arXiv research

Study shows short rate can explode to infinity in HJM model, impacting Eurodollar futures.

New framework assesses regularization norms in ill-posed problems, revealing L2 instability and proposing adaptive fractional RKHS solutions.

A new data-adaptive prior stabilizes kernel learning in operators.

The paper studies large deviation principles for stochastic volatility models with reflection, focusing on binary barrier options and call prices.

DSM on manifolds removes singularities and computes small-noise expansions.

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

Graph-based semi-supervised learning is the problem of propagating labels from a small number of labelled data points to a larger set of unlabelled data. This paper is concerned with the consistency of optimization-based techniques for such problems, in the limit where the labels have small noise and the underlying unl…

This paper studies least-square regression penalized with partly smooth convex regularizers. This class of functions is very large and versatile allowing to promote solutions conforming to some notion of low-complexity. Indeed, they force solutions of variational problems to belong to a low-dimensional manifold (the so…

Unified analysis of stochastic ADMM variants via SME.

A new clustering method for vector time series using autoregressive dynamics.

Large learning rates prevent memorization in denoising score matching.

Study large deviations for hypoelliptic diffusion on sub-Riemannian manifolds.

Logit regularization induces logit clustering, affecting classifier performance.

Optimal B-robust estimate is constructed for multidimensional parameter in drift coefficient of diffusion type process with small noise. Optimal mean-variance robust (optimal V -robust) trading strategy is find to hedge in mean-variance sense the contingent claim in incomplete financial market with arbitrary informatio…

Study large deviations in fractional volatility models with non-Gaussian volatility.

We expand volatility models for rough stochastic volatility.

Deep learning solves dynamic programming with recursive utility.

Optimizes variance reduction in Heston model using large and moderate deviations.

We find stationary distributions in a financial model with trends and mean-reversion.

In the compagnion paper [Marginal density expansions for diffusions and stochastic volatility, part I] we discussed density expansions for multidimensional diffusions $(X^1,...,X^d)$, at fixed time $T$ and projected to their first $l$ coordinates, in the small noise regime. Global conditions were found which replace th…

Density expansions for hypoelliptic diffusions $(X^1,...,X^d)$ are revisited. In particular, we are interested in density expansions of the projection $(X_T^1,...,X_T^l)$, at time $T>0$, with $l \leq d$. Global conditions are found which replace the well-known "not-in-cutlocus" condition known from heat-kernel asymptot…

This paper optimizes importance sampling for rare-event options pricing under the Heston model.

A new method uses deep learning to predict rare events in complex systems.

Study volatility models with rough paths, focusing on large deviations and option behavior.

Reduces learning periodic neural networks to lattice problems, proving hardness under cryptographic assumptions.

The one-dimensional SDE with non Lipschitz diffusion coefficient $dX_{t} = b(X_{t})dt + σX_{t}^γ dB_{t}, \ X_{0}=x, \ γ<1$ is widely studied in mathematical finance. Several works have proposed asymptotic analysis of densities and implied volatilities in models involving instances of this equation, based on a careful i…

New method improves nonlinear filtering accuracy with reduced computation.

New method uses MRI data to improve PET tomography uncertainty quantification.

We develop the mathematical foundations of the stochastic modified equations (SME) framework for analyzing the dynamics of stochastic gradient algorithms, where the latter is approximated by a class of stochastic differential equations with small noise parameters. We prove that this approximation can be understood math…

New method analyzes volatility models for option prices, especially in rough volatility.

We present a new methodology to analyze large classes of (classical and rough) stochastic volatility models, with special regard to short-time and small noise formulae for option prices. Our main tool is the theory of regularity structures, which we use in the form of [Bayer et al; A regularity structure for rough vola…

We present ErasureHead, a new approach for distributed gradient descent (GD) that mitigates system delays by employing approximate gradient coding. Gradient coded distributed GD uses redundancy to exactly recover the gradient at each iteration from a subset of compute nodes. ErasureHead instead uses approximate gradien…

In phase retrieval we want to recover an unknown signal $\boldsymbol x\in\mathbb C^d$ from $n$ quadratic measurements of the form $y_i = |\langle{\boldsymbol a}_i,{\boldsymbol x}\rangle|^2+w_i$ where $\boldsymbol a_i\in \mathbb C^d$ are known sensing vectors and $w_i$ is measurement noise. We ask the following weak rec…

Motivated by marginals-mimicking results for Itô processes via SDEs and by their applications to volatility modeling in finance, we discuss the weak convergence of the law of a hypoelliptic diffusions conditioned to belong to a target affine subspace at final time, namely $\mathcal{L}(Z_t|Y_t = y)$ if $X_{\cdot}=(Y_\cd…

We consider the learning from noisy labels (NL) problem which emerges in many real-world applications. In addition to the widely-studied synthetic noise in the NL literature, we also consider the pseudo labels in semi-supervised learning (Semi-SL) as a special case of NL. For both types of noise, we argue that the gene…

People belong to multiple communities, words belong to multiple topics, and books cover multiple genres; overlapping clusters are commonplace. Many existing overlapping clustering methods model each person (or word, or book) as a non-negative weighted combination of "exemplars" who belong solely to one community, with …

New spectral clustering method using LASSO regularization for robust graph partitioning.

We study fractional stochastic volatility models in which the volatility process is a positive continuous function $σ$ of a continuous Gaussian process $\widehat{B}$. Forde and Zhang established a large deviation principle for the log-price process in such a model under the assumptions that the function $σ$ is globally…

Noise in RNNs promotes flatter minima and more stable dynamics.

We study the problem of robust subspace recovery (RSR) in the presence of adversarial outliers. That is, we seek a subspace that contains a large portion of a dataset when some fraction of the data points are arbitrarily corrupted. We first examine a theoretical estimator that is intractable to calculate and use it to …

New method solves group synchronization with cycle-edge message passing.

Improved function approximation for noisy data.

Bayesian SSR on graphs improves regression with noisy labels.

TRA detects causal direction from bivariate data using geometric shapes.

Non-negative matrix factorization (NMF) is a natural model of admixture and is widely used in science and engineering. A plethora of algorithms have been developed to tackle NMF, but due to the non-convex nature of the problem, there is little guarantee on how well these methods work. Recently a surge of research have …

New framework embeds generalization in learning dynamics using large deviation theory.

Rectified linear units, or ReLUs, have become the preferred activation function for artificial neural networks. In this paper we consider two basic learning problems assuming that the underlying data follow a generative model based on a ReLU-network -- a neural network with ReLU activations. As a primarily theoretical …

Analytic expressions for deep neural network output under stochastic training.