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Uniform sampling of bipartite graphs and hypergraphs with given degree sequences is necessary for building null models to statistically evaluate their topology. Because these graphs can be represented as binary matrices, the problem is equivalent to uniformly sampling $r \times c$ binary matrices with fixed row and column sums. The trade algorithm, which includes both the curveball and fastball implementations, is the state-of-the-art for performing such sampling. Its mixing time is currently unknown, although $5r$ is currently used as a heuristic. In this paper we propose a new distribution-based approach that not only provides an estimation of the mixing time, but also actually returns a sample of matrices that are guaranteed (within a user-chosen error tolerance) to be uniformly randomly sampled. In numerical experiments on matrices that vary by size, fill, and row and column sum distributions, we find that the upper bound on mixing time is at least $10r$, and that it increases as a function of both $c$ and the fraction of cells containing a 1.

Standardness is a popular assumption in the literature on set estimation. It also appears in statistical approaches to topological data analysis, where it is common to assume that the data were sampled from a probability measure that satisfies the standard assumption. Relevant results in this field, such as rates of convergence and confidence sets, depend on the standardness parameter, which in practice may be unknown. In this paper, we review the notion of standardness and its connection to other geometrical restrictions. We prove the almost sure consistency of a plug-in type estimator for the so-called standardness constant, already studied in the literature. We propose a method to correct the bias of the plug-in estimator and corroborate our theoretical findings through a small simulation study. We also show that it is not possible to determine, based on a finite sample, whether a probability measure satisfies the standard assumption.

Groundwater contamination caused by Dense Non-Aqueous Phase Liquid (DNAPL) has an adverse impact on human health and environment. Remediation techniques, such as the in-situ injection of nano Zero Valent Iron (nZVI) particles, are widely used in mitigating DNAPL-induced groundwater contamination. However, an effective remediation strategy requires predictive insights and understanding of the physiochemical interaction of nZVI and contamination along with the porous media properties. While several stand-alone models are widely used for predictive modeling, the integration of these models for better scalability and accuracy is still rarely utilized. This study presents an end-to-end integrated modeling framework for the remediation of DNAPL-contaminated aquifers using nZVI. The framework simulates the migration pathway of DNAPL and subsequently its dissolution in groundwater resulting in an aqueous contaminant plume. Additionally, the framework includes simulations of nZVI mobility, transport, and reactive behavior, allowing for the prediction of the radius of influence and efficiency of nZVI for contaminant degradation. The framework has been applied to a hypothetical 2-dimensional and heterogeneous silty sand aquifer, considering trichloroethylene (TCE) as the DNAPL contaminant and carboxymethyl cellulose (CMC) coated nZVI for remediation. The results demonstrate the framework's capability to provide comprehensive insights into the contaminant's behavior and the effectiveness of the remediation strategy. The proposed modeling framework serves as a reference for future studies and can be expanded to incorporate real field data and complex geometries for upscaled applications.

We present new min-max relations in digraphs between the number of paths satisfying certain conditions and the order of the corresponding cuts. We define these objects in order to capture, in the context of solving the half-integral linkage problem, the essential properties needed for reaching a large bramble of congestion two (or any other constant) from the terminal set. This strategy has been used ad-hoc in several articles, usually with lengthy technical proofs, and our objective is to abstract it to make it applicable in a simpler and unified way. We provide two proofs of the min-max relations, one consisting in applying Menger's Theorem on appropriately defined auxiliary digraphs, and an alternative simpler one using matroids, however with worse polynomial running time. As an application, we manage to simplify and improve several results of Edwards et al. [ESA 2017] and of Giannopoulou et al. [SODA 2022] about finding half-integral linkages in digraphs. Concerning the former, besides being simpler, our proof provides an almost optimal bound on the strong connectivity of a digraph for it to be half-integrally feasible under the presence of a large bramble of congestion two (or equivalently, if the directed tree-width is large, which is the hard case). Concerning the latter, our proof uses brambles as rerouting objects instead of cylindrical grids, hence yielding much better bounds and being somehow independent of a particular topology. We hope that our min-max relations will find further applications as, in our opinion, they are simple, robust, and versatile to be easily applicable to different types of routing problems in digraphs.

The mean width of a convex body is the average distance between parallel supporting hyperplanes when the normal direction is chosen uniformly over the sphere. The Simplex Mean Width Conjecture (SMWC) is a longstanding open problem that says the regular simplex has maximum mean width of all simplices contained in the unit ball and is unique up to isometry. We give a self contained proof of the SMWC in $d$ dimensions. The main idea is that when discussing mean width, $d+1$ vertices $v_i\in\mathbb{S}^{d-1}$ naturally divide $\mathbb{S}^{d-1}$ into $d+1$ Voronoi cells and conversely any partition of $\mathbb{S}^{d-1}$ points to selecting the centroids of regions as vertices. We will show that these two conditions are enough to ensure that a simplex with maximum mean width is a regular simplex.

