Smooth Embeddings Research Articles

ScSwinFormer: A Transformer-Based Cell-Type Annotation Method for scRNA-Seq Data Using Smooth Gene Embedding and Global Features.

Single-cell omics techniques have made it possible to analyze individual cells in biological samples, providing us with a more detailed understanding of cellular heterogeneity and biological systems. Accurate identification of cell types is critical for single-cell RNA sequencing (scRNA-seq) analysis. However, scRNA-seq data are usually high dimensional and sparse, posing a great challenge to analyze scRNA-seq data. Existing cell-type annotation methods are either constrained in modeling scRNA-seq data or lack consideration of long-term dependencies of characterized genes. In this work, we developed a Transformer-based deep learning method, scSwinFormer, for the cell-type annotation of large-scale scRNA-seq data. Sequence modeling of scRNA-seq data is performed using the smooth gene embedding module, and then, the potential dependencies of genes are captured by the self-attention module. Subsequently, the global information inherent in scRNA-seq data is synthesized using the Cell Token, thereby facilitating accurate cell-type annotation. We evaluated the performance of our model against current state-of-the-art scRNA-seq cell-type annotation methods on multiple real data sets. ScSwinFormer outperforms the current state-of-the-art scRNA-seq cell-type annotation methods in both external and benchmark data set experiments.

Discretized gradient flow for manifold learning

Gradient descent, or negative gradient flow, is a standard technique in optimization to find the minima of functions. Many implementations of gradient descent rely on discretized versions, i.e. moving in the gradient direction for a set step size, recomputing the gradient and continuing. In this paper, we present an approach to manifold learning where gradient descent takes place in the infinite-dimensional space [Formula: see text] of smooth embeddings [Formula: see text] of a manifold [Formula: see text] into [Formula: see text]. Implementing a discretized version of gradient descent for [Formula: see text], a penalty function that scores an embedding [Formula: see text], requires estimating how far we can move in a fixed direction — the direction of one gradient step — before leaving the space of smooth embeddings. Our main result is to give an explicit lower bound for this step length in terms of the Riemannian geometry of [Formula: see text]. In particular, we consider the case when the gradient of [Formula: see text] is pointwise normal to the embedded manifold [Formula: see text]. We prove this case arises when [Formula: see text] is invariant under diffeomorphisms of [Formula: see text], a natural condition in manifold learning.

Pullback Exponential Attractors with Explicit Fractal Dimensions for Non-Autonomous Partial Functional Differential Equations

The aim of this paper is to propose a new method to construct pullback exponential attractors with explicit fractal dimensions for non-autonomous infinite-dimensional dynamical systems in Banach spaces. The approach is established by combining the squeezing properties and the covering of finite subspace of Banach spaces, which generalize the method established for autonomous systems in Hilbert spaces (Eden A, Foias C, Nicolaenko B, and Temam R Exponential attractors for dissipative evolution equations, Wiley, New York, 1994). The method is especially effective for non-autonomous partial functional differential equations for which phase space decomposition based on the exponential dichotomy of the linear part or variation techniques are available for proving squeezing property. The theoretical results are illustrated by applications to several specific non-autonomous partial functional differential equations, including a retarded reaction–diffusion equation, a retarded 2D Navier–Stokes equation and a retarded semilinear wave equation. The constructed exponential attractors possess explicit fractal dimensions which do not depend on the entropy number but only on some inner characteristics of the studied equations including the spectra of the linear part and the Lipschitz constants of the nonlinear terms and hence do not require the smooth embedding between two spaces in the previous work.

Rigidity and flexibility of isometric extensions

In this paper we consider the rigidity and flexibility of C^{1, \theta} isometric extensions. We show that the Hölder exponent \theta_0=\frac12 is critical in the following sense: if u\in C^{1,\theta} is an isometric extension of a smooth isometric embedding of a codimension one submanifold \Sigma and \theta> \frac12 , then the tangential connection agrees with the Levi-Civita connection along \Sigma . On the other hand, for any \theta<\frac12 we can construct C^{1,\theta} isometric extensions via convex integration which violate such property. As a byproduct we get moreover an existence theorem for C^{1, \theta} isometric embeddings, \theta<\frac12 , of compact Riemannian manifolds with C^1 metrics and sharper amount of codimension.

