arXiv research

MuRiT efficiently computes multi-parameter persistence barcodes.

New metrics and coordinates for barcode space using group theory.

We propose to study neural networks' loss surfaces by methods of topological data analysis. We suggest to apply barcodes of Morse complexes to explore topology of loss surfaces. An algorithm for calculations of the loss function's barcodes of local minima is described. We have conducted experiments for calculating barc…

We define notions of differentiability for maps from and to the space of persistence barcodes. Inspired by the theory of diffeological spaces, the proposed framework uses lifts to the space of ordered barcodes, from which derivatives can be computed. The two derived notions of differentiability (respectively from and t…

Entropy measures geodesic flow complexity.

Quantum-enhanced barcode decoding and pattern recognition outperforms classical methods.

We introduce a new feature map for barcodes that arise in persistent homology computation. The main idea is to first realize each barcode as a path in a convenient vector space, and to then compute its path signature which takes values in the tensor algebra of that vector space. The composition of these two operations …

This article introduces an application of Ghrist barcodes in the study of persistent Betti numbers derived from vortex nerve complexes found in triangulations of video frames. A Ghrist barcode is a topology of data pictograph useful in representing the persistence of the features of changing shapes. The basic approach …

Paper introduces stable vectorization for multiparameter PH using signed barcodes.

Introduces TSI, a variance-based measure for persistence barcodes.

We consider a strictly pathwise setting for Delta hedging exotic options, based on Föllmer's pathwise Itō calculus. Price trajectories are $d$-dimensional continuous functions whose pathwise quadratic variations and covariations are determined by a given local volatility matrix. The existence of Delta hedging strategie…

This work proposes using zero-variance control variates to reduce variance in pathwise gradient estimators for variational inference.

Pathwise uniqueness shown for specific stochastic equations.

Homotopy types of Vietoris-Rips metric thickenings of the circle confirmed.

Stable topological summary captures evolving dependency structure in dynamic Bayesian networks.

Develops pathwise analysis for log-optimal portfolios using rough paths theory.

Develops portfolio theory without probabilistic analysis, focusing on pathwise decomposition.

We consider different notions of equivalence for Morse functions on the sphere in the context of persistent homology, and introduce new invariants to study these equivalence classes. These new invariants are as simple, but more discerning than existing topological invariants, such as persistence barcodes and Reeb graph…

A new approach to continuous-time universal portfolios using pathwise Itô calculus.

Paper stabilizes persistent homology rank functions for statistical inference.

The pathwise coordinate optimization is one of the most important computational frameworks for high dimensional convex and nonconvex sparse learning problems. It differs from the classical coordinate optimization algorithms in three salient features: {\it warm start initialization}, {\it active set updating}, and {\it …

We use pathwise Itô calculus to prove two strictly pathwise versions of the master formula in Fernholz' stochastic portfolio theory. Our first version is set within the framework of Föllmer's pathwise Itô calculus and works for portfolios generated from functions that may depend on the current states of the market port…

This paper develops a mathematical framework for the analysis of continuous-time trading strategies which, in contrast to the classical setting of continuous-time mathematical finance, does not rely on stochastic integrals or other probabilistic notions. Our purely analytic framework allows for the derivation of a path…

This paper simplifies hedge ratios in financial models using pathwise algorithmic differentiation.

The Monte Carlo pathwise sensitivities approach is well established for smooth payoff functions. In this work, we present a new Monte Carlo algorithm that is able to calculate the pathwise sensitivities for discontinuous payoff functions. Our main tool is to combine the one-step survival idea of Glasserman and Staum wi…

This paper gives several simple constructions of the pathwise Ito integral $\int_0^tφdω$ for an integrand $φ$ and a price path $ω$ as integrator, with $φ$ and $ω$ satisfying various topological and analytical conditions. The definitions are purely pathwise in that neither $φ$ nor $ω$ are assumed to be paths of stochast…

This dissertation advances scalable Gaussian processes using iterative methods and pathwise conditioning.

This work introduces efficient sampling methods for Gaussian processes by focusing on pathwise conditioning.

We exploit the link between the transport equation and derivatives of expectations to construct efficient pathwise gradient estimators for multivariate distributions. We focus on two main threads. First, we use null solutions of the transport equation to construct adaptive control variates that can be used to construct…

We study the use of the multilevel Monte Carlo technique in the context of the calculation of Greeks. The pathwise sensitivity analysis differentiates the path evolution and reduces the payoff's smoothness. This leads to new challenges: the inapplicability of pathwise sensitivities to non-Lipschitz payoffs often makes …

We observe that gradients computed via the reparameterization trick are in direct correspondence with solutions of the transport equation in the formalism of optimal transport. We use this perspective to compute (approximate) pathwise gradients for probability distributions not directly amenable to the reparameterizati…

We investigate whether it is possible to formulate option pricing and hedging models without using probability. We present a model that is consistent with two notions of volatility: a historical volatility consistent with statistical analysis, and an implied volatility consistent with options priced with the model. The…

Symplectic homology matches dual capacities for convex domains.

Efficient estimators for smooth Hilbert-valued parameters with theoretical guarantees.

The paper optimizes bridge-type estimators for sparse models using pathwise methods.

Floer theory constructs filtrations on quantum cohomology for symplectic manifolds.

We consider a class $\mathscr{X}$ of continuous functions on $[0,1]$ that is of interest from two different perspectives. First, it is closely related to sets of functions that have been studied as generalizations of the Takagi function. Second, each function in $\mathscr{X}$ admits a linear pathwise quadratic variatio…

A new method in finance without probabilities or integrals.

Study shows how market firm capitalization models converge to stochastic PDE solutions.

Following a hedging based approach to model free financial mathematics, we prove that it should be possible to make an arbitrarily large profit by investing in those one-dimensional paths which do not possess local times. The local time is constructed from discrete approximations, and it is shown that it is $α$-Hölder …

Exact simulation method for market impact estimation under various execution strategies.

NM-PPG optimizes adaptive feature acquisition in POMDPs for better predictions.

We apply the barcodes of persistent homology theory to the Chekanov-Eliashberg algebra of a Legendrian submanifold to deduce displacement energy bounds for arbitrary Legendrians. We do not require the full Chekanov-Eliashberg algebra to admit an augmentation as we linearize the algebra only below a certain action level…

New method reduces errors in pricing and sensitivities for discontinuous payoffs.

Detection of rare variants by resequencing is important for the identification of individuals carrying disease variants. Rapid sequencing by new technologies enables low-cost resequencing of target regions, although it is still prohibitive to test more than a few individuals. In order to improve cost trade-offs, it has…

Develops a method for solving optimal stopping problems with multiple exercise rights.

We provide a unifying treatment of pathwise moderate deviations for models commonly used in financial applications, and for related integrated functionals. Suitable scaling allows us to transfer these results into small-time, large-time and tail asymptotics for diffusions, as well as for option prices and realised vari…

Develops a fast algorithm for high-dimensional LASSO penalized quantile regression.