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Improved global performance guarantees of second-order methods in convex minimization
Get PDFIn this paper, we attempt to compare two distinct branches of research on second-order optimization methods. The first one studies self-concordant functions and barriers, the main assumption being that the third derivative of the objective is bounded by the second derivative. The second branch studies cubic regularized Newton methods (CRNMs) with the main assumption that the second derivative is Lipschitz continuous. We develop a new theoretical analysis for a path-following scheme (PFS) for general self-concordant functions, as opposed to the classical path-following scheme developed for self-concordant barriers. We show that the complexity bound for this scheme is better than that of the Damped Newton Method (DNM) and show that our method has global superlinear convergence. We propose also a new predictor-corrector path-following scheme (PCPFS) that leads to further improvement of constant factors in the complexity guarantees for minimizing general self-concordant functions. We also apply path-following schemes to different classes of constrained optimization problems and obtain the resulting complexity bounds. Finally, we analyze an important subclass of general self-concordant functions, namely a class of strongly convex functions with Lipschitz continuous second derivative, and show that for this subclass CRNMs give even better complexity bounds
Generalized Identifiability Bounds for Mixture Models With Grouped Samples
No full textRecent work has shown that finite mixture models with m components are identifiable, while making no assumptions on the mixture components, so long as one has access to groups of samples of size 2m−1 which are known to come from the same mixture component. In this work we generalize that result and show that, if every subset of k mixture components of a mixture model are linearly independent, then that mixture model is identifiable with only (2m−1)/(k−1) samples per group. We further show that this value cannot be improved. We prove an analogous result for a stronger form of identifiability known as “determinedness” along with a corresponding lower bound. This independence assumption almost surely holds if mixture components are chosen randomly from a k -dimensional space. We describe some implications of our results for multinomial mixture models and topic modeling
Modelling the Age Distribution of Longevity Leaders
No full textHuman longevity leaders with remarkably long lifespan play a crucial role in the advancement of longevity research. In this paper, we propose a stochastic model to describe the evolution of the age of the oldest person in the world by a Markov process, in which we assume that the births of the individuals follow a Poisson process with increasing intensity, lifespans of individuals are independent and can be characterized by a gamma-Gompertz distribution with time-dependent parameters. We utilize a dataset of the world's oldest person title holders since 1955, and we compute the maximum likelihood estimate for the parameters iteratively by numerical integration. Based on our preliminary estimates, the model provides a good fit to the data and shows that the age of the oldest person alive increases over time in the future. The estimated parameters enable us to describe the distribution of the age of the record holder process at a future time point
Precision fMRI and Cluster‐Failure in the Individual Brain
No full textAdvances in neuroimaging acquisition protocols and denoising techniques, along with increasing magnetic field strengths, have dramatically improved the temporal signal-to-noise ratio (tSNR) in functional magnetic resonance imaging (fMRI). This permits spatial resolution with submillimeter voxel sizes and ultrahigh temporal resolution and opens a route toward performing precision fMRI in the brains of individuals. Yet ultrahigh spatial and temporal resolution comes at a cost: it reduces tSNR and, therefore, the sensitivity to the blood oxygen level-dependent (BOLD) effect and other functional contrasts across the brain. Here we investigate the potential of various smoothing filters to improve BOLD sensitivity while preserving the spatial accuracy of activated clusters in single-subject analysis. We introduce adaptive-weight smoothing with optimized metrics (AWSOM), which addresses this challenge extremely well. AWSOM employs a local inference approach that is as sensitive as cluster-corrected inference of data smoothed with large Gaussian kernels, but it preserves spatial details across multiple tSNR levels. This is essential for examining whole-brain fMRI data because tSNR varies across the entire brain, depending on the distance of a brain region from the receiver coil, the type of setup, acquisition protocol, preprocessing, and resolution. We found that cluster correction in single subjects results in inflated family-wise error and false positive rates. AWSOM effectively suppresses false positives while remaining sensitive even to small clusters of activated voxels. Furthermore, it preserves signal integrity, that is, the relative activation strength of significant voxels, making it a valuable asset for a wide range of fMRI applications. Here we demonstrate these features and make AWSOM freely available to the research community for download
