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Adapting the step size locally in the no-U-turn sampler (NUTS) is challenging because the step-size and path-length tuning parameters are interdependent. The determination of an optimal path length requires a predefined step size, while the ideal step size must account for errors along the selected path. Ensuring reversibility further complicates this tuning problem. In this paper, we present a method for locally adapting the step size in NUTS that is an instance of the Gibbs self-tuning (GIST) framework. Our approach guarantees reversibility with an acceptance probability that depends exclusively on the conditional distribution of the step size. We validate our step-size-adaptive NUTS method on Neal's funnel density and a high-dimensional normal distribution, demonstrating its effectiveness in challenging scenarios., Comment: for companion code, see https://github.com/bob-carpenter/adaptive-hmc

Three approaches for adaptively tuning diagonal scale matrices for HMC are discussed and compared. The common practice of scaling according to estimated marginal standard deviations is taken as a benchmark. Scaling according to the mean log-target gradient (ISG), and a scaling method targeting that the frequency of when the underlying Hamiltonian dynamics crosses the respective medians should be uniform across dimensions, are taken as alternatives. Numerical studies suggest that the ISG method leads in many cases to more efficient sampling than the benchmark, in particular in cases with strong correlations or non-linear dependencies. The ISG method is also easy to implement, computationally cheap and would be relatively simple to include in automatically tuned codes as an alternative to the benchmark practice.

We propose a generic approach for numerically efficient simulation from analytically intractable distributions with constrained support. Our approach relies upon Generalized Randomized Hamiltonian Monte Carlo (GRHMC) processes and combines these with a randomized transition kernel that appropriately adjusts the Hamiltonian flow at the boundary of the constrained domain, ensuring that it remains within the domain. The numerical implementation of this constrained GRHMC process exploits the sparsity of the randomized transition kernel and the specific structure of the constraints so that the proposed approach is numerically accurate, computationally fast and operational even in high-dimensional applications. We illustrate this approach with posterior distributions of several Bayesian models with challenging parameter domain constraints in applications to real-word data sets. Building on the capability of GRHMC processes to efficiently explore otherwise challenging and high-dimensional posteriors, the proposed method expands the set of Bayesian models that can be analyzed by using the standard Markov-Chain Monte-Carlo (MCMC) methodology, As such, it can advance the development and use of Bayesian models with useful constrained priors, which are difficult to handle with existing methods. The article is accompanied by an R-package (\url{https://github.com/torekleppe/pdmphmc}), which allows for automatically implementing GRHMC processes for arbitrary target distributions and domain constraints.

Let $\varphi: F\longrightarrow G$ be a graded morphism between free $R$-modules of rank $t$ and $t+c-1$, respectively, and let $I_j(\varphi)$ be the ideal generated by the $j \times j$ minors of a matrix representing $\varphi$. In this short note: (1) We show that the canonical module of $R/I_j(\varphi)$ is up to twist equal to a suitable Schur power $\Sigma ^I M$ of $M=\coker (\varphi ^*)$; thus equal to $\wedge ^{t+1-j}M$ if $c=2$ in which case we find a minimal free $R$-resolution of $\wedge ^{t+1-j}M$ for any $j$, (2) For $c = 3$, we construct a free $R$-resolution of $\wedge ^2M$ which starts almost minimally (i.e. the first three terms are minimal up to a precise summand), and (3) For $c \ge 4$, we construct under a certain depth condition the first three terms of a free $R$-resolution of $\wedge ^2M$ which are minimal up to a precise summand. As a byproduct we answer the first open case of a question posed by Buchsbaum and Eisenbud.

We introduce $5/2$- and $7/2$-order $L^2$-accurate randomized Runge-Kutta-Nystr\"{o}m methods, tailored for approximating Hamiltonian flows within non-reversible Markov chain Monte Carlo samplers, such as unadjusted Hamiltonian Monte Carlo and unadjusted kinetic Langevin Monte Carlo. We establish quantitative $5/2$-order $L^2$-accuracy upper bounds under gradient and Hessian Lipschitz assumptions on the potential energy function. The numerical experiments demonstrate the superior efficiency of the proposed unadjusted samplers on a variety of well-behaved, high-dimensional target distributions.

