Space Of Linear Functionals Research Articles

Approximation Spaces of Deep Neural Networks

We study the expressivity of deep neural networks. Measuring a network's complexity by its number of or by its number of neurons, we consider the class of functions for which the error of best approximation with networks of a given complexity decays at a certain rate when increasing the complexity budget. Using results from classical approximation theory, we show that this class can be endowed with a (quasi)-norm that makes it a linear function space, called approximation space. We establish that allowing the networks to have certain types of skip connections does not change the resulting approximation spaces. We also discuss the role of the network's nonlinearity (also known as activation function) on the resulting spaces, as well as the role of depth. For the popular ReLU nonlinearity and its powers, we relate the newly constructed spaces to classical Besov spaces. The established embeddings highlight that some functions of very low Besov smoothness can nevertheless be well approximated by neural networks, if these networks are sufficiently deep.

Zero duality gap conditions via abstract convexity

Using tools provided by the theory of abstract convexity, we extend conditions for zero duality gap to the context of non-convex and nonsmooth optimization. Mimicking the classical setting, an abstract convex function is the upper envelope of a family of abstract affine functions (being conventional vertical translations of the abstract linear functions). We establish new conditions for zero duality gap under no topological assumptions on the space of abstract linear functions. In particular, we prove that the zero duality gap property can be fully characterized in terms of an inclusion involving (abstract) -subdifferentials. This result is new even for the classical convex setting. Endowing the space of abstract linear functions with the topology of pointwise convergence, we extend several fundamental facts of functional/convex analysis. This includes (i) the classical Banach–Alaoglu–Bourbaki theorem (ii) the subdifferential sum rule, and (iii) a constraint qualification for zero duality gap which extends a fact established by Borwein, Burachik and Yao (2014) for the conventional convex case. As an application, we show with a specific example how our results can be exploited to show zero duality for a family of non-convex, non-differentiable problems.

Symmetric spectral synthesis

According to the famous and pioneering result of Laurent Schwartz, any closed translation invariant linear space of continuous functions on the reals is synthesizable from its exponential monomials. Due to a result of D. I. Gurevič there is no straightforward extension of this result to higher dimensions. Following Székelyhidi (Acta Math Hungar 153(1):120–142, 2017), with the aid of Gelfand pairs and K-spherical functions, K-synthesizability of K-varieties can be described. In this paper we contribute to this direction in the special case when K is the symmetric group of order d.

Saddle point least squares for the reaction–diffusion problem

We consider a mixed variational formulation for the reaction–diffusion problem based on a saddle point least square approach with an optimal test norm and nonconforming trial spaces. An Uzawa type iterative process for solving the discrete mixed formulations is proposed and choices for discrete stable spaces are provided. The implementation requires a nodal basis only for the test space, and assembly of a global saddle point system is avoided. For the test space, we use piecewise linear spaces of functions on Shishkin type meshes that provide almost optimal approximation in the standard symmetric elliptic formulation. Our saddle point least squares method has the advantage that the order of approximation of the solution in a balanced norm is improved if compared with the standard variational approach. Numerical results are included to support the proposed method.

Rayleigh quotient and orthogonality in the linear space of boundary functions, finding accurate upper bounds of natural frequencies of non-uniform beams

Instead of the traditional method of utilizing the extremal values of Rayleigh quotient, in this research we develop an upper bound theory to determine the natural frequencies in a linear space of boundary functions. The boundary function will have the following restrictions: boundary conditions are satisfied, fourth-order polynomial at least with unit leading coefficient. It proves that the maximum value of Rayleigh quotient in terms of kth-order boundary function builds a good upper bound when the $$(k - 3)$$th-order natural frequency is approximated. There are four boundary conditions of the beam, so the fourth-order boundary function is unique without having any parameter. On the contrary, the kth-order boundary function has $$k-4$$ free parameters, which leave us a chance to optimize them. We employ the orthogonality to derive the higher-order optimal boundary functions from the lower-order optimal boundary functions exactly, which are constructed sequentially by starting from the fourth-order boundary function. We examine the upper bound theory for uniform beams under three types of supports, and the single- and double- tapered beams under cantilevered support. Comparison is made between the first four natural frequencies with the exact/numerical ones, which proves the usefulness of the upper bound theory.

