General Uniqueness Research Articles

The generic uniqueness of AI imagery: A critical approach to Dall-E as semiotic technology

The generic uniqueness of AI imagery: A critical approach to Dall-E as semiotic technology

Generic uniqueness for the Plateau problem

Given a complete Riemannian manifold M⊂Rd which is a Lipschitz neighborhood retract of dimension m+n, of class Ch,β and an oriented, closed submanifold Γ⊂M of dimension m−1, which is a boundary in integral homology, we construct a complete metric space B of Ch,α-perturbations of Γ inside M, with α<β, enjoying the following property. For the typical element b∈B, in the sense of Baire categories, there exists a unique m-dimensional integral current in M which solves the corresponding Plateau problem and it has multiplicity one.

Generic uniqueness of saddle point for two-person zero-sum differential games

Abstract The generic uniqueness of saddle point for two-person zero-sum differential games, within the class of open-loop, against the perturbation of the right-hand side function of the control system is investigated. By employing set-valued mapping theory, it is proved that the majority of the two-person zero-sum differential games have unique saddle point in the sense of Baire’s category.

On the Generic Uniqueness of Pareto-Efficient Solutions of Vector Optimization Problems

In this paper, the generic uniqueness of Pareto weakly efficient solutions, especially Pareto-efficient solutions, of vector optimization problems is studied by using the nonlinear and linear scalarization methods, and some further results on the generic uniqueness are proved. These results present that, for most of the vector optimization problems in the sense of the Baire category, the Pareto weakly efficient solution, especially the Pareto-efficient solution, is unique. Furthermore, based on these results, the generic Tykhonov well-posedness of vector optimization problems is given.

Generic uniqueness and stability for the mixed ray transform

We consider the mixed ray transform of tensor fields on a three-dimensional compact simple Riemannian manifold with boundary. We prove the injectivity of the transform, up to natural obstructions, and establish stability estimates for the normal operator on generic three dimensional simple manifold in the case of 1 + 1 1+1 and 2 + 2 2+2 tensors fields. We show how the anisotropic perturbations of averaged isotopic travel-times of q S qS -polarized elastic waves provide partial information about the mixed ray transform of 2 + 2 2+2 tensors fields. If in addition we include the measurement of the shear wave amplitude, the complete mixed ray transform can be recovered. We also show how one can obtain the mixed ray transform from an anisotropic perturbation of the \text{Dirichlet-to-Neumann} map of an isotropic elastic wave equation on a smooth and bounded domain in three dimensional Euclidean space.

Erratum: Fatter Attraction: Anthropometric and Socioeconomic Matching on the Marriage Market

