arXiv research

A theorem connects integral of second-order derivatives to function rise.

The paper studies kernel smoothing and mean shift for directional data, deriving convergence rates and mode estimation.

In the bordered Floer theory, gluing thickened torus of positive meridional Dehn twist to the boundary of a knot complement result in the knot complement of increased framing. For a fixed knot K, we construct a direct system of positively framed knot complements and study the direct limit. We also study the morphism sp…

This paper deals with the enumeration of the higher order non-trivial compositions of the differential operations and the directional derivative in the space $R^n$ ($n \geq 3$). We present the recurrences for a counting the higher order non-trivial compositions.

Gradient-free method solves infinite-dimensional optimization problems.

New stochastic gradient descent with random search directions improves efficiency and convergence.

New CRB derived for curved models using extrinsic geometry.

New IBP formulae for rough stochastic Volterra processes.

Estimation of density derivatives is a versatile tool in statistical data analysis. A naive approach is to first estimate the density and then compute its derivative. However, such a two-step approach does not work well because a good density estimator does not necessarily mean a good density-derivative estimator. In t…

We provide a correction to the expression for scoring Gaussian directed acyclic graphical models derived in Geiger and Heckerman [Ann. Statist. 30 (2002) 1414-1440] and discuss how to evaluate the score efficiently.

Forward gradients improve neural network training without backpropagation issues.

A direct, bundle-theoretic method for defining and extending local isometries out of curvature data is developed. As a by-product, conceptual direct proofs of a classical result of Singer and a recent result of the authors are derived.

In this paper, we show that the derivative of the genus-1 Virasoro conjecture for Gromov-Witten invariants along the direction of quantum volume element holds for all smooth projective varieties. This result provides new evidence for the Virasoro conjecture.

In this paper we develop a Morse theory for the uniform energy. We use the one-sided directional derivative of the distance function to study the minimizing properties of variations through closed geodesics. This derivative is then used to define a one-sided directional derivative for the uniform energy which allows us…

New method computes affine normal directions efficiently for sparse polynomials.

This paper considers the problem of embedding directed graphs in Euclidean space while retaining directional information. We model a directed graph as a finite set of observations from a diffusion on a manifold endowed with a vector field. This is the first generative model of its kind for directed graphs. We introduce…

We study a class of Riemannian manifolds with respect to the covariant derivative of their curvature tensors. We introduce geometrically the class of directed Riemannian manifolds of pointwise constant relative sectional curvature and give a tensor characterization for such manifolds. We prove that all rotational hyper…

The study extracts market direction from transaction data.

New PG losses improve decision optimization in misspecified models.

Efficiently approximates higher-order derivatives for generative models.

Paper proves linear convergence of SCMS algorithm for directional data.

Variational Optimization forms a differentiable upper bound on an objective. We show that approaches such as Natural Evolution Strategies and Gaussian Perturbation, are special cases of Variational Optimization in which the expectations are approximated by Gaussian sampling. These approaches are of particular interest …

Spectral clustering for directed graphs using likelihood estimation.

GANs excel at learning high dimensional distributions, but they can update generator parameters in directions that do not correspond to the steepest descent direction of the objective. Prominent examples of problematic update directions include those used in both Goodfellow's original GAN and the WGAN-GP. To formally d…

We present a unified derivation of covariant time derivatives, which transform as tensors under a time-dependent coordinate change. Such derivatives are essential for formulating physical laws in a frame-independent manner. Three specific derivatives are described: convective, corotational, and directional. The covaria…

Uniform bounds are developed for derivatives of solutions of the $2$-dimensional constant negative curvature equation and the Weil-Petersson metric for the Teichmüller and moduli spaces. The dependence of the bounds on the geometry of the underlying Riemann surface is studied. The comparisons between the $C^0$, $C^{2,α…

DAGSurv uses deep neural networks to analyze survival data based on causal graphs.

In optimization, the negative gradient of a function denotes the direction of steepest descent. Furthermore, traveling in any direction orthogonal to the gradient maintains the value of the function. In this work, we show that these orthogonal directions that are ignored by gradient descent can be critical in equilibri…

Constructs Lie algebras from labeled directed graphs and identifies properties of these algebras.

A new metric based on hitting probabilities for directed graphs and Markov chains.

New method clusters directed graphs using Koopman operators.

Unified routing and arbitrage with concave continuation.

The location-based social network, Foursquare, reflects the human activities of a city. The mobility dynamics inferred from Foursquare helps us understanding urban social events like crime In this paper, we propose a directed graph from the aggregated movement between regions using Foursquare data. We derive region ris…

We relate Ambrosio-Kirchheim metric currents to Alberti representations and Weaver derivations. In particular, given a metric current $T$, we show that if the module $\mathscr{X}(\|T\|)$ of Weaver derivations is finitely generated, then $T$ can be represented in terms of derivations; this extends previous results of Wi…

Study Loday algebroids, prove splitting theorem, and linearize problems.

Paper proves equivalence of derivatives for maps between Carnot groups.

We explore the geometrical interpretation of the PCA based clustering algorithm Principal Direction Divisive Partitioning (PDDP). We give several examples where this algorithm breaks down, and suggest a new method, gap partitioning, which takes into account natural gaps in the data between clusters. Geometric features …

Gradient-only line searches (GOLS) adaptively determine step sizes along search directions for discontinuous loss functions resulting from dynamic mini-batch sub-sampling in neural network training. Step sizes in GOLS are determined by localizing Stochastic Non-Negative Associated Gradient Projection Points (SNN-GPPs) …

New RL approach infers optimal policies via variational inference.

Paper defines a new dimension to measure self-directed learning complexity.

In this paper, we consider derivative free optimization problems, where the objective function is smooth but is computed with some amount of noise, the function evaluations are expensive and no derivative information is available. We are motivated by policy optimization problems in reinforcement learning that have rece…

We prove the differentiability of Lipschitz maps X-->V, where X is a complete metric measure space satisfying a doubling condition and a Poincaré inequality, and V is a Banach space with the Radon Nikodym Property (RNP). The proof depends on a new characterization of the differentiable structure on such metric measure …

Cross-sectional "Information Coefficient" (IC) is a widely and deeply accepted measure in portfolio management. The paper gives an insight into IC in view of high-dimensional directional statistics: IC is a linear operator on the components of a centralizing-unitizing standardized random vector of next-period cross-sec…

In a fibre bundle, natural derivatives of a section are defined as tangent vector fields on the image of a section of the fibre bundle. A local extension to vector fields in the tangent bundle leads to a direct proof of the formula expressing the curvature of a connection in terms of covariant derivatives. The result i…

The sizes of Markov equivalence classes of directed acyclic graphs play important roles in measuring the uncertainty and complexity in causal learning. A Markov equivalence class can be represented by an essential graph and its undirected subgraphs determine the size of the class. In this paper, we develop a method to …

TERA method speeds up derivative Gaussian processes in high dimensions.

ConMeZO speeds up zeroth-order optimization for large language models.

Alternative method for derivatives pricing using quantum computers.