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Du Nguyen

Publications and source records attributed to Du Nguyen.

15 recordsLinked to original sources
arXiv2024

Parallel transport on matrix manifolds and Exponential Action

We express parallel transport for several common matrix Lie groups with a family of pseudo-Riemannian metrics in terms of matrix exponential and exponential actions. The metrics are constructed from a deformation of a bi-invariant metric and are naturally reductive. There is a similar picture for homogeneous spaces when taking quotients satisfying a general condition. In particular, for a Stiefel manifold of orthogonal matrices of size $n\times d$, we give an expression for parallel transport along a geodesic from time zero to $t$, that could be computed with time complexity of $O(n d^2)$ for small $t$, and of $O(td^3)$ for large $t$, contributing a step in a long-standing open problem in matrix manifolds. A similar result holds for {\it flag manifolds} with the canonical metric. We also show the parallel transport formulas for the {\it general linear group} and the {\it special orthogonal group} under these metrics.

math.NA↗
arXiv2024

Second-order differential operators, stochastic differential equations and Brownian motions on embedded manifolds

We specify the conditions when a manifold M embedded in an inner product space E is an invariant manifold of a stochastic differential equation (SDE) on E, linking it with the notion of second-order differential operators on M. When M is given a Riemannian metric, we derive a simple formula for the Laplace-Beltrami operator in terms of the gradient and Hessian on E and construct the Riemannian Brownian motions on M as solutions of conservative Stratonovich and Ito SDEs on E. We derive explicitly the SDE for Brownian motions on several important manifolds in applications, including left-invariant matrix Lie groups using embedded coordinates. Numerically, we propose three simulation schemes to solve SDEs on manifolds. In addition to the stochastic projection method, to simulate Riemannian Brownian motions, we construct a second-order tangent retraction of the Levi-Civita connection using a given E-tubular retraction. We also propose the retractive Euler-Maruyama method to solve a SDE, taking into account the second-order term of a tangent retraction. We provide software to implement the methods in the paper, including Brownian motions of the manifolds discussed. We verify numerically that on several compact Riemannian manifolds, the long-term limit of Brownian simulation converges to the uniform distributions, suggesting a method to sample Riemannian uniform distributions

math.PR↗
arXiv2024

Families of costs with zero and nonnegative MTW tensor in optimal transport and the c-divergences

We study the information geometry of $\bcc$-divergences from families of costs of the form $\mathsf{c}(x, \barx) =\mathsf{u}(x^{\mathfrak{t}}\barx)$ through the optimal transport point of view. Here, $\mathsf{u}$ is a scalar function with inverse $\mathsf{s}$, $x^{\ft}\barx$ is a nondegenerate bilinear pairing of vectors $x, \barx$ belonging to an open subset of $\mathbb{R}^n$. We compute explicitly the MTW tensor (or cross curvature) for the optimal transport problem on $\mathbb{R}^n$ with this cost. The condition that the MTW-tensor vanishes on null vectors under the Kim-McCann metric is a fourth-order nonlinear ODE, which could be reduced to a linear ODE of the form $\mathsf{s}^{(2)} - S\mathsf{s}^{(1)} + P\mathsf{s} = 0$ with constant coefficients $P$ and $S$. The resulting inverse functions include {\it Lambert} and {\it generalized inverse hyperbolic\slash trigonometric} functions. The square Euclidean metric and $\log$-type costs are equivalent to instances of these solutions. The optimal map may be written explicitly in terms of the potential function. For cost functions of a similar form on a hyperboloid model of the hyperbolic space and unit sphere, we also express this tensor in terms of algebraic expressions in derivatives of $\mathsf{s}$ using the Gauss-Codazzi equation, obtaining new families of strictly regular costs for these manifolds, including new families of {\it power function costs}. We express the divergence geometry of the $\mathsf{c}$-divergence in terms of the Kim-McCann metric, including a $\mathsf{c}$-Crouzeix identity and a formula for the primal connection. We analyze the $\sinh$-type hyperbolic cost, providing examples of $\mathsf{c}$-convex functions, which are used to construct a new \emph{local form} of the $\alpha$-divergences on probability simplices. We apply the optimal maps to sample the multivariate $t$-distribution.

