The stability of persistent homology has led to wide applications of the persistence diagram as a trusted topological descriptor in the presence of noise. However, with the increasing demand for high-dimension and low-sample-size data processing in modern science, it is questionable whether persistence diagrams retain their reliability in the presence of high-dimensional noise. This work aims to study the reliability of persistence diagrams in the high-dimension low-sample-size data setting. By analyzing the asymptotic behavior of persistence diagrams for high-dimensional random data, we show that persistence diagrams are no longer reliable descriptors of low-sample-size data under high-dimensional noise perturbations. We refer to this loss of reliability of persistence diagrams in such data settings as the curse of dimensionality on persistence diagrams. Next, we investigate the performance of applying normalized principal component analysis for addressing the curse of dimensionality and show that utilizing this method yields a theoretical improvement, although there remain challenges in fully recovering the original topological features.

We study sublevel set and superlevel set persistent homology on discrete functions through the perspective of finite ordered sets of both linearly and cyclically ordered domains. Finite ordered sets also serve as the codomain of our functions making all arguments finite and discrete. We prove duality of filtrations of sublevel sets and superlevel sets that undergirds a range of duality results of sublevel set persistent homology without the need to invoke complications of continuous functions or classical Morse theory. We show that Morse-like behavior can be achieved for flat extrema without assuming genericity. Additionally, we show that with inversion of order, one can compute sublevel set persistence from superlevel set persistence, and vice versa via a duality result that does not require the boundary to be treated as a special case. Furthermore, we discuss aspects of barcode construction rules, surgery of circular and linearly ordered sets, as well as surgery on auxiliary structures such as box snakes, which segment the ordered set by extrema and monotones.

Obtaining meaningful solutions for inverse problems has been a major challenge with many applications in science and engineering. Recent machine learning techniques based on proximal and diffusion-based methods have shown promising results. However, as we show in this work, they can also face challenges when applied to some exemplary problems. We show that, similar to previous works on over-complete dictionaries, it is possible to overcome these shortcomings by embedding the solution into higher dimensions. The novelty of the work proposed is that we jointly design and learn the embedding and the regularizer for the embedding vector. We demonstrate the merit of this approach on several exemplary and common inverse problems.

By approximating posterior distributions with weighted samples, particle filters (PFs) provide an efficient mechanism for solving non-linear sequential state estimation problems. While the effectiveness of particle filters has been recognised in various applications, their performance relies on the knowledge of dynamic models and measurement models, as well as the construction of effective proposal distributions. An emerging trend involves constructing components of particle filters using neural networks and optimising them by gradient descent, and such data-adaptive particle filtering approaches are often called differentiable particle filters. Due to the expressiveness of neural networks, differentiable particle filters are a promising computational tool for performing inference on sequential data in complex, high-dimensional tasks, such as vision-based robot localisation. In this paper, we review recent advances in differentiable particle filters and their applications. We place special emphasis on different design choices for key components of differentiable particle filters, including dynamic models, measurement models, proposal distributions, optimisation objectives, and differentiable resampling techniques.

We introduce a Bayesian framework for confocal fluorescence spectroscopy that addresses key limitations of traditional fluorescence correlation spectroscopy. By jointly modeling molecular motion, optical effects, and photon detection, the method enables direct inference of molecular trajectories from photon count data. A superresolution parameter further refines trajectory estimation beyond the rate of data acquisition. To explore the resulting high-dimensional posterior distribution, we develop Hamiltonian Monte Carlo (HMC) algorithms adapted to the correlation structure of spectroscopy data. This structure gives rise to a stiff ordinary differential equation involving a two-scale discrete Laplacian. Analysis of its spectral properties yields a CFL-type stability condition for the commonly used Störmer–Verlet integrator, revealing instability at practical step sizes. We overcome this limitation with a semi-implicit (IMEX) scheme that separates stiff and non-stiff terms while exploiting the Laplacian's sparsity for efficiency. Numerical experiments demonstrate that IMEX maintains second-order accuracy and energy stability while permitting larger step sizes. This framework establishes a scalable foundation for future models of surface-constrained or multi-mode molecular motion.