High-temperature superconductivity has been observed in binary hydrides such as LaH10 at pressures above 150 GPa. Hydrogen cage structures have been identified as a common motif beneficial for high critical temperatures Tc. Efforts are now focused on finding hydride high-temperature superconductors at lower pressures. We present evidence for high-temperature superconductivity in binary La4H23 with A15-type structure featuring hydrogen cages at a pressure of P = 95 GPa. We synthesise La4H23 from a lanthanum film capped with a palladium catalyst promoting the dissociation of hydrogen. In resistance measurements, we observe superconductivity with a transition temperature Tc = 90 K. In X-ray diffraction on the same sample, we identify the A15-type cubic body centred structure of the lanthanum sublattice. From comparison with earlier XRD and structure prediction studies, we identify this phase with La4H23. Our study reinforces the concept of hydrogen cages for high-temperature superconductivity., Comment: 18 Pages

A metric tensor for Riemann manifold Monte Carlo particularly suited for non-linear Bayesian hierarchical models is proposed. The metric tensor is built from symmetric positive semidefinite log-density gradient covariance (LGC) matrices, which are also proposed and further explored here. The LGCs generalize the Fisher information matrix by measuring the joint information content and dependence structure of both a random variable and the parameters of said variable. Consequently, positive definite Fisher/LGC-based metric tensors may be constructed not only from the observation likelihoods as is current practice, but also from arbitrarily complicated non-linear prior/latent variable structures, provided the LGC may be derived for each conditional distribution used to construct said structures. The proposed methodology is highly automatic and allows for exploitation of any sparsity associated with the model in question. When implemented in conjunction with a Riemann manifold variant of the recently proposed numerical generalized randomized Hamiltonian Monte Carlo processes, the proposed methodology is highly competitive, in particular for the more challenging target distributions associated with Bayesian hierarchical models.

Macaulay Duality, between quotients of a polynomial ring over a field, annihilated by powers of the variables, and finitely generated submodules of the ring's graded dual, is generalized over any Noetherian ring, and used to provide isomorphisms between the subschemes of the Hilbert scheme parameterizing various sorts of these quotients, and the corresponding subschemes of the Quot scheme of the dual. Thus notably the locus of recursively compressed algebras of permissible socle type is proved to be covered by open subschemes, each one isomorphic to an open subscheme of a certain affine space. Moreover, the polynomial variables are weighted, the polynomial ring is replaced by a graded module, and attention is paid to induced filtrations and gradings. Furthermore, a similar theory is developed for (relatively) maximal quotients of a graded Gorenstein Artinian algebra., Comment: Clarifications were made following the many thoughtful suggestions of the referee of the Collino Memorial Volume, including new indices of terminology and notation and the observation (justified more in this version than the one in press) that a general quotient of socle type bounded by t(-) is RECURSIVELY compressed of socle type t(-)

Contributions of the participants of the 24th International Workshop "What Comes Beyond the Standard Models", July 5 to July 11, 2021, Bled, Slovenia, presented either with the arxiv number of each contribution or with the webside of the Proceedings published by DMFA, ZALOlZNISTVO LJUBLJANA, DECEMBER 2021., Comment: The whole Proceedings has 350 pages

While the importance of automatic image analysis is continuously increasing, recent meta-research revealed major flaws with respect to algorithm validation. Performance metrics are particularly key for meaningful, objective, and transparent performance assessment and validation of the used automatic algorithms, but relatively little attention has been given to the practical pitfalls when using specific metrics for a given image analysis task. These are typically related to (1) the disregard of inherent metric properties, such as the behaviour in the presence of class imbalance or small target structures, (2) the disregard of inherent data set properties, such as the non-independence of the test cases, and (3) the disregard of the actual biomedical domain interest that the metrics should reflect. This living dynamically document has the purpose to illustrate important limitations of performance metrics commonly applied in the field of image analysis. In this context, it focuses on biomedical image analysis problems that can be phrased as image-level classification, semantic segmentation, instance segmentation, or object detection task. The current version is based on a Delphi process on metrics conducted by an international consortium of image analysis experts from more than 60 institutions worldwide., Comment: Shared first authors: Annika Reinke and Minu D. Tizabi. This is a dynamic paper on limitations of commonly used metrics. It discusses metrics for image-level classification, semantic and instance segmentation, and object detection. For missing use cases, comments or questions, please contact a.reinke@dkfz.de. Substantial contributions to this document will be acknowledged with a co-authorship

agtboost is an R package implementing fast gradient tree boosting computations in a manner similar to other established frameworks such as xgboost and LightGBM, but with significant decreases in computation time and required mathematical and technical knowledge. The package automatically takes care of split/no-split decisions and selects the number of trees in the gradient tree boosting ensemble, i.e., agtboost adapts the complexity of the ensemble automatically to the information in the data. All of this is done during a single training run, which is made possible by utilizing developments in information theory for tree algorithms {\tt arXiv:2008.05926v1 [stat.ME]}. agtboost also comes with a feature importance function that eliminates the common practice of inserting noise features. Further, a useful model validation function performs the Kolmogorov-Smirnov test on the learned distribution., Comment: 16 pages, 5 figures, 2 tables

