On the local Kan structure and differentiation of simplicial manifolds

Fractal structure of 4D euclidean simplicial manifold

Recently there has been a lot of interest in geometrically motivated approaches dealing with data in high dimensional spaces. We consider the case where data is sampled from a low dimensional manifold which is embedded in high dimensional Euclidean space. In this paper, we propose a novel unsupervised linear subspace learning algorithm called Local and Global Manifold Preserving Embedding (LGMPE). Different from existing manifold learning based linear subspace learning algorithms which aims at preserving either single kind of local manifold structure or single kind of global manifold structure on the data manifold, LGMPE can preserve different local and global manifold structures simultaneously in the graph embedding framework. Several experiments on real face datasets demonstrate the effectiveness of the proposed algorithm.

Unsupervised hyperspectral image (HSI) band selection methods have been attracting ever-increasing attention. However, the local structural features captured by most of the existing methods suffer from a certain degree of risk of feature ambiguity. Moreover, these methods typically have difficulty preserving the latent manifold structure of the HSI in selected subsets of the bands. To address these problems, firstly, the graph approximate representation learning (GARL) model is proposed in this article by incorporating graph-regularized sparse coding into self-representation learning to preserve the local manifold structure information of HSI in the selected subset of bands. Meanwhile, to alleviate the ambiguity of the local graph structures, we design a local-global feature relationship expression model to construct the latent long-distance contextual connectivity (LDCC) graphs between the obtained local graph structures. Then, to preserve HSI’s local and non-local manifold structure information as much as possible in the selected subset of bands, a multigraph approximate representation learning (MGARL) model is proposed in this article by incorporating the obtained LDCC graph into the proposed GARL model. Next, we design a solution method to solve the proposed formulation. Finally, extensive experimental results on four datasets reveal the promising performance of MGARL over some state-of-the-art methods.

We define secondary theories and characteristic classes for simplicial smooth manifolds generalizing Karoubi's multiplicative K-theory and multiplicative cohomology groups for smooth manifolds. As a special case we get versions of the groups of differential characters of Cheeger and Simons for simplicial smooth manifolds. Special examples include classifying spaces of Lie groups and Lie groupoids.

This paper presents a method to quantify the spectral characteristics of reduction in speech. Hämäläinen et al. (2009) proposes a measure of spectral reduction which is able to predict a substantial amount of the variation in duration that linguistically motivated variables do not account for. In this paper, we continue studying acoustic reduction in speech by developing a new acoustic measure of reduction, based on local manifold structure in speech. We show that this measure yields significantly improved statistical models for predicting variation in duration.

We present a semiclassical technique that relies on replacing complicated classical manifold structure with simpler manifolds, which are then evaluated by the usual semiclassical rules. Under circumstances where the original manifold structure gives poor or useless results semiclassically the replacement manifolds can yield remarkable accuracy. We give several working examples to illustrate the theory presented here.

We prove a number of new restrictions on the enumerative properties of homology manifolds and semi-Eulerian complexes and posets. These include a determination of the affine span of the fine h -vector of balanced semi-Eulerian complexes and the toric h -vector of semi-Eulerian posets. The lower bounds on simplicial homology manifolds, when combined with higher dimensional analogues of Walkup’s 3-dimensional constructions [47], allow us to give a complete characterization of the f -vectors of arbitrary simplicial triangulations of S^1 \times S^3 , ℂP^2, K3 surfaces, and (S^2 \times S^2) \# (S^2 \times S^2) . We also establish a principle which leads to a conjecture for homology manifolds which is almost logically equivalent to the g -conjecture for homology spheres. Lastly, we show that with sufficiently many vertices, every triangulable homology manifold without boundary of dimension three or greater can be triangulated in a 2-neighborly fashion.

Multiple kernel methods based on k-means aims to integrate a group of kernels to improve the performance of kernel k-means clustering. However, we observe that most existing multiple kernel k-means methods exploit the nonlinear relationship within kernels, whereas the local manifold structure among multiple kernel space is not sufficiently considered. In this paper, we adopt the manifold adaptive kernel, instead of the original kernel, to integrate the local manifold structure of kernels. Thus, the induced multiple manifold adaptive kernels not only reflect the nonlinear relationship but also the local manifold structure. We then perform multiple kernel clustering within the multiple kernel k-means clustering framework. It has been verified that the proposed method outperforms several state-of-the-art baseline methods on a variety of data sets.

