Representation Theorem Research Articles

Fuzzy-System Kernel Machines: A Kernel Method Based on the Connections Between Fuzzy Inference Systems and Kernel Machines

This article introduces the <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">fuzzy-system kernel machines</i> —a class of machine learning models based on the connection between fuzzy inference systems and kernel machines. For the connection, we observed a relationship between the representer theorem of kernel methods and the functional representation of nonsingleton fuzzy systems. We found that the <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">nonsingleton kernel on fuzzy sets</i> —a kernel defined in this article—is the core element allowing this two-way connection perspective. Consequently, a fuzzy system trained with the kernel method can be regarded as a kernel machine, whereas a kernel machine trained with a <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">nonsingleton kernel on fuzzy sets</i> can be interpreted as a fuzzy system. We conducted several experiments in supervised classification to understand the generalization power and properties of the proposed fuzzy-system kernel machines.

Primary decomposition of modules: A computational differential approach

We study primary submodules and primary decompositions from a differential and computational point of view. Our main theoretical contribution is a general structure theory and a representation theorem for primary submodules of an arbitrary finitely generated module over a polynomial ring. We characterize primary submodules in terms of differential operators and punctual Quot schemes. Moreover, we introduce and implement an algorithm that computes a minimal differential primary decomposition for a module.

Para-linearity as the Nonassociative Counterpart of Linearity

In an octonionic Hilbert space H, the octonionic inner product induces maps which fail to be octonionic linear. This fact motivates us to introduce a new notion of the octonionic para-linearity instead. To do this we encounter an insurmountable obstacle. That is, the axiom $$\begin{aligned} \left\langle pu ,u\right\rangle =p\left\langle u ,u\right\rangle \end{aligned}$$for any octonion p and element \(u\in H\) introduced by Goldstine and Horwitz in 1964 can not be interpreted as a property to be obeyed by the octonionic para-linear maps. In this article, we solve this critical problem by showing that this axiom is in fact non-independent from others. This enables us to initiate the study of octonionic para-linear maps. We can thus establish the octonionic Riesz representation theorem which, up to isomorphism, identifies two octonionic Hilbert spaces, one of which is the dual of the other. The dual space consists of continuous left para-linear functionals and it becomes a right octonionic module under the multiplication defined in terms of the second associators which measure the failure of octonionic linearity. This right multiplication has an alternative expression $$\begin{aligned} {(f\odot p)(x)}=pf(p^{-1}x)p, \end{aligned}$$which is a generalized Moufang identity. Remarkably, the multiplication is compatible with the canonical norm, i.e., $$\begin{aligned} \left| \left| f\odot p\right| \right| =\left| \left| f\right| \right| \left| p\right| . \end{aligned}$$Our final conclusion is that para-linearity is the nonassociative counterpart of linearity.

An iterative data-driven turbulence modeling framework based on Reynolds stress representation

Data-driven turbulence modeling studies have reached such a stage that the basic framework is settled, but several essential issues remain that strongly affect the performance. Two problems are studied in the current research: (1) the processing of the Reynolds stress tensor and (2) the coupling method between the machine learning model and flow solver. For the Reynolds stress processing issue, we perform the theoretical derivation to extend the relevant tensor arguments of Reynolds stress. Then, the tensor representation theorem is employed to give the complete irreducible invariants and integrity basis. An adaptive regularization term is employed to enhance the representation performance. For the coupling issue, an iterative coupling framework with consistent convergence is proposed and then applied to a canonical separated flow. The results have high consistency with the direct numerical simulation true values, which proves the validity of the current approach.

Optimality of Independently Randomized Symmetric Policies for Exchangeable Stochastic Teams with Infinitely Many Decision Makers

