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12th International Conference on Learning Representations, ICLR 2024

作者： Geadah, Victor Pillow, Jonathan W. Program in Applied and Computational Mathematics Princeton University United States Princeton Neuroscience Institute Princeton University United States

Unsupervised methods for dimensionality reduction of neural activity and behavior have provided unprecedented insights into the underpinnings of neural information processing. One popular approach involves the recurrent switching linear dynamical system (rSLDS) model, which describes the latent dynamics of neural spike train data using discrete switches between a finite number of low-dimensional linear dynamical systems. However, a few properties of rSLDS model limit its deployability on trial varying data, such as a fixed number of states over trials, and no latent structure nor organization of states. Here we overcome these limitations by endowing the rSLDS model with a semi-Markov discrete state process, with latent geometry, that captures key properties of stochastic processes over partitions with flexible state cardinality. We leverage partial differential equations (PDE) theory to derive an efficient, semi-parametric formulation for dynamical sufficient statistics to the discrete states. This process, combined with switching dynamics, defines our infinite recurrent switching linear dynamical system (irSLDS) model class. We first validate and demonstrate the capabilities of our model on synthetic data. Next, we turn to the analysis of mice electrophysiological data during decision-making, and uncover strong non-stationary processes underlying both within-trial and trial-averaged neural activity. © 2024 12th International Conference on Learning Representations, ICLR 2024. All rights reserved.

关键词： Stochastic systems

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Fat Arcs for Implicitly Defined Curves

Fat Arcs for Implicitly Defined Curves

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7th International Conference on Mathematical Methods for Curves and Surfaces

作者： Bela, Szilvia Juettler, Bert Doctoral Program in Computational Mathematics Austria Institute of Applied Geometry Johannes Kepler University Linz 4040 Altenberger Str. 69 Austria

ISBN: (纸本)9783642116193

We present an algorithm generating a collection of fat arcs which bound the zero set of a given bivariate polynomial in Bernstein-Bezier representation. We demonstrate the performance of the algorithm (in particular the convergence rate) and we apply the results to the computation of intersection curves between implicitly defined algebraic surfaces and rational parametric surfaces.

关键词： Polynomial approximation

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ISING-MODEL SIMULATIONS ON THE MANIFOLDS OF 2-DIMENSIONAL QUANTUM-GRAVITY

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MODERN PHYSICS LETTERS A 1991年第17期6卷 1615-1628页

作者： AGISHTEIN, ME BAILLIE, CF Program in Applied and Computational Mathematics Princeton University Fine Hall Princeton NJ 08544-1000 USA Physics Department University of Colorado Boulder CO 80309 USA

The Ising model is stimulated on the manifolds of 2-dimensional quantum gravity, which are represented by fixed random triangulations (so-called quenched Ising model). Unlike the case of the Ising model on a dynamical random triangulation, there is no analytical prediction for the quenched case, since these manifolds do not have internal Hausdorff dimension and the problem cannot be formulated in matrix model language. The recursive sampling technique is used to generate the triangulations, lattice sizes being up to ten thousand triangles. The Metropolis algorithm was used for the spin update in order to obtain the initial estimation of the Curie point. After that we used the Wolff cluster algorithm in the critical region. We observed a second order phase transition, similar to that for the Ising model on a regular 2-dimensional lattice, and measured the critical exponents.

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Factor Analysis for Spectral Estimation

Factor Analysis for Spectral Estimation

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International Conference on Sampling Theory and Applications

作者： Joakim Anden Amit Singer Program in Applied and Computational Mathematics Princeton University Department of Mathematics and Program in Applied and Computational Mathematics Princeton University

ISBN: (纸本)9781538615669

Power spectrum estimation is an important tool in many applications, such as the whitening of noise. The popular multitaper method enjoys significant success, but fails for short signals with few samples. We propose a statistical model where a signal is given by a random linear combination of fixed, yet unknown, stochastic sources. Given multiple such signals, we estimate the subspace spanned by the power spectra of these fixed sources. Projecting individual power spectrum estimates onto this subspace increases estimation accuracy. We provide accuracy guarantees for this method and demonstrate it on simulated and experimental data from cryo-electron microscopy.

关键词： Spectrum Analysis power spectra Subspace STATIONARY SOURCES Factor analysis Cryoelectron Microscopy Linear Combination statistical models (nuclear) signals Individual Power simulating

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Nonlinear Oscillations, Dynamical Systems, and Bifurcations of Vector Fields 1

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丛书名： applied Mathematical Sciences

1983年

作者： John Guckenheimer Philip Holmes

ISBN: (数字)9781461211402

ISBN: (纸本)9780387908199;9781461270201

"This book is concerned with the application of methods from dynamical systems and bifurcation theories to the study of nonlinear oscillations. Chapter 1 provides a review of basic results in the theory of dynamical systems, covering both ordinary differential equations and discrete mappings. Chapter 2 presents 4 examples from nonlinear oscillations. Chapter 3 contains a discussion of the methods of local bifurcation theory for flows and maps, including center manifolds and normal forms. Chapter 4 develops analytical methods of averaging and perturbation theory. Close analysis of geometrically defined two-dimensional maps with complicated invariant sets is discussed in chapter 5. Chapter 6 covers global homoclinic and heteroclinic bifurcations. The final chapter shows how the global bifurcations reappear in degenerate local bifurcations and ends with several more models of physical problems which display these behaviors." #;#1 "An attempt to make research tools concerning `strange attractors' developed in the last 20 years available to applied scientists and to make clear to research mathematicians the needs in applied works. Emphasis on geometric and topological solutions of differential equations. Applications mainly drawn from nonlinear oscillations." #;#2

关键词： Analysis

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4D seismic monitoring of CO2 sequestration based upon spectral-element and adjoint methods: Comparison of acoustic, elastic and poroelastic theories 80

4D seismic monitoring of CO2 sequestration based upon spectr...