math.AP↗
arXiv2023

Geometry in global coordinates in mechanics and optimal transport

For a manifold embedded in an inner product space, we express geometric quantities such as {\it Hamilton vector fields, affine and Levi-Civita connections, curvature} in global coordinates. Instead of coordinate indices, the global formulas for most quantities are expressed as {\it operator-valued} expressions, using an {\it affine projection} to the tangent bundle. For a submersion image of an embedded manifold, we introduce {\it liftings} of Hamilton vector fields, allowing us to use embedded coordinates on horizontal bundles. We derive a {\it Gauss-Codazzi equation} for affine connections on vector bundles. This approach allows us to evaluate geometric expressions globally, and could be used effectively with modern numerical frameworks in applications. Examples considered include rigid body mechanics and Hamilton mechanics on Grassmann manifolds. We show explicitly the cross-curvature (MTW-tensor) for the {\it Kim-McCann} metric with a reflector antenna-type cost function on the space of positive-semidefinite matrices of fixed rank has nonnegative cross-curvature, while the corresponding cost could have negative cross-curvature on Grassmann manifolds, except for projective spaces.

math.DG↗
arXiv2022

Road Damages Detection and Classification with YOLOv7

Maintaining the roadway infrastructure is one of the essential factors in enabling a safe, economic, and sustainable transportation system. Manual roadway damage data collection is laborious and unsafe for humans to perform. This area is poised to benefit from the rapid advance and diffusion of artificial intelligence technologies. Specifically, deep learning advancements enable the detection of road damages automatically from the collected road images. This work proposes to collect and label road damage data using Google Street View and use YOLOv7 (You Only Look Once version 7) together with coordinate attention and related accuracy fine-tuning techniques such as label smoothing and ensemble method to train deep learning models for automatic road damage detection and classification. The proposed approaches are applied to the Crowdsensing-based Road Damage Detection Challenge (CRDDC2022), IEEE BigData 2022. The results show that the data collection from Google Street View is efficient, and the proposed deep learning approach results in F1 scores of 81.7% on the road damage data collected from the United States using Google Street View and 74.1% on all test images of this dataset.

cs.CV↗
arXiv2022

Understanding Public Opinion on Using Hydroxychloroquine for COVID-19 Treatment via Social Media

Hydroxychloroquine (HCQ) is used to prevent or treat malaria caused by mosquito bites. Recently, the drug has been suggested to treat COVID-19, but that has not been supported by scientific evidence. The information regarding the drug efficacy has flooded social networks, posting potential threats to the community by perverting their perceptions of the drug efficacy. This paper studies the reactions of social network users on the recommendation of using HCQ for COVID-19 treatment by analyzing the reaction patterns and sentiment of the tweets. We collected 164,016 tweets from February to December 2020 and used a text mining approach to identify social reaction patterns and opinion change over time. Our descriptive analysis identified an irregularity of the users' reaction patterns associated tightly with the social and news feeds on the development of HCQ and COVID-19 treatment. The study linked the tweets and Google search frequencies to reveal the viewpoints of local communities on the use of HCQ for COVID-19 treatment across different states. Further, our tweet sentiment analysis reveals that public opinion changed significantly over time regarding the recommendation of using HCQ for COVID-19 treatment. The data showed that high support in the early dates but it significantly declined in October. Finally, using the manual classification of 4,850 tweets by humans as our benchmark, our sentiment analysis showed that the Google Cloud Natural Language algorithm outperformed the Valence Aware Dictionary and sEntiment Reasoner in classifying tweets, especially in the sarcastic tweet group.

cs.IR↗
arXiv2021

Curvatures of Stiefel manifolds with deformation metrics

We compute curvatures of a family of tractable metrics on Stiefel manifolds, introduced recently by H{\"u}per, Markina and Silva Leite, which includes the well-known embedded and canonical metrics on Stiefel manifolds as special cases. The metrics could be identified with the Cheeger deformation metrics. We identify parameter values in the family to make a Stiefel manifold an Einstein manifold and show Stiefel manifolds always carry an Einstein metric. We analyze the sectional curvature range and identify the parameter range where the manifold has non-negative sectional curvature. We provide the exact sectional curvature range when the number of columns in a Stiefel matrix is $2$, and a conjectural range for other cases. We derive the formulas from two approaches, one from a global curvature formula derived in our recent work, another using curvature formulas for left-invariant metrics. The second approach leads to curvature formulas for Cheeger deformation metrics on normal homogeneous spaces.