Discretization of an unsteady spatial model describing the solidification process of an ingot in a mold of a continuous casting machine

Continuous casting machines (CCM) provide high quality metal. Optimization of technological processes, at present, cannot be based only on an empirical approach based on a generalization of production experience. This is due to the fact that each caster is a complex technical system with its own specifics and features. The improvement of continuous casting technology is based, first of all, on the creation of mathematical models describing technological processes taking into account many technological and structural factors. When modeling crystallization processes, most researchers, in order to simplify the solution of the problem, consider stationary plane-parallel models. Such models cannot describe the studied phenomenon well in principle and therefore have a limited scope. Therefore, the development of methods for solving the unsteady spatial model of continuous casting is of great scientific interest and expands the possibilities of solving very complex boundary value problems with phase transitions. The finite element method (FEM) is one of the most universal numerical methods for solving differential and integral equations and their systems. FEM leads to a system of algebraic equations. The article describes the use of FEM for numerical modeling of the solidification process of an ingot in a continuous casting mold with the aim of conducting computational experiments. The discretization process of the system of differential equations and the problem solution domain is presented. In addition, an algorithm of finite-difference approximation of non-stationary members of the equations is described. To sample the region, three-dimensional finite elements in the form of a parallelepiped with nodes located at the vertices of the elements are used. The solution is in the space of piecewise linear functions.

Linear invariants of complex manifolds and their plurisubharmonic variations

For a bounded domain D and a real number p>0, denote by Ap(D) the linear space of Lp integrable holomorphic functions on D, equipped with the Lp-pseudonorm. We prove that two bounded hyperconvex domains D1⊂Cn and D2⊂Cm are biholomorphic (in particular n=m) if there is a linear isometry between Ap(D1) and Ap(D2) for some 0<p<2. The same result holds for p>2,p≠2,4,⋯, provided that the p-Bergman kernels on D1 and D2 are exhaustive. With this as a motivation, we show that, for all p>0, the p-Bergman kernel on a strongly pseudoconvex domain with C2 boundary or a simply connected homogeneous regular domain is exhaustive. These results show that spaces of pluricanonical sections of complex manifolds equipped with canonical pseudonorms are important linear invariants of complex manifolds. The second part of the present work devotes to studying variations of these invariants. We show that the direct image sheaf of the twisted relative pluricanonical bundle associated to a holomorphic family of Stein manifolds or compact Kähler manifolds is positively curved, with respect to the canonical singular Finsler metric.

Arbitrary Lagrangian–Eulerian finite element method for curved and deforming surfaces: I. General theory and application to fluid interfaces

An arbitrary Lagrangian–Eulerian (ALE) finite element method for arbitrarily curved and deforming two-dimensional materials and interfaces is presented here. An ALE theory is developed by endowing the surface with a mesh whose in-plane velocity need not depend on the in-plane material velocity, and can be specified arbitrarily. A finite element implementation of the theory is formulated and applied to curved and deforming surfaces with in-plane incompressible flows. Numerical inf–sup instabilities associated with in-plane incompressibility are removed by locally projecting the surface tension onto a discontinuous space of piecewise linear functions. The general isoparametric finite element method, based on an arbitrary surface parametrization with curvilinear coordinates, is tested and validated against several numerical benchmarks. A new physical insight is obtained by applying the ALE developments to cylindrical fluid films, which are computationally and analytically found to be stable to non-axisymmetric perturbations, and unstable with respect to long-wavelength axisymmetric perturbations when their length exceeds their circumference. A Lagrangian scheme is attained as a special case of the ALE formulation. Though unable to model fluid films with sustained shear flows, the Lagrangian scheme is validated by reproducing the cylindrical instability. However, relative to the ALE results, the Lagrangian simulations are found to have spatially unresolved regions with few nodes, and thus larger errors.

Optimal Execution with Rough Path Signatures

We present a method for obtaining approximate solutions to the problem of optimal execution, based on a signature method. The framework is general, only requiring that the price process is a geometric rough path and the price impact function is a continuous function of the trading speed. Following an approximation of the optimization problem, we calculate an optimal solution for the trading speed in the space of linear functions on a truncation of the signature of the price process. We provide strong numerical evidence illustrating the accuracy and flexibility of the approach. Our numerical investigation both examines cases where exact solutions are known, demonstrating that the method accurately approximates these solutions, and models where closed-form solutions of the optimal trading speed are not known. In the latter case, we obtain favorable comparisons with standard execution strategies.