Hyperspectral image classification with discriminative manifold broad learning system

In this paper, we show that discrete torsion phases in string orbifold partition functions, and membrane discrete torsion phases, are topological actions on the simplicial manifolds associated to orbifold group actions. For this purpose, we introduce an integration theory of smooth Deligne cohomology on a general simplicial manifold, and prove that the integration induces a well-defined paring between the smooth Deligne cohomology and the singular cycles.

We explore in this paper an efficient algorithmic solution to single image super-resolution (SR). We propose the gCLSR, namely graph-Constrained Least Squares Regression, to super-resolve a high-resolution (HR) image from a single low-resolution (LR) observation. The basic idea of gCLSR is to learn a projection matrix mapping the LR image patch to the HR image patch space while preserving the intrinsic geometric structure of the original HR image patch manifold. Even if gCLSR resembles other manifold learning-based SR methods in preserving the local geometric structure of HR and LR image patch manifolds, the innovation of gCLSR lies in that it preserves the intrinsic geometric structure of the original HR image patch manifold rather than the LR image patch manifold, which may be contaminated by image degeneration (e.g., blurring, down-sampling and noise). Upon acquiring the projection matrix, the target HR image can be simply super-resolved from a single LR image without the need of HR-LR training pairs, which favors resource-limited applications. Experiments on images from the public database show that gCLSR method can achieve competitive quality as state-of-the-art methods, while gCLSR is much more efficient in computation than some state-of-the-art methods.

Spectral clustering requires the time-consuming decomposition of the Laplacian matrix of the similarity graph, thus limiting its applicability to large datasets. To improve the efficiency of spectral clustering, a top-down approach was recently proposed, which first divides the data into several micro-clusters (granular-balls), then splits these micro-clusters when they are not ``compact'', and finally uses these micro-clusters as nodes to construct a similarity graph for more efficient spectral clustering. However, this top-down approach is challenging to adapt to unevenly distributed or structurally complex data. This is because constructing micro-clusters as a rough ball struggles to capture the shape and structure of data in a local range, and the simplistic splitting rule that solely targets ``compactness'' is susceptible to noise and variations in data density and leads to micro-clusters with varying shapes, making it challenging to accurately measure the similarity between them. To resolve these issues and improve spectral clustering, this paper first proposes to start from local structures to obtain micro-clusters, such that the complex structural information inside local neighborhoods is well captured by them. Moreover, by noting that Euclidean distance is more suitable for convex sets, this paper further proposes a data splitting rule that couples local density and data manifold structures, so that the similarities of the obtained micro-clusters can be easily characterized. A novel similarity measure between micro-clusters is then proposed for the final spectral clustering. A series of experiments based on synthetic and real-world datasets demonstrate that the proposed method has better adaptability to structurally complex data than granular-ball based methods.

Construction of a reliable similarity matrix is fundamental for graph-based clustering methods. However, most of the current work is built upon some simple manifold structure, whereas limited work has been conducted on nonlinear data sets where data reside in a union of manifolds rather than a union of subspaces. Therefore, we construct a similarity graph to capture both global and local manifold structures of the input data set. The global structure is exploited based on the self-expressive property of data in an implicit feature space using kernel methods. Since the similarity graph computation is independent of the subsequent clustering, the final results may be far from optimal. To overcome this limitation, we simultaneously learn similarity graph and clustering structure in a principled way. Experimental studies demonstrate that our proposed algorithms deliver consistently superior results to other state-of-the-art algorithms.

We discuss two generalizations of Lie groupoids. One consists of Lie n-groupoids defined as simplicial manifolds with trivial π k≥ n+1 . The other consists of stacky Lie groupoids with a differentiable stack. We build a 1-1 correspondence between Lie 2-groupoids and stacky Lie groupoids up to a certain Morita equivalence. We prove this in a general setup so that the statement is valid in both differential and topological categories. Hypercovers of higher groupoids in various categories are also described.

Joint Projected Fuzzy Neighborhood Preserving C-means Clustering with Local Adaptive Learning