Image steganography based on smooth cycle-consistent adversarial learning

Steganography is the process of concealing a secret message in ordinary digital media by making small modifications that try to preserve cover statistics. In this paper, a novel unified steganographic framework is introduced inspired by image-to-image translation networks and adversarial learning. We consider image steganography as an image-to-image translation which is a transformation from cover distribution to stego distribution. The image-to-image translation models can exploit imperceptible perturbations to embed useful input information into the target image so that the input image can be retrieved more accurately. So, we adapt the CycleGAN as a popular translator to conceal the secret message in the form of an imperceptible, high-frequency signal in the container image called stego, while the cyclic consistency constraint helps to extract the secret message. As deep neural networks are sensitive to small adversarial perturbations, adversarial learning is exploited to embed the secret message more semantically to preserve the statistical characteristics of the cover as much as possible. According to the competitive relation of steganography and steganalysis, the steganographic methods act as an adversarial attack to mislead the steganalysis network toward determining the input images as the cover images. Therefore, in the steganographic concept, a CNN-based steganalyzer is considered as the discriminator such that the steganographer and the steganalyzer alternately adjust their weights according to the updated knowledge of one another. On the other hand, we append the smoothness constraint into the objective function to encourage the model to encode the secret message as smooth embedding modifications to improve the secrecy of generated image. This constraint limits the range of local differences and enforces a stronger correlation between the cover and the corresponding stego image. Smoothing modifications mitigate the statistical impact and enhance the robustness against steganalyzers. Experiment results show the efficiency of our proposed framework which achieves an appropriate trade-off between secrecy and capacity and outperforms the existing works.

Homology Spheres Bounding Acyclic Smooth Manifolds and Symplectic Fillings

In this paper, we collect various structural results to determine when an integral homology 3-sphere bounds an acyclic smooth 4-manifold, and when this can be upgraded to a Stein manifold. In a different direction, we study whether a smooth embedding of connected sums of lens spaces in C2 can be upgraded to a Stein embedding, and we have determined that this never happens.

Synthesis of sandwich-like structured carbon spheres @ MXene as anode for high-performance potassium-ion batteries

Potassium-ion batteries are receiving increasing attention owing to their rich potassium storage and cheap. However, relatively large radius of K+ hinder it smooth embedding and extraction from the electrode material, leading to poor electrode dynamics and structural instability. Due to the disadvantages of insufficient large layer spacing for Ti3C2 MXene, we develop a novel carbon spheres coating and embedding on the surface and interlayer of the Ti3C2 MXene composite (CSs@Ti3C2) with the freeze drying and annealing method. The carbon spheres embedding on the interlayer of the Ti3C2, which can enlarge the lattice distance and improve potassium ion insertion/extraction the Ti3C2. Furthermore, the carbon spheres can provide high surface area and many active sites for storage the K+, obtaining high specific capacity. Therefore, the CSs@Ti3C2 exhibits splendid cycling property (195.8 mAhg−1 at 100 mAg−1 after 200 cycles) and long-life charge and discharge (143.3 mAhg−1 at 500 mAg−1 after 1000 charge and discharge). This study offers a new method for achieving high property MXene as anodic material for storage system.

Stability of concordance embeddings

We prove a stability theorem for spaces of smooth concordance embeddings. From it we derive various applications to spaces of concordance diffeomorphisms and homeomorphisms.