Quasi-Herglotz functions and convex optimization.

We introduce the set of quasi-Herglotz functions and demonstrate that it has properties useful in the modelling of non-passive systems. The linear space of quasi-Herglotz functions constitutes a natural extension of the convex cone of Herglotz functions. It consists of differences of Herglotz functions and we show that several of the important properties and modelling perspectives are inherited by the new set of quasi-Herglotz functions. In particular, this applies to their integral representations, the associated integral identities or sum rules (with adequate additional assumptions), their boundary values on the real axis and the associated approximation theory. Numerical examples are included to demonstrate the modelling of a non-passive gain medium formulated as a convex optimization problem, where the generating measure is modelled by using a finite expansion of B-splines and point masses.

Duality and Free Measures in Vector Spaces, the Spectral Theory of Actions of Non-Locally Compact Groups

The paper presents a general duality theory for vector measure spaces taking its origin in author’s papers written in the 1960s. The main result establishes a direct correspondence between the geometry of a measure in a vector space and properties of the space of measurable linear functionals on this space regarded as closed subspaces of an abstract space of measurable functions. An example of useful new features of this theory is the notion of a free measure and its applications.

Nonparametric registration to low-dimensional function spaces

Registration aims to decompose amplitude and phase variation of samples of curves. Phase variation is captured by warping functions which monotonically transform the domains. Resulting registered curves should then only exhibit amplitude variation. Most existing methods assume that all sample functions exhibit a typical sequence of shape features like peaks or valleys, and registration focuses on aligning these features. A more general perspective is adopted which goes beyond feature alignment. A registration method is introduced where warping functions are defined in such a way that the resulting registered curves span a low dimensional linear function space. The approach may be used as a tool for analyzing any type of functional data satisfying a structural regularity condition called bounded shape variation. Problems of identifiability are discussed in detail, and connections to established registration procedures are analyzed. The method is applied to real and simulated data.

Hypercontractivity of Spherical Averages in Hamming Space

Consider the linear space of functions on the binary hypercube and the linear operator $S_\delta$ acting by averaging a function over a Hamming sphere of radius $\delta n$ around every point. It is...

Convolution algebras and the deformation theory of infinity-morphisms

Given a coalgebra C over a cooperad, and an algebra A over an operad, it is often possible to define a natural homotopy Lie algebra structure on hom(C,A), the space of linear maps between them, called the convolution algebra of C and A. In the present article, we use convolution algebras to define the deformation complex for infinity-morphisms of algebras over operads and coalgebras over cooperads. We also complete the study of the compatibility between convolution algebras and infinity-morphisms of algebras and coalgebras. We prove that the convolution algebra bifunctor can be extended to a bifunctor that accepts infinity-morphisms in both slots and which is well defined up to homotopy, and we generalize and take a new point of view on some other already known results. This paper concludes a series of works by the two authors dealing with the investigation of convolution algebras.

A Functional Representation for Graph Matching.

Graph matching is an important and persistent problem in computer vision and pattern recognition for finding node-to-node correspondence between graphs. However, graph matching that incorporates pairwise constraints can be formulated as a quadratic assignment problem (QAP), which is NP-complete and results in intrinsic computational difficulties. This paper presents a functional representation for graph matching (FRGM) that aims to provide more geometric insights on the problem and reduce the space and time complexities. To achieve these goals, we represent each graph by a linear function space equipped with a functional such as inner product or metric, that has an explicit geometric meaning. Consequently, the correspondence matrix between graphs can be represented as a linear representation map. Furthermore, this map can be reformulated as a new parameterization for matching graphs in Euclidean space such that it is consistent with graphs under rigid or nonrigid deformations. This allows us to estimate the correspondence matrix and geometric deformations simultaneously. We use the representation of edge-attributes rather than the affinity matrix to reduce the space complexity and propose an efficient optimization strategy to reduce the time complexity. The experimental results on both synthetic and real-world datasets show that the FRGM can achieve state-of-the-art performance.

