K-dimensional Manifold Research Articles

Autoencoders for a manifold learning problem with a jacobian rank constraint

We formulate the manifold learning problem as the problem of finding an operator that maps any point to a close neighbor that lies on a “hidden” k-dimensional manifold. We call this operator the correcting function. Under this formulation, autoencoders can be viewed as a tool to approximate the correcting function. Given an autoencoder whose Jacobian has rank k, we deduce from the classical Constant Rank Theorem that its range has a structure of a k-dimensional manifold. A k-dimensionality of the range can be forced by the architecture of an autoencoder (by fixing the dimension of the code space), or alternatively, by an additional constraint that the rank of the autoencoder mapping is not greater than k. This constraint is included in the objective function as a new term, namely a squared Ky-Fan k-antinorm of the Jacobian function. We claim that this constraint is a factor that effectively reduces the dimension of the range of an autoencoder, additionally to the reduction defined by the architecture. We also add a new curvature term into the objective. To conclude, we experimentally compare our approach with the CAE+H method on synthetic and real-world datasets.

Efficient integration of generative topic models into discriminative classifiers using robust probabilistic kernels

We propose an alternative to the generative classifier that usually models both the class conditionals and class priors separately, and then uses the Bayes theorem to compute the posterior distribution of classes given the training set as a decision boundary. Because SVM (support vector machine) is not a probabilistic framework, it is really difficult to implement a direct posterior distribution-based discriminative classifier. As SVM lacks in full Bayesian analysis, we propose a hybrid (generative–discriminative) technique where the generative topic features from a Bayesian learning are fed to the SVM. The standard latent Dirichlet allocation topic model with its Dirichlet (Dir) prior could be defined as Dir–Dir topic model to characterize the Dirichlet placed on the document and corpus parameters. With very flexible conjugate priors to the multinomials such as generalized Dirichlet (GD) and Beta-Liouville (BL) in our proposed approach, we define two new topic models: the BL–GD and GD–BL. We take advantage of the geometric interpretation of our generative topic (latent) models that associate a K-dimensional manifold (K is the size of the topics) embedded into a V-dimensional feature space (word simplex) where V is the vocabulary size. Under this structure, the low-dimensional topic simplex (the subspace) defines a document as a single point on its manifold and associates each document with a single probability. The SVM, with its kernel trick, performs on these document probabilities in classification where it utilizes the maximum margin learning approach as a decision boundary. The key note is that points or documents that are close to each other on the manifold must belong to the same class. Experimental results with text documents and images show the merits of the proposed framework.

Equipment of Sets with Cardinality of the Continuum by Structures of Polish Groups with Haar Measures

It is introduced a certain approach for equipment of sets with cardinality of the continuum by structures of Polish groups with two-sided (left or right) invariant Haar measures. By using this approach we answer positively Maleki’s certain question (2012) what are the real k-dimensional manifolds with at least two different Lie group structures that have the same Haar measure. It is demonstrated that for each diffused Borel probability measure defined in a Polish space (G;ρ;Bρ(G)) without isolated points there exist a metric ρ1and a group operation ⊙ in G such that Bρ(G) = Bρ1(G) and (G;ρ1;Bρ1(G);⊙) stands a compact Polish group with a two-sided (left or right) invariant Haar measure μ , where Bρ(G) and Bρ1(G) denote Borel σ-algebras of subsets of G generated by metrics ρ and ρ1, respectively. Similar approach is used for a construction of locally compact non-compact or non-locally compact Polish groups equipped with two-sided (left or right) invariant quasi-finite Borel measures.