The smooth extension embedding method with Chebyshev polynomials

AbstractWe propose an implementation of the smooth extension embedding method (SEEM), first described by Agress and Guidotti's study, in the setting of Chebyshev polynomials. SEEM is a hybrid fictitious domain/collocation method which solves general boundary value problems in complex domains by recasting them as constrained optimization problems in a simple encompassing set. Previously, SEEM was introduced and implemented using a periodic box (read a torus) using Fourier series; here, it is implemented on a (non‐periodic) rectangle using Chebyshev polynomial expansions. This implementation has faster convergence on smaller grids. Numerical experiments will demonstrate that the method provides a simple, robust, efficient, and high order fictitious domain method which can solve elliptic and parabolic problems in complex geometries, with non‐constant coefficients, and for general boundary conditions. We consider applications to two and three dimensional boundary value problems as well as an initial boundary value problem via a genuinely space–time discretization.

Extending periodic maps on surfaces over the 4-sphere

Let [Formula: see text] be the closed orientable surface of genus [Formula: see text]. We address the problem to extend torsion elements of the mapping class group [Formula: see text] over the 4-sphere [Formula: see text]. Let [Formula: see text] be a torsion element of maximum order in [Formula: see text]. Results including: (1) For each [Formula: see text], [Formula: see text] is periodically extendable over [Formula: see text] for some non-smooth embedding [Formula: see text], and not periodically extendable over [Formula: see text] for any smooth embedding [Formula: see text]. (2) For each [Formula: see text], [Formula: see text] is extendable over [Formula: see text] for some smooth embedding [Formula: see text] if and only if [Formula: see text]. (3) Each torsion element of order [Formula: see text] in [Formula: see text] is extendable over [Formula: see text] for some smooth embedding [Formula: see text] if either (i) [Formula: see text] and [Formula: see text] is even; or (ii) [Formula: see text] and [Formula: see text]; or (iii) [Formula: see text]. Moreover, the conditions on [Formula: see text] in (i) and (ii) cannot be removed.

Coding of Knots by T-Graphs

In this paper, knots are considered as smooth embeddings of a circle into ℝ3 defined by their flat diagrams. We propose a new method of coding knots by T-graphs describing the torsion structure on a flat diagram. For this method of coding, we introduce conceptions of a cycle and a block and describe transformations of T-graphs under the first and the third Reidemeister moves applied to the flat diagram of a knot.

Configuration spaces, multijet transversality, and the square-peg problem

We prove a transversality “lifting property” for compactified configuration spaces as an application of the multijet transversality theorem: given a submanifold of configurations of points on an embedding of a compact manifold M in Euclidean space, we can find a dense set of smooth embeddings of M for which the corresponding configuration space of points is transverse to any submanifold of the configuration space of points in Euclidean space, as long as the two submanifolds of compactified configuration space are boundary-disjoint. We use this setup to provide an attractive proof of the square-peg problem: there is a dense family of smoothly embedded circles in the plane where each simple closed curve has an odd number of inscribed squares, and there is a dense family of smoothly embedded circles in Rn where each simple closed curve has an odd number of inscribed square-like quadrilaterals.

Lifting Generic Maps to Embeddings. Triangulation and Smoothing

We show that if a non-degenerate PL map f : N → M lifts to a topological embedding in then it lifts to a PL embedding in there. We also show that if a stable smooth map Nn → Mm, m ≥ n, lifts to a topological embedding in , then it lifts to a smooth embedding in there.

Doubly Slice Montesinos Links

In this paper, we compare the notions of double sliceness for links. The main result is showing that a large family of 2-component Montesinos links are not strongly doubly slice despite being weakly doubly slice and having doubly slice components. Our principal obstruction to strong double slicing comes by considering branched double covers. To this end, we prove a result classifying Seifert fibered spaces that admit a smooth embedding into integer homology S1×S3s by maps inducing surjections on the first homology group. A number of other results and examples pertaining to doubly slice links are also given.

Embedding Heegaard Decompositions

A smooth embedding of a closed $3$-manifold $M$ in $\mathbb{R}^4$ may generically be composed with projection to the fourth coordinate to determine a Morse function on $M$ and hence a Heegaard splitting $M=X\cup_\Sigma Y$. However, starting with a Heegaard splitting, we find an obstruction coming from the geometry of the curve complex $C(\Sigma)$ to realizing a corresponding embedding $M\hookrightarrow \mathbb{R}^4$.