We study stochastic teams (known also as decentralized stochastic control problems or identical interest stochastic dynamic games) with large or countably infinite numbers of decision makers and characterize the existence and structural properties of (globally) optimal policies. We consider both static and dynamic nonconvex teams where the cost function and dynamics satisfy an exchangeability condition. To arrive at existence and structural results for optimal policies, we first introduce a topology on control policies, which involves various relaxations given the decentralized information structure. This is then utilized to arrive at a de Finetti–type representation theorem for exchangeable policies. This leads to a representation theorem for policies that admit an infinite exchangeability condition. For a general setup of stochastic team problems with N decision makers, under exchangeability of observations of decision makers and the cost function, we show that, without loss of global optimality, the search for optimal policies can be restricted to those that are N-exchangeable. Then, by extending N-exchangeable policies to infinitely exchangeable ones, establishing a convergence argument for the induced costs, and using the presented de Finetti–type theorem, we establish the existence of an optimal decentralized policy for static and dynamic teams with countably infinite numbers of decision makers, which turns out to be symmetric (i.e., identical) and randomized. In particular, unlike in prior work, convexity of the cost in policies is not assumed. Finally, we show the near optimality of symmetric independently randomized policies for finite N-decision-maker teams and thus establish approximation results for N-decision-maker weakly coupled stochastic teams. Funding: This work was supported by Natural Sciences and Engineering Research Council of Canada.

A representation theorem approach to probabilistic interpretation for viscosity solutions of Isaacs equations

The main objective of this paper is to show a certain representation theorem for generators of generalized backward stochastic differential equations (GBSDEs) and its application to probabilistic interpretation for viscosity solutions of Hamilton-Jacobi-Bellman-Isaacs equations with nonlinear Neumann boundary value problems. In the representation theorem a time change method is adopted to treat the difficulty brought from the random measure contained in GBSDEs. By means of the representation theorem the viscosity solution of Isaacs equations with nonlinear Neumann problems is proved to be the value function of a two-player zero-sum stochastic differential game problem with state constraints and recursive cost functionals.

Target-enclosing inversion using an interferometric objective function

SUMMARY Full waveform inversion is a high-resolution subsurface imaging technique, in which full seismic waveforms are used to infer subsurface physical properties. We present a novel, target-enclosing, full-waveform inversion framework based on an interferometric objective function. This objective function exploits the equivalence between the convolution and correlation representation formulas, using data from a closed boundary around the target area of interest. Because such equivalence is violated when the knowledge of the enclosed medium is incorrect, we propose to minimize the mismatch between the wavefields independently reconstructed by the two representation formulas. The proposed method requires only kinematic knowledge of the subsurface model, specifically the overburden for redatuming and does not require prior knowledge of the model below the target area. In this sense it is truly local: sensitive only to the medium parameters within the chosen target, with no assumptions about the medium or scattering regime outside the target. We present the theoretical framework and derive the gradient of the new objective function via the adjoint-state method and apply it to a synthetic example with exactly redatumed wavefields. A comparison with FWI of surface data and target-oriented FWI based on the convolution representation theorem only shows the superiority of our method both in terms of the quality of target recovery and reduction in computational cost.

Risk-sensitive credit portfolio optimization under partial information and contagion risk

This paper investigates the finite horizon risk-sensitive portfolio optimization in a regime-switching credit market with physical and information-induced default contagion. It is assumed that the underlying regime-switching process has countable states and is unobservable. The stochastic control problem is formulated under partial observations of asset prices and sequential default events. By establishing a martingale representation theorem based on incomplete and phasing out filtration, we connect the control problem to a quadratic BSDE with jumps, in which the driver term is nonstandard and carries the conditional filter as an infinite-dimensional parameter. By proposing some truncation techniques and proving uniform a priori estimates, we obtain the existence of a solution to the BSDE using the convergence of solutions associated to some truncated BSDEs. The verification theorem can be concluded with the aid of our BSDE results, which in turn yields the uniqueness of the solution to the BSDE.

Quotients of Span Categories that are Allegories and the Representation of Regular Categories

We consider the ordinary category \(\mathsf {Span}({\mathcal {C}})\) of (isomorphism classes of) spans of morphisms in a category \(\mathcal {C}\) with finite limits as needed, composed horizontally via pullback, and give a general criterion for a quotient of \(\mathsf {Span}({\mathcal {C}})\) to be an allegory. In particular, when \({\mathcal {C}}\) carries a pullback-stable, but not necessarily proper, \(({\mathcal {E}},{\mathcal {M}})\)-factorization system, we establish a quotient category \(\mathsf {Span}_{{\mathcal {E}}}({\mathcal {C}})\) that is isomorphic to the category \(\mathsf {Rel}_{{\mathcal {M}}}({\mathcal {C}})\) of \({\mathcal {M}}\)-relations in \({\mathcal {C}}\), and show that it is a (unitary and tabular) allegory precisely when \({\mathcal {M}}\) is a class of monomorphisms in \({\mathcal {C}}\). Without the restriction to monomorphisms, one can still find a least pullback-stable and composition-closed class \({\mathcal {E}}_{\bullet }\) containing \(\mathcal E\) such that \(\mathsf {Span}_{{\mathcal {E}}_{\bullet }}({\mathcal {C}})\) is a unitary and tabular allegory. In this way one obtains a left adjoint to the 2-functor that assigns to every unitary tabular allegory the regular category of its Lawverian maps. With the Freyd-Scedrov Representation Theorem for regular categories, we conclude that every finitely complete category with a stable factorization system has a reflection into the 2-category of all regular categories.