Solutions of Semilinear Elliptic Equations in Tubes

Given a smooth compact k-dimensional manifold Λ embedded in ℝ m , with m≥2 and 1≤k≤m−1, and given ϵ>0, we define B ϵ (Λ) to be the geodesic tubular neighborhood of radius ϵ about Λ. In this paper, we construct positive solutions of the semilinear elliptic equation $$\begin{cases} \Delta u + u^p = 0 &\mbox{in } B_{\epsilon}(\varLambda) \\ u = 0 & \mbox{on } \partial B_{\epsilon}(\varLambda) , \end{cases} $$ when the parameter ϵ is chosen small enough. In this equation, the exponent p satisfies either p>1 when n:=m−k≤2 or $p\in(1, \frac{n+2}{n-2})$ when n>2. In particular, p can be critical or supercritical in dimension m≥3. As ϵ tends to 0, the solutions we construct have Morse index tending to infinity. Moreover, using a Pohozaev type argument, we prove that our result is sharp in the sense that there are no positive solutions for $p>\frac{n+2}{n-2}$ , n≥3, if ϵ is sufficiently small.

On face numbers of manifolds with symmetry

Necessary conditions on the face numbers of Cohen–Macaulay simplicial complexes admitting a proper action of the cyclic group Z/p Z of a prime order are given. This result is extended further to necessary conditions on the face numbers and the Betti numbers of Buchsbaum simplicial complexes with a proper Z/p Z -action. Adin's upper bounds on the face numbers of Cohen–Macaulay complexes with symmetry are shown to hold for all ( d−1)-dimensional Buchsbaum complexes with symmetry on n⩾3 d−2 vertices. A generalization of Kühnel's conjecture on the Euler characteristic of 2 k-dimensional manifolds and Sparla's analog of this conjecture for centrally symmetric 2 k-manifolds are verified for all 2 k-manifolds on n⩾6 k+3 vertices. Connections to the Upper Bound Theorem are discussed and its new version for centrally symmetric manifolds is established.

THE MINIMIZING TOTAL VARIATION FLOW WITH MEASURE INITIAL CONDITIONS

In this paper we obtain existence and uniqueness of solutions for the Cauchy problem for the minimizing total variation flow when the initial condition is a Radon measure in ℝN. We study limit solutions obtained by weakly approximating the initial measure μ by functions in L1(ℝN). We are able to characterize limit solutions when the initial condition μ=h+μs, where h∈L1(ℝN)∩L∞(ℝN), and μs=αℋk⌊ S,α≥0,k is an integer and S is a k-dimensional manifold with bounded curvatures. In case k<N-1 we prove that the singular part of the solution does not move, it remains equal to μsfor all t≥0. In particular, u(t)=δ0when u(0)=δ0. In case k=N-1 we prove that the singular part of the limit solution is [Formula: see text] and we also characterize its absolutely continuous part. This explicit behaviour permits to characterize limit solutions. We also give an entropy condition characterization of the solution which is more satisfactory when k<N-1. Finally, we describe some distributional solutions which do not have the behaviour characteristic of limit solutions.

Geometric formulas for Smale invariants of codimension two immersions

We give three formulas expressing the Smale invariant of an immersion f of a (4 k−1)-sphere into (4 k+1)-space. The terms of the formulas are geometric characteristics of any generic smooth map g of any oriented 4 k-dimensional manifold, where g restricted to the boundary is an immersion regularly homotopic to f in (6 k−1)-space. The formulas imply that if f and g are two non-regularly homotopic immersions of a (4 k−1)-sphere into (4 k+1)-space then they are also non-regularly homotopic as immersions into (6 k−1)-space. Moreover, any generic homotopy in (6 k−1)-space connecting f to g must have at least a k (2 k−1)! cusps, where a k =2 if k is odd and a k =1 if k is even.

Topological quantum field theory in quaternionic geometry

A topological quantum field theory on a 4 k-dimensional manifold M admitting an almost quaternionic structure is proposed. Expectation values of certain operators on M are proved to be independent of the choice of an almost quaternionic structure used in calculations and thus carry only smooth information about M. These invariants are explicitly expressed as integrals of differential forms over the instanton moduli space associated with a chosen almost quaternionic structure. When M admits a hyperKähler structure the topological field theory has three additional supersymmetries which induce three complex structures on the associated instanton moduli space proving thus that the latter is a hypercomplex manifold. In this case an analogue of the four-dimensional Donaldson map is constructed which provides a number of candidates for new invariants of 4 k-dimensional hypercomplex structures. In the case k = 1 the proposed topological theory on an almost quaternionic manifold reproduces the Witten interpretation of the four-dimensional Donaldson invariants.

