Dual Space Research Articles

Mirror Descent of Hopfield Model

Abstract Mirror descent is an elegant optimization technique that leverages a dual space of parametric models to perform gradient descent. While originally developed for convex optimization, it has increasingly been applied in the field of machine learning. In this study, we propose a novel approach for using mirror descent to initialize the parameters of neural networks. Specifically, we demonstrate that by using the Hopfield model as a prototype for neural networks, mirror descent can effectively train the model with significantly improved performance compared to traditional gradient descent methods that rely on random parameter initialization. Our findings highlight the potential of mirror descent as a promising initialization technique for enhancing the optimization of machine learning models.

Qualitative Analysis for Solving a Fractional Integro-Differential Equation of Hyperbolic Type with Numerical Treatment Using the Lerch Matrix Collocation Method

In this research, we present a qualitative analysis for studying a new modification of a nonlinear hyperbolic fractional integro-differential equation (NHFIDEq) in dual Banach space CEE, J. Under some suitable conditions, the existence and uniqueness of a solution are demonstrated with the use of fixed-point theorems. The verification of the offered method has been conducted by applying the Lerch matrix collocation (LMC) method as a numerical treatment. The major motivation for selecting the LMC approach is that it reduces the solution of the given NHFIDEq to a matrix representation form corresponding to a linear system of algebraic equations; additionally, to demonstrate that the proposed strategy has better precision than alternative numerical methods, we study the error and the convergence analysis. Finally, we introduce numerical examples illustrating comparisons between the exact solutions and numerical solutions for different values of the Lerch parameters λ and time t as well as how the absolute error in each example is calculated.

Feedback control for fluid mixing via advection

This work is concerned with nonlinear feedback control design for the problem of fluid mixing via advection. The overall dynamics is governed by the transport and Stokes equations in an open bounded and connected domain Ω⊂Rd, with d=2 or d=3. The feedback laws are constructed based on the ideas of instantaneous control as well as a direct approximation of the optimality system derived from an optimal open-loop control problem. It can be shown that under appropriate numerical discretization schemes, two approaches generate the same sub-optimal feedback law. On the other hand, different discretization schemes may result in feedback laws of different regularity, which determine different mixing results. The Sobolev norm of the dual space (H1(Ω))′ of H1(Ω) is used as the mix-norm to quantify mixing based on the known property of weak convergence. The major challenge is encountered in the analysis of the asymptotic behavior of the closed-loop systems due to the absence of diffusion in the transport equation together with its nonlinear coupling with the flow equations. To address these issues, we first establish the decay properties of the velocity, which in turn help obtain the estimates on scalar mixing and its long-time behavior. Finally, mixed continuous Galerkin (CG) and discontinuous Galerkin (DG) methods are employed to discretize the closed-loop system. Numerical experiments are conducted to demonstrate our ideas and compare the effectiveness of different feedback laws.

A variational multiscale method derived from an adaptive stabilized conforming finite element method via residual minimization on dual norms

This paper interprets the stabilized finite element method via residual minimization as a variational multiscale method. We approximate the solution to the partial differential equations using two discrete spaces that we build on a triangulation of the domain; we denote these spaces as coarse and enriched spaces. Building on the adaptive stabilized finite element method via residual minimization, we find a coarse-scale approximation in a continuous space by minimizing the residual on a dual discontinuous Galerkin norm; this process allows us to compute a robust error estimate to construct an on-the-fly adaptive method. We reinterpret the residual projection using the variational multiscale framework to derive a fine-scale approximation. As a result, on each mesh of the adaptive process, we obtain stable coarse- and fine-scale solutions derived from a symmetric saddle-point formulation and an a-posteriori error indicator to guide automatic adaptivity. We test our framework in several challenging scenarios for linear and nonlinear convection-dominated diffusion problems to demonstrate the framework’s performance in providing stability in the solution with optimal convergence rates in the asymptotic regime and robust performance in the pre-asymptotic regime. Lastly, we introduce a heuristic dual-term contribution in the variational form to improve the full-scale approximation for symmetric formulations (e.g., diffusion problem).

