arXiv research

New sparse Gaussian process method tackles unconstrained regression problems.

Let $\hat{L}$ be the projective completion of an ample line bundle $L$ over $D$, a smooth projective manifold. Hwang-Singer \cite{HwangS} have constructed complete CSCK metric on $\hat{L}\backslash D$. When the corresponding \kahler form is in the cohomology class of a rational divisor $A$ and when $L$ has negative CSC…

We show that the if a sequence of normalized polynomials gives rise to a positive basis of the skein algebra of a surface, then it is sandwiched between the two types of Chebyshev polynomials. For the closed torus, we show that the normalized sequence of Chebyshev polynomials of type one $(\hat{T}_n)$ is the only one w…

We introduce three non-trivial 2-cocycles $c_k$, k=0,1,2, on the Lie algebra $S^3H=Map(S^3,H)$ with the aid of the corresponding basis vector fields on $S^3$, and extend them to 2-cocycles on the Lie algebra $S^3gl(n,H)=S^3H \otimes gl(n,C)$. Then we have the corresponding central extension $S^3gl(n,H)\oplus \oplus_k (…

Here we consider the discrete time dynamics described by a transformation $T:M \to M$, where $T$ is the shift and $M=\{1,2,...,d\}^\mathbb{N}$. It is known that the infinite-dimensional manifold $\mathcal{N}$ of Hölder equilibrium probabilities is an analytical manifold and carries a natural Riemannian metric. Given a …

This paper concerns the performance of the LASSO (also knows as basis pursuit denoising) for recovering sparse signals from undersampled, randomized, noisy measurements. We consider the recovery of the signal $x_o \in \mathbb{R}^N$ from $n$ random and noisy linear observations $y= Ax_o + w$, where $A$ is the measuremen…

Let $(M,g)$ be a complete non-compact Riemannian manifold. We consider operators of the form $Δ_g + V$, where $Δ_g$ is the non-negative Laplacian associated with the metric $g$, and $V$ a locally integrable function. Let $ρ: (\hat{M},\hat{g}) \to (M,g)$ be a Riemannian covering, with Laplacian $Δ_{\hat{g}}$ and potenti…

Let $(\hat{X}, T^{1,0} \hat{X})$ be a compact orientable CR embeddable three dimensional strongly pseudoconvex CR manifold, where $T^{1,0} \hat{X}$ is a CR structure on $\hat{X}$. Fix a point $p \in \hat{X}$ and take a global contact form $\hatθ$ so that $\hatθ$ is asymptotically flat near $p$. Then $(\hat{X}, T^{1,0} …

Differential forms on the Fréchet manifold F(S,M) of smooth functions on a compact k-dimensional manifold S can be obtained in a natural way from pairs of differential forms on M and S by the hat pairing. Special cases are the transgression map associating (p-k)-forms on F(S,M) to p-forms on M (hat pairing with a const…

Let $M$ be a differentiable manifold and $TM$ be its tangent bundle. A $C^0$-Finsler structure on $M$ is a continuous function $F: TM \rightarrow \mathbb R$ such that its restriction to each tangent space is a norm. In this work we present a large family of projectively equivalent $C^0$-Finsler manifolds $(\hat M \cong…

Given a time function $τ$ on a spacetime $M$, we define a `null distance function', $\hat{d}_τ$, built from and closely related to the causal structure of $M$. In basic models with timelike $\nabla τ$, we show that 1) $\hat{d}_τ$ is a definite distance function, which induces the manifold topology, 2) the causal struct…

In the total least squares problem, one is given an $m \times n$ matrix $A$, and an $m \times d$ matrix $B$, and one seeks to "correct" both $A$ and $B$, obtaining matrices $\hat{A}$ and $\hat{B}$, so that there exists an $X$ satisfying the equation $\hat{A}X = \hat{B}$. Typically the problem is overconstrained, meanin…

Let $t_1,\ldots,t_n$ be $\ell$-group terms in the variables $X_1,\ldots,X_m$. Let $\hat t_1,\ldots,\hat t_n$ be their associated piecewise homogeneous linear functions. Let $G$ be the $\ell$-group generated by $\hat t_1, \ldots,\hat t_n$ in the free $m$-generator $\ell$-group $\mathcal A_m.$ We prove: (i) the problem …

A financial contract's value is determined by a quantum measurement outcome, and a pricing state exists to value it.

The paper proves uniform Temple charts and applies them to null distance metrics.

We consider spacetimes $N$ satisfying some structural conditions, which are still fairly general, and prove convergence results for the leaves of an inverse mean curvature flow. Moreover, we define a new spacetime $\hat N$ by switching the light cone and using reflection to define a new time function, such that the two…

Given a weighted graph with $N$ vertices, consider a real-valued regression problem in a semi-supervised setting, where one observes $n$ labeled vertices, and the task is to label the remaining ones. We present a theoretical study of $\ell_p$-based Laplacian regularization under a $d$-dimensional geometric random graph…

In order to cluster or partition data, we often use Expectation-and-Maximization (EM) or Variational approximation with a Gaussian Mixture Model (GMM), which is a parametric probability density function represented as a weighted sum of $\hat{K}$ Gaussian component densities. However, model selection to find underlying …

Study how past radiation determines present matter in Penrose's cyclic cosmology.

The study explores vector flows on manifolds, focusing on polynomial constraints and equivalence relations.

