arXiv research

In this paper, we study stable weighted minimal hypersurfaces in manifolds with nonnegative Bakry-Emery Ricci curvature. We will give some geometric and topological applications. In particular, we give some partial classification of complete 3-manifolds with nonnegative Bakry-Emery Ricci curvature assuming that $f$ is …

New proof of splitting theorem and finite ends of minimal hypersurfaces in nonnegative curvature manifolds.

Fixed points of nonnegative neural networks are analyzed using fixed point theory.

New NMF algorithm uses Toeplitz matrix for facial recognition.

In this paper, we introduce the weighted mixed (sectional, Ricci and scalar) curvature of a foliated (and almost-product) Riemannian manifold $(M,g)$ equipped with a vector field $X$. We define several functions ($q$th Ricci type curvatures), which "interpolate" between the weighed sectional and Ricci curvatures. The n…

We investigate the distribution of eigenvalues of the weighted Laplacian on closed weighted Riemannian manifolds of nonnegative Bakry-Émery Ricci curvature. We derive some universal inequalities among eigenvalues of the weighted Laplacian on such manifolds. These inequalities are quantitative versions of the previous t…

We show that one-dimensional circle is the only case for closed smooth metric measure spaces with nonnegative Bakry-Émery Ricci curvature whose spectrum of the weighted Laplacian has an optimal positive upper bound. This result extends the work of Hang-Wang in the manifold case (Int. Math. Res. Not. 18 (2007), Art. ID …

Proposes WM-NMF for better multi-view clustering.

We prove a weighted Sobolev inequality and a Hardy inequality on manifolds with nonnegative Ricci curvature satisfying an inverse doubling volume condition. It enables us to obtain rigidity results for Ricci flat manifolds, generalizing earlier work of Bando, Kasue and Nakajima.

On a closed weighted Riemannian manifold with nonnegative Bakry-Émery Ricci curvature, it is shown that the ratio of the $k$-th to first eigenvalues of the weighted Laplacian is dominated by $641k^2$, using an argument via the Cheeger constant. While improving the previous exponential upper bound, the order of $k$ here…

Nonnegative matrix factorization (NMF) is a linear dimensionality reduction technique for analyzing nonnegative data. A key aspect of NMF is the choice of the objective function that depends on the noise model (or statistics of the noise) assumed on the data. In many applications, the noise model is unknown and difficu…

The paper proves geometric inequalities for hypersurfaces in weighted manifolds.

We consider a complete noncompact smooth metric measure space $(M^n,g,e^{-f} dv)$ and the associated drifting Laplacian. We find sufficient conditions on the geometry of the space so that every nonnegative $f$-subharmonic function with bounded weighted $L^1$ norm is constant.

We demonstrate a new deep learning autoencoder network, trained by a nonnegativity constraint algorithm (NCAE), that learns features which show part-based representation of data. The learning algorithm is based on constraining negative weights. The performance of the algorithm is assessed based on decomposing data into…

The backpropagation algorithm for calculating gradients has been widely used in computation of weights for deep neural networks (DNNs). This method requires derivatives of objective functions and has some difficulties finding appropriate parameters such as learning rate. In this paper, we propose a novel approach for c…

New method for sparse data using L1-NMF with improved sparsity control.

We prove that $n$-dimensional ($n\geqslant3$) complete and non-compact metric measure spaces with non-negative weighted Ricci curvature in which some Caffarelli-Kohn-Nirenberg type inequality holds are close to the model metric measure $n$-space (i.e., the Euclidean metric $n$-space).

Study nonnegative solutions on Riemannian manifolds using fractional porous medium equation.

The paper proves inequalities for scalar curvature on various manifolds.

It was conjectured by Escobar [J. Funct. Anal. 165 (1999), 101-116] that for an $n$-dimensional ($n\geq 3$) smooth compact Riemannian manifold with boundary, which has nonnegative Ricci curvature and boundary principal curvatures bounded below by $c>0$, the first nonzero Steklov eigenvalue is greater than or equal to $…

Researchers solve porous medium equation on noncompact manifolds with Ricci curvature.

We investigate the structure of a Finsler manifold of nonnegative weighted Ricci curvature including a straight line, and extend the classical Cheeger-Gromoll-Lichnerowicz splitting theorem. Such a space admits a diffeomorphic, measure-preserving splitting in general. As for a special class of Berwald spaces, we can pe…

New theorem splits weighted Lorentz-Finsler manifolds into simpler parts.