Previous articleNext article FreeErratumErratum: Fatter Attraction: Anthropometric and Socioeconomic Matching on the Marriage MarketPierre-André Chiappori, Sonia Oreffice, and Climent Quintana-DomequePierre-André ChiapporiColumbia University Search for more articles by this author , Sonia OrefficeUniversity of Exeter and Institute for Labor Economics Search for more articles by this author , and Climent Quintana-DomequeUniversity of Exeter and Institute for Labor Economics Search for more articles by this author Original articleFatter Attraction: Anthropometric and Socioeconomic Matching on the Marriage MarketPDFPDF PLUSFull Text Add to favoritesDownload CitationTrack CitationsPermissionsReprints Share onFacebookTwitterLinked InRedditEmailQR Code SectionsMoreAlfred Galichon pointed out to us an error in our paper “Fatter Attraction: Anthropometric and Socioeconomic Matching on the Marriage Market” (Chiappori, Oreffice, and Quintana-Domeque 2012). The properties derived in the theory section (sec. III) are not sufficient to validate the empirical strategy developed in the following section; the latter requires more specific assumptions.The issue can easily be described in the TU (transferable-utility) case (sec. III.B). We use the same notation as in the initial paper. In particular, women (men) are characterized by a vector (X,ε)∈ℝL×ℝK ((Y,η)∈ℝK×ℝL), where X (Y) is a vector of observable female (male) characteristics and ε (η) is a random vector reflecting female (male) unobservable attributes. Proposition 2 actually implies that, for any stable matching, the conditional distribution of the female index I(X), given the male characteristics Y, depends only on the male index J(Y), and conversely. This property can be used to empirically estimate these indexes even in the most general framework, a possibility explored in forthcoming work.However, under the hypothesis made in the initial paper, this property does not necessarily extend to individual characteristics themselves. The issue here is that in this context, generic uniqueness of the stable matching does not hold for index models. The argument goes as follows. Consider a stable matching, and take any X¯ matched with some Y¯; define the sets 𝕏¯={XsuchthatI(X)=I(X¯)} and 𝕐¯={YsuchthatJ(Y)=J(Y¯)} (which, under standard smoothness assumptions, are included within (L−1)- or (K−1)-dimensional manifolds). Then any one-to-one, measure-preserving mapping from 𝕏¯ to 𝕐¯ generates a stable matching. In other words, while the index structure constrains the mapping between indices, the matching within sets of individuals with the same index is fully indeterminate. In particular, it is always possible to find stable matchings for which the conditional distribution of some (or all) of the components of vector Y, given X, does not depend only on I(X).To be valid in a continuum context, the empirical strategy adopted in the paper requires more specific assumptions regarding the stochastic structure of the model. For instance, one may use the normal quadratic model of Chiappori, McCann, and Pass (2017), which can be estimated through ordinary least squares regressions (or SURs [seemingly unrelated regressions]) of individual characteristics on the spouse’s. Alternatively, one may omit the stochastic vector η, take K=L, and assume that the surplus takes the additive form,S(X,ε;Y)=s(X,Y)+Y′ε,where the function s(X, Y) is, moreover, linear in Y:(1)s(X,Y)=Y′⋅Ψ(X)for some mapping Ψ from ℝK to ℝK. Then the utility v(Y) of male Y satisfies(2)DYv(Y)=Ψ(X)+ε,where DYv denotes the gradient of v. If the mapping Φ:Y→DYv, from ℝK to ℝK, is invertible, then equation (2) can be written as(3)Y=Φ−1(Ψ(X)+ε),which is a multidimensional transformation model (see, e.g., Chiappori, Komunjer, and Kristensen 2015). In the specific version where the surplus s is bilinear and the distributions of (X, ε) and Y are normal, then both Ψ and Φ are, moreover, linear:Ψ(X)=Ψ¯X, Φ(Y)=Φ¯Y,where Ψ¯ and Φ¯ are K×K matrices. The relationship (3) becomesY=AX+Φ¯−1ε,where A=Φ¯−1Ψ¯.If ε is independent of X, this relationship, which is a particular case of Chiappori, McCann, and Pass (2017), can be estimated by SUR.Finally, if the surplus is indexed in X,s(X,Y)=Y′⋅ψ(I(X)),then equation (3) becomesY=Φ−1(ψ(I(X))+ε).We still have a transformation model that moreover depends on X only through I(X). In particular, the conditional distribution of Y, given X, depends only on I(X); and in the linear case, the matrix A is of rank 1.NotesWe thank Alfred Galichon and Bernard Salanié for their comments. Errors are ours.ReferencesChiappori, Pierre-André, Ivana Komunjer, and Dennis Kristensen. 2015. “Nonparametric Identification and Estimation of Transformation Models.” J. Econometrics 188 (1): 22–39.First citation in articleCrossrefGoogle ScholarChiappori, Pierre-André, Robert McCann, and Brendan Pass. 2017. “Multidimensional Matching.” Working paper, Columbia Univ.First citation in articleGoogle ScholarChiappori, Pierre-André, Sonia Oreffice, and Climent Quintana-Domeque. 2012. “Fatter Attraction: Anthropometric and Socioeconomic Matching on the Marriage Market.” J.P.E. 120 (4): 659–95.First citation in articleLinkGoogle Scholar Previous articleNext article DetailsFiguresReferencesCited by Journal of Political Economy Volume 128, Number 12December 2020 Article DOIhttps://doi.org/10.1086/710838 Views: 2026Total views on this site HistoryPublished online October 22, 2020 © 2020 by The University of Chicago. All rights reserved.PDF download Crossref reports no articles citing this article.Related articlesFatter Attraction: Anthropometric and Socioeconomic Matching on the Marriage Market19 Jul 2015Journal of Political Economy

