Subspace Of Space Research Articles

Accelerating one-shot neural architecture search via constructing a sparse search space

Neural Architecture Search (NAS) has garnered significant attention for its ability to automatically design high-quality deep neural networks (DNNs) tailored to various hardware platforms. The major challenge for NAS is the time-consuming network estimation process required to select optimal networks from a large pool of candidates. Rather than training each candidate from scratch, recent one-shot NAS methods accelerate the estimation process by only training a supernet and sampling sub-networks from it, inheriting partial network architectures and weights. Despite significant acceleration, the supernet training with a large search space (i.e., the number of candidate sub-networks) still requires thousands of GPU hours to support high-quality sub-network sampling. In this work, we propose SparseNAS, an approach for one-shot NAS acceleration by reducing the redundancy of the search space. We observe that many sub-networks in the space are underperforming, with significant performance disparity to high-performance sub-networks. Crucially, this disparity can be observed early in the beginning of the supernet training. Therefore, we train an early predictor to learn this disparity and filter out high-quality networks in advance. Then, the supernet training will be conducted in this space sub-space. Compared to the state-of-the-art one-shot NAS, our SparseNAS reports a 3.1× training speedup with comparable network performance on the ImageNet dataset. Compared to the state-of-the-art acceleration method, SparseNAS reports a maximum of 1.5% higher Top-1 accuracy and 28% training cost reduction with a 7× bigger search space. Extensive experiment results demonstrated that SparseNAS achieves better trade-offs between efficiency and performance than state-of-the-art one-shot NAS.

A study on various generalizations of generalized centers (GC) in Banach spaces

In [Generalized centers of finite sets in Banach spaces. Acta Math Univ Comenian (N.S.). 1997;66(1):83–115], Veselý developed the idea of generalized centers for finite sets in Banach spaces. In this work, we explore the concept of restricted F -center property for a triplet ( X , Y , F ( X ) ) , where Y is a subspace of a Banach space X and F ( X ) is the family of finite subsets of X. In addition, we generalize the analysis to include all closed, bounded subsets of X. Similar to how Lindenstrauss characterized n .2 . I . P . , we characterize n . X . I . P . . So, it is possible to figure out that Y has n . X . I . P . in X for all natural numbers n if and only if rad Y ( F ) = rad X ( F ) for all finite subsets F of Y. It then turns out that, for all continuous, monotone functions f, the f-radii viz. rad Y f ( F ) , rad X f ( F ) are same whenever the generalized radii viz. rad Y ( F ) , rad X ( F ) are also same, for all finite subsets F of Y. We establish a variety of characterizations of central subspaces of Banach spaces. With reference to an appropriate subfamily of closed and bounded subsets, it appears that a number of function spaces and subspaces exhibit the restricted weighted Chebyshev center property.

Extended state space for describing renormalized Fock spaces in QFT

We revisit the non-perturbative renormalization of a class of simple polaron models with resting fermions. The considered dispersion relations and form factors are allowed to be highly singular, such that infinite self-energies and wave function renormalizations may occur and the diagonalizing dressing transformations might not be implementable on Fock space. Instead of taking cutoffs, we rigorously interpret the self-energies and wave function renormalizations as elements of suitable vector spaces, as well as a field extension of the complex numbers. Moreover, we define two extended state vector spaces (ESSs) over this field extension, which contain a dense subspace of Fock space, but also incorporate non-square-integrable wave functions. Elements of these ESSs can be seen as states of virtual particles, described in a mathematically rigorous way. The Hamiltonian without cutoffs, formally infinite counterterms, and the dressing transformation can then be defined as linear operators between certain subspaces of the two Fock space extensions. This way, we obtain a renormalized Hamiltonian which can be realized as a densely defined self-adjoint operator on Fock space.

Fractional Proportional Linear Control Systems: A Geometric Perspective on Controllability and Observability

The paper presents a detailed analysis of control and observation of generalized Caputo proportional fractional time-invariant linear systems. The focus is on identifying controllable states and observable systems within the controllable subspace, null space, and unobservable subspace of the proposed system. The necessary conditions for the controllable subspace and the necessary and sufficient conditions for observability criteria are firmly established. The controllable subspace is treated geometrically as the set of controllable states, while the observable system is characterized by a zero unobservable subspace. The results are reinforced by examples and will immensely benefit future studies on fractional-order control systems.

