Research

Currently pursuing a PhD under Prof. Farzad Kamalabadi, my research is focused on the implications of commonly used model assumptions on traditional estimation techniques.

My first papers were focused on the powerful implications of a space-invariant prior, resulting in a linear improvement in estimation quality as a function of the number of parameters being estimated. While still working on that direction, my current research is focused on dynamical system forecasting.

Publications

H. Naumer and F. Kamalabadi, “Dimensionality Collapse: Optimal Measurement Selection for Low-Error Infinite-Horizon Forecasting,” Proceedings of The 26th International Conference on Artificial Intelligence and Statistics, 2023, pp. 6166-6198, url: https://proceedings.mlr.press/v206/naumer23a.html.
Abstract: This work introduces a method to select linear functional measurements of a vector-valued time series optimized for forecasting distant time-horizons. By formulating and solving the problem of sequential linear measurement design as an infinite-horizon problem with the time-averaged trace of the Cramér-Rao lower bound (CRLB) for forecasting as the cost, the most informative data can be collected irrespective of the eventual forecasting algorithm. By introducing theoretical results regarding measurements under additive noise from natural exponential families, we construct an equivalent problem from which a local dimensionality reduction can be derived. This alternative formulation is based on the future collapse of dimensionality inherent in the limiting behavior of many differential equations and can be directly observed in the low-rank structure of the CRLB for forecasting. Implementations of both an approximate dynamic programming formulation and the proposed alternative are illustrated using an extended Kalman filter for state estimation, with results on simulated systems with limit cycles and chaotic behavior demonstrating a linear improvement in the CRLB as a function of the number of collapsing dimensions of the system.

H. Naumer and F. Kamalabadi, “Prediction of Shock Formation From Boundary Measurements,” 2023 IEEE Statistical Signal Processing Workshop (SSP), 2023, pp. 41-45, doi: 10.1109/SSP53291.2023.10207929.
Abstract: We propose a method for the Bayesian prediction of shocks in scalar partial differential equations (PDEs) representing conservation equations from noisy observations of the boundary conditions. By considering the implicit transformation from boundary conditions to shocks, we construct an arrival process interpretation of shocks as well as an associated arrival rate function. We then introduce a Monte Carlo method to approximate the arrival rate of shocks based on the probability of a sufficiently large range of values in an epsilon ball conditioned on noisy boundary measurements. We illustrate the method with simulations of Burgers’ equation with initial conditions set by Brownian motion. Despite the non-smooth boundary, our proposed method constructs a sparse and readily interpretable probabilistic structure of shock arrival and propagation.

P. Karimi, H. Naumer and F. Kamalabadi, “A Continuous Representation Of Switching Linear Dynamic Systems For Accurate Tracking,” 2023 IEEE Statistical Signal Processing Workshop (SSP), 2023, pp. 339-343, doi: 10.1109/SSP53291.2023.10207936.
Abstract: We propose a method for tracking linear representations of a nonlinear dynamic system with time-varying parameters based on a continuous representation of its switching linear dynamic system (SLDS) model. Given approximate linear representations for a finite set of unknown intrinsic parameters of the dynamics, a combination of autoencoder-based dimensionality reduction and cubic curve-fitting are applied to learn the continuous manifold of dynamics embedded in the evolution operator. This representation enables a significant reduction of the squared Frobenius norm of error in maximum likelihood (ML) system identification relative to that of the original SLDS model. Numerical experiments also verify this result.

H. Naumer, Y. Lu and F. Kamalabadi, “Marginalization on Bifurcation Diagrams: A New Perspective on Infinite-Horizon Prediction,” 2021 IEEE Statistical Signal Processing Workshop (SSP), 2021, pp. 431-435, doi: 10.1109/SSP49050.2021.9513863.
Abstract: This work addresses the problem of accurate infinite-horizon forecasting of dynamical systems with uncertain parameters. We introduce an intermediate representation of the probability distribution though the marginalization onto the sparse bifurcation structure of an ordinary differential equation (ODE) through integration over regions of attraction. With operations on this representation naturally completed in the space of parameter and state, metrics for the space of limiting behavior can be directly applied. Both limit points and limit cycles are investigated using the Hausdorff distance to treat them in a unified manner. The technique is further applied to stability detection, resulting in a likelihood ratio test.

