arXiv research

Let $C$ be a smooth projective curve of genus $2$. Following a method by O' Grady, we construct a semismall desingularization $\tilde{\mathcal{M}}_{Dol}^G$ of the moduli space $\mathcal{M}_{Dol}^G$ of semistable $G$-Higgs bundles of degree 0 for $G=GL(2,\mathbb{C}), SL(2,\mathbb{C})$. By the decomposition theorem by Be…

Constructs diffeological moduli stacks for Higgs and flat bundles on Kähler manifolds

A theoretical framework for non-negative matrix factorization based on generalized dual Kullback-Leibler divergence, which includes members of the exponential family of models, is proposed. A family of algorithms is developed using this framework and its convergence proven using the Expectation-Maximization algorithm. …

We offer mathematical tractability and new insights for a framework of exponential utility with non-negative consumption, a constraint often omitted in the literature giving rise to economically unviable solutions. Specifically, using the Kuhn-Tucker theorem and the notion of aggregate state price density (Malamud and …

New gradient flows for non-negative and probability measures combining optimal transport and interaction forces.

A common challenge in estimating parameters of probability density functions is the intractability of the normalizing constant. While in such cases maximum likelihood estimation may be implemented using numerical integration, the approach becomes computationally intensive. The score matching method of Hyvärinen [2005] …

Establishes exponential contraction in Wasserstein distance on manifolds and flows.

A new model BGAR(1) improves temporal NMF for time series data.

Study computes invariants on six-dimensional solvmanifolds, providing symplectic structure obstructions.

The past decade has seen substantial work on the use of non-negative matrix factorization and its probabilistic counterparts for audio source separation. Although able to capture audio spectral structure well, these models neglect the non-stationarity and temporal dynamics that are important properties of audio. The re…

Study Hilbert's projective metric for bounded growth functions leading to Sinkhorn's algorithm convergence.

We study the sparse non-negative least squares (S-NNLS) problem. S-NNLS occurs naturally in a wide variety of applications where an unknown, non-negative quantity must be recovered from linear measurements. We present a unified framework for S-NNLS based on a rectified power exponential scale mixture prior on the spars…

Study on MMV in jump-diffusion models resolves MV's non-monotonicity issues.

In this paper we study a generalization of the Kahler-Ricci flow, in which the Ricci form is twisted by a closed, non-negative (1,1)-form. We show that when a twisted Kahler-Einstein metric exists, then this twisted flow converges exponentially. This generalizes a result of Perelman on the convergence of the Kahler-Ric…

The study proves inequalities and curvature properties for Markov chains.

In this paper we study the problem of stopping a Brownian bridge $X$ in order to maximise the expected value of an exponential gain function. In particular, we solve the stopping problem $\sup_{0\le τ\le 1}\mathsf{E}[\mathrm{e}^{X_τ}]$ which was posed by Ernst and Shepp in their paper [Commun. Stoch. Anal., 9 (3), 20…

In this paper we examine the Riemannian geometry of the group of contactomorphisms of a compact contact manifold. We compute the sectional curvature of $\mathcal{D}_θ(M)$ in the sections containing the Reeb field and show that it is non-negative. We also solve explicitly the Jacobi equation along the geodesic correspon…

Extend classical theory of affine processes to path-dependent setting

Study Ricci flows on manifolds, proving they behave like self-similar solutions and confirming a conjecture.

Here we investigate some geometric properties of the contactomorphism group $\mathcal{D}_θ(M)$ of a compact contact manifold with the $L^2$ metric on the stream functions. Viewing this group as a generalization to the $\mathcal{D}(S^1)$, the diffeomorphism group of the circle, we show that its sectional curvature is al…

In this paper, we study gradient Ricci expanding solitons $(X,g)$ satisfying $Rc=cg+D^2f,$ where $Rc$ is the Ricci curvature, $c<0$ is a constant, and $D^2f$ is the Hessian of the potential function $f$ on $X$. We show that for a gradient expanding soliton $(X,g)$ with non-negative Ricci curvature, the scalar curva…

DeepMP improves non-negative sparse recovery performance.

Sharp inequality in spaces with non-negative Ricci curvature.

Odd GKM-manifolds with non-negative curvature split cohomology.

Non-negative curvature affects Markov chains' mixing and expansion properties.

We consider the problem of estimating the parameters of a $d$-dimensional rectified Gaussian distribution from i.i.d. samples. A rectified Gaussian distribution is defined by passing a standard Gaussian distribution through a one-layer ReLU neural network. We give a simple algorithm to estimate the parameters (i.e., th…

Formal manifolds with non-negative Ricci curvature have formal covers.

In this paper we study non-negatively curved and rationally elliptic GKM$_4$ manifolds and orbifolds. We show that their rational cohomology rings are isomorphic to the rational cohomology of certain model orbifolds. These models are quotients of isometric actions of finite groups on non-negatively curved torus orbifol…

Study Kähler metrics on complex tori with almost non-negative scalar curvature.

Sharp inequality for submanifolds in manifolds with non-negative Ricci curvature.

The study examines symmetries in spaces with positive or non-negative curvature.

The paper proves conjectures and classifies metrics on 3D manifolds.

Survey on rigidity and almost rigidity of Green functions in non-negative Ricci curvature spaces.

We study spaces and moduli spaces of Riemannian metrics with non-negative Ricci or non-negative sectional curvature on closed and open manifolds. We construct, in particular, the first classes of manifolds for which these moduli spaces have non-trivial rational homotopy, homology and cohomology groups. We also show tha…

Study on rigidity with non-negative intermediate curvature on low-dimensional manifolds.

We prove that if a closed, smooth, simply-connected 4-manifold with a circle action admits an almost non-negatively curved sequence of invariant Riemannian metrics, then it also admits a non-negatively curved Riemannian metric invariant with respect to the same action. The same is shown for torus actions of higher rank…

We introduce and demonstrate the variational autoencoder (VAE) for probabilistic non-negative matrix factorisation (PAE-NMF). We design a network which can perform non-negative matrix factorisation (NMF) and add in aspects of a VAE to make the coefficients of the latent space probabilistic. By restricting the weights i…

Lower bound on minimum vertex degree for non-negative Lin-Lu-Yau curvature on graphs.

Upper bounds on Laplacian eigenvalues on manifolds with non-negative curvature.

Optimal diameter estimates for 3D spaces with non-negative Ricci curvature.

In this paper, we construct a set of new functionals of Ricci curvature on any Kaehler manifolds which are invariant under holomorphic transfermations in Kaehler Einstein manifolds and essentially decreasing under the Kaehler Ricci flow. Moreover, if the initial metric has non-negative bisectional curvature, using Tian…

New Bochner technique for foliations with non-negative Ricci curvature.

Paper proves edge-connectivity equals minimum degree for graphs with non-negative curvature.

nUDEs use neural networks to model biology without negative values.

Non-negative $L_1$-approximating polynomials for Gaussian distributions are proven for certain classes of sets.

NCL improves interpretability of deep features by enforcing non-negativity.

Compact Kähler orbifolds with non-negative Ricci curvature are simply connected.

Vanishing theorem on CR manifolds with non-negative curvature.