The Dyadic Representation Theorem Using Smooth Wavelets with Compact Support

The representation of a general Calderón–Zygmund operator in terms of dyadic Haar shift operators first appeared as a tool to prove the A_2 theorem, and it has found a number of other applications. In this paper we prove a new dyadic representation theorem by using smooth compactly supported wavelets in place of Haar functions. A key advantage of this is that we achieve a faster decay of the expansion when the kernel of the general Calderón–Zygmund operator has additional smoothness.

Kripke Contexts, Double Boolean Algebras with Operators and Corresponding Modal Systems

The notion of a context in formal concept analysis and that of an approximation space in rough set theory are unified in this study to define a Kripke context. For any context (G,M,I), a relation on the set G of objects and a relation on the set M of properties are included, giving a structure of the form ((G,R), (M,S), I). A Kripke context gives rise to complex algebras based on the collections of protoconcepts and semiconcepts of the underlying context. On abstraction, double Boolean algebras (dBas) with operators and topological dBas are defined. Representation results for these algebras are established in terms of the complex algebras of an appropriate Kripke context. As a natural next step, logics corresponding to classes of these algebras are formulated. A sequent calculus is proposed for contextual dBas, modal extensions of which give logics for contextual dBas with operators and topological contextual dBas. The representation theorems for the algebras result in a protoconcept-based semantics for these logics.

Consequence operators, interior operators and fuzzy relations

In this article the relationship between fuzzy consequence and fuzzy interior operators with fuzzy relations is analyzed. While the situation has been studied in depth if the fuzzy relation is a ⁎-preorder it is not the case for arbitrary fuzzy relations. If R is an arbitrary fuzzy relation, a fuzzy consequence γR and a fuzzy interior δR operators can be derived from R. Using the Representation Theorem for ⁎-preorders, from the columns and rows of R two ⁎-preorders, Rcol and Rrow, can be generated that coincide with the ⁎-preorders generated by γR and δR respectively. A duality result between them and the ⁎-coherence of γR are also shown.

Representation Theorem and Functional CLT for RKHS-Based Function-on-Function Regressions

We investigate a nonparametric, varying coefficient regression approach for modeling and estimating the regression effects caused by two functionally correlated datasets. Due to modern biomedical technology to measure multiple patient features during a time interval or intermittently at several discrete time points to review underlying biological mechanisms, statistical models that do not properly incorporate interventions and their dynamic responses may lead to biased estimates of the intervention effects. We propose a shared parameter change point function-on-function regression model to evaluate the pre- and post-intervention time trends and develop a likelihood-based method for estimating the intervention effects and other parameters. We also propose new methods for estimating and hypothesis testing regression parameters for functional data via reproducing kernel Hilbert space. The estimators of regression parameters are closed-form without computation of the inverse of a large matrix, and hence are less computationally demanding and more applicable. By establishing a representation theorem and a functional central limit theorem, the asymptotic properties of the proposed estimators are obtained, and the corresponding hypothesis tests are proposed. Application and the statistical properties of our method are demonstrated through an immunotherapy clinical trial of advanced myeloma and simulation studies.

Periodic homogenization in the context of structured deformations

An energy for first-order structured deformations in the context of periodic homogenization is obtained. This energy, defined in principle by relaxation of an initial energy of integral-type featuring contributions of bulk and interfacial terms, is proved to possess an integral representation in terms of relaxed bulk and interfacial energy densities. These energy densities, in turn, are obtained via asymptotic cell formulae defined by suitably averaging, over larger and larger cubes, the bulk and surface contributions of the initial energy. The integral representation theorem, the main result of this paper, is obtained by mixing blow-up techniques, typical in the context of structured deformations, with the averaging process underlying the theory of homogenization.

