arXiv subjects

Ilya Chevyrev

Publications and source records attributed to Ilya Chevyrev.

45 records · Page 3Linked to original sources

arXiv2018

Persistence paths and signature features in topological data analysis

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 - barcode to path, path to tensor series - results in a feature map that has several desirable properties for statistical learning, such as universality and characteristicness, and achieves state-of-the-art results on common classification benchmarks.

stat.ML↗

arXiv2018

A support and density theorem for Markovian rough paths

We establish two results concerning a class of geometric rough paths $\mathbf{X}$ which arise as Markov processes associated to uniformly subelliptic Dirichlet forms. The first is a support theorem for $\mathbf{X}$ in $α$-Hölder rough path topology for all $α\in (0,1/2)$, which answers in the positive a conjecture of Friz-Victoir (2010). The second is a Hörmander-type theorem for the existence of a density of a rough differential equation driven by $\mathbf{X}$, the proof of which is based on analysis of (non-symmetric) Dirichlet forms on manifolds.

math.PR↗

arXiv2018

An isomorphism between branched and geometric rough paths

We exhibit an explicit natural isomorphism between spaces of branched and geometric rough paths. This provides a multi-level generalisation of the isomorphism of Lejay-Victoir (2006) as well as a canonical version of the Itô-Stratonovich correction formula of Hairer-Kelly (2015). Our construction is elementary and uses the property that the Grossman-Larson algebra is isomorphic to a tensor algebra. We apply this isomorphism to study signatures of branched rough paths. Namely, we show that the signature of a branched rough path is trivial if and only if the path is tree-like, and construct a non-commutative Fourier transform for probability measures on signatures of branched rough paths. We use the latter to provide sufficient conditions for a random signature to be determined by its expected value, thus giving an answer to the uniqueness moment problem for branched rough paths.

math.PR↗

arXiv2018

Canonical RDEs and general semimartingales as rough paths

In the spirit of Marcus canonical stochastic differential equations, we study a similar notion of rough differential equations (RDEs), notably dropping the assumption of continuity prevalent in the rough path literature. A new metric is exhibited in which the solution map is a continuous function of the driving rough path and a so-called path function, which directly models the effect of the jump on the system. In a second part, we show that general multidimensional semimartingales admit canonically defined rough path lifts. An extension of Lépingle's BDG inequality to this setting is given, and in turn leads to a number of novel limit theorems for semimartingale driven differential equations, both in law and in probability, conveniently phrased via Kurtz-Protter's uniformly-controlled-variations (UCV) condition. A number of examples illustrate the scope of our results.

math.PR↗

arXiv2017

Characteristic functions of measures on geometric rough paths

We define a characteristic function for probability measures on the signatures of geometric rough paths. We determine sufficient conditions under which a random variable is uniquely determined by its expected signature, thus partially solving the analogue of the moment problem. We furthermore study analyticity properties of the characteristic function and prove a method of moments for weak convergence of random variables. We apply our results to signature arising from Lévy, Gaussian and Markovian rough paths.

math.PR↗

arXiv2017

Random walks and Lévy processes as rough paths

We consider random walks and Lévy processes in a homogeneous group $G$. For all $p > 0$, we completely characterise (almost) all $G$-valued Lévy processes whose sample paths have finite $p$-variation, and give sufficient conditions under which a sequence of $G$-valued random walks converges in law to a Lévy process in $p$-variation topology. In the case that $G$ is the free nilpotent Lie group over $\mathbb{R}^d$, so that processes of finite $p$-variation are identified with rough paths, we demonstrate applications of our results to weak convergence of stochastic flows and provide a Lévy-Khintchine formula for the characteristic function of the signature of a Lévy process. At the heart of our analysis is a criterion for tightness of $p$-variation for a collection of càdlàg strong Markov processes.

math.PR↗

arXiv2017

Examples of renormalized SDEs

We demonstrate two examples of stochastic processes whose lifts to geometric rough paths require a renormalisation procedure to obtain convergence in rough path topologies. Our first example involves a physical Brownian motion subject to a magnetic force which dominates over the friction forces in the small mass limit. Our second example involves a lead-lag process of discretised fractional Brownian motion with Hurst parameter $H \in (1/4,1/2)$, in which the stochastic area captures the quadratic variation of the process. In both examples, a renormalisation of the second iterated integral is needed to ensure convergence of the processes, and we comment on how this procedure mimics negative renormalisation arising in the study of singular SPDEs and regularity structures.

math.PR↗

arXiv2014

Constructing supersingular elliptic curves with a given endomorphism ring

Let O be a maximal order in the quaternion algebra B_p over Q ramified at p and infinity. The paper is about the computational problem: Construct a supersingular elliptic curve E over F_p such that End(E) = O. We present an algorithm that solves this problem by taking gcds of the reductions modulo p of Hilbert class polynomials. New theoretical results are required to determine the complexity of our algorithm. Our main result is that, under certain conditions on a rank three sublattice O^T of O, the order O is effectively characterized by the three successive minima and two other short vectors of O^T. The desired conditions turn out to hold whenever the j-invariant j(E), of the elliptic curve with End(E) = O, lies in F_p. We can then prove that our algorithm terminates with running time O(p^{1+ε}) under the aforementioned conditions. As a further application we present an algorithm to simultaneously match all maximal order types with their associated j-invariants. Our algorithm has running time O(p^{2.5+ε}) operations and is more efficient than Cervino's algorithm for the same problem.

math.NT↗

arXiv2011

On the Number of Facets of Polytopes Representing Comparative Probability Orders

Fine and Gill (1973) introduced the geometric representation for those comparative probability orders on n atoms that have an underlying probability measure. In this representation every such comparative probability order is represented by a region of a certain hyperplane arrangement. Maclagan (1999) asked how many facets a polytope, which is the closure of such a region, might have. We prove that the maximal number of facets is at least F_{n+1}, where F_n is the nth Fibonacci number. We conjecture that this lower bound is sharp. Our proof is combinatorial and makes use of the concept of flippable pairs introduced by Maclagan. We also obtain an upper bound which is not too far from the lower bound.

math.CO↗