Related papers: Probabilistic Analysis of Multiparameter Persisten…
Diffusion models are a remarkably effective way of learning and sampling from a distribution $p(x)$. In posterior sampling, one is also given a measurement model $p(y \mid x)$ and a measurement $y$, and would like to sample from $p(x \mid…
We extend the persistence algorithm, viewed as an algorithm computing the homology of a complex of free persistence or graded modules, to complexes of modules that are not free. We replace persistence modules by their presentations and…
In this article, we focus on the invariance property of Morse homology on noncompact manifolds. We expect to apply outcomes of this article to several types of Floer homology, thus we define Morse homology purely axiomatically and…
A univariate polynomial f over a field is decomposable if it is the composition f = g(h) of two polynomials g and h whose degree is at least 2. We determine the dimension (over an algebraically closed field) of the set of decomposables, and…
Given a probability measure $\mu$ on the real line, there exists a semigroup $\mu_t$ with real parameter $t>1$ which interpolates the discrete semigroup of measures $\mu_n$ obtained by iterating its free convolution. It was shown in…
Algorithms for persistent homology and zigzag persistent homology are well-studied for persistence modules where homomorphisms are induced by inclusion maps. In this paper, we propose a practical algorithm for computing persistence under…
For nonstationary, strongly mixing sequences of random variables taking their values in a finite-dimensional Euclidean space, with the partial sums being normalized via matrix multiplication, with certain standard conditions being met, the…
Bivariate generating functions for various subsets of the class of permutations containing no descending sequence of length three or more are determined. The notion of absolute indecomposability of a permutation is introduced, and used in…
Recent research reveals that machine learning (ML) models are highly sensitive to minor changes in their training procedure, such as the inclusion or exclusion of a single data point, leading to conflicting predictions on individual data…
Let $X$ be a quasi-projective variety over a number field, admitting (after passage to $\mathbb{C}$) a geometric variation of Hodge structure whose period mapping has zero-dimensional fibers. Then the integral points of $X$ are sparse: the…
Persistence diagrams, which summarize the birth and death of homological features extracted from data, are employed as stable signatures for applications in image analysis and other areas. Besides simply considering the multiset of…
We prove a multiple recurrence result for arbitrary measure-preserving transformations along polynomials in two variables of the form $m+p_i(n)$, with rationally independent $p_i$'s with zero constant term. This is in contrast to the single…
Indecomposable continua with one composant are $\textit{large}$ in the sense of being non-metrisable. We adapt the method of Smith $[18]$ to construct an example which is $\textit{small}$ in the sense of being separable.
We propose a new way of thinking about one parameter persistence. We believe topological persistence is fundamentally not about decomposition theorems but a central role is played by a choice of metrics. Choosing a pseudometric between…
Persistent cycles, especially the minimal ones, are useful geometric features functioning as augmentations for the intervals in a purely topological persistence diagram (also termed as barcode). In our earlier work, we showed that computing…
In this paper we examine the use of topological methods for multivariate statistics. Using persistent homology from computational algebraic topology, a random sample is used to construct estimators of persistent homology. This estimation…
An ideal invariant for multiparameter persistence would be discriminative, computable and stable. In this work we analyse the discriminative power of a stable, computable invariant of multiparameter persistence modules: the fibered bar…
The convergence of simultaneous and marginal predictive classifiers under partition exchangeability in supervised classification is obtained. The result shows the asymptotic convergence of these classifiers under infinite amount of training…
We consider the well-studied problem of decomposing a vector time series signal into components with different characteristics, such as smooth, periodic, nonnegative, or sparse. We describe a simple and general framework in which the…
The most conspicuous property of a semiflexible polymer is its persistence length, defined as the decay length of tangent correlations along its contour. Using an efficient stochastic growth algorithm to sample polymers embedded in a…