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In this paper, a projected primal-dual gradient flow of augmented Lagrangian is presented to solve convex optimization problems that are not necessarily strictly convex. The optimization variables are restricted by a convex set with…
The unsupervised task of aligning two or more distributions in a shared latent space has many applications including fair representations, batch effect mitigation, and unsupervised domain adaptation. Existing flow-based approaches estimate…
This paper studies minimax optimization problems defined over infinite-dimensional function classes of overparameterized two-layer neural networks. In particular, we consider the minimax optimization problem stemming from estimating linear…
This paper considers non-smooth optimization problems where we seek to minimize the pointwise maximum of a continuously parameterized family of functions. Since the objective function is given as the solution to a maximization problem,…
In this paper, we analyze several methods for approximating gradients of noisy functions using only function values. These methods include finite differences, linear interpolation, Gaussian smoothing and smoothing on a sphere. The methods…
We study the implicit bias of gradient flow (i.e., gradient descent with infinitesimal step size) on linear neural network training. We propose a tensor formulation of neural networks that includes fully-connected, diagonal, and…
This paper presents a groundbreaking approach to causal inference by integrating continuous normalizing flows (CNFs) with parametric submodels, enhancing their geometric sensitivity and improving upon traditional Targeted Maximum Likelihood…
This article is concerned with the existence of nonnegative weak solutions to a particular fourth-order partial differential equation: it is a formal gradient flow with respect to a generalized Wasserstein transportation distance with…
We have studied the gradient-flow equations in information geometry from a point-particle perspective. Based on the motion of a null (or light-like) particle in a curved space, we have rederived the Hamiltonians which describe the…
Particle filters flexibly represent multiple posterior modes nonparametrically, via a collection of weighted samples, but have classically been applied to tracking problems with known dynamics and observation likelihoods. Such generative…
The gradient-flow dynamics of an arbitrary geometric quantity is derived using a generalization of Darcy's Law. We consider flows in both Lagrangian and Eulerian formulations. The Lagrangian formulation includes a dissipative modification…
Popular Bayes filters often apply linearization techniques, such as Taylor expansion or stochastic linear regression, to enable the use of the Kalman filter structure, but this can lead to large errors in strongly nonlinear systems. The…
A new class of integro-partial differential equation models is derived for the prediction of granular flow dynamics. These models are obtained using a novel limiting averaging method (inspired by techniques employed in the derivation of…
In the field of machine learning, comprehending the intricate training dynamics of neural networks poses a significant challenge. This paper explores the training dynamics of neural networks, particularly whether these dynamics can be…
Accurate state estimation for robotic systems evolving on Lie group manifolds, such as legged robots, is a prerequisite for achieving agile control. However, this task is challenged by nonlinear observation models defined on curved…
Variational inference (VI) can be cast as an optimization problem in which the variational parameters are tuned to closely align a variational distribution with the true posterior. The optimization task can be approached through vanilla…
Bayesian filtering approximates the true underlying behavior of a time-varying system by inverting an explicit generative model to convert noisy measurements into state estimates. This process typically requires either storage, inversion,…
In this paper, we present the detailed mathematical derivation of the gradient and Hessian matrix for the Vora-Value based colorimetric filter optimization. We make a full recapitulation of the steps involved in differentiating the…
Consider a probability measure on a Hilbert space defined via its density with respect to a Gaussian. The purpose of this paper is to demonstrate that an appropriately defined Markov chain, which is reversible with respect to the measure in…
We prove that the sequence of marginals obtained from the iterations of the Sinkhorn algorithm or the iterative proportional fitting procedure (IPFP) on joint densities, converges to an absolutely continuous curve on the $2$-Wasserstein…