On the Dual Narrative Space of the Film "Assassin in Red"

The construction of the story world in movies is mainly achieved through the expressive function of space. In the movie "Assassin in Red", the dual narrative space of surrealism creates a fantasy aesthetic intertwined with reality and the alien world. The two narrative spaces are full of connections and interactions, greatly breaking through the audience's life experience and aesthetic cognition, becoming a quite eye-catching presence during the Spring Festival period. This article will focus on exploring the parallel world of the movie "Assassination of the Novelist", where the vertical Chongqing mountain city structure in the real space serves as a metaphor for power and class, and the closed imperial capital space has a utopian illusion function, making the novel world a reflexive reflection on the real world, and the real world a reference for the novel world. By interpreting the dual space, we will better explore the inspiration this film brings to our society.

Hamiltonian structure of rational isomonodromic deformation systems

The Hamiltonian approach to isomonodromic deformation systems is extended to include generic rational covariant derivative operators on the Riemann sphere with irregular singularities of arbitrary Poincaré rank. The space of rational connections with given pole degrees carries a natural Poisson structure corresponding to the standard classical rational R-matrix structure on the dual space L*gl(r) of the loop algebra Lgl(r). Nonautonomous isomonodromic counterparts of isospectral systems generated by spectral invariants are obtained by identifying deformation parameters as Casimir elements on the phase space. These are shown to coincide with higher Birkhoff invariants determining local asymptotics near to irregular singular points, together with the pole loci. Pairs consisting of Birkhoff invariants, together with the corresponding dual spectral invariant Hamiltonians, appear as “mirror images” matching, at each pole, the negative power coefficients in the principal part of the Laurent expansion of the fundamental meromorphic differential on the associated spectral curve with the corresponding positive power terms in the analytic part. Infinitesimal isomonodromic deformations are shown to be generated by the sum of the Hamiltonian vector field and an explicit derivative vector field that is transversal to the symplectic foliation. The Casimir elements serve as coordinates complementing those along the symplectic leaves, defining a local symplectomorphism between them. The explicit derivative vector fields preserve the Poisson structure and define a flat transversal connection, spanning an integrable distribution whose leaves may be identified as the orbits of a free Abelian local group action. The projection of infinitesimal isomonodromic deformation vector fields to the quotient manifold under this action gives commuting Hamiltonian vector fields corresponding to the spectral invariants dual to the Birkhoff invariants and the pole loci.

Poincaré complex diagonals and the Bass trace conjecture

For a finitely dominated Poincaré duality space $M$ , we show how the first author's total obstruction $\mu _M$ to the existence of a Poincaré embedding of the diagonal map $M \to M \times M$ in [17] relates to the Reidemeister trace of the identity map of $M$ . We then apply this relationship to show that $\mu _M$ vanishes when suitable conditions on the fundamental group of $M$ are satisfied.

Differential forms on orbifolds with corners

Motivated by symplectic geometry, we give a detailed account of differential forms and currents on orbifolds with corners, the pull-back and push-forward operations, and their fundamental properties. We work within the formalism where the category of orbifolds with corners is obtained as a localization of the category of étale proper groupoids with corners. Constructions and proofs are formulated in terms of the structure maps of the groupoids, avoiding the use of orbifold charts. The Fréchet space of differential forms on an orbifold and the dual space of currents are shown to be independent of which étale proper groupoid is chosen to represent the orbifold.

Fractional nonhomogeneous system with Hardy-Littlewood-Sobolev critical nonlinearity

In this paper, we consider the following fractional elliptic system of Choquard type in where , N>2s, , is the upper critical exponent in the Hardy-Littlewood-Sobolev inequality and are nonnegative functionals in the dual space of . When , we establish the uniqueness of the solution to the above problem. On the other hand, when and are nonnegative functionals with , the multiplicity of solutions to the above problem is also shown. Moreover, we obtain the global compactness result by using (PS) decomposition.

Inner–outer dual space protection of free-standing MoS2 anodes via electrospinning for stable lithium-ion storage

Although significant achievements in improving the stability of MoS2 anodes have been made, the cycling life in most studies is still less than 1000 cycles. This is because MoS2 anodes directly contact the electrolyte and generate byproducts, leading to the loss of active mass and capacity decay. Herein, the inner–outer dual space protection of MoS2 fibers is realized by regulating the surface and interface structure of electrospinning precursors (noted as X-MoS2/CNFs). Inside the fibers, Mo–N covalent bond is constructed to anchor the active material, preventing MoS2 from falling off the matrix after multiple cycles. Simultaneously, surface of the fibers, a stable solid electrolyte interface layer is induced to prevent contact between active materials and electrolytes. In addition, the initial Coulombic efficiency is enhanced as high as 84.4%. The profound investigations of morphological evolution and internal real-time resistance confirm the double structural protection of 800-MoS2/CNFs. As a result, a decent cycling performance (408.9 mAh g−1 at 1000 mA g−1 for 2000 cycles) and the satisfied rate capacities (100–1000 mA g−1) are achieved. This work provides a new idea for the preparation of stable anodes for alkali metal ion secondary batteries.

