arXiv research

We consider the quasiconformal dilatation of projective transformations of the real projective plane. For non-affine transformations, the contour lines of dilatation form a hyperbolic pencil of circles, and these are the only circles that are mapped to circles. We apply this result to analyze the dilatation of the circ…

Aggregates predictions from multiple regression models using random projections and kernel methods.

In this article, we introduce symbol calculus on a projective scheme. Using holomorphic Poisson structures, we construct deformations of ring structures for structure sheaves on projective spaces.

This paper is motivated by the real symplectic isotopy problem : does there exists a nonsingular real pseudoholomorphic curve not isotopic in the projective plane to any real algebraic curve of the same degree? Here, we focus our study on symmetric real curves on the projective plane. We give a classification of real s…

The method of random projections has become a standard tool for machine learning, data mining, and search with massive data at Web scale. The effective use of random projections requires efficient coding schemes for quantizing (real-valued) projected data into integers. In this paper, we focus on a simple 2-bit coding …

The paper describes hyperkähler geometry of cotangent bundles using rank-1 projections.

We describe the natural geometry of Hilbert schemes of curves in ${\mathbb P}^3$ and, in some cases, in ${\mathbb P}^n$ , $n\geq 4$.

Low-rank structure have been profoundly studied in data mining and machine learning. In this paper, we show a dense matrix $X$'s low-rank approximation can be rapidly built from its left and right random projections $Y_1=XA_1$ and $Y_2=X^TA_2$, or bilateral random projection (BRP). We then show power scheme can further…

We study codimension one holomorphic distributions on the projective three-space, analyzing the properties of their singular schemes and tangent sheaves. In particular, we provide a classification of codimension one distributions of degree at most 2 with locally free tangent sheaves, and show that codimension one distr…

Improved statistical computation through efficient matrix sampling.

We investigate Fano schemes of conditionally generic intersections, i.e. of hypersurfaces in projective space chosen generically up to additional conditions. Via a correspondence between generic properties of algebraic varieties and events in probability spaces that occur with probability one, we use the obtained resul…

Researchers find Busemann functions in Wasserstein space, enabling efficient projections and distances.

Consider convex optimization problems subject to a large number of constraints. We focus on stochastic problems in which the objective takes the form of expected values and the feasible set is the intersection of a large number of convex sets. We propose a class of algorithms that perform both stochastic gradient desce…

Large sectors of the recent optimization literature focused in the last decade on the development of optimal stochastic first order schemes for constrained convex models under progressively relaxed assumptions. Stochastic proximal point is an iterative scheme born from the adaptation of proximal point algorithm to nois…

We show that the Debarre-de Jong conjecture that the Fano scheme of lines on a smooth hypersurface of degree at most n in n-dimensional projective space must have its expected dimension, and the Beheshti-Starr conjecture that bounds the dimension of the Fano scheme of lines for hypersurfaces of degree at least n in n-d…

Via the transverse Hilbert scheme construction, we associate a holomorphic completely integrable system to a surface $S$ endowed with a holomorphic symplectic form $ω$ and a projection onto $\mathbb{C}$. We provide a full characterization of the completely integrable systems that arise in this way.

We propose a natural discretisation scheme for classical projective minimal surfaces. We follow the classical geometric characterisation and classification of projective minimal surfaces and introduce at each step canonical discrete models of the associated geometric notions and objects. Thus, we introduce discrete ana…

We present algorithms for topic modeling based on the geometry of cross-document word-frequency patterns. This perspective gains significance under the so called separability condition. This is a condition on existence of novel-words that are unique to each topic. We present a suite of highly efficient algorithms based…

We define and calculate signature and nullity invariants for complex schemes for curves in the real projective plane. We use an analog of the Murasugi-Tristram inequality to prohibit certain schemes from being realized by real algebraic curves. We give new formulas for Casson-Gordon invariants of graph manifolds, and s…

New schemes for SDEs on manifolds keep solutions close to the manifold.

An unobstructedness theorem is proved for deformations of compact holomorphic Poisson manifolds and applied to a class of examples. These include certain rational surfaces and Hilbert schemes of points on Poisson surfaces. We study in particular the Hilbert schemes of the projective plane and show that a generic deform…

Defines and classifies toric co-Higgs bundles on projective toric varieties.

Unified theory and debiasing framework for random oblique projections in high dimensions.

The study classifies and analyzes two-dimensional holomorphic distributions on a four-dimensional projective space.

We construct an explicit scheme to associate to any potential symbol an operator acting between sections of natural bundles (associated to irreducible representations) for a so-called AHS-structure. Outside of a finite set of critical (or resonant) weights, this procedure gives rise to a quantization, which is intrinsi…

Network embedding, which learns low-dimensional vector representation for nodes in the network, has attracted considerable research attention recently. However, the existing methods are incapable of handling billion-scale networks, because they are computationally expensive and, at the same time, difficult to be accele…

We discuss some differential geometry pertaining to continuum mechanics and the route recently taken by D.N. Arnold, R.S. Falk, and R. Winther in deriving new improved finite element schemes in linear elasticity from constructions in projective geometry.