Generic uniqueness of expanders with vanishing relative entropy

We define a relative entropy for two self-similarly expanding solutions to mean curvature flow of hypersurfaces, asymptotic to the same cone at infinity. Adapting work of White (Indiana Univ Math J 36(3):567–602, 1987) and using recent results of Bernstein (Asymptotic structure of almost eigenfunctions of drift laplacians on conical ends) and Bernstein-Wang (The space of asymptotically conical self-expanders of mean curvature flow), we show that expanders with vanishing relative entropy are unique in a generic sense. This also implies that generically locally entropy minimising expanders are unique.

Uniqueness for the inverse fixed angle scattering problem

Abstract We present a uniqueness result in dimensions 3 for the inverse fixed angle scattering problem associated to the Schrödinger operator - Δ + q {-\Delta+q} , where q is a small real-valued potential with compact support in the Sobolev space W β , 2 {W^{\beta,2}} , with β &gt; 0 . {\beta&gt;0.} This result improves the known result [P. Stefanov, Generic uniqueness for two inverse problems in potential scattering, Comm. Partial Differential Equations 17 1992, 55–68], in the sense that almost no regularity is required for the potential. The uniqueness result still holds in dimension 4, but for more regular potentials in W β , 2 {W^{\beta,2}} , with β &gt; 2 / 3 {\beta&gt;2/3} . The proof is a consequence of the reconstruction method presented in our previous work, [J. A. Barceló, C. Castro, T. Luque and M. C. Vilela, A new convergent algorithm to approximate potentials from fixed angle scattering data, SIAM J. Appl. Math. 78 2018, 2714–2736].

The Generic Uniqueness and Well-Posedness of Nash Equilibria for Stable Population Games

This paper aims at studying a new kind of stable population games introduced by J. Hofbauer and H. Sandholm in 2009. We first construct a complete distance space M consisting of stable population games and show that most of stable population games have unique Nash equilibrium point that according to Baire’s category theorem. It implies that every stable population game that possesses more than one Nash equilibrium can be approached arbitrarily by a sequence of the stable population game each of which has a unique Nash equilibrium. Then, we construct a bounded rationality function and deduce some results on the generic well-posedness implying Tikhonov well-posedness and Hadamard well-posedness for stable population games.

Simple Markovian equilibria in dynamic spatial legislative bargaining

The paper proves, by construction, the existence of Markovian equilibria in a dynamic spatial legislative bargaining model. Players bargain over policies in an infinite horizon. In each period, a sequential protocol of proposal-making and voting, with random proposer recognitions and a simple majority, produces a policy that becomes the next period's status-quo; the status-quo is endogenous. The construction relies on simple strategies determined by strategic bliss points computed by the algorithm we provide. A strategic bliss point, the dynamic utility ideal, is a moderate policy relative to a bliss point, the static utility ideal. Moderation is strategic and germane to the dynamic environment; players moderate in order to constrain the future proposals of opponents. Moderation is a strategic substitute; when a player's opponents do moderate, she does not, and when they do not moderate, she does. We provide conditions under which the simple strategies induced by the strategic bliss points computed by the algorithm deliver a Stationary Markov Perfect equilibrium, and we prove its existence in generic games with impatient players and in symmetric games. Because the algorithm constructs all equilibria in simple strategies, we provide their general characterization, and we show their generic uniqueness.

A relative entropy for expanders of the harmonic map flow

In this article, we focus on the uniqueness question for (expanding) solutions of the harmonic map flow coming out of smooth 0-homogeneous maps with values into a closed Riemannian manifold. We introduce a relative entropy for two purposes. On the one hand, we prove the existence of two expanding solutions associated to any suitable solution coming out of a 0-homogeneous map by a blow-up and a blow-down process. On the other hand, generic uniqueness of expanding solutions coming out of the same 0-homogeneous map of 0 relative entropy is proved.