The rectangular peg problem

For every smooth Jordan curve $\gamma$ and rectangle $R$ in the Euclidean plane, we show that there exists a rectangle similar to $R$ whose vertices lie on $\gamma$. The proof relies on the theorem of Shevchishin and Nemirovski that the Klein bottle does not admit a smooth Lagrangian embedding in $\mathbb{C}^2$.

Smooth Embedding and Word Sampling Research Based on Transformer Pointer Generation Network

Smooth Embedding and Word Sampling Research Based on Transformer Pointer Generation Network

Projected and Near-Projected Embeddings

A stable smooth map f : N → M is called k-realizable if its composition with the inclusion M ⊂ M × ℝk is C0-approximable by smooth embeddings; and a k-prem if the same composition is C∞-approximable by smooth embeddings, or, equivalently, if f lifts vertically to a smooth embedding N ↪ M × ℝk. It is obvious that if f is a k-prem, then it is k-realizable. We refute the so-called “prem conjecture” that the converse holds. Namely, for each n = 4k+3 ≥ 15 there exists a stable smooth immersion Sn ↬ ℝ2n−7 that is 3-realizable but is not a 3-prem. We also prove the converse in a wide range of cases. A k-realizable stable smooth fold map Nn → M2n−q is a k-prem if q ≤ n and q ≤ 2k−3; or if q < n/2 and k = 1; or if q ∈ {2k−1, 2k−2} and k ∈ {2, 4, 8} and n is sufficiently large.

Topologies on sets of polynomial knots and the homotopy types of the respective spaces

A polynomial knot in \(\mathbb {R}^n\) is a smooth embedding of \(\mathbb {R}\) in \(\mathbb {R}^n\) such that the component functions are real polynomials. In the earlier paper with Mishra, we have studied the space \(\mathcal {P}\) of polynomial knots in \(\mathbb {R}^3\) with the inductive limit topology coming from the spaces \(\mathcal {O}_d\) for \(d\ge 3\), where \(\mathcal {O}_d\) is the space of polynomial knots in \(\mathbb {R}^3\) with degree d and having some conditions on the degrees of the component polynomials. In the same paper, we have proved that the space of polynomial knots in \(\mathbb {R}^3\) has the same homotopy type as \(S^2\). The homotopy type of the space is the mere consequence of the topology chosen. If we have another topology on \(\mathcal {P}\), the homotopy type may change. With this in mind, we consider in general the set \(\mathcal {L}^n\) of polynomial knots in \(\mathbb {R}^n\) with various topologies on it and study the homotopy type of the respective spaces. Let \(\mathcal {L}\) be the union of the sets \(\mathcal {L}^n\) for \(n\ge 1\). We also explore the homotopy type of the space \(\mathcal {L}\) with some natural topologies on it.

The Smooth Extension Embedding Method

A new approach to the solution of boundary value problems within the so-called fictitious domain method philosophy is proposed, which avoids well-known shortcomings of other methods of this type, including the need to generate extensions of the data. The salient feature of the novel method, which we refer to as SEEM (smooth extension embedding method), is that it reduces the whole boundary value problem to a linear constraint for an appropriate optimization problem formulated in a larger, simpler set containing the domain on which the boundary value problem is posed and which allows for the use of straightforward discretizations. It can also be viewed as a fully discrete meshfree method, which uses a novel class of basis functions and thus builds a bridge between fictitious domain and meshfree methods. The proposed method, in essence, computes a (discrete) extension of the solution to the boundary value problem by selecting it as a smooth element of the complete affine family of solutions of the original equations now yielding an underdetermined problem for an unknown defined in the whole fictitious domain. The actual regularity of this extension is determined by that of the analytic solution and by the choice of objective functional. Numerical experiments demonstrate that it is stable enough to efficiently solve boundary value problems on general geometries and that it produces solutions of tunable (and high) accuracy.