An information theoretic approach to learning the complexity of classification and regression trees and the number of trees in gradient tree boosting is proposed. The optimism (test loss minus training loss) of the greedy leaf splitting procedure is shown to be the maximum of a Cox-Ingersoll-Ross process, from which a generalization-error based information criterion is formed. The proposed procedure allows fast local model selection without cross validation based hyper parameter tuning, and hence efficient and automatic comparison among the large number of models performed during each boosting iteration. Relative to xgboost, speedups on numerical experiments ranges from around 10 to about 1400, at similar predictive-power measured in terms of test-loss.

Let $Hilb ^{p(t)}(P^n)$ be the Hilbert scheme of closed subschemes of $P^n$ with Hilbert polynomial $p(t) \in Q[t]$, and let $W:= \overline{W(\underline{b};\underline{a};r)}$ be the closure of the locus in $Hilb ^{p(t)}(P^n)$ of determinantal schemes defined by the vanishing of the $(t-r+1)\times (t - r+1)$ minors of some matrix $\mathcal A$ of size $t\times (t+c-1)$ with $ij$-enty a homogeneous form of degree $a_j-b_i$ and with $r$ satisfying $\max\{1,2-c\} \le r < t$. $W$ is an irreducible algebraic set. First of all, we compute an upper $r$-independent bound for the dimension of $W$ in terms of $a_j$ and $b_i$ which is sharp for $r=1$. In the linear case ($a_j = 1, b_i=0$) and cases sufficiently close, we conjecture and to a certain degree prove that this bound is achieved for all $r$. Then, we study to what extent $W$ is a generically smooth component of $Hilb ^{p(t)}(P^n)$. Under some weak numerical assumptions on the integers $a_j$ and $b_i$ (or under some depth conditions) we conjecture and often prove that $W$ is a generically smooth component. Moreover, we also study the depth of the normal module of the homogeneous coordinate ring of $(X)\in W$ and of a closely related module. We conjecture, and in some cases prove, that their codepth is often 1 (resp. $r$). These results extend previous results on standard determinantal schemes to determinantal schemes; i.e. previous results of the authors on $W(\underline{b};\underline{a};1)$ to $W$ with $1\le r < t$ and $c\ge 2-r$. Finally, deformations of exterior powers of the cokernel of the map determined by $\mathcal A$ are studied and proven to be given as deformations of $X \subset P^n$ if $\dim X \ge 3$. The work contains many examples which illustrate the results obtained and a considerable number of open problems; some of them are collected as conjectures in the final section., Comment: This version (v2) is a postprint (AAM) of an article published in the Memoirs of the American Mathematical Society, with very minor changes from version v1. The final authenticated version is available online at DOI: 10.1090/memo/1418. Remark 3.2 is, however, extended and the new extension in (3) and (4) is not included in the published paper. Remark 3.2(4) makes a correction to [33]

Numerical Generalized Randomized Hamiltonian Monte Carlo is introduced, as a robust, easy to use and computationally fast alternative to conventional Markov chain Monte Carlo methods for continuous target distributions. A wide class of piecewise deterministic Markov processes generalizing Randomized HMC (Bou-Rabee and Sanz-Serna, 2017) by allowing for state-dependent event rates is defined. Under very mild restrictions, such processes will have the desired target distribution as an invariant distribution. Secondly, the numerical implementation of such processes, based on adaptive numerical integration of second order ordinary differential equations (ODEs) is considered. The numerical implementation yields an approximate, yet highly robust algorithm that, unlike conventional Hamiltonian Monte Carlo, enables the exploitation of the complete Hamiltonian trajectories (hence the title). The proposed algorithm may yield large speedups and improvements in stability relative to relevant benchmarks, while incurring numerical biases that are negligible relative to the overall Monte Carlo errors. Granted access to a high-quality ODE code, the proposed methodology is both easy to implement and use, even for highly challenging and high-dimensional target distributions.

We propose a state-space model (SSM) for commodity prices that combines the competitive storage model with a stochastic trend. This approach fits into the economic rationality of storage decisions, and adds to previous deterministic trend specifications of the storage model. Parameters are estimated using a particle Markov chain Monte Carlo procedure. Empirical application to four commodity markets shows that the stochastic trend SSM is favored over deterministic trend specifications. The stochastic trend SSM identifies structural parameters that differ from those for deterministic trend specifications. In particular, the estimated price elasticities of demand are significantly larger under the stochastic trend SSM.