We study the weak recovery problem on the $r$-uniform hypergraph stochastic block model ($r$-HSBM) with two balanced communities. In HSBM a random graph is constructed by placing hyperedges with higher density if all vertices of a hyperedge share the same binary label, and weak recovery asks to recover a non-trivial fraction of the labels. We introduce a multi-terminal version of strong data processing inequalities (SDPIs), which we call the multi-terminal SDPI, and use it to prove a variety of impossibility results for weak recovery. In particular, we prove that weak recovery is impossible below the Kesten-Stigum (KS) threshold if $r=3,4$, or a strength parameter $\lambda$ is at least $\frac 15$. Prior work Pal and Zhu (2021) established that weak recovery in HSBM is always possible above the KS threshold. Consequently, there is no information-computation gap for these cases, which (partially) resolves a conjecture of Angelini et al. (2015). To our knowledge this is the first impossibility result for HSBM weak recovery. As usual, we reduce the study of non-recovery of HSBM to the study of non-reconstruction in a related broadcasting on hypertrees (BOHT) model. While we show that BOHT's reconstruction threshold coincides with KS for $r=3,4$, surprisingly, we demonstrate that for $r\ge 7$ reconstruction is possible also below KS. This shows an interesting phase transition in the parameter $r$, and suggests that for $r\ge 7$, there might be an information-computation gap for the HSBM. For $r=5,6$ and large degree we propose an approach for showing non-reconstruction below KS, suggesting that $r=7$ is the correct threshold for onset of the new phase.

This paper is concerned with variational and Bayesian approaches to neuro-electromagnetic inverse problems (EEG and MEG). The strong indeterminacy of these problems is tackled by introducing sparsity inducing regularization/priors in a transformed domain, namely a spatial wavelet domain. Sparsity in the wavelet domain allows to reach ''data compression'' in the cortical sources domain. Spatial wavelets defined on the mesh graph of the triangulated cortical surface are used, in combination with sparse regression techniques, namely LASSO regression or sparse Bayesian learning, to provide localized and compressed estimates for brain activity from sensor data. Numerical results on simulated and real MEG data are provided, which outline the performances of the proposed approach in terms of localization.

We consider isogeometric discretizations of the Poisson model problem, focusing on high polynomial degrees and strong hierarchical refinements. We derive a posteriori error estimates by equilibrated fluxes, i.e., vector-valued mapped piecewise polynomials lying in the $\boldsymbol{H}({\rm div})$ space which appropriately approximate the desired divergence constraint. Our estimates are constant-free in the leading term, locally efficient, and robust with respect to the polynomial degree. They are also robust with respect to the number of hanging nodes arising in adaptive mesh refinement employing hierarchical B-splines. Two partitions of unity are designed, one with larger supports corresponding to the mapped splines, and one with small supports corresponding to mapped piecewise multilinear finite element hat basis functions. The equilibration is only performed on the small supports, avoiding the higher computational price of equilibration on the large supports or even the solution of a global system. Thus, the derived estimates are also as inexpensive as possible. An abstract framework for such a setting is developed, whose application to a specific situation only requests a verification of a few clearly identified assumptions. Numerical experiments illustrate the theoretical developments.

For many applications involving a sequence of linear systems with slowly changing system matrices, subspace recycling, which exploits relationships among systems and reuses search space information, can achieve huge gains in iterations across the total number of linear system solves in the sequence. However, for general (i.e., non-identity) shifted systems with the shift value varying over a wide range, the properties of the linear systems vary widely as well, which makes recycling less effective. If such a sequence of systems is embedded in a nonlinear iteration, the problem is compounded, and special approaches are needed to use recycling effectively. In this paper, we develop new, more efficient, Krylov subspace recycling approaches for large-scale image reconstruction and restoration techniques that employ a nonlinear iteration to compute a suitable regularization matrix. For each new regularization matrix, we need to solve regularized linear systems, ${\bf A} + \gamma_\ell {\bf E}_k$, for a sequence of regularization parameters, $\gamma_\ell$, to find the optimally regularized solution that, in turn, will be used to update the regularization matrix. In this paper, we analyze system and solution characteristics to choose appropriate techniques to solve each system rapidly. Specifically, we use an inner-outer recycling approach with a larger, principal recycle space for each nonlinear step and smaller recycle spaces for each shift. We propose an efficient way to obtain good initial guesses from the principle recycle space and smaller shift-specific recycle spaces that lead to fast convergence. Our method is substantially reduces the total number of matrix-vector products that would arise in a naive approach. Our approach is more generally applicable to sequences of shifted systems where the matrices in the sum are positive semi-definite.

This paper considers a variant of zero-sum matrix games where at each timestep the row player chooses row $i$, the column player chooses column $j$, and the row player receives a noisy reward with mean $A_{i,j}$. The objective of the row player is to accumulate as much reward as possible, even against an adversarial column player. If the row player uses the EXP3 strategy, an algorithm known for obtaining $\sqrt{T}$ regret against an arbitrary sequence of rewards, it is immediate that the row player also achieves $\sqrt{T}$ regret relative to the Nash equilibrium in this game setting. However, partly motivated by the fact that the EXP3 strategy is myopic to the structure of the game, O'Donoghue et al. (2021) proposed a UCB-style algorithm that leverages the game structure and demonstrated that this algorithm greatly outperforms EXP3 empirically. While they showed that this UCB-style algorithm achieved $\sqrt{T}$ regret, in this paper we ask if there exists an algorithm that provably achieves $\text{polylog}(T)$ regret against any adversary, analogous to results from stochastic bandits. We propose a novel algorithm that answers this question in the affirmative for the simple $2 \times 2$ setting, providing the first instance-dependent guarantees for games in the regret setting. Our algorithm overcomes two major hurdles: 1) obtaining logarithmic regret even though the Nash equilibrium is estimable only at a $1/\sqrt{T}$ rate, and 2) designing row-player strategies that guarantee that either the adversary provides information about the Nash equilibrium, or the row player incurs negative regret. Moreover, in the full information case we address the general $n \times m$ case where the first hurdle is still relevant. Finally, we show that EXP3 and the UCB-based algorithm necessarily cannot perform better than $\sqrt{T}$.