相关论文: Applications of a finite-dimensional duality princ…
Constraints are found on the spatial variation of finite-time Lyapunov exponents of two and three-dimensional systems of ordinary differential equations. In a chaotic system, finite-time Lyapunov exponents describe the average rate of…
We develop a unified theory of augmented Lagrangians for nonconvex optimization problems that encompasses both duality theory and convergence analysis of primal-dual augmented Lagrangian methods in the infinite dimensional setting. Our goal…
Datasets containing both categorical and continuous variables are frequently encountered in many areas, and with the rapid development of modern measurement technologies, the dimensions of these variables can be very high. Despite the…
The polynomial method has been used recently to obtain many striking results in combinatorial geometry. In this paper, we use affine Hilbert functions to obtain an estimation theorem in finite field geometry. The most natural way to state…
We investigate the theory of finite observables, i.e., resolutions of the finite-dimensional identity by means of positive operators, that have a physical interpretation in terms of measurement schemes. We focus on extremal and rank-one…
We study balancing weight estimators, which reweight outcomes from a source population to estimate missing outcomes in a target population. These estimators minimize the worst-case error by making an assumption about the outcome model. In…
About forty years ago it was realized by several researchers that the essential features of certain objects of Probability theory, notably Gaussian processes and limit theorems, may be better understood if they are considered in settings…
The missing data problem has been broadly studied in the last few decades and has various applications in different areas such as statistics or bioinformatics. Even though many methods have been developed to tackle this challenge, most of…
Modern statistical analysis often encounters high-dimensional problems but with a limited sample size. It poses great challenges to traditional statistical estimation methods. In this work, we adopt auxiliary learning to solve the…
Duality between estimation and optimal control is a problem of rich historical significance. The first duality principle appears in the seminal paper of Kalman-Bucy, where the problem of minimum variance estimation is shown to be dual to a…
We consider linear regression problems with a varying number of random projections, where we provably exhibit a double descent curve for a fixed prediction problem, with a high-dimensional analysis based on random matrix theory. We first…
We show that the main results of the expected utility and dual utility theories can be derived in a unified way from two fundamental mathematical ideas: the separation principle of convex analysis, and integral representations of continuous…
Time reversal of vast classes of phenomena has direct implications with predictability, causality and the second principle of thermodynamics. We analyze in detail time reversibility of a paradigmatic dissipative nonlinear dynamical system,…
A new approach to linear programming duality is proposed which relies on quadratic penalization, so that the relation between solutions to the penalized primal and dual problems becomes affine. This yields a new proof of Levin's duality…
We analyze, mainly using bifurcation methods, an elliptic superlinear problem in one-dimension with periodic boundary conditions. One of the main novelties is that we follow for the first time a bifurcation approach, relying on a…
We give the first dimensionality reduction methods for the overconstrained Tukey regression problem. The Tukey loss function $\|y\|_M = \sum_i M(y_i)$ has $M(y_i) \approx |y_i|^p$ for residual errors $y_i$ smaller than a prescribed…
Most inverse optimization models impute unspecified parameters of an objective function to make an observed solution optimal for a given optimization problem with a fixed feasible set. We propose two approaches to impute unspecified…
We introduce a two-dimensional (2D) early exit strategy that coordinates layer-wise and sentence-wise exiting for classification tasks in large language models. By processing input incrementally sentence-by-sentence while progressively…
We introduce estimation and test procedures through divergence optimization for discrete or continuous parametric models. This approach is based on a new dual representation for divergences. We treat point estimation and tests for simple…
Random sets are used to get a continuous partition of the cardinality of the union of many overlapping sets. The formalism uses M\"obius transforms and adapts Shapley's methodology in cooperative game theory, into the context of set theory.…