H. Naumer and F. Kamalabadi, “Linear Space-Invariant System Identification And Mismatch Bounds For Estimation Of Dynamical Images,” 2020 IEEE International Conference on Image Processing (ICIP), 2020, pp. 2920-2924, doi: 10.1109/ICIP40778.2020.9191061.
Abstract: For linear space-invariant temporal systems, we provide a lower bound on the penalty incurred by approximating system dynamics in a Kalman filter by a random walk model, a common model when dynamics are unknown. We then present a computationally tractable algorithm for system identification of high-dimensional linear space-invariant dynamical systems, whereby the circulant structure of the state transition operator yields an estimate of the governing dynamics from a small number of temporal steps. By completing all operations in the frequency domain, we efficiently provide an estimate of the system dynamics and the state of the system. The estimation of system dynamics greatly improves the state estimation over the random walk model, suggesting classical estimators may remain applicable in modern imaging tasks.

H. Naumer and F. Kamalabadi, “Estimation of Linear Space-Invariant Dynamics,” in IEEE Signal Processing Letters, vol. 27, pp. 2154-2158, 2020, doi: 10.1109/LSP.2020.3040941.
Abstract: We propose a computationally efficient estimator for multi-dimensional linear space-invariant system dynamics with periodic boundary conditions that attains low mean squared error from very few temporal steps. By exploiting the inherent redundancy found in many real-world spatiotemporal systems, the estimator performance improves with the dimensionality of the system. This paper provides a detailed analysis of maximum likelihood estimation of the state transition operator in linear space-invariant systems driven by Gaussian noise. The key result of this work is that, by incorporating the space-invariance prior, the mean squared error of a estimator normalized to the number of parameters is upper bounded by N⁻¹M⁻¹ + O(N⁻¹M⁻²), where N is the number of spatial points, and M is the number of observed timesteps after the initial value.

Under Review

H. Naumer and F. Kamalabadi, “Statistically Optimal ODE Forecasting Through Explicit Trajectory Optimization.”
Abstract: This work introduces a method to enable accurate forecasting of time series governed by ordinary differential equations (ODE) through the usage of cost functions explicitly dependent on the future trajectory rather than the past measurement times. We prove that the space of solutions of an N-dimensional, smooth, Lipschitz ODE on any given finite time horizon is an N-dimensional Riemmanian manifold embedded in the space of square integrable continuous functions. This finite dimensional manifold structure enables the application of common statistical objectives such as maximum likelihood (ML), maximum a posteriori (MAP), and minimum mean squared error (MMSE) estimation directly in the space of feasible ODE solutions. The restriction to feasible trajectories of the system limits known issues such as oversmoothing seen in unconstrained MMSE forecasting. We demonstrate that direct optimization of trajectories reduces error in forecasting when compared to estimating initial conditions or minimizing empirical error. Beyond theoretical justifications, we provide Monte Carlo simulations evaluating the performance of the optimal solutions of six different objective functions: ML, MAP state estimation, MMSE state estimation, MAP trajectory estimation, MMSE trajectory estimation over all square integrable functions, and MMSE trajectory estimation over solutions of the differential equation.

Teaching

University of Illinois Urbana-Champaign

Massachusetts Institute of Technology

Mentorship

Service

Organizational

Reviewer

Graduate Level Coursework

University of Illinois Urbana-Champaign

Spring 2023

Fall 2022

Fall 2021

Spring 2021

Fall 2020

Spring 2020

Fall 2019

Massachusetts Institute of Technology

Spring 2019

Fall 2018

Spring 2018

Fall 2017

Past Research Jobs

MIT Lincoln Laboratory — Summer Intern

Group 62, Advanced RF Techniques and Systems — Summer 2018, Summer 2019

MIT — Undergraduate Researcher

Novel Electronic Systems Group under Prof. Max Shulaker — Fall 2016 to Spring 2019

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