math.DG↗
arXiv2021

Riemannian Geometry with differentiable ambient space and metric operator

We show Riemannian geometry could be studied by identifying the tangent bundle of a Riemannian manifold $\mathcal{M}$ with a subbundle of the trivial bundle $\mathcal{M} \times \mathcal{E}$, obtained by embedding $\mathcal{M}$ differentiably in a Euclidean space $\mathcal{E}$. Given such an embedding, we can extend the metric tensor on $\mathcal{M}$ to a (positive-definite) operator-valued function acting on $\mathcal{E}$, giving us an embedded ambient structure. The formulas for the Christoffel symbols and Riemannian curvature in local coordinates have simple generalizations to this setup. For a Riemannian submersion $\mathfrak{q}:\mathcal{M}\to \mathcal{B}$ from an embedded manifold $\mathcal{M}\subset \mathcal{E}$, we define a submersed ambient structure and obtain similar formulas, with the O'Neil tensor expressed in terms of the projection to the horizontal bundle $\mathcal{H}\mathcal{M}$. Using this framework, we provide the embedded and submersed ambient structures for the double tangent bundle $\mathcal{T}\mathcal{T}\mathcal{M}$ and the tangent of the horizontal bundle $\mathcal{T}\mathcal{H}\mathcal{M}$, describe the fibration of a horizontal bundle over the tangent bundle of the base manifold and extend the notion of a canonical flip to the submersion case. We obtain a formula for horizontal lifts of Jacobi fields, and a new closed-form formula for Jacobi fields of naturally reductive homogeneous spaces. We construct natural metrics on these double tangent bundles, in particular, extending Sasaki and other natural metrics to the submersion case. We illustrate by providing explicit calculations for several manifolds.

math.DG↗
arXiv2021

Closed-form geodesics and trust-region method to calculate Riemannian logarithms on Stiefel and its quotient manifolds

We provide two closed-form geodesic formulas for a family of metrics on Stiefel manifold, parameterized by two positive numbers, having both the embedded and canonical metrics as special cases. The closed-form formulas allow us to compute geodesics by matrix exponential in reduced dimension for low-rank manifolds. Combining with the use of Fr{\'e}chet derivatives to compute the gradient of the square Frobenius distance between a geodesic ending point to a given point on the manifold, we show the logarithm map and geodesic distance between two endpoints on the manifold could be computed by {\it minimizing} this square distance by a {\it trust-region} solver. This leads to a new framework to compute the geodesic distance for manifolds with known geodesic formula but no closed-form logarithm map. We show the approach works well for Stiefel as well as flag manifolds. The logarithm map could be used to compute the Riemannian center of mass for these manifolds equipped with the above metrics. We also deduce simple trigonometric formulas for the Riemannian exponential and logarithm maps on the Grassmann manifold.

math.DG↗
arXiv2020

Operator-valued formulas for Riemannian Gradient and Hessian and families of tractable metrics

We provide an explicit formula for the Levi-Civita connection and Riemannian Hessian for a Riemannian manifold that is a quotient of a manifold embedded in an inner product space with a non-constant metric function. Together with a classical formula for projection, this allows us to evaluate Riemannian gradient and Hessian for several families of metrics on classical manifolds, including a family of metrics on Stiefel manifolds connecting both the constant and canonical ambient metrics with closed-form geodesics. Using these formulas, we derive Riemannian optimization frameworks on quotients of Stiefel manifolds, including flag manifolds, and a new family of complete quotient metrics on the manifold of positive-semidefinite matrices of fixed rank, considered as a quotient of a product of Stiefel and positive-definite matrix manifold with affine-invariant metrics. The method is procedural, and in many instances, the Riemannian gradient and Hessian formulas could be derived by symbolic calculus. The method extends the list of potential metrics that could be used in manifold optimization and machine learning.

math.OC↗
arXiv2019

A theorem of Kalman and minimal state-space realization of Vector Autoregressive Models