Online modeling method for composite load model including EVs and battery storage based on measurement data

Load models have a significant influence on power system simulation. However, current load modeling approaches can hardly satisfy the diversity and time-varying characteristics of loads [including electric vehicles (EVs) and battery storage] in terms of model accuracy and computing efficiency. An online modeling method for composite load models based on measurement information is proposed in this paper. Firstly, the dominant factors in load model output are analyzed based on the active subspace of parameter space. Then the clustering algorithm is applied to cluster the large number of underlying loads based on the characteristics of load daily output curves. Finally, the underlying loads are equivalently aggregated from the low voltage levels to the high voltage levels to construct the composite load model. Simulation results obtained based on PSCAD/EMTDC demonstrate that the load model constructed by the proposed approach can accurately reflect the actual load characteristics of a power system.

Shift invariant subspaces in growth spaces and sets of finite entropy

AbstractWe investigate shift invariant subspaces within the realm of analytic functions in the unit disc, whose radial growth is determined by a majorant . Our main result offers a complete characterization of the shift invariant subspaces generated by Nevanlinna class functions within the aforementioned class of growth spaces. Our description is expressed in terms of a certain ‐entropy condition, which naturally arises in connection with boundary zero sets for analytic functions in the unit disc, having modulus of continuity not exceeding on the unit circle. Furthermore, we utilize our findings to establish a structural theorem on approximation in model spaces.

Relevant sampling in a reproducing kernel subspace of Orlicz space

In this paper, we aim to provide a general paradigm for dealing with the sampling and random sampling problem in a reproducing kernel subspace of Orlicz space [Formula: see text]. We consider the function space [Formula: see text] as the image of an idempotent integral operator on [Formula: see text], where the integral kernel satisfies certain off-diagonal decay and regularity conditions. The model example of such reproducing kernel subspace of [Formula: see text] includes the finitely generated shift-invariant space and signal space with a finite rate of innovation. We show that a signal in [Formula: see text] can be stably reconstructed from its samples at distinct points separated by a sufficiently small gap. Next, we deduce that the random sampling inequality holds with a high probability for the class of functions in [Formula: see text] concentrated on a cube [Formula: see text], when the samples collected at i.i.d. random points are drawn on [Formula: see text] of order [Formula: see text].

On the reconstruction of unknown driving forces from low-mode observations in the 2D Navier–Stokes equations

This article is concerned with the problem of determining an unknown source of non-potential, external time-dependent perturbations of an incompressible fluid from large-scale observations on the flow field. A relaxation-based approach is proposed for accomplishing this, which makes use of a nonlinear property of the equations of motions to asymptotically enslave small scales to large scales. In particular, an algorithm is introduced that systematically produces approximations of the flow field on the unobserved scales in order to generate an approximation to the unknown force; the process is then repeated to generate an improved approximation of the unobserved scales, and so on. A mathematical proof of convergence of this algorithm is established in the context of the two-dimensional Navier–Stokes equations with periodic boundary conditions under the assumption that the force belongs to the observational subspace of phase space; at each stage in the algorithm, it is shown that the model error, represented as the difference between the approximating and true force, asymptotically decreases to zero in a geometric fashion provided that sufficiently many scales are observed and certain parameters of the algorithm are appropriately tuned.

Realization of unitary representations of the Lorentz group on de Sitter space

This paper builds on our previous work in which we showed that, for all connected semisimple linear Lie groups G acting on a non-compactly causal symmetric space M=G/H, every irreducible unitary representation of G can be realized by boundary value maps of holomorphic extensions in distributional sections of a vector bundle over M. In the present paper we discuss this procedure for the connected Lorentz group G=SO1,d(R)e acting on de Sitter space M=dSd. We show in particular that the previously constructed nets of real subspaces satisfy the locality condition. Following ideas of Bros and Moschella from the 1990’s, we show that the matrix-valued spherical function that corresponds to our extension process extends analytically to a large domain Gℂcut in the complexified group Gℂ=SO1,d(ℂ), which for d=1 specializes to the complex cut plane ℂ∖(−∞,0]. A number of special situations is discussed specifically: (a) The case d=1, which closely corresponds to standard subspaces in Hilbert spaces, (b) the case of scalar-valued functions, which for d>2 is the case of spherical representations, for which we also describe the jump singularities of the holomorphic extensions on the cut in de Sitter space, (c) the case d=3, where we obtain rather explicit formulas for the matrix-valued spherical functions.

