arXiv research

We study the use of "sign $α$-stable random projections" (where $0<α\leq 2$) for building basic data processing tools in the context of large-scale machine learning applications (e.g., classification, regression, clustering, and near-neighbor search). After the processing by sign stable random projections, the inner pr…

The fields of compressed sensing (CS) and matrix completion have shown that high-dimensional signals with sparse or low-rank structure can be effectively projected into a low-dimensional space (for efficient acquisition or processing) when the projection operator achieves a stable embedding of the data by satisfying th…

FROCC uses random projections for fast one-class classification.

NodeSig efficiently computes binary node embeddings for scalable graph analysis.

Sparse random projection (RP) is a popular tool for dimensionality reduction that shows promising performance with low computational complexity. However, in the existing sparse RP matrices, the positions of non-zero entries are usually randomly selected. Although they adopt uniform sampling with replacement, due to lar…

The study restricts stable minimal immersions in product spaces to specific configurations.

The study classifies semiaffine stable planes into affine, projective, or punctured projective planes.

Stable planes are locally isomorphic to classical projective planes.

The study classifies stable submanifolds in product spaces of projective spaces.

Projective varieties remain stable under close polarizations, extending to Kähler cones.

In this note we show that Hamiltonian stable minimal Lagrangian submanifolds of projective space need not have parallel second fundamental form.

We investigate random complex dynamics of rational or polynomial maps on the Riemann sphere. We show that regarding random complex dynamics of polynomials, generically, the chaos of the averaged system disappears at any point in the Riemann sphere due to the automatic coopeartion of many kinds of maps in the system, ev…

New methods tackle statistical inverse problems with random data.

Random Fourier features improve tabular deep learning convergence.

Assume that a projective variety together with a polarization is uniformly K-stable. If the polarization is canonical or anti-canonical, then the projective variety is uniformly K-stable with respects to any polarization sufficiently close to the original polarization.

This paper surveys various methods for dimensionality reduction and nearest neighbor search.

Estimation of low-rank matrices is of significant interest in a range of contemporary applications. In this paper, we introduce a rank-one projection model for low-rank matrix recovery and propose a constrained nuclear norm minimization method for stable recovery of low-rank matrices in the noisy case. The procedure is…

The Corlette-Donaldson-Hitchin-Simpson's correspondence states that, on a compact Kähler manifold $(X, ω)$, there is a one-to-one correspondence between the moduli space of semisimple flat complex vector bundles and the moduli space of poly-stable Higgs bundles with vanishing Chern numbers. In this paper, we extend thi…

We study primitive stable representations of free groups into higher rank semisimple Lie groups and their properties. Let $Σ$ be a compact, connected, orientable surface (possibly with boundary) of negative Euler characteristic. We first verify the $σ_{mod}$-regularity for convex projective structures and positive repr…

We model the logarithm of the price (log-price) of a financial asset as a random variable obtained by projecting an operator stable random vector with a scaling index matrix $\underline{\underline{E}}$ onto a non-random vector. The scaling index $\underline{\underline{E}}$ models prices of the individual financial asse…

We show that the moduli space of genus zero stable maps is a real projective variety if the target space is a smooth convex real projective variety. We show that evaluation maps, forgetful maps are real morphisms. We analyze the real part of the moduli space.

For a complex projective space the inertia group, the homotopy inertia group and the concordance inertia group are isomorphic. In complex dimension 4n+1, these groups are related to computations in stable cohomotopy. Using stable homotopy theory, we make explicit computations to show that the inertia group is non-trivi…

The paper studies how norms of random vectors are preserved by random projections.

Sharp inequality for complex projective space's systoles under scalar curvature bounds.

Paper studies binary random projections with controllable sparsity patterns for computational and accuracy advantages.

Free Random Projection enhances reinforcement learning by naturally incorporating hierarchical structure.

Extremal metrics exist if uniformly $K$-stable over models.

We investigate Chow stability of projective bundles P(E) where E is a strictly Gieseker stable bundle over a base manifold that has constant scalar curvature. We show that, for suitable polarisations L, the pair (P(E),L) is Chow stable and give examples for which it is not asymptotically Chow stable.

Characterizes W-congruences to study their stable umbilical points.

Random projections have been applied in many machine learning algorithms. However, whether margin is preserved after random projection is non-trivial and not well studied. In this paper we analyse margin distortion after random projection, and give the conditions of margin preservation for binary classification problem…

Random matrix theory is used to assess the significance of weak correlations and is well established for Gaussian statistics. However, many complex systems, with stock markets as a prominent example, exhibit statistics with power-law tails, that can be modelled with Levy stable distributions. We review comprehensively …

New bound on partition function proves Kähler-Einstein stability.

New projection complex shows some surface homeomorphisms have positive commutator length.

Tensorized random projections reduce high-dimensional tensor size efficiently.

Following similar results in arXiv:1301.5934 for flat tori and round spheres, in this paper is presented a proof of the fact that, for "arbitrary" initial conditions $f_0$, the solution $f_t$ at time $t$ of the heat equation on real or complex projective spaces eventually becomes (and remains) a minimal Morse function.…

A theorem of Lawson and Simons states that the only stable minimal submanifolds in complex projective spaces are complex submanifolds. We generalize their result to the cases of quaternionic and octonionic projective spaces. Our approach gives a unified viewpoint towards conformal and projective geometries.

We show that a projective manifold is stable if and only if the Mabuchi energy is proper on the space of algebraic metrics. We show that stability implies finite automorphism group.

Random projections help in representing sparse graphs efficiently.

Given a one parameter flat family of polarized algebraic varieties, we show that any K-stable limit is unique. In particular, moduli spaces of K-stable polarized varieties are automatically Hausdorff when they exist. We also give a characterization of K-stable limits in terms of the CM line bundle, and some application…

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…

Study shows double descent curve in high-dimensional linear regression with random projections.

New invariants derived from random matrices for words in free groups.

The note proves a metric equivalence for stable bundles on surfaces.

Moduli space of genus zero stable maps to the projective three-space naturally carries a real structure such that the fixed locus is a moduli space for real rational spatial curves with real marked points. The latter is a normal projective real variety. The singular locus being in codimension at least two, a first Stie…

New projection operators for multipatch spaces with stable properties.

The paper proves a Donaldson-Uhlenbeck-Yau theorem for normal varieties and semistable bundles.

A new framework for dimension reduction using ensemble of random projections.

Study on volumes of random inscribed polytopes in projective geometries.