arXiv research

The large-N limit of Segal-Bargmann transform on spheres is studied.

Study of large mass limits of G2 and Calabi-Yau monopoles on specific manifolds.

Infinitesimal boosting converges to a deterministic process in large sample limit.

New insights into tSNE for large datasets.

Study of limits of Einstein-Bogomol'nyi metrics on P^1 in two regimes.

The study connects K-stability and large complex structure limits in mirror symmetry.

We analyze multi-layer neural networks in the asymptotic regime of simultaneously (A) large network sizes and (B) large numbers of stochastic gradient descent training iterations. We rigorously establish the limiting behavior of the multi-layer neural network output. The limit procedure is valid for any number of hidde…

Scalings in which the graph Laplacian approaches a differential operator in the large graph limit are used to develop understanding of a number of algorithms for semi-supervised learning; in particular the extension, to this graph setting, of the probit algorithm, level set and kriging methods, are studied. Both optimi…

Approximations to utility indifference prices are provided for a contingent claim in the large position size limit. Results are valid for general utility functions on the real line and semi-martingale models. It is shown that as the position size approaches infinity, the utility function's decay rate for large negative…

Study connects knot contact homology to Chern-Simons theory's large N limit.

New limits of minimal surface systems have surprising large interior parts.

We prove a law of large numbers for the loss from default and use it for approximating the distribution of the loss from default in large, potentially heterogenous portfolios. The density of the limiting measure is shown to solve a non-linear SPDE, and the moments of the limiting measure are shown to satisfy an infinit…

We rigorously prove a central limit theorem for neural network models with a single hidden layer. The central limit theorem is proven in the asymptotic regime of simultaneously (A) large numbers of hidden units and (B) large numbers of stochastic gradient descent training iterations. Our result describes the neural net…

The paper studies stochastic optimization on matrices and its limits as dimensions grow.

The paper examines how loss aversion impacts multi-armed bandit decisions over long periods.

This paper evaluates LLMs on large graph property estimation tasks.

The MBO scheme for data clustering is analyzed in the large data limit, proving convergence to optimal partition problems.

CKA with Gaussian RBF kernels converges linearly as bandwidth increases.

In this paper, we study the large time behavior of the heat kernel on complete Riemannian manifolds with nonnegative Ricci curvature, which was studied by P. Li with additional maximum volume growth assumption. Following Y. Ding's original strategy, by blowing down the metric, using Cheeger and Colding's theory about l…

Researchers compute invariants for knots and links in lens spaces using large N and k limits.

Researchers approximate partition functions on Riemannian spaces in the large N limit.

For large genus, spectral gaps on hyperbolic surfaces approach a limit.

This paper derives explicit formulas for both the small and large time limits of the implied volatility in the minimal market model. It is shown that interest rates do impact on the implied volatility in the long run even though they are negligible in the short time limit.

Motivated by the classical statements of Mirror Symmetry, we study certain Kahler metrics on the complexified Kahler cone of a Calabi-Yau threefold, conjecturally corresponding to approximations to the Weil-Petersson metric near large complex structure limit for the mirror. In particular, the naturally defined Riemanni…

Theory explains neural network scaling with dataset and model size.

Proves a special case of the Gaussian kinematic formula using large sphere limits.

Study on kernel methods in large-scale machine learning problems.

GNTK reveals convergence of GNNs on large graphs.

Lectures on deep learning properties in infinite and large-width networks.

Study eigenvector overlaps in large Gaussian matrices, simplifying for GOE.

As it is known in the finance risk and macroeconomics literature, risk-sharing in large portfolios may increase the probability of creation of default clusters and of systemic risk. We review recent developments on mathematical and computational tools for the quantification of such phenomena. Limiting analysis such as …

PASHA optimizes model tuning for large datasets with limited resources.

In this work we obtain the limit of the Hawking energy of a large class of foliations along general null hypersurfaces $Ω$ satisfying a weak notion of asymptotic flatness. The foliations are not required to be either geodesic or approaching large spheres at infinity. The limit is obtained in terms of a reference backgr…

We study the dynamics of the limit order book of liquid stocks after experiencing large intra-day price changes. In the data we find large variations in several microscopical measures, e.g., the volatility the bid-ask spread, the bid-ask imbalance, the number of queuing limit orders, the activity (number and volume) of…

Large corporate credit models may be adapted for small business risk assessment.

Using particle system methodologies we study the propagation of financial distress in a network of firms facing credit risk. We investigate the phenomenon of a credit crisis and quantify the losses that a bank may suffer in a large credit portfolio. Applying a large deviation principle we compute the limiting distribut…

Graph Neural Networks struggle on random graphs without node identifiers.

Large learning rates work surprisingly well in standard parameterization, contrary to theory.

Optimal stock trading strategy with market orders and limit orders in a risky market.

We propose a novel time discretization for the log-normal SABR model which is a popular stochastic volatility model that is widely used in financial practice. Our time discretization is a variant of the Euler-Maruyama scheme. We study its asymptotic properties in the limit of a large number of time steps under a certai…

We study the limiting behaviour of the empirical measure of a system of diffusions interacting through their ranks when the number of diffusions tends to infinity. We prove that the limiting dynamics is given by a McKean-Vlasov evolution equation. Moreover, we show that in a wide range of cases the evolution of the cum…

The paper shows Gaussian fluctuations in eigenvalue statistics of random hyperbolic surfaces.

Study on heavy tails in closing auction returns, explaining imbalance through limit order submission.

ResNets converge to a limit model with improved error rates.

Derives ideal train/test split for ridge regression in large data limit.

The paper calculates large genus limits for quadratic differential volumes and constants.

Study on order book dynamics with uniform catastrophes, explaining volatility and trends.

This paper explores the limits of large batch sizes in deep learning.