arXiv subjects

Eric Shen

Publications and source records attributed to Eric Shen.

12 recordsLinked to original sources

arXiv2026

Rayleigh-Residual Flow I: Normal Matrices

We study the squared Rayleigh-residual functional on complex projective space and its negative gradient flow. For an arbitrary complex matrix, every positive-residual critical point has a strictly descending tangent direction; consequently, the local minima are precisely the eigenlines. For normal matrices, we describe the critical set in terms of circles through spectral points and identify the amplitude dynamics with a replicator equation whose payoff matrix has rank at most four. This gives explicit logarithmic first integrals and a reduced representation of interior trajectories. We compute transverse escape rates near critical components and analyze how polynomial reweighting changes unstable coordinates. Finally, we distinguish local saddle trapping from concentration of Haar-distributed initial states and derive the density of the filtered ensemble. Numerical examples with random spectra and a Jordan block illustrate convergence, nonmonotone Rayleigh motion, and the improvement of finite-time spectral exploration by low-degree random polynomial filters.

math.FA↗

arXiv2026

The Zaporozhets-Tarasov Conjecture on Mean Distances

For a convex body $K \subset \mathbb R^d$ let $Δ(K)$ be the expected distance between two independent uniform points of $K$, and let $θ(K)$ be the corresponding expectation for normalized surface measure on $\partial K$. The Zaporozhets-Tarasov conjecture asserts $Δ(K) \le θ(K)$. We prove this conjecture in case $d=2$. In addition, we give a six-vertex convex polytope in $\mathbb R^3$ for which the reverse strict inequality holds, and obtain counterexamples in every dimension $d \ge 3$ by taking products with segments. Finally, we show that $θ(K)> \frac{\operatorname{per}K}{6}$ for every planar convex body.

math.MG↗

arXiv2026

The Roman Coronagraph Community Participation Program: corgisim - a simulation suite for the Nancy Grace Roman Space Telescope Coronagraph Instrument

NASA's Roman Space Telescope will feature a pathfinder Coronagraph Instrument to demonstrate advanced high-contrast imaging from space, paving the way for future missions like the Habitable Worlds Observatory. The Coronagraph Instrument could obtain imaging, polarimetry and spectroscopy of Jupiter analogs in reflected visible light for the first time. We present the development of an open-source simulation package ``corgisim'' as part of the Roman Coronagraph Community Participate Program. Built on established optical propagation libraries including PROPER and CGISim, corgisim provides a user-friendly, publicly available Python framework for end-to-end simulations of the Coronagraph Instrument observations. The package produces high-fidelity, format-compliant data for pre-launch calibration, pipeline testing, and community applications such as target selection and observation planning. We will give an overview of corgisim's infrastructure, functionalities, and current implementation across planned imaging, polarimetry, and spectroscopy modes, including the ability to simulate host stars, injected companions, and extended disks. We will also highlight suitable applications of corgisim and provide guidance on how users can access and employ the software.

astro-ph.IM↗

arXiv2026

The Roman Coronagraph Community Participation Program: calibration strategy for the Mueller matrix using on-sky sources

The Nancy Grace Roman Space Telescope Coronagraph Instrument will provide space-based polarimetric observations of circumstellar disks and exoplanetary systems. Accurate reconstruction of the linear polarization fraction requires calibration of the instrumental Mueller matrix using polarized and weakly polarized standard stars. We constructed a candidate catalog by combining published optical polarimetry with Gaia DR3 astrometry and photometry and selected separate samples for coronagraphic calibration observations and observations using a neutral-density filter. Precursor $VRI$-band polarimetry of 18 faint candidates was obtained with HONIR on the 1.5-m Kanata telescope. The wavelength dependence of their normalized Stokes parameters was modeled using the Serkowski law to predict their polarization properties in \cgi\ Bands~1 and 4, and 2 sets of 3 calibrators were selected for the 2 calibration scenarios. We then estimated the achievable LPF reconstruction accuracy using Monte Carlo simulations that include uncertainties in the calibrator polarization properties, photometric noise, and residual detector-response errors. A dithered observing configuration was also simulated to reduce differential detector-response errors among the calibrators. The current estimates indicate LPF reconstruction errors at the few-percentage-point level, with a small bias arising from treating a weakly polarized calibrator as unpolarized. Finally, we present progress toward an end-to-end test using \texttt{corgisim} and \texttt{corgiDRP}, including successful processing of simulated datasets from Level~1 through Level~2b.

astro-ph.IM↗

arXiv2026

The Roman Coronagraph Community Participation Program: data reduction pipeline design and implementation

The Roman Space Telescope Coronagraph Instrument will demonstrate a series of technologies and techniques to enable the direct detection of reflected-light planets with space-based observatories. To characterize and validate the performance of the Coronagraph Instrument, the Community Participation Program is developing corgidrp, an open-source Python-based data reduction pipeline. The pipeline can process data from the required and best-effort observing modes and their associated calibration sequences into calibrated science-ready data products. We present the software design and implementation of corgidrp and the motivation behind specific design decisions. We describe the software architecture, data flow, processing steps, automation tools, testing framework, and development philosophy. We also outline future development plans in preparation for on-sky data.

astro-ph.IM↗

arXiv2026

The Roman Coronagraph Community Participation Program: Observation planning and data reduction for polarimetric mode