TeethGNN: Semantic 3D Teeth Segmentation with Graph Neural Networks.

In this paper, we present TeethGNN, a novel 3D tooth segmentation method based on graph neural networks (GNNs). Given a mesh-represented 3D dental model in non-Euclidean domain, our method outputs accurate and fine-grained separation of each individual tooth robust to scanning noise, foreign matters (e.g., bubbles, dental accessories, etc.), and even severe malocclusion. Unlike previous CNN-based methods that bypass handling non-Euclidean mesh data by reshaping hand-crafted geometric features into regular grids, we explore the non-uniform and irregular structure of mesh itself in its dual space and exploit graph neural networks for effective geometric feature learning. To address the crowded teeth issues and incomplete segmentation that commonly exist in previous methods, we design a two-branch network, one of which predicts a segmentation label for each facet while the other regresses each facet an offset away from its tooth centroid. Clustering are later conducted on offset-shifted locations, enabling both the separation of adjoining teeth and the adjustment of incompletely segmented teeth. Exploiting GNN for directly processing mesh data frees us from extracting hand-crafted feature, and largely speeds up the inference procedure. Extensive experiments have shown that our method achieves the new state-of-the-art results for teeth segmentation and outperforms previous methods both quantitatively and qualitatively.

Duality of Three-parameter Hardy Spaces Associated with a Sum of Two Flag Singular Integrals

Duality of Three-parameter Hardy Spaces Associated with a Sum of Two Flag Singular Integrals

Least squares solvers for ill-posed PDEs that are conditionally stable

This paper is concerned with the design and analysis of least squares solvers for ill-posed PDEs that are conditionally stable. The norms and the regularization term used in the least squares functional are determined by the ingredients of the conditional stability assumption. We are then able to establish a general error bound that, in view of the conditional stability assumption, is qualitatively the best possible, without assuming consistent data. The price for these advantages is to handle dual norms which reduces to verifying suitable inf-sup stability. This, in turn, is done by constructing appropriate Fortin projectors for all sample scenarios. The theoretical findings are illustrated by numerical experiments.

Spectral properties of generalized Cesàro operators in sequence spaces

The generalized Cesàro operators C_t, for tin [0,1], were first investigated in the 1980s. They act continuously in many classical Banach sequence spaces contained in {{mathbb {C}}}^{{{mathbb {N}}}_0}, such as ell ^p, c_0, c, bv_0, bv and, as recently shown in Curbera et al. (J Math Anal Appl 507:31, 2022) [26], also in the discrete Cesàro spaces ces(p) and their (isomorphic) dual spaces d_p. In most cases C_t (tnot =1) is compact and its spectra and point spectrum, together with the corresponding eigenspaces, are known. We study these properties of C_t, as well as their linear dynamics and mean ergodicity, when they act in certain non-normable sequence spaces contained in {{mathbb {C}}}^{{{mathbb {N}}}_0}. Besides {{mathbb {C}}}^{{{mathbb {N}}}_0} itself, the Fréchet spaces considered are ell (p+), ces(p+) and d(p+), for 1le p<infty , as well as the (LB)-spaces ell (p-), ces(p-) and d(p-), for 1<ple infty .