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Society of Exploration Geophysicists International Exposition and 80th Annual Meeting 2010, SEG 2010

作者： Morency, Christina Luo, Yang Tromp, Jeroen Department of Geosciences Princeton University United States Department of Geosciences and Program in Applied and Computational Mathematics Princeton University United States

ISBN: (纸本)9781617389801

The key issues in CO2 sequestration monitoring involve accurate monitoring, from the injection stage to prediction & verification, of CO2 movement over time for environmental considerations. A natural non-intrusive monitoring technique is referred to as "4D seismics", which involves 3D time-lapse seismic surveys. The success of monitoring the CO2 movement is subject to a proper description of the physical properties of the structure. We realize time-lapse migrations comparing acoustic, elastic, and poroelastic simulations of 4D seismic imaging to characterize the storage zone, based solely upon the first arrival traveltime anomaly arising from the injection of CO2. This approach highlights the influence of using different physical theories on interpreting seismic data, and, more importantly, on extracting the CO2 signature from the seismic wave field. Simulations are performed using a spectral-element method, which allows for highly accurate results. Biot? Equations are implemented to account for poroelastic effects. The sensitivity of observables to the model parameters is quantified based upon finite-frequency sensitivity kernels calculated using an adjoint method. © 2010 SEG.

关键词： Carbon dioxide

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Recovery of frequency-sparse signals from compressive measurements

Recovery of frequency-sparse signals from compressive measur...

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48th Annual Allerton Conference on Communication, Control, and Computing, Allerton 2010

作者： Duarte, Marco F. Baraniuk, Richard G. Program in Applied and Computational Mathematics Princeton University United States Department of Electrical and Computer Engineering Rice University United States

ISBN: (纸本)9781424482146

Compressive sensing (CS) is a new approach to simultaneous sensing and compression for sparse and compressible signals. While the discrete Fourier transform has been widely used for CS of frequency-sparse signals, it provides optimal sparse representations only for signals with components at integral frequencies. There exist redundant frames that provide compressible representations for frequency-sparse signals, but such frames are highly coherent and severely affect the performance of standard CS recovery. In this paper, we show that by modifying standard CS recovery algorithms to prevent coherent frame elements from being present in the signal estimate, it is possible to bypass the shortcomings introduced by the coherent frame. The resulting algorithm comes with theoretical guarantees and is shown to perform significantly better for frequency-sparse signal recovery than its standard counterparts. The algorithm can also be extended to similar settings that use coherent frames. ©2010 IEEE.

关键词： Compressed sensing

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Coarse-grained spectral projection: A deep learning assisted approach to quantum unitary dynamics

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Physical Review B 2021年第2期103卷 024304-024304页

作者： Pinchen Xie Weinan E Department of Mathematics and Program in Applied and Computational Mathematics Princeton University New Jersey 08544 USA

We propose the coarse-grained spectral projection method (CGSP), a deep learning assisted approach for tackling quantum unitary dynamic problems with an emphasis on quench dynamics. We show that CGSP can extract spectral components of many-body quantum states systematically with a sophisticated neural network quantum ansatz. CGSP fully exploits the linear unitary nature of the quantum dynamics and is potentially superior to other quantum Monte Carlo methods for ergodic dynamics. Preliminary numerical results on one-dimensional XXZ models with periodic boundary conditions are carried out to demonstrate the practicality of CGSP.

关键词： Quantum quench Quantum statistical mechanics Nonequilibrium lattice models Machine learning Monte Carlo methods

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C^0DISCONTINUOUS GALERKIN METHODS FOR A PLATE FRICTIONAL CONTACT PROBLEM

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Journal of computational mathematics 2019年第2期37卷 184-200页

作者： Fei Wang Tianyi Zhang Weimin Han School of Mathematics and Statistics Xi 'an Jiaotong University Xi'an 710049 China Program in Applied Mathematical and Computational Sciences University of Iowa Iowa City IA 52242 USA School of Mathematics and Statistics Xi 'an Jiaotong University Xi'an 710049 China Departinent of Mathematics University of Iowa Iowa City. IA 52242 USA

Numerous C^0 discontinuous Galerkin (DG) schemes for the Kirchhoff plate bending problem are extended to solve a plate frictional contact problem, which is a fourth-order elliptic variational inequality of the second kind. This variational inequality contains a nondifferentiable term due to the frictional contact. We prove that these C^0 DG methods are consis tent and st able, and derive optimal order error estima tes for the quadratic element. A numerical example is presented to show the performance of the C^0 DG methods;and the numerical convergence orders confirm the theoretical prediction.

关键词： Variational inequality of fourth-order Discontinuous Galerkin method Plate frictional contact problem Optimal order error estimate

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Renormalization theory for eddy diffusivity in turbulent transport

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Physical Review Letters 1992年第20期68卷 3028-3028页

作者： Marco Avellaneda Andrew J. Majda Department of Mathematics Program in Applied and Computational Mathematics Princeton University Princeton New Jersey 08544

We examine the derivation of eddy-diffusivity equations for transport of passive scalars in a turbulent velocity field. Our main contention is that, in the long-time–large-distance limit, the eddy-diffusivity equations can take very different forms according to the statistical properties of the subgrid velocity, and that these equations depend very sensitively on the interplay between spatial and temporal velocity fluctuations. Such crossovers can be represented in a ‘‘phase diagram’’ involving two relevant statistical parameters. Strikingly, the Kolmogorov-Obukhov statistical theory is shown to lie on a phase-transition boundary.

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