We expound on some known lower bounds of the quadratic Wasserstein distance between random vectors in $ \mathbb{R}^n $ with an emphasis on affine transformations that have been used in manifold learning of data in Wasserstein space. In particular, we give concrete lower bounds for rotated copies of random vectors in $ \mathbb{R}^2 $ by computing the Bures metric between the covariance matrices. We also derive upper bounds for compositions of affine maps which yield a fruitful variety of diffeomorphisms applied to an initial data measure. We apply these bounds to various distributions including those lying on a 1-dimensional manifold in $ \mathbb{R}^2 $ and illustrate the quality of the bounds. Finally, we give a framework for mimicking handwritten digit or alphabet datasets that can be applied in a manifold learning framework.

The λ-exponential family has recently been proposed to generalize the exponential family. While the exponential family is well-understood and widely used, this is not the case yet for the λ-exponential family. However, many applications require models that are more general than the exponential family, and the λ-exponential family is often a good alternative. In this work, we propose a theoretical and algorithmic framework to solve variational inference and maximum likelihood estimation problems over the λ-exponential family. We give new sufficient optimality conditions for variational inference problems. Our conditions take the form of generalized moment-matching conditions and generalize existing similar results for the exponential family. We exhibit novel characterizations of the solutions of maximum likelihood estimation problems, that recover optimality conditions in the case of the exponential family. For the resolution of both problems, we propose novel proximal-like algorithms that exploit the geometry underlying the λ-exponential family. These new theoretical and methodological insights are tested on numerical examples, showcasing their usefulness and interest, especially on heavy-tailed target distributions.

Path homology proposed by S.-T.Yau and his co-workers provides a new mathematical model for directed graphs and networks. Persistent path homology (PPH) extends the path homology with filtration to deal with asymmetry structures. However, PPH is constrained to purely topological persistence and cannot track the homotopic shape evolution of data during filtration. To overcome the limitation of PPH, persistent path Laplacian (PPL) is introduced to capture the shape evolution of data. PPL's harmonic spectra fully recover PPH's topological persistence and its non-harmonic spectra reveal the homotopic shape evolution of data during filtration.

The problem of interpreting or aggregating multiple rankings is common to many real-world applications. Perhaps the simplest and most common approach is a weighted rank aggregation, wherein a (convex) weight is applied to each input ranking and then ordered. This paper describes a new tool for visualizing and displaying ranking information for the weighted rank aggregation method. Traditionally, the aim of rank aggregation is to summarize the information from the input rankings and provide one final ranking that hopefully represents a more accurate or truthful result than any one input ranking. While such an aggregated ranking is, and clearly has been, useful to many applications, it also obscures information. In this paper, we show the wealth of information that is available for the weighted rank aggregation problem due to its structure. We apply weight set decomposition to the set of convex multipliers, study the properties useful for understanding this decomposition, and visualize the indifference regions. This methodology reveals information–that is otherwise collapsed by the aggregated ranking–into a useful, interpretable, and intuitive decision support tool. Included are multiple illustrative examples, along with heuristic and exact algorithms for computing the weight set decomposition.

<p style='text-indent:20px;'>Nowadays, neural networks are widely used in many applications as artificial intelligence models for learning tasks. Since typically neural networks process a very large amount of data, it is convenient to formulate them within the mean-field and kinetic theory. In this work we focus on a particular class of neural networks, i.e. the residual neural networks, assuming that each layer is characterized by the same number of neurons <inline-formula><tex-math id="M1">\begin{document}$ N $\end{document}</tex-math></inline-formula>, which is fixed by the dimension of the data. This assumption allows to interpret the residual neural network as a time-discretized ordinary differential equation, in analogy with neural differential equations. The mean-field description is then obtained in the limit of infinitely many input data. This leads to a Vlasov-type partial differential equation which describes the evolution of the distribution of the input data. We analyze steady states and sensitivity with respect to the parameters of the network, namely the weights and the bias. In the simple setting of a linear activation function and one-dimensional input data, the study of the moments provides insights on the choice of the parameters of the network. Furthermore, a modification of the microscopic dynamics, inspired by stochastic residual neural networks, leads to a Fokker-Planck formulation of the network, in which the concept of network training is replaced by the task of fitting distributions. The performed analysis is validated by artificial numerical simulations. In particular, results on classification and regression problems are presented.</p>