Lifelong Person Re-identification via Knowledge Refreshing and Consolidation

Lifelong person re-identification (LReID) is in significant demand for real-world development as a large amount of ReID data is captured from diverse locations over time and cannot be accessed at once inherently. However, a key challenge for LReID is how to incrementally preserve old knowledge and gradually add new capabilities to the system. Unlike most existing LReID methods, which mainly focus on dealing with catastrophic forgetting, our focus is on a more challenging problem, which is, not only trying to reduce the forgetting on old tasks but also aiming to improve the model performance on both new and old tasks during the lifelong learning process. Inspired by the biological process of human cognition where the somatosensory neocortex and the hippocampus work together in memory consolidation, we formulated a model called Knowledge Refreshing and Consolidation (KRC) that achieves both positive forward and backward transfer. More specifically, a knowledge refreshing scheme is incorporated with the knowledge rehearsal mechanism to enable bi-directional knowledge transfer by introducing a dynamic memory model and an adaptive working model. Moreover, a knowledge consolidation scheme operating on the dual space further improves model stability over the long-term. Extensive evaluations show KRC’s superiority over the state-of-the-art LReID methods with challenging pedestrian benchmarks. Code is available at https://github.com/cly234/LReID-KRKC.

Heuristic Search in Dual Space for Constrained Fixed-Horizon POMDPs with Durative Actions

Heuristic Search in Dual Space for Constrained Fixed-Horizon POMDPs with Durative Actions

Continuity of families of Calderón projections

We consider a continuous family of linear elliptic differential operators of arbitrary order over a smooth compact manifold with boundary. Assuming constant dimension of the spaces of inner solutions, we prove that the orthogonalized Calderón projections of the underlying family of elliptic operators form a continuous family of projections. Hence, its images (the Cauchy data spaces) form a continuous family of closed subspaces in the relevant Sobolev spaces. We use only elementary tools and classical results: basic manipulations of operator graphs and other closed subspaces in Banach spaces, elliptic regularity, Green's formula and trace theorems for Sobolev spaces, well-posed boundary conditions, duality of spaces and operators in Hilbert space, and the interpolation theorem for operators in Sobolev spaces.

The Concept of "Opinion Leader" Awakening and "Dual Discourse Space" Reconstruction

In the era of financial media, the audience's opinions have wider channels, more forms and stronger willingness. With social media platform and the media traffic, occupies an important position in the traffic competition, stick to the dominance of information dissemination, value transmission, the mainstream media is also trying to launch with traffic cohesion, topic guiding force, the audience attractive mirror reporters to cope with the status quo of the mainstream media dominance, trying to use "big flow" drive "positive energy". According to this phenomenon, based on the two theoretical perspectives of "opinion leader" and "dual discourse space", we discuss the identity change of journalists in the era of financial media, namely, why the identity change; the transition in terms of skills and responsibilities; and the difficulties and corresponding suggestions in the current identity change. Thus looking for the theory behind the reporter avatar and breakthrough, finally the research point: a reporter as melting media era new "opinion leader", need to construct more perfect "dual discourse space", from journalists monotonous "appearance" role, more non-obligation or even non-functional responsibility.

Some elliptic second order problems and neural network solutions: Existence and error estimates

Recently some neural networks have been proposed for computing approximate solutions to partial differential equations. For some second order elliptic or parabolic PDEs, error estimates are proved between the solution and the computed one with neural networks, assuming this one minimizes a Lp norm or a dual norm of the residual, or an abstract loss function. In this article, for some second order elliptic PDEs, thanks to a gradient flow strategy, we prove the existence of a neural network solution minimizing the loss function with respect to the neural network parameters and we give an error estimate between the solution and the computed one with neural networks. For some nonsymmetric elliptic PDEs, the problem is expressed in form of a MinMax problem which is approximating with a double NN. Thanks to a diagonal extraction process a result of convergence is established with respect to the parameters of NNs and errors estimates are also given.

Quantum algorithm for the linear Vlasov equation with collisions

The Vlasov equation is a nonlinear partial differential equation that provides a first-principles description of the dynamics of plasmas. Its linear limit is routinely used in plasma physics to investigate plasma oscillations and stability. In this paper, we present a quantum algorithm that simulates the linearized Vlasov equation with and without collisions, in the one-dimensional electrostatic limit. Rather than solving this equation in its native spatial and velocity phase space, we adopt an efficient representation in the dual space yielded by a Fourier-Hermite expansion. For a given simulation time, the Fourier-Hermite representation is exponentially more compact, thus yielding a classical algorithm that can match the performance of a previously proposed quantum algorithm for this problem. This representation results in a system of linear ordinary differential equations (ODEs) which can be solved with well-developed quantum algorithms: a Hamiltonian simulation in the collisionless case, and quantum ODE solvers in the collisional case. In particular, we demonstrate that a quadratic speedup in system size is attainable.