arXiv research

Tree++ graph kernel captures similarities at multiple granularities.

The paper efficiently estimates parameters from truncated Gaussian and linear models.

New methods for scalable causal discovery from complex data.

The basin of infinity of a polynomial map $f : {\bf C} \arrow {\bf C}$ carries a natural foliation and a flat metric with singularities, making it into a metrized Riemann surface $X(f)$. As $f$ diverges in the moduli space of polynomials, the surface $X(f)$ collapses along its foliation to yield a metrized simplicial t…

In this paper we analyze, evaluate, and improve the performance of training Random Forest (RF) models on modern CPU architectures. An exact, state-of-the-art binary decision tree building algorithm is used as the basis of this study. Firstly, we investigate the trade-offs between using different tree building algorithm…

New method for constructing truncated vine copulas.

Low-dimensional probability models for local distribution functions in a Bayesian network include decision trees, decision graphs, and causal independence models. We describe a new probability model for discrete Bayesian networks, which we call an embedded Bayesian network classifier or EBNC. The model for a node $Y$ g…

Derives a general derivative identity for conditional mean in Gaussian noise.

Algorithm recovers spike direction from modulo-reduced measurements in high dimensions.

'Kuranishi spaces' were introduced in the work of Fukaya, Oh, Ohta and Ono in symplectic geometry (see e.g. arXiv:1106.4882), as the geometric structure on moduli spaces of $J$-holomorphic curves. An alternative to Kuranishi spaces is the 'polyfolds' of Hofer, Wysocki and Zehnder (see e.g. arXiv:1407.3185). Finding a s…

Gieseker-Nakajima moduli spaces $M_{k}(n)$ parametrize the charge $k$ noncommutative $U(n)$ instantons on ${\bf R}^{4}$ and framed rank $n$ torsion free sheaves $\mathcal{E}$ on ${\bf C\bf P}^{2}$ with ${\rm ch}_{2}({\mathcal{E}}) = k$. They also serve as local models of the moduli spaces of instantons on general four-…

TVineSynth generates synthetic data to balance privacy and utility.

The paper classifies algebraic curves in 4-balls and their boundaries.

The paper studies conditions for Cannon-Thurston maps in trees of hyperbolic spaces.

We describe a new instance-based learning algorithm called the Boundary Forest (BF) algorithm, that can be used for supervised and unsupervised learning. The algorithm builds a forest of trees whose nodes store previously seen examples. It can be shown data points one at a time and updates itself incrementally, hence i…

In this paper we survey methods and results of classification of $k$-forms (resp. $k$-vectors on ${\bf R}^n$), understood as description of the orbit space of the standard ${\bf GL}(n, {\bf R})$-action on $Λ^k {\bf R}^{n*}$ (resp. on $Λ^k {\bf R}^n$). We discuss the existence of related geometry defined by differential…

Robustifies tree learning algorithms for corrupted data.

In this paper, we prove properness of the action of the reparametrization group $PSL(n+1, {\bf C})$ on the space of $v$-stable $L_k^p$-maps on ${\bf P}^n$ as well as related results.

This is a survey of the author's paper arXiv:1409.6908 and in-progress book. 'Kuranishi spaces' were introduced in the work of Fukaya, Oh, Ohta and Ono in symplectic geometry (see e.g. arXiv:1503.07631), as the geometric structure on moduli spaces of $J$-holomorphic curves. We propose a new definition of Kuranishi spac…

We give algorithms for estimating the expectation of a given real-valued function $φ:X\to {\bf R}$ on a sample drawn randomly from some unknown distribution $D$ over domain $X$, namely ${\bf E}_{{\bf x}\sim D}[φ({\bf x})]$. Our algorithms work in two well-studied models of restricted access to data samples. The first o…

Study BF invariants using simple type concepts.

We consider the class of locally boost isotropic spacetimes in arbitrary dimension. For any spacetime with boost isotropy, the corresponding curvature tensor and all of its covariant derivatives must be simultaneously of alignment type ${\bf D}$ relative to some common null frame. Such spacetimes are known as type ${\b…

TRF uses ternary random features to improve ML performance without extra computation.

Classifies actions of SL(n,R) and SL(n,Z) on closed n-manifolds.

We propose a general framework to study the stability of the subspace spanned by $P$ consecutive eigenvectors of a generic symmetric matrix ${\bf H}_0$, when a small perturbation is added. This problem is relevant in various contexts, including quantum dissipation (${\bf H}_0$ is then the Hamiltonian) and risk control …

We consider nonconvex-concave minimax optimization problems of the form $\min_{\bf x}\max_{\bf y\in{\mathcal Y}} f({\bf x},{\bf y})$, where $f$ is strongly-concave in $\bf y$ but possibly nonconvex in $\bf x$ and ${\mathcal Y}$ is a convex and compact set. We focus on the stochastic setting, where we can only access an…

Abstract: Rational decomposition of homeomorphism spaces for manifolds.