In this note, we present a new averaging technique for the projected stochastic subgradient method. By using a weighted average with a weight of t+1 for each iterate w_t at iteration t, we obtain the convergence rate of O(1/t) with both an easy proof and an easy implementation. The new scheme is compared empirically to…

We consider projective varieties with degenerate Gauss image whose focal hypersurfaces are non-reduced schemes. Examples of this situation are provided by the secant varieties of Severi and Scorza varieties. The Severi varieties are moreover characterized by a uniqueness property.

In this paper, we study rational sections of the relative Picard scheme of a linear system on a smooth projective variety. We prove that if the linear system is basepoint-free and the locus of non-integral divisors has codimension at least two, then all rational sections of the relative Picard scheme come from restrict…

A simple and computationally efficient scheme for tree-structured vector quantization is presented. Unlike previous methods, its quantization error depends only on the intrinsic dimension of the data distribution, rather than the apparent dimension of the space in which the data happen to lie.

This paper develops quantization algorithms for random Fourier features, simplifying the process and improving performance.

This is a sequel of our paper [arXiv:1809.08425] on the Quot-scheme limit and variational properties of Donaldson's functional, which established its coercivity for slope stable holomorphic vector bundles over smooth projective varieties. Assuming that the coercivity is uniform in a certain sense, we provide a new proo…

Study evaluates UK CDC schemes, finding intergenerational cross-subsidies in flat-accrual schemes and dynamic-accrual schemes can reduce but not eliminate them.

We give two characterizations of varieties whose universal cover is a bounded symmetric domain without ball factors in terms of the existence of a holomorphic endomorphism \s of the tensor product T\otimes T' of the tangent bundle T with the cotangent bundle T'. To such a curvature type tensor \s one associates the fir…

Data privacy is an important concern in machine learning, and is fundamentally at odds with the task of training useful learning models, which typically require the acquisition of large amounts of private user data. One possible way of fulfilling the machine learning task while preserving user privacy is to train the m…

Consider $E$ a holomorphic vector bundle over a projective manifold $X$ polarized by an ample line bundle $L$. Fix $k$ large enough, the holomorphic sections $H^0(E\otimes L^k)$ provide embeddings of $X$ in a Grassmanian space. We define the \textit{balancing flow for bundles} as a flow on the space of projectively equ…

Study analyzes FIT schemes under market and regulatory uncertainty.

Distributed sensors compress and send features to a fusion center for linear regression.

Feature hashing and other random projection schemes are commonly used to reduce the dimensionality of feature vectors. The goal is to efficiently project a high-dimensional feature vector living in $\mathbb{R}^n$ into a much lower-dimensional space $\mathbb{R}^m$, while approximately preserving Euclidean norm. These sc…

New method improves MMD estimation without convexity assumptions.

In this paper we study the problem of recovering a structured but unknown parameter ${\bfθ}^*$ from $n$ nonlinear observations of the form $y_i=f(\langle {\bf{x}}_i,{\bfθ}^*\rangle)$ for $i=1,2,\ldots,n$. We develop a framework for characterizing time-data tradeoffs for a variety of parameter estimation algorithms when…

Efficiently simulates the Heston model with large time steps using a novel method.

An asymptotic formula for the Tian-Paul CM-line of a flat family blown-up at a flat closed sub-scheme is given. As an application we prove that the blow-up of a polarized manifold along a (relatively) Chow-unstable submanifold admits no (extremal) constant scalar curvature Kahler metrics in classes making the exception…

In this work, we revisit fast dimension reduction approaches, as with random projections and random sampling. Our goal is to summarize the data to decrease computational costs and memory footprint of subsequent analysis. Such dimension reduction can be very efficient when the signals of interest have a strong structure…

We realise Stroppel's extended arc algebra in the Fukaya-Seidel category of a natural Lefschetz fibration on the generic fiber of the adjoint quotient map on a type $A$ nilpotent slice with two Jordan blocks, and hence obtain a symplectic interpretation of certain parabolic two-block versions of Bernstein-Gelfan'd-Gelf…

In this paper we present qualitative and quantitative comparison of various analytical and numerical approximation methods for calculating a position of the early exercise boundary of the American put option paying zero dividends. First we analyze their asymptotic behavior close to expiration. In the second part of the…

Let $X \rightarrow S$ be a smooth projective surjective morphism, where $X$ and $S$ are integral schemes over complex numbers. Let L_0, L_1, .... L_{n-1}, L_{n} be line bundles over $X$. There is a natural isomorphism of the Deligne pairing $<L_{0},...,L_{n}>$ with the determinant line bundle ${\rm Det}(\otimes_{i=0}^{…