GENERIC UNLABELED GLOBAL RIGIDITY

Let $\mathbf{p}$ be a configuration of $n$ points in $\mathbb{R}^{d}$ for some $n$ and some $d\geqslant 2$ . Each pair of points has a Euclidean distance in the configuration. Given some graph $G$ on $n$ vertices, we measure the point-pair distances corresponding to the edges of $G$ . In this paper, we study the question of when a generic $\mathbf{p}$ in $d$ dimensions will be uniquely determined (up to an unknowable Euclidean transformation) from a given set of point-pair distances together with knowledge of $d$ and $n$ . In this setting the distances are given simply as a set of real numbers; they are not labeled with the combinatorial data that describes which point pair gave rise to which distance, nor is data about $G$ given. We show, perhaps surprisingly, that in terms of generic uniqueness, labels have no effect. A generic configuration is determined by an unlabeled set of point-pair distances (together with $d$ and $n$ ) if and only if it is determined by the labeled distances.

An accretive operator approach to ergodic zero-sum stochastic games

We study some ergodicity property of zero-sum stochastic games with a finite state space and possibly unbounded payoffs. We formulate this property in operator-theoretical terms, involving the solvability of an optimality equation for the Shapley operators (i.e., the dynamic programming operators) of a family of perturbed games. The solvability of this equation entails the existence of the uniform value, and its solutions yield uniform optimal stationary strategies. We first provide an analytical characterization of this ergodicity property, and address the generic uniqueness, up to an additive constant, of the solutions of the optimality equation. Our analysis relies on the theory of accretive mappings, which we apply to maps of the form $Id - T$ where $T$ is nonexpansive. Then, we use the results of a companion work to characterize the ergodicity of stochastic games by a geometrical condition imposed on the transition probabilities. This condition generalizes classical notion of ergodicity for finite Markov chains and Markov decision processes.

Fiber Sampling Approach to Canonical Polyadic Decomposition and Application to Tensor Completion

Tensor decompositions play an important role in a variety of applications, such as signal processing and machine learning. In practice, the tensor can be incomplete or very large, making it difficult to analyze and process using conventional tensor techniques. In this paper we focus on the basic canonical polyadic decomposition (CPD). We propose an algebraic framework for finding the CPD of tensors that have missing fibers. This includes extensions of multilinear algebraic as well as generic uniqueness conditions originally developed for the CPD of fully observed tensors. Computationally, we reduce the CPD of a tensor with missing fibers to relatively simple matrix completion problems via a matrix eigenvalue decomposition (EVD). Under the given conditions, the EVD-based algorithm is guaranteed to return the exact CPD. The derivation establishes connections with so-called coupled CPDs, an emerging concept that has proven to be of great interest in a range of array processing and signal processing applications. It will become clear that relatively few fibers are needed in order to compute the CPD. This makes fiber sampling interesting for large scale tensor decompositions. Numerical experiments show that the algebraic framework may significantly speed up more common optimization-based computation schemes for the estimation of the CPD of incomplete noisy data tensors.

Double coupled canonical polyadic decomposition of third-order tensors: Algebraic algorithm and relaxed uniqueness conditions

Double coupled canonical polyadic decomposition (DC-CPD) decomposes multiple tensors with coupling in the first two modes, into minimal number of rank-1 tensors that also admit the double coupling structure. It has a particular interest in joint blind source separation (J-BSS) applications. In a preceding paper, we proposed an algebraic algorithm for underdetermined DC-CPD, of which the factor matrices in the first two modes of each tensor may have more columns than rows. It uses a pairwise coupled rank-1 detection mapping to transform a possibly underdetermined DC-CPD into an overdetermined DC-CPD, which can be solved algebraically via generalized eigenvalue decomposition (GEVD). In this paper, we generalize the pairwise or second-order coupled rank-1 detection mapping to an arbitrary order K≥2. Based on this generalized coupled rank-1 detection mapping, we propose a broad framework for the algebraic computation of DC-CPD, which consists of a series of algorithms with more relaxed working assumptions, each corresponding to a fixed order K≥2. Deterministic and generic uniqueness conditions are provided. We will show through analysis and numerical results that our new uniqueness conditions for DC-CPD are more relaxed than the existing results for DC-CPD and CPD. We will further show, through simulation results, the performance of the proposed algebraic DC-CPD framework in approximate DC-CPD and a J-BSS application, in comparison with existing DC-CPD and CPD algorithms.