Gelfand–Naimark theorems for ordered -algebras

Abstract The classical Gelfand–Naimark theorems provide important insight into the structure of general and of commutative $C^*$ -algebras. It is shown that these can be generalized to certain ordered $^*$ -algebras. More precisely, for $\sigma $ -bounded closed ordered $^*$ -algebras, a faithful representation as operators is constructed. Similarly, for commutative such algebras, a faithful representation as complex-valued functions is constructed if an additional necessary regularity condition is fulfilled. These results generalize the Gelfand–Naimark representation theorems to classes of $^*$ -algebras larger than $C^*$ -algebras, and which especially contain $^*$ -algebras of unbounded operators. The key to these representation theorems is a new result for Archimedean ordered vector spaces V: If V is $\sigma $ -bounded, then the order of V is induced by the extremal positive linear functionals on V.

An analysis of (0,1,2;0) polynomial interpolation including interpolation on boundary points of interval [-1,1

In this paper, we survey an interpolation on polynomials with Hermite conditions on the zeros of ultraspherical polynomials at intervals [-1,1]. Our aim is to demonstrate the existence, uniqueness, explicit representation, and convergence theorem of the interpolatory polynomials, which are the zeros of the polynomials Pn(k)(x) and Pn−1(k+1)(x) respectively, where Pn(k)(x) is the ultraspherical polynomial of degree n.

Introducing discrepancy values of matrices with application to bounding norms of commutators

We introduce discrepancy values, quantities inspired by the notion of spectral spread of Hermitian matrices. We define them as the discrepancy between two consecutive Ky-Fan-like seminorms. As a result, discrepancy values share many properties with singular values and eigenvalues, yet are substantially different to merit their own study. We describe key properties of discrepancy values, and establish tools such as representation theorems, majorization inequalities, convex formulations, etc., for working with them. As an important application, we illustrate the role of discrepancy values in deriving tight bounds on the norms of commutators.

Kernel Structure Design for Data-Driven Probabilistic Load Flow Studies

In power system analysis, probabilistic load flow (PLF) accounts for uncertainties stemming from power entities such as renewable generation systems and consumer demands. In this paper, we present a Gaussian Process (GP) emulator for enabling data-driven PLF on practical distribution networks (DNs) without requiring knowledge of underlying system parameters. The main novelty of our proposal lies in the kernel design process. An ideal kernel allows for greater efficiency during training and inferencing stages without being subject to common issues that include overfitting to the training dataset and poor accuracies. In this proposal, the kernel selection process is formulated as a bi-level optimization problem. This is a very difficult problem to solve as it is composed of discrete variables and non-convex constraints. We overcome these issues by proposing a best-response strategy refinement process to identify an efficient kernel configuration in an iterative manner. The convergence properties of this iterative algorithm and the approximating capabilities of the resulting GP emulator are established using potential game theoretic constructs, universal approximation theorem and representer theorem. The performance of the proposed emulator is then showcased via comprehensive simulations conducted on practical DNs and comparisons with the state-of-the-art.

Applications of α , β -Symmetrical Functions on a Certain Class of Spirallike Functions

In this note, we use the notions of α , β -symmetrical, generalized Janowski-type and spirallike functions to define a new class S α , β λ N , M , μ defined in the open unit disk. In particular, we obtain a structural formula, a representation theorem, Marx-Strohacker inequality. Our results continue to hold the covering and distortion properties.

Stochastic optimal control in infinite dimensions with state constraints

We consider the state-constrained stochastic optimal control problem in infinite-dimensional separable Hilbert spaces, where the state process is driven by the Q-Wiener process and the (possibly unbounded) linear operator. By applying the stochastic target theory and the backward reachability approach, we show that the original (possibly discontinuous) value function can be represented by the zero-level set of the auxiliary (continuous) value function. The auxiliary value function is obtained from the penalized unconstrained stochastic control problem (in infinite dimensions) that includes an additional control variable as a consequence of the (infinite-dimensional) martingale representation theorem. We then prove that the auxiliary value function is a unique (continuous) viscosity solution to the associated Hamilton–Jacobi–Bellman (HJB) equation in infinite dimensions. Note that the viscosity analysis developed in our paper generalizes that presented in the existing literature, since the corresponding infinite-dimensional HJB equation includes an additional operator-valued control variable in the Hamiltonian maximization and depends on an additional initial state variable.