Pressure-Robust Approximation of the Incompressible Navier--Stokes Equations in a Rotating Frame of Reference
No full textA pressure-robust space discretization of the incompressible Navier--Stokes equations in a rotating frame of reference is considered. The discretization employs divergence-free, H1-conforming mixed finite element methods like Scott--Vogelius pairs. An error estimate for the velocity is derived that tracks the dependency of the error bound on the coefficients of the problem, in particular on the angular velocity. Numerical examples illustrate the theoretical results
On the computation of high dimensional Voronoi diagrams
Get PDFWe investigate a recently implemented new algorithm for the computation of a Voronoi diagram in high dimensions and generalize it to N nodes in general or non-general position using a geometric characterization of edges merging in a given vertex. We provide a mathematical proof that the algorithm is exact, convergent and has computational costs of O(E*nn(N)), where E is the number of edges and nn(N) is the computational cost to calculate the nearest neighbor among N points. We also provide data from performance tests in the recently developed Julia package ,,HighVoronoi.jl''
Pricing High-Dimensional Bermudan Options with Hierarchical Tensor Formats
No full textAn efficient compression technique based on hierarchical tensors for popular option pricing methods is presented. It is shown that the “curse of dimensionality” can be alleviated for the computation of Bermudan option prices with the Monte Carlo least-squares approach as well as the dual martingale method, both using high-dimensional tensorized polynomial expansions. Complexity estimates are provided as well as a description of the optimization procedures in the tensor train format. Numerical experiments illustrate the favorable accuracy of the proposed methods. The dynamical programming method yields results comparable to recent neural network based methods
Bayesian estimation of laser linewidth from delayed self-heterodyne measurements
Get PDFWe present a statistical inference approach to estimate the frequency noise characteristics of ultra-narrow linewidth lasers from delayed self-heterodyne beat note measurements using Bayesian inference. Particular emphasis is on estimation of the intrinsic (Lorentzian) laser linewidth. The approach is based on a statistical model of the measurement process, taking into account the effects of the interferometer as well as the detector noise. Our method therefore yields accurate results even when the intrinsic linewidth plateau is obscured by detector noise. The regression is performed on periodogram data in the frequency domain using a Markov-chain Monte Carlo method. By using explicit knowledge about the statistical distribution of the observed data, the method yields good results already from a single time series and does not rely on averaging over many realizations, since the information in the available data is evaluated very thoroughly. The approach is demonstrated for simulated time series data from a stochastic laser rate equation model with 1 / textitf-type non-Markovian noise
On the long-time behaviour of reversible interacting particle systems in one and two dimensions
Get PDFBy refining Holley's free energy technique, we show that, under quite general assumptions on the dynamics, the attractor of a (possibly non-translation-invariant) interacting particle system in one or two spatial dimensions is contained in the set of Gibbs measures if the dynamics admits a reversible Gibbs measure. In particular, this implies that there can be no reversible interacting particle system that exhibits time-periodic behaviour and that every reversible interacting particle system is ergodic if and only if the reversible Gibbs measure is unique. In the special case of non-attractive stochastic Ising models this answers a question due to Liggett
Stability of Deep Neural Networks via Discrete Rough Paths
Get PDFUsing rough path techniques, we provide a priori estimates for the output of deep residual neural networks in terms of both the input data and the (trained) network weights. As trained network weights are typically very rough when seen as functions of the layer, we propose to derive stability bounds in terms of the total p -variation of trained weights for any p∈[1,3] . Unlike the C1 -theory underlying the neural ODE literature, our estimates remain bounded even in the limiting case of weights behaving like Brownian motions, as suggested in [A.-S. Cohen, R. Cont, A. Rossier, and R. Xu, Proceedings of the 38th International Conference on Machine Learning, JMLR, Cambridge, MA, 2021, pp. 2039–2048]. Mathematically, we interpret residual neural network as solutions to (rough) difference equations, and analyze them based on recent results of discrete-time signatures and rough path theory