The study shows ends of shrinking gradient $ρ$-Einstein solitons are non-parabolic.

The paper proves inequalities for hypersurfaces in weighted manifolds.

Stacked regressions improve predictive accuracy by combining estimators.

The paper extends spin geometry to weighted manifolds and defines a new mass for Ricci flow.

We generalize a classification result for self-shrinkers of the mean curvature flow with nonnegative mean curvature, which was obtained by T. Colding and W. Minicozzi, replacing the assumption on polynomial volume growth with a weighted $L^2$ condition on the norm of the second fundamental form. Our approach adopt the …

The paper studies Finsler manifolds with a new curvature concept.

The paper explores inequalities for strongly-convex sets in weighted Riemannian manifolds.

In this paper, we prove the local gradient estimate for harmonic functions on complete, noncompact Finsler measure spaces under the condition that the weighted Ricci curvature has a lower bound. As applications, we obtain Liouville type theorem on Finsler manifolds with nonnegative Ricci curvature.

The purpose is to study the CR-manifold with a contact structure conformal to the Heisenberg group. In our previous work \cite{WY}, we have proved that if the $Q'$-curvature is nonnegative, and the integral of $Q'$-curvature is below the dimensional bound $c_1'$, then we have the isoperimetric inequality. In this paper…

Paper proves a sharp weighted Isoperimetric inequality for substatic manifolds.

No stable discrete maps into certain curved spaces exist.

We investigate the $m$-relative entropy, which stems from the Bregman divergence, on weighted Riemannian and Finsler manifolds. We prove that the displacement $K$-convexity of the $m$-relative entropy is equivalent to the combination of the nonnegativity of the weighted Ricci curvature and the $K$-convexity of the weig…

Under an infinitesimal version of the Bishop-Gromov relative volume comparison condition for a measure on an Alexandrov space, we prove a topological splitting theorem of Cheeger-Gromoll type. As a corollary, we prove an isometric splitting theorem for Riemannian manifolds with singularities of nonnegative (Bakry-Emery…

In this paper, we prove that if a metric measure space satisfies the volume doubling condition and the Gagliardo-Nirenberg inequality with the same exponent $n$ $(n\geq 2)$, then it has exactly the $n$-dimensional volume growth. Besides, two interesting applications have also been given. The one is that we show that if…

New functional proves mass positivity for ALE metrics.

Paper introduces SMM for forecasting multiple time series with missing values.

New lower bounds of the first nonzero eigenvalue of the weighted $p$-Laplacian are established on compact smooth metric measure spaces with or without boundaries. Under the assumption of positive lower bound for the $m$-Bakry--Émery Ricci curvature, the Escober--Lichnerowicz--Reilly type estimates are proved; under the…

Submodular functions have many applications. Matchings have many applications. The bitext word alignment problem can be modeled as the problem of maximizing a nonnegative, monotone, submodular function constrained to matchings in a complete bipartite graph where each vertex corresponds to a word in the two input senten…

The paper proves a Wulff inequality for minimal submanifolds with boundary in Euclidean space.

We represent an exchange economy in terms of statistical ensembles for complex networks by introducing the concept of market configuration. This is defined as a sequence of nonnegative discrete random variables $\{w_{ij}\}$ describing the flow of a given commodity from agent $i$ to agent $j$. This sequence can be arran…

We prove some old and new isoperimetric inequalities with the best constant using the ABP method applied to an appropriate linear Neumann problem. More precisely, we obtain a new family of sharp isoperimetric inequalities with weights (also called densities) in open convex cones of $\mathbb{R}^n$. Our result applies to…

We study geometry of complete Riemannian manifolds endowed with a weighted measure, where the weight function is of quadratic growth. Assuming the associated Bakry-Emery curvature is bounded from below, we derive a new Laplacian comparison theorem and establish various sharp volume upper and lower bounds. We also obtai…

We introduce a class of generalized relative entropies (inspired by the Bregman divergence in information theory) on the Wasserstein space over a weighted Riemannian or Finsler manifold. We prove that the convexity of all the entropies in this class is equivalent to the combination of the nonnegative weighted Ricci cur…

Given a Coxeter system $(W,S)$ and a multiparameter $\mathbf{q}$ of real numbers indexed by $S$, one can define the weighted $L^2$-cohomology groups and associate to them a nonnegative real number called the weighted $L^2$-Betti number. We show that for ranges of $\mathbf{q}$ depending on certain subgroups of $W$, the …

KATA improves associative recall by optimizing feature maps derived from nonnegative attention weights.