On the p-approximate property for hypersurfaces of ℝn

Let S(ℝn) be the space of Schwartz class functions and S'(ℝn) be the dual space of S(ℝn). Given a k-dimensional manifold M in ℝn with area (in case k = 1, M is a curve with length), then for 1 ≤ p ≤ ∞ we say that M has the p-approximate property of for each TεS'(ℝn) with supp T⊆M, and εLp(ℝn), we can find a sequence of measures {Tj,j = 1,2,…} on M, absolutely continuous with respect to the area measure on M, such that ∥j-∥p → 0 as j → ∞.

Invariant manifolds with asymptotic phase

THIS PAPER is concerned with invariant manifolds allowing a generic approach to solutions (of the underlying autonomous differential system) only. This means. that any solution approaching such an invariant manifold M ultimately behaves like a particular solution on J4, in the sense that the difference n-(t) i(l) of the two solutions tends to 0 as t + CQ. In other words, only solutions on the stable manifold of M can approach M as f+ cc. Invariant manifolds with that property are said to have an asymptotic phase. The simplest nontrivial example of an invariant manifold with asymptotic phase is the orbit of an isolated periodic solution whose Floquet multipliers lie all but one in the open unit disc of the complex plane. Malkin [12] proved a manifold of stationary solutions to have an asymptotic phase if each of these solutions has all eigenvalues with nonpositive real parts and if in addition the number of eigenvalues on the imaginary axis equals the dimension of the manifold. More generally, a k-dimensional manifold generated by a family of periodic solutions has an asymptotic phase if all but k Floquet multipliers have modulus less than 1 (see Hale & Stokes [9]). Conditions for a torus to be asymptotically stable with asymptotic phase are given by Samoilenko [13] and Coppel[5]. Invariant manifolds without any specified structure are considered by Fenichel [6,7] and in Kirchgraber’s note [ 111. Moreover, each center manifold has an asymptotic phase if the corresponding equilibrium is stable (see Carr & Al-Amood [4] and Henry [lo]). In any case the condition guaranteeing the existence of an asymptotic phase amounts to the fact that the decay rate of solutions toward the manifold is greater than the decay rate of the solutions within the manifold. In each of the cases quoted so far the invariant manifold is asymptotically stable and thus the question arises whether an asymptotic phase can exist only in connection with asymptotic stability. A negative answer to this question is given by the author in [l, 21 where hyperbolic manifolds of periodic solutions are proved to have an asymptotic phase. In [l] the manifold is supposed to be compact while the noncompact case is treated in [2] under the restriction that the generating family of periodic solutions has an amplitude independent period. Recently Hale & Massatt [8] extended the corresponding result for manifolds of stationary solutions to a certain class of partial differential equations. In this paper we drop the periodicity assumption of the flow on the given manifold and prove the following result: A compact invariant manifold has an asymptotic phase if the manifold is hyperbolic (assumption Hl) and carries a flow parallel in the sense of our assumption H2. This includes most of the aforementioned results.

A classification of immersions of the two-sphere

An immersion of one C' differentiable manifold in another is a regular map (a C' map whose Jacobian is of maximum rank) of the first into the second. A homotopy of an immersion is called regular if at each stage it is regular and if the induced homotopy of the tangent bundle is continuous. Little is known about the general problem of classification of immersions under regular homotopy. Whitney [5] has shown that two immersions of a k-dimensional manifold in an n-dimensional manifold, n>2k+2, are regularly homotopic if and only if they are homotopic. The Whitney-Graustein Theorem [4] classifies immersions of the circle S in the plane E2. In my thesis [3] this theorem is extended to the case where E2 is replaced byany C2 manifold Mn, n> 1. As far as I know, these are the only known results. In this paper we give a classification of immersions of the 2-sphere S2 in Euclidean n-space E , n>2, with respect to regular homotopy. Let V.,2 be the Stiefel manifold of all 2-frames in En. If f and g are two immersions of S2 in En, an invariant Q(f, g) Cw2(Vn,2) is defined.