We introduce a concept of $autoregressive$ (AR)state-space realization that could be applied to all transfer functions $\boldsymbol{T}(L)$ with $\boldsymbol{T}(0)$ invertible. We show that a theorem of Kalman implies each Vector Autoregressive model (with exogenous variables) has a minimal $AR$-state-space realization of form $\boldsymbol{y}_t = \sum_{i=1}^p\boldsymbol{H}\boldsymbol{F}^{i-1}\boldsymbol{G}\boldsymbol{x}_{t-i}+\boldsymbol{\epsilon}_t$ where $\boldsymbol{F}$ is a nilpotent Jordan matrix and $\boldsymbol{H}, \boldsymbol{G}$ satisfy certain rank conditions. The case $VARX(1)$ corresponds to reduced-rank regression. Similar to that case, for a fixed Jordan form $\boldsymbol{F}$, $\boldsymbol{H}$ could be estimated by least square as a function of $\boldsymbol{G}$. The likelihood function is a determinant ratio generalizing the Rayleigh quotient. It is unchanged if $\boldsymbol{G}$ is replaced by $\boldsymbol{S}\boldsymbol{G}$ for an invertible matrix $\boldsymbol{S}$ commuting with $\boldsymbol{F}$. Using this invariant property, the search space for maximum likelihood estimate could be constrained to equivalent classes of matrices satisfying a number of orthogonal relations, extending the results in reduced-rank analysis. Our results could be considered a multi-lag canonical-correlation-analysis. The method considered here provides a solution in the general case to the polynomial product regression model of Velu et. al. We provide estimation examples. We also explore how the estimates vary with different Jordan matrix configurations and discuss methods to select a configuration. Our approach could provide an important dimensional reduction technique with potential applications in time series analysis and linear system identification. In the appendix, we link the reduced configuration space of $\boldsymbol{G}$ with a geometric object called a vector bundle.

math.ST↗
arXiv2019

Vector Autoregressive Moving Average Model with Scalar Moving Average

We show Vector Autoregressive Moving Average models with scalar Moving Average components could be estimated by generalized least square (GLS) for each fixed moving average polynomial. The conditional variance of the GLS model is the concentrated covariant matrix of the moving average process. Under GLS the likelihood function of these models has similar format to their VAR counterparts. Maximum likelihood estimate can be done by optimizing with gradient over the moving average parameters. These models are inexpensive generalizations of Vector Autoregressive models. We discuss a relationship between this result and the Borodin-Okounkov formula in operator theory.

stat.ME↗
arXiv2019

Rayleigh Quotient Iteration, cubic convergence, and second covariant derivative

We generalize the Rayleigh Quotient Iteration (RQI) to the problem of solving a nonlinear equation where the variables are divided into two subsets, one satisfying additional equality constraints and the other could be considered as (generalized nonlinear Lagrange) multipliers. This framework covers several problems, including the (linear\slash nonlinear) eigenvalue problems, the constrained optimization problem, and the tensor eigenpair problem. Often, the RQI increment could be computed in two equivalent forms. The classical Rayleigh quotient algorithm uses the {\it Schur form}, while the projected Hessian method in constrained optimization uses the {\it Newton form}. We link the cubic convergence of these iterations with a {\it constrained Chebyshev term}, showing it is related to the geometric concept of {\it second covariant derivative}. Both the generalized Rayleigh quotient and the {\it Hessian of the retraction} used in the RQI appear in the Chebyshev term. We derive several cubic convergence results in application and construct new RQIs for matrix and tensor problems.

math.OC↗
arXiv2016

An Explicit Formula for Likelihood Function for Gaussian Vector Autoregressive Moving-Average Model Conditioned on Initial Observables with Application to Model Calibration

We derive an explicit formula for likelihood function for Gaussian VARMA model conditioned on initial observables where the moving-average (MA) coefficients are scalar. For fixed MA coefficients the likelihood function is optimized in the autoregressive variables $Φ$'s by a closed form formula generalizing regression calculation of the VAR model with the introduction of an inner product defined by MA coefficients. We show the assumption of scalar MA coefficients is not restrictive and this formulation of the VARMA model shares many nice features of VAR and MA model. The gradient and Hessian could be computed analytically. The likelihood function is preserved under the root invertion maps of the MA coefficients. We discuss constraints on the gradient of the likelihood function with moving average unit roots. With the help of FFT the likelihood function could be computed in $O((kp+1)^2T +ckT\log(T))$ time. Numerical calibration is required for the scalar MA variables only. The approach can be generalized to include additional drifts as well as integrated components. We discuss a relationship with the Borodin-Okounkov formula and the case of infinite MA components.

q-fin.ST↗