arXiv research

SGD and stochastic gradient descent converge at optimal rates for certain non-convex functions.

The geometric Lévy model (GLM) is a natural generalisation of the geometric Brownian motion model (GBM) used in the derivation of the Black-Scholes formula. The theory of such models simplifies considerably if one takes a pricing kernel approach. In one dimension, once the underlying Lévy process has been specified, th…

In high dimensions, the mean and geometric median are nearly identical.

Characterizes distribution-free rates in unbalanced classification problems.

Estimates mean curvature flow with geometric bounds.

For binary classification we establish learning rates up to the order of $n^{-1}$ for support vector machines (SVMs) with hinge loss and Gaussian RBF kernels. These rates are in terms of two assumptions on the considered distributions: Tsybakov's noise assumption to establish a small estimation error, and a new geometr…

Investigates geometric mean reversion process using Lie symmetry method.

Paper studies CLT rates for dependent data in Wasserstein-p distance.

A novel approach models rating transitions using Lie groups and Deep Learning.

In this paper, we extend the geometric descent method recently proposed by Bubeck, Lee and Singh to tackle nonsmooth and strongly convex composite problems. We prove that our proposed algorithm, dubbed geometric proximal gradient method (GeoPG), converges with a linear rate $(1-1/\sqrtκ)$ and thus achieves the optimal …

Study of a generalized geometric Brownian motion with varying entry and exit rates.

I present the technique which can analyse some interest rate models: Constantinides-Ingersoll, CIR-model, geometric CIR and Geometric Brownian Motion. All these models have the unified structure of Whittaker function. The main focus of this text is closed-form solutions of the zero-coupon bond value in these models. In…

Paper tackles dynamic pricing in a geometrically decaying environment, achieving better occupancy with lower rates.

Asymptotic Laplace transform for geometric Brownian motion applied to bond pricing.

Study on convergence rate of weighted Yamabe flow.

This paper investigates the supervised learning problem with observations drawn from certain general stationary stochastic processes. Here by \emph{general}, we mean that many stationary stochastic processes can be included. We show that when the stochastic processes satisfy a generalized Bernstein-type inequality, a u…

We give another proof for a result of Brick stating that the simple connectivity at infinity is a geometric property of finitely presented groups. This allows us to define the rate of vanishing of $\p1i$ for those groups which are simply connected at infinity. Further we show that this rate is linear for cocompact latt…

The rate of convergence of weighted kernel herding (WKH) and sequential Bayesian quadrature (SBQ), two kernel-based sampling algorithms for estimating integrals with respect to some target probability measure, is investigated. Under verifiable conditions on the chosen kernel and target measure, we establish a near-geom…

Study minimax rates for binary classifier estimation with margin conditions.

Establishes geometric convergence of iterative optimization algorithms.

Two algorithms converge to dictionary learning with geometric rate for non-uniform data.

Geometric analysis improves convergence of variational inference.

Stochastic (sub)gradient methods require step size schedule tuning to perform well in practice. Classical tuning strategies decay the step size polynomially and lead to optimal sublinear rates on (strongly) convex problems. An alternative schedule, popular in nonconvex optimization, is called \emph{geometric step decay…

Bayesian methods estimate regression functions on submanifolds using graph Laplacian eigenbasis.

We characterize geometrically the Lyapunov exponents of a cocycle (of arbitrary rank) with respect to a harmonic current defined on a hyperbolic Riemann surface lamination. Our characterizations are formulated in terms of the expansion rates of the cocycle along geodesic rays.

New rates for GLD and SGLD in infinite-dimensional spaces without dimensionality issues.

Foundation models fail to preserve continuous geometry, identified as the Geometric Alignment Tax.

Neuroscientific studies of drawing-like movements usually analyze neural representation of either geometric (eg. direction, shape) or temporal (eg. speed) features of trajectories rather than trajectory's representation as a whole. This work is about empirically supported mathematical ideas behind splitting and merging…

The state-of-the-art performance of deep learning algorithms has led to a considerable increase in the utilization of machine learning in security-sensitive and critical applications. However, it has recently been shown that a small and carefully crafted perturbation in the input space can completely fool a deep model.…

Geometric observables detect financial regime shifts with high accuracy.

Estimates mixing coefficients of geometrically ergodic Markov processes from a single sample path.

This study examines Gaussian processes on Riemannian manifolds and proves contraction rates.

Study Brownian loops on hyperbolic surfaces, linking to Selberg zeta function.

Optimal dividend strategy in dual risk model is well studied in the literatures. But to the best of our knowledge, all the previous works assumes deterministic interest rate. In this paper, we study the optimal dividends strategy in dual risk model, under a stochastic interest rate, assuming the discounting factor foll…

In this paper, we study the problem of sparse multiple kernel learning (MKL), where the goal is to efficiently learn a combination of a fixed small number of kernels from a large pool that could lead to a kernel classifier with a small prediction error. We develop an efficient algorithm based on the greedy coordinate d…

Modeling bone microarchitecture adaptation using geometric flows.

Researchers prove inner product recovery is impossible in latent space models.

Autoencoder-based geometric shaping is proposed that includes optimizing bit mappings. Up to 0.2 bits/QAM symbol gain in GMI is achieved for a variety of data rates and in the presence of transceiver impairments. The gains can be harvested with standard binary FEC at no cost w.r.t. conventional BICM.

The study examines a financial model with sticky prices and finds no arbitrage when interest rate is zero.

Deep networks improve by progressively refining approximations at each layer.

The choice of constellations largely affects the performance of communication systems. When designing constellations, both the locations and probability of occurrence of the points can be optimized. These approaches are referred to as geometric and probabilistic shaping, respectively. Usually, the geometry of the const…

Study on efficiency of Dutch auctions on blockchains considering various parameters.

Upper bound on CRN reaction rates derived using information geometry.

CR-AIS improves AIS efficiency by constant rate annealing.

Develops a method to estimate the shadow riskless rate from empirical data.

Paper compares stock price prediction models using Heston and Geometric Brownian Motion.

Geometric Mean Market Makers super-hedge impermanent loss without models.

The Gauss-Newton method is analyzed for neural networks using Riemannian optimization techniques.