The $\hat B_n^{(1)}$-hierarchy is constructed from the standard splitting of the affine Kac-Moody algebra $\hat B_n^{(1)}$, the Drinfeld-Sokolov $\hat B_n^{(1)}$-KdV hierarchy is obtained by pushing down the $\hat B_n^{(1)}$-flows along certain gauge orbit to a cross section of the gauge action. In this paper, we (1) u…

The paper extends graph metrics to include new points and distances.

Null distance encodes causal structure in spacetimes.

Generalizes results for Riemannian manifolds with boundary to those without.

Combines linear and spectral estimators for signal recovery in generalized linear models.

We give a bordered extension of involutive HF-hat and use it to give an algorithm to compute involutive HF-hat for general 3-manifolds. We also explain how the mapping class group action on HF-hat can be computed using bordered Floer homology. As applications, we prove that involutive HF-hat satisfies a surgery exact t…

Let ${\mathcal S}_m$ be the set of all $m\times m$ density matrices (Hermitian positively semi-definite matrices of unit trace). Consider a problem of estimation of an unknown density matrix $ρ\in {\mathcal S}_m$ based on outcomes of $n$ measurements of observables $X_1,\dots, X_n\in {\mathbb H}_m$ (${\mathbb H}_m$ bei…

Researchers found the $\hat{Z}$-invariant for $SU(N)/\mathbb{Z}_m$ is constant regardless of $m$.

New formulas connect knot invariants with theta functions.

Let G be a connected compact Lie group acting on a manifold M and let D be a transversally elliptic operator on M. The multiplicity of the index of D is a function on the set of irreducible representations of G. Let T be a maximal torus of G with Lie algebra Lie(T). We construct a finite number of piecewise polynomial …

We extend the definition of Bockstein basis $σ(G)$ to nilpotent groups $G$. A metrizable space $X$ is called a {\it Bockstein space} if $\dim_G(X) = \sup\{\dim_H(X) | H\in σ(G)\}$ for all Abelian groups $G$. Bockstein First Theorem says that all compact spaces are Bockstein spaces. Here are the main results of the pape…

New $\hat{Z}$ invariant for plumbed 3-manifolds with detailed computations and conjectures.

On a manifold of dimension at least six, let $(g,τ)$ be a pair consisting of a Kähler metric g which is locally Kähler irreducible, and a nonconstant smooth function $τ$. Off the zero set of $τ$, if the metric $\hat{g}=g/τ^2$ is a gradient Ricci soliton which has soliton function $1/τ$, we show that $\hat{g}$ is Kähler…

Let \hat{S} be the algebraic universal cover of a closed surface of genus >1, T(\hat{S}) its Teichmuller space, M(\hat{S}) the group of mapping classes stabilizing a fixed leaf l. The L^1 Ehrenpreis conjecture asserts that M(\hat{S}) on T(\hat{S}) with dense orbits with respect to the L^1 topology (the topology induced…

The paper explores a duality between conformally flat metrics and hyperbolic geometry.

We classify hypersurfaces of rank two of Euclidean space $\R^{n+1}$ that admit genuine isometric deformations in $\R^{n+2}$. That an isometric immersion $\hat f\colon\,M^n\to\R^{n+2}$ is a genuine isometric deformation of a hypersurface $f\colon\, M^n\to\R^{n+1}$ means that $\hat f$ is nowhere a composition $\hat f=\ha…

Natural coordinates for SL3-webs on surfaces are shown to be consistent under triangulation changes.

We consider differential operators acting on densities of arbitrary weights on manifold $M$ identifying pencils of such operators with operators on algebra of densities of all weights. This algebra can be identified with the special subalgebra of functions on extended manifold $\hat M$. On one hand there is a canonical…

A Z-structure on a group G, defined by M. Bestvina, is a pair (\hat{X}, Z) of spaces such that \hat{X} is a compact ER, Z is a Z-set in \hat{X}, G acts properly and cocompactly on X=\hat{X}\Z, and the collection of translates of any compact set in X forms a null sequence in \hat{X}. It is natural to ask whether a given…

Kernel extreme learning machine (KELM) is a novel feedforward neural network, which is widely used in classification problems. To some extent, it solves the existing problems of the invalid nodes and the large computational complexity in ELM. However, the traditional KELM classifier usually has a low test accuracy when…

Ricci-positive manifolds span the kernel of the $\hat A$-genus in rational Spin bordism.

Minimizing a convex, quadratic objective of the form $f_{\mathbf{A},\mathbf{b}}(x) := \frac{1}{2}x^\top \mathbf{A} x - \langle \mathbf{b}, x \rangle$ for $\mathbf{A} \succ 0$ is a fundamental problem in machine learning and optimization. In this work, we prove gradient-query complexity lower bounds for minimizing conv…

The paper provides guarantees for learning nonlinear representations from multiple non-identically distributed data sources.

New relation found between 3-manifold Witt invariants and $\hat{Z}$-invariants.

Study calibrates high-dimensional binary classifiers using angle between estimator and true weights.

In this note, we prove the following generalization of a theorem of Shi and Tam \cite{ShiTam02}: Let $(Ω, g)$ be an $n$-dimensional ($n \geq 3$) compact Riemannian manifold, spin when $n>7$, with non-negative scalar curvature and mean convex boundary. If every boundary component $Σ_i$ has positive scalar curvature and …

In this paper we show that for a generalized Berger metric $\hat{g}$ on $S^3$ close to the round metric, the conformally compact Einstein (CCE) manifold $(M, g)$ with $(S^3, [\hat{g}])$ as its conformal infinity is unique up to isometries. For the high-dimensional case, we show that if $\hat{g}$ is an $\text{SU}(k+1)$-…

Develops a TQFT framework to compute $\hat{Z}$ invariants of three-manifolds.