A Banach spaces-based fully mixed virtual element method for the stationary two-dimensional Boussinesq equations

In this paper we extend recent results obtained for the Navier–Stokes equations to propose and analyze a new fully mixed virtual element method (mixed-VEM) for the stationary two-dimensional Boussinesq equations appearing in non-isothermal flow phenomena. The model consists of a Navier–Stokes type system, modeling the velocity and the pressure of the fluid, coupled to an advection-diffusion equation for the temperature. The variational formulation is based on the introduction of the additional unknowns given by a modified pseudostress tensor, which depends on the pressure, and the diffusive and convective terms of the fluid, and the pseudoheat vector, which involves the temperature, its gradient, and the velocity. As a consequence of the former, the pressure is eliminated from the system, but computed afterwards via a post-processing formula. In turn, for the Galerkin approximation we follow the approach employed in a previous work introducing for the first time an Lp spaces-based mixed-VEM for the Navier–Stokes equations, and couple it with a similar mixed-VEM for the convection–diffusion equation modeling the temperature. The solvability analysis of the resulting coupled discrete scheme is carried out by using appropriate fixed-point arguments, along with the discrete versions of the Babuška–Brezzi theory and the Banach-Nečas-Babuška theorem, both in subspaces of Banach spaces. The first Strang lemma is applied to derive the a priori error estimates for the virtual element solution as well as for the fully computable approximation of the pseudostress tensor, the pseudoheat vector, and the post-processed pressure. Finally, several numerical results, illustrating the performance of the mixed-VEM scheme and confirming the rates of convergence predicted by the theory, are reported.

Refining Graph Structure for Incomplete Multi-View Clustering.

As a challenging problem, incomplete multi-view clustering (MVC) has drawn much attention in recent years. Most of the existing methods contain the feature recovering step inevitably to obtain the clustering result of incomplete multi-view datasets. The extra target of recovering the missing feature in the original data space or common subspace is difficult for unsupervised clustering tasks and could accumulate mistakes during the optimization. Moreover, the biased error is not taken into consideration in the previous graph-based methods. The biased error represents the unexpected change of incomplete graph structure, such as the increase in the intra-class relation density and the missing local graph structure of boundary instances. It would mislead those graph-based methods and degrade their final performance. In order to overcome these drawbacks, we propose a new graph-based method named Graph Structure Refining for Incomplete MVC (GSRIMC). GSRIMC avoids recovering feature steps and just fully explores the existing subgraphs of each view to produce superior clustering results. To handle the biased error, the biased error separation is the core step of GSRIMC. In detail, GSRIMC first extracts basic information from the precomputed subgraph of each view and then separates refined graph structure from biased error with the help of tensor nuclear norm. Besides, cross-view graph learning is proposed to capture the missing local graph structure and complete the refined graph structure based on the complementary principle. Extensive experiments show that our method achieves better performance than other state-of-the-art baselines.

Commutants and complex symmetry of finite Blaschke product multiplication operator in

AbstractConsider the multiplication operator MB in $L^2(\mathbb{T})$, where the symbol B is a finite Blaschke product. In this article, we characterize the commutant of MB in $L^2(\mathbb{T})$. As an application of this characterization result, we explicitly determine the class of conjugations commuting with $M_{z^2}$ or making $M_{z^2}$ complex symmetric by introducing a new class of conjugations in $L^2(\mathbb{T})$. Moreover, we analyse their properties while keeping the whole Hardy space, model space and Beurling-type subspaces invariant. Furthermore, we extended our study concerning conjugations in the case of finite Blaschke products.