An R(x)-orthonormal theory for the vibration performance of a non-smooth symmetric composite beam with complex interface

A composite beam is symmetric if both the material property and support are symmetric with respect to the middle point. In order to study the free vibration performance of the symmetric composite beams with different complex non-smooth/discontinuous interfaces, we develop an R(x)-orthonormal theory, where R(x) is an integrable flexural rigidity function. The R(x)-orthonormal bases in the linear space of boundary functions are constructed, of which the second-order derivatives of the boundary functions are asked to be orthonormal with respect to the weight function R(x). When the vibration modes of the symmetric composite beam are expressed in terms of the R(x)-orthonormal bases we can derive an eigenvalue problem endowed with a special structure of the coefficient matrix $${{\varvec{A}}}:=[a_{ij}]$$ , $$a_{ij}=0$$ if $$i+j$$ is odd. Based on the special structure we can prove two new theorems, which indicate that the characteristic equation of $${{\varvec{A}}}$$ can be decomposed into the product of the characteristic equations of two sub-matrices with dimensions half lower. Hence, we can sequentially solve the natural frequencies in closed-form owing to the specialty of $${{\varvec{A}}}$$ . We use this powerful new theory to analyze the free vibration performance and the vibration modes of symmetric composite beams with three different interfaces.

On The Spaces of Linear Operators Acting Between Asymmetric Cone Normed Spaces

An asymmetric norm is a positive sublinear functional p on a real vector space X satisfying \(x=\theta _X\) whenever \(p(x)=p(-x)=0\). Since the space of all lower semi-continuous linear functionals of an asymmetric normed space is not a linear space, the theory is different in the asymmetric case. The main purpose of this study is to define bounded and continuous linear operators acting between asymmetric cone normed spaces. After examining the differences with symmetric case, we give some results related to Baire’s characterization of completeness in asymmetric cone normed spaces.

Locally Convex Limit Spaces of Measurable Functions with Order Units and Its Duals

We consider linear normed spaces of measurable functions dominated by positive measurable function powered by real positive parameter. Also, we consider its dual and predual, and we propose a method for constructing a limit spaces of these functional spaces taken by power parameter. We prove that these limit spaces are (LF)-spaces and also prove that the limit spaces presume the relation of duality, i.e., the limit space of predual spaces is predual for the limit space of dominated functions, and the limit space of duals is dual for it. Also, the limit space of predual spaces is embedded into the limit space of dual spaces.

Geometric sequences and zero-free region of the zeta function

Let N be the linear space of functions ∑k=1nakρ(θk/x) with a condition ∑k=1nakθk=0 for 0<θk≤1. Here ρ(x) denotes the fractional part of x. Beurling pointed out that the problem of how well a constant function can be approximated by functions in N is closely related to the zero-free region of the Riemann zeta function. More precisely, Báez-Duarte gave a zero-free region related to a Lp-norm estimation of a constant function by using the Dirichlet series for the zeta function. In this paper, we consider the L∞-norm estimation of a constant function and give a wider zero-free region than that of the Báez-Duarte result.

Linearity in higher-order recursion schemes

Higher-order recursion schemes (HORS) have recently emerged as a promising foundation for higher-order program verification. We examine the impact of enriching HORS with linear types. To that end, we introduce two frameworks that blend non-linear and linear types: a variant of the λY -calculus and an extension of HORS, called linear HORS (LHORS). First we prove that the two formalisms are equivalent and there exist polynomial-time translations between them. Then, in order to support model-checking of (trees generated by) LHORS, we propose a refined version of alternating parity tree automata, called LNAPTA, whose behaviour depends on information about linearity. We show that the complexity of LNAPTA model-checking for LHORS depends on two type-theoretic parameters: linear order and linear depth. The former is in general smaller than the standard notion of order and ignores linear function spaces. In contrast, the latter measures the depth of linear clusters inside a type. Our main result states that LNAPTA model-checking of LHORS of linear order n is n-EXPTIME-complete, when linear depth is fixed. This generalizes and improves upon the classic result of Ong, which relies on the standard notion of order. To illustrate the significance of the result, we consider two applications: the MSO model-checking problem on variants of HORS with case distinction (RSFD and HORSC) on a finite domain and a call-by-value resource verification problem. In both cases, decidability can be established by translation into HORS, but the implied complexity bounds will be suboptimal due to increases in type order. In contrast, we show that the complexity bounds derived by translations into LHORS and appealing to our result are optimal in that they match the respective hardness results.