arXiv research

Characterizes intrinsic Lorentzian spaces using midpoint properties.

The paper analyzes the randomized midpoint method for Langevin diffusions, revealing biases and asymptotic properties.

Solitons are special polygon midpoints under affine transformations.

We say that a pair of points x and y is secure if there exist a finite set of blocking points such that any geodesic between x and y passes through one of the blocking points. The main point of this paper is to exhibit new examples of blocking phenomena both in the manifold and the billiard table setting. As an approac…

Poisson Midpoint Method improves Langevin Dynamics for diffusion models.

The paper develops methods for sampling from log-concave distributions with constraints.

One of the most fundamental concepts in statistics is the concept of sample mean. Properties of the sample mean that are well-defined in Euclidean spaces become unwieldy or even unclear in graph spaces. Open problems related to the sample mean of graphs include: non-existence, non-uniqueness, statistical inconsistency,…

New sampling method improves efficiency for diffusion models.

Let $M$ be either the 2-sphere $\SS^2 \subset\RR^3$ or the hyperbolic plane $\HH^2 \subset \RR^3$. If $Δ(abc)$ is a geodesic triangle on $M$ with corners at $a,b,c\in M$, we denote by $α, β, γ\in M$ the midpoints of their sides. If $Ω$ denotes the oriented area of this triangle on $M$, it satisfies the relations: $$ \s…

Develops a framework for valuing Asian options with market impact.

Optimizes Lipschitz estimates for partitions of unity and characterizes spaces with Assouad-Nagata dimension.

New analysis shows ROI's predictive power for stock returns weakens significantly.

We show that for any point $p$ in a closed Riemannian manifold $M$, there exists at least one point $q\in M$ such that $p$ is critical for the distance function from $q$. We also show that such a point $q$ cannot always be reached with geodesic loops based at $q$ with midpoint $p$.

New set class preserves Fourier series terms for planar ovals, leading to isoperimetric inequalities.

We discuss multiscale representations of discrete manifold-valued data. As it turns out that we cannot expect general manifold-analogues of biorthogonal wavelets to possess perfect reconstruction, we focus our attention on those constructions which are based on upscaling operators which are either interpolating or midp…

Study on symmetry defects of complete intersections in complex space.

A new framework for hyperbolic neural networks using the Klein model is introduced.

We study the exponential map of connected symmetric spaces and characterize, in terms of midpoints and of infinitesimal conditions, when it is a diffeomorphism, generalizing the Dixmier-Saito theorem for solvable Lie groups. We then give a geometric characterization of the (strongly) exponential solvable symmetric spac…

A new method for sampling from posterior distributions in Bayesian inverse problems.

Study compares 5 ODE solvers on 3 case studies, finding varying accuracy.

In this article, we analyze the microlocal properties of the linearized forward scattering operator $F$ and the normal operator $F^{*}F$ (where $F^{*}$ is the $L^{2}$ adjoint of $F$) which arises in Synthetic Aperture Radar imaging for the common midpoint acquisition geometry. When $F^{*}$ is applied to the scattered d…

For a polygon in Euclidean space we consider a transformation T which is obtained by applying the midpoints polygon construction twice and using an index shift. For a closed polygon this is a curve shortening process. A polygon is called (affine) soliton of the transformation T if its image under T is an affine image o…

Motivated by the problem of optimal portfolio liquidation under transient price impact, we study the minimization of energy functionals with completely monotone displacement kernel under an integral constraint. The corresponding minimizers can be characterized by Fredholm integral equations of the second type with cons…

New method finds arbitrage opportunities in fluctuating asset bands.

We present a large scale hyperbolic recommender system. We discuss why hyperbolic geometry is a more suitable underlying geometry for many recommendation systems and cover the fundamental milestones and insights that we have gained from its development. In doing so, we demonstrate the viability of hyperbolic geometry f…

Study of J-Hermitian matrices and geometric mean definition.

Develops methods to simulate rare transitions in molecular systems.

The price impact for a single trade is estimated by the immediate response on an event time scale, i.e., the immediate change of midpoint prices before and after a trade. We work out the price impacts across a correlated financial market. We quantify the asymmetries of the distributions and of the market structures of …

Study of $q$-rationals and their geometric properties, including deformed Farey triangulation and Springborn operations.

New algorithms tackle latent bandit problems with lower regret.

New tiles allow efficient knot mosaics for small knots.

This research analyzes and accelerates score-based diffusion models using discretization and Hessian information.

The paper explores discrete isothermic nets using checkerboard patterns in quadrilateral nets.

There is a perceived trade-off between machine learning code that is easy to write, and machine learning code that is scalable or fast to execute. In machine learning, imperative style libraries like Autograd and PyTorch are easy to write, but suffer from high interpretive overhead and are not easily deployable in prod…

The paper proves inequalities for hyperbolic sets and curves.

The paper defines and studies the geometric mean for tensors and its associated Riemannian geometry.

New method proves dimension-free convergence for ULD in KL divergence.

New method improves sampling efficiency for complex distributions.

This paper provides a practical method to extract caplet volatilities from quoted data.

A simpler edge-based discretization method without dual volumes.

Mixup helps learn both features in multi-view classification problems.

Random forests have become an important tool for improving accuracy in regression and classification problems since their inception by Leo Breiman in 2001. In this paper, we revisit a historically important random forest model originally proposed by Breiman in 2004 and later studied by Gérard Biau in 2012, where a feat…

Study on hyperbolic triangles and once-punctured torus groups, focusing on group relations and deformations.

We study the cause of large fluctuations in prices in the London Stock Exchange. This is done at the microscopic level of individual events, where an event is the placement or cancellation of an order to buy or sell. We show that price fluctuations caused by individual market orders are essentially independent of the v…

Study Zoll manifolds with boundary, showing unique geodesic properties.

Sampling from log-concave distributions is a well researched problem that has many applications in statistics and machine learning. We study the distributions of the form $p^{*}\propto\exp(-f(x))$, where $f:\mathbb{R}^{d}\rightarrow\mathbb{R}$ has an $L$-Lipschitz gradient and is $m$-strongly convex. In our paper, we p…

A protocol reduces transaction costs for portfolio managers.

Any two equivalent discrete curves must have the same invariants at the corresponding points under an affine transformation. In this paper, we construct the moving frame and invariants for the discrete centroaffine curves, which could be used to discriminate the same discrete curves from different graphics, and estimat…