Linear systems with a canonical polyadic decomposition constrained solution: Algorithms and applications

SummaryReal‐life data often exhibit some structure and/or sparsity, allowing one to use parsimonious models for compact representation and approximation. When considering matrix and tensor data, low‐rank models such as the (multilinear) singular value decomposition, canonical polyadic decomposition (CPD), tensor train, and hierarchical Tucker model are very common. The solution of (large‐scale) linear systems is often structured in a similar way, allowing one to use compact matrix and tensor models as well. In this paper, we focus on linear systems with a CPD‐constrained solution (LS‐CPD). Our main contribution is the development of optimization‐based and algebraic methods to solve LS‐CPDs. Furthermore, we propose a condition that guarantees generic uniqueness of the obtained solution. We also show that LS‐CPDs provide a broad framework for the analysis of multilinear systems of equations. The latter are a higher‐order generalization of linear systems, similar to tensor decompositions being a generalization of matrix decompositions. The wide applicability of LS‐CPDs in domains such as classification, multilinear algebra, and signal processing is illustrated.

Coupled Canonical Polyadic Decompositions and Multiple Shift Invariance in Array Processing

The canonical polyadic decomposition (CPD) plays an important role for signal separation in array processing. The CPD model requires arrays composed of several displaced but identical subarrays. Consequently, it is less appropriate for more complex array geometries. In this paper, we explain that coupled CPD allows a much more flexible modeling that can handle multiple shift-invariance structures, i.e., arrays that can be decomposed into multiple but not identical displaced subarrays. Both deterministic and generic identifiability conditions are presented. We also point out that, under mild conditions, the signal separation problem can in the exact case be solved by means of an eigenvalue decomposition. This is similar to ESPRIT, although the working conditions are much more relaxed. Borrowing tools from algebraic geometry, we derive generic uniqueness bounds for L-shaped, frame-shaped, and triangular-shaped arrays that come close to bounds that are necessary for uniqueness. Recognizing multiple shift invariance can be a bit of an art by itself. We explain that any centrosymmetric array processing problem can be interpreted in terms of a coupled CPD. In addition, we demonstrate that the coupled CPD model allows us to significantly relax the far-field assumption commonly used in CPD-based array processing.

Computing the unique CANDECOMP/PARAFAC decomposition of unbalanced tensors by homotopy method

The Candecomp/Parafac decomposition (CPD) of the tensor whose maximal dimension is greater than its rank is considered. We derive the upper bound of rank under which computing the CPD is equivalent to solving a structured polynomial system that is determined by the full rank factorization of the matricization of the tensor. Under the generic uniqueness conditions, the CPD solutions of the system are isolated so that these solutions can be achieved by homotopy method.

A generic uniqueness result for an interpolation problem for the join of a tangential variety gamma(x) and several copies of the variety X in P^2

Let X ⊂ ℙr be an integral and non-degenerate variety non-singular in codimension 1. Let τ(X,2) =τ(X) ⊂ ℙr be the tangential variety of X. Set n := dim(X). For any integer b&gt;2 let τ(X,b)$ be the join of τ(X) and b-2 copies of X. Mimicking the notion of weakly k-degenerate varieties introduced by Chiantini and Ciliberto we give a conditions on X assuring that a general q ∈ τ(X,b) is in the linear span of a unique scheme Z ⊂ X with Z = v ∪ {p1, ... , pb-2} with v connected of degree 2 and v ⊂ Xreg.

An approximation theorem and generic convergence for equilibrium problems

In this paper, we prove an approximation theorem for equilibrium problems and provide theoretical support for many algorithms. Simon’s bounded rationality is illustrated by an approximation theorem, that is, bounded rationality is approaching full rationality as its ultimate goal. Furthermore, by the methods of set-valued analysis, we obtain the generic uniqueness and generic convergence of the solutions of monotone equilibrium problems in the sense of Baire category. As applications, we investigate the optimization problem, variational inequality problem and saddle point problem as special cases.