We prove a Li-Yau gradient estimate for positive solutions to the heat equation, with Neumann boundary conditions, on a compact Riemannian submanifold with boundary ${\bf M}^n\subseteq {\bf N}^n$, satisfying the integral Ricci curvature assumption: \begin{equation} D^2 \sup_{x\in {\bf N}} \left( \oint_{B(x,D)} |Ric^-|^…

In conventional Differential Geometry one studies manifolds, locally modelled on ${\mathbb R}^n$, manifolds with boundary, locally modelled on $[0,\infty)\times{\mathbb R}^{n-1}$, and manifolds with corners, locally modelled on $[0,\infty)^k\times{\mathbb R}^{n-k}$. They form categories ${\bf Man}\subset{\bf Man^b}\sub…

Introduces a new relation between BF theory and gravity.

An irreducible open 3-manifold $W$ is {\bf R}$^2$-irreducible if every proper plane in $W$ splits off a halfspace. In this paper it is shown that if such a $W$ is the universal cover of a connected, {\bf P}$^2$-irreducible open 3-manifold $M$ with finitely generated fundamental group, then either $W$ is homeomorphic to…

In this paper, we study locally projectively flat Finsler metrics with constant flag curvature ${\bf K}$. We prove those are totally determined by their behaviors at the origin by solving some nonlinear PDEs. The classifications when ${\bf K}=0$, ${\bf K}=-1$ and ${\bf K} =1$ are given respectively in an algebraic way.…

Balloons are two-dimensional spheres. Hoops are one dimensional loops. Knotted Balloons and Hoops (KBH) in 4-space behave much like the first and second homotopy groups of a topological space - hoops can be composed as in π_1, balloons as in π_2, and hoops "act" on balloons as π_1 acts on π_2. We observe that ordinary …

We extend our family rigidity and vanishing theorems in [{\bf LiuMaZ}] to the Spin^c case. In particular, we prove a K-theory version of the main results of [{\bf H}], [{\bf Liu1}, Theorem B] for a family of almost complex manifolds.

The paper studies weak singular Hermite-Einstein structures on homogeneous vector bundles.

Let $({\bf X},ω_{\bf X}^*)$ be a separated, $-2$-shifted symplectic derived $\mathbb C$-scheme, in the sense of Pantev, Toen, Vezzosi and Vaquie arXiv:1111.3209, of complex virtual dimension ${\rm vdim}_{\mathbb C}{\bf X}=n\in\mathbb Z$, and $X_{\rm an}$ the underlying complex analytic topological space. We prove that …

The paper introduces a new method to characterize cosmological models using observer-based invariants.

For any given $C^\infty$ immersion ${\bf r}: S^1\to \mathbb{R}^2$ such that the set $\mathcal{NS}_{\bf r}=\mathbb{R}^2-\cup_{s\in S^1}\left({\bf r}(s)+d{\bf r}_{s}(T_s(S^1))\right)$ is not empty, a simple geometric model of crystal growth is constructed. It is shown that our geometric model of crystal growth never form…

This paper shows that the Grassmann Manifolds $G_{\bf F}(n,N)$ can all be imbedded in an Euclidean space $M_{\bf F}(N)$ naturally and the imbedding can be realized by the eigenfunctions of Laplacian $\triangle$ on $G_{\bf F}(n,N)$. They are all minimal submanifolds in some spheres of $M_{\bf F}(N)$ respectively. Using …

BF-VI improves posterior approximation in complex models.

Decision stumps accurately screen variables in nonparametric models.

Groups with semistable peripheral subgroups are semistable.

Let ${\bf X}$ and ${\bf X}$ be two $n$-dimensional elliptical random vectors, we establish an identity for $E[f({\bf Y})]-E[f({\bf X})]$, where $f: \Bbb{R}^n \rightarrow \Bbb{R}$ fulfilling some regularity conditions. Using this identity we provide a unified derivation of sufficient and necessary conditions for classif…

Efficiently infers interventional distributions from observational data.

We introduce and study deformation $T_{{\bf b},φ}$ of Minkowski norms in $\mathbb{R}^n$, determined by a set ${\bf b}=(β_1,\ldots,β_p)$ of linearly independent 1-forms and a smooth positive function $φ$ of $p$ variables. In particular, the $T_{{\bf b},φ}$-image of a Euclidean norm $α$ is a Minkowski norm, whose indicat…

Let ${{\bf H}_{\mathbb H}}^n$ denote the $n$-dimensional quaternionic hyperbolic space. The linear group ${\rm{Sp}}(n,1)$ acts by the isometries of ${{\bf H}_{\mathbb H}}^n$. A subgroup $G$ of ${\rm {Sp}}(n,1)$ is called \emph{Zariski dense} if it does not fix a point on ${{\bf H}_{\mathbb H}}^n \cup \partial {{\bf H}_…

Expanding FCCO to non-smooth weakly-convex problems, improving deep learning performance.

The paper proves existence of solutions to Kazdan-Warner equations on finite graphs.