Features of the geometry of the five-dimensional pseudo-Euclidean space of index two

The article is devoted to the study of the geometry of subspaces of a five-dimensional pseudo-Euclidean space. This space is attractive because all kinds of semi-Euclidean, semi-pseudo-Euclidean, hyperbolic three-dimensional spaces with projective metrics are realized in its subspaces. In the sphere of the imaginary radius of space, de Sitter space is realized. Here there is a space with projective metrics in the sense of Cayley-Klein. It is a three-dimensional space with a metric that preserves space on itself when mapped linearly. The corresponding linear transformation is called the motion of this space. An interpretation of de Sitter space in a four-dimensional pseudo-Euclidean space is proved. Studies have confirmed that in subspaces of space , in addition to elliptic spaces, there is a geometry of three-dimensional spaces with projective metrics. De Sitter space of the second kind is also realized in the sphere of imaginary radius. De Sitter space is a geodesic mapping in four-dimensional Minkowski space.

On subspaces of Orlicz spaces spanned by independent copies of a mean zero function

We study the subspaces of the Orlicz spaces $L_M$ spanned by independent copies $f_k$, $k=1,2,…$, of a function $f\in L_M$, $\int_0^1 f(t) dt=0$. Any such a subspace $H$ is isomorphic to some Orlicz sequence space $\ell_\psi$. In terms of dilations of the function $f$, a description of strongly embedded subspaces of this type is obtained, and conditions guaranteeing that the unit ball of such a subspace consists of functions with equicontinuous norms in $L_M$ are found. In particular, we prove that there is a wide class of Orlicz spaces $L_M$ (containing the $L^p$-spaces, $1\le p< 2$), for which each of the above properties of $H$ holds if and only if the Matuszewska-Orlicz indices of the functions $M$ and $\psi$ satisfy $\alpha_\psi^0>\beta_M^\infty$.

Submanifolds with ample normal bundle

AbstractWe construct germs of complex manifolds of dimension along projective submanifolds of dimension with ample normal bundle and without nonconstant meromorphic functions whenever . We also show that our methods do not allow the construction of similar examples when by establishing an algebraicity criterion for foliations on projective spaces that generalizes a classical result by Van den Ven characterizing linear subspaces of projective spaces as the only submanifolds with split tangent sequence.

Characterizations of the metric and generalized metric projections on subspaces of Banach spaces

The variational principles of the metric projection onto closed and convex sets are commonly used as the basis for characterizations of metric projections onto subspaces of Hilbert spaces. However, it is important to realize that the variational principles do not hold as necessary and sufficient conditions in general Banach spaces to characterize the metric projection and its related extensions. This technical handicap prevents the study of metric projection onto subspaces of general Banach spaces from being fully explored. Motivated by this existing lacuna in the literature, we investigate the metric projection and the generalized metric projection onto subspaces of general Banach spaces in this study. Focusing on concrete Banach spaces, we show that even though there are no general variational principles in such spaces for the projection onto subspaces, specific elements satisfy a variational characterization. We provide concrete examples to illustrate various notions of projections in general Banach spaces.

Quasicomplemented Subspaces of Banach Spaces and Separable Quotients

We present a result which generalizes the theorems of Gurariy-Kadets and Lindenstrauss-Rosenthal on quasicomplemented subspaces of Banach spaces. We also provide a solution in wide classes of Banach spaces of a problem posed by Ivan Singer concerning the existence of a w^*-basic sequence in the dual of a Banach space which is biorthogonal to a given basic sequence (or more generally, a given minimal sequence) in that space, and other related problems.

Determining homology of an unknown space from a sample

The homology of an unknown subspace of Euclidean space can be determined from the intrinsic Čech complex of a sample of points in the subspace, without reference to the ambient Euclidean space. More precisely, given a subspace X of Euclidean space and a sample A of points in X, we give conditions for the homology of X to be isomorphic to a certain persistent homology group of the intrinsic Čech complex.

Special families of subspaces of topological spaces and their applications

In the present article we introduce and study the following types of families: a P-metrizable family, a family of A(P)-subspaces, a family of p-subspaces and a family of strict p-subspaces It is shown that every family of A(k)-subspaces is an open continuous image of a k-metrizable family (Corollary 6.9). Other relations between these families are shown, and some selections theorems are obtained (Theorem 7.1 and Corollary 7.4).

Embeddings into countably compact Hausdorff spaces

UDC 515.122 We consider the problem of characterization of topological spaces embedded into countably compact Hausdorff topological spaces. We study the separation axioms for subspaces of Hausdorff countably compact topological spaces and construct an example of a regular separable scattered topological space that cannot be embedded into a Urysohn countably compact topological space.