Reflected-light polarimetry of exoplanets constrains and resolves degeneracies in atmospheric properties, while polarized light observations of debris disks enable the characterization of dust-grain properties. The best-effort polarimetric mode of the Roman Coronagraph Instrument will be able to perform multi-wavelength observations of planetary systems using both the Hybrid Lyot Coronagraph (HLC) and the Shaped Pupil Coronagraph (SPC). This paper presents an overview of observation planning, simulations, and data reduction procedures for the polarimetric mode of the Roman Coronagraph. As an initial test of simulation and data reduction, a dataset of polarimetric observing sequences for the debris disk HD 172555 in HLC mode was generated using corgisim with estimated observation parameters, and data reduction was performed using corgidrp, incorporating all relevant noise factors and calibration products. Currently, mock calibration products are used in corgidrp; these will be replaced with simulated calibration products in future updates

astro-ph.IM↗

arXiv2026

Learning Gaussian Graphical Models from a Glauber Trajectory Without Mixing

We study the task of learning the structure of a $d$-sparse Gaussian graphical model on $n$ variables from a single trajectory of Glauber dynamics. Beyond algorithmic considerations, many applications present temporally correlated observations rather than i.i.d.\ samples. In the classical i.i.d.\ setting, under comparably general sparsity and minimum edge-strength assumptions, sublinear-in-$n$ sample guarantees are known, but achieving them in polynomial-time remains open. Motivated in part by this gap, we give a polynomial-time algorithm that recovers the conditional-independence graph from a single Glauber trajectory, with a trajectory-length guarantee that does not depend on the mixing time. Technically, our algorithm has three components. First, we estimate the conditional variances and rescale the trajectory to reduce to the unit-diagonal case, without changing the underlying graph. Second, we design a local edge test that extracts adjacency information from short update windows by isolating pairwise influence. Third, we aggregate these local statistics using a robust median-based estimator, and prove accuracy despite temporal dependence arising from a single trajectory.

cs.LG↗

arXiv2026

On circular Kippenhahn curves and the Gau-Wang-Wu conjecture about nilpotent partial isometries

We study linear operators on a finite-dimensional space whose Kippenhahn curves consist of concentric circles centered at the origin. We say that such operators have Circularity property. One class of examples is rotationally invariant operators. To every operator with norm at most one, we associate an infinite sequence of partial isometries and study when Circularity property can be passed back and forth along that sequence. In particular, we introduce a class of operators for which every partial isometry in the aforementioned sequence has Circularity property, and show that this class is broader than the class of rotationally invariant operators. As a consequence, every such an operator provides a counterexample to the Gau--Wang--Wu conjecture about nilpotent partial isometries. We also discuss possible refinements of the conjecture. Finally, we propose a way to check whether a matrix is rotationally invariant, suitable for numerical experiments.

math.FA↗

arXiv2026

On Kippenhahn curves of low rank partial isometries

Conditions are established for rank three partial isometries to have circular components contained in their Kippenhahn curves. In particular, such matrices with circular numerical ranges are described. It is also established that the Gau-Wang-Wu conjecture holds for matrices under consideration.

math.FA↗

arXiv2025

Tighter Bounds on the Expected Absorbing Time of Ungarian Markov Chains

In $2023$, Defant and Li defined the Ungarian Markov chain $\mathbf{U}_L$ associated to a finite lattice $L$. This Markov chain has state space $L$, and from any state $x \in L$ transitions to the meet of $\{x\} \cup T$, where $T$ is a randomly selected subset of the elements of $L$ covered by $x$. For any lattice $L$, let $\mathcal{E}(L)$ be the expected number of steps until the maximal element of $L$ transitions into the minimal element in the Ungarian Markov chain. We show that $\mathcal{E}(L)$ is linear in $n$ when $L$ is the weak order on the symmetric group $S_n$, and satisfies an $n^{1-o(1)}$ lower bound when $L$ is the $n^\text{th}$ Tamari lattice. This completely resolves a conjecture by Defant and Li and partially resolves another.

math.CO↗

arXiv2023

On Effective Sato-Tate Distributions for Surfaces Arising from Products of Elliptic Curves

We prove, with an unconditional effective error bound, the Sato-Tate distributions for two families of surfaces arising from products of elliptic curves, namely a one-parameter family of K3 surfaces and double quadric surfaces. To prove these effective Sato-Tate distributions, we prove an effective form of the joint Sato-Tate distribution for two twist-inequivalent elliptic curves, along with an effective form of the Sato-Tate distribution for an elliptic curve for primes in arithmetic progressions. The former completes the previous work of Thorner by including the cases in which one of the elliptic curves has CM.

math.NT↗

arXiv2015

Probabilistic Model-Based Approach for Heart Beat Detection

Nowadays, hospitals are ubiquitous and integral to modern society. Patients flow in and out of a veritable whirlwind of paperwork, consultations, and potential inpatient admissions, through an abstracted system that is not without flaws. One of the biggest flaws in the medical system is perhaps an unexpected one: the patient alarm system. One longitudinal study reported an 88.8% rate of false alarms, with other studies reporting numbers of similar magnitudes. These false alarm rates lead to a number of deleterious effects that manifest in a significantly lower standard of care across clinics. This paper discusses a model-based probabilistic inference approach to identifying variables at a detection level. We design a generative model that complies with an overview of human physiology and perform approximate Bayesian inference. One primary goal of this paper is to justify a Bayesian modeling approach to increasing robustness in a physiological domain. We use three data sets provided by Physionet, a research resource for complex physiological signals, in the form of the Physionet 2014 Challenge set-p1 and set-p2, as well as the MGH/MF Waveform Database. On the extended data set our algorithm is on par with the other top six submissions to the Physionet 2014 challenge.

cs.AI↗