrated to improve the performance of HMC. In many applications, one wishes to sample from a distribution restricted to a constrained set, often manifested as an embedded manifold (for example, the surface of a sphere). We introduce magnetic manifold HMC, an HMC algorithm on embedded manifolds motivated by the physics of particles constrained to a manifold and moving under magnetic field forces. We discuss the theoretical properties of magnetic Hamiltonian dynamics on manifolds, and introduce a reversible and symplectic integrator for the HMC updates. We demonstrate that magnetic manifold HMC produces favorable sampling behaviors relative to the canonical variant of manifold-constrained HMC. |

Lederman, Roy R; Talmon, Ronen; Wu, Hau-tieng; Lo, Yu-Lun; Coifman, Ronald R Alternating diffusion for common manifold learning with application to sleep stage assessment Inproceedings 2015 IEEE International Conference on Acoustics, Speech and Signal Processing (ICASSP), pp. 5758–5762, 2015, (ISSN: 2379-190X). Abstract | Links | BibTeX | Tags: Alternating Diffusion, Common variable, diffusion maps, Kernel, learning (artificial intelligence), Manifolds, multimodal, multimodal respiratory signals, multimodal signal processing, Physiology, Sensitivity, Sensor phenomena and characterization, signal processing, sleep, sleep stage assessment, standard manifold learning method, time series @inproceedings{lederman_alternating_2015, title = {Alternating diffusion for common manifold learning with application to sleep stage assessment}, author = {Roy R Lederman and Ronen Talmon and Hau-tieng Wu and Yu-Lun Lo and Ronald R Coifman}, doi = {10.1109/ICASSP.2015.7179075}, year = {2015}, date = {2015-01-01}, booktitle = {2015 IEEE International Conference on Acoustics, Speech and Signal Processing (ICASSP)}, pages = {5758--5762}, abstract = {In this paper, we address the problem of multimodal signal processing and present a manifold learning method to extract the common source of variability from multiple measurements. This method is based on alternating-diffusion and is particularly adapted to time series. We show that the common source of variability is extracted from multiple sensors as if it were the only source of variability, extracted by a standard manifold learning method from a single sensor, without the influence of the sensor-specific variables. In addition, we present application to sleep stage assessment. We demonstrate that, indeed, through alternating-diffusion, the sleep information hidden inside multimodal respiratory signals can be better captured compared to single-modal methods.}, note = {ISSN: 2379-190X}, keywords = {Alternating Diffusion, Common variable, diffusion maps, Kernel, learning (artificial intelligence), Manifolds, multimodal, multimodal respiratory signals, multimodal signal processing, Physiology, Sensitivity, Sensor phenomena and characterization, signal processing, sleep, sleep stage assessment, standard manifold learning method, time series}, pubstate = {published}, tppubtype = {inproceedings} } In this paper, we address the problem of multimodal signal processing and present a manifold learning method to extract the common source of variability from multiple measurements. This method is based on alternating-diffusion and is particularly adapted to time series. We show that the common source of variability is extracted from multiple sensors as if it were the only source of variability, extracted by a standard manifold learning method from a single sensor, without the influence of the sensor-specific variables. In addition, we present application to sleep stage assessment. We demonstrate that, indeed, through alternating-diffusion, the sleep information hidden inside multimodal respiratory signals can be better captured compared to single-modal methods. |