We present a generalization of the discrete Lehmann representation (DLR) to three-point correlation and vertex functions in imaginary time and Matsubara frequency. The representation takes the form of a linear combination of judiciously chosen exponentials in imaginary time, and products of simple poles in Matsubara frequency, which are universal for a given temperature and energy cutoff. We present a systematic algorithm to generate compact sampling grids, from which the coefficients of such an expansion can be obtained by solving a linear system. We show that the explicit form of the representation can be used to evaluate diagrammatic expressions involving infinite Matsubara sums, such as polarization functions or self-energies, with controllable, high-order accuracy. This collection of techniques establishes a framework through which methods involving three-point objects can be implemented robustly, with a substantially reduced computational cost and memory footprint.
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This note discusses a simple modification of cross-conformal prediction inspired by recent work on e-values. The precursor of conformal prediction developed in the 1990s by Gammerman, Vapnik, and Vovk was also based on e-values and is called conformal e-prediction in this note. Replacing e-values by p-values led to conformal prediction, which has important advantages over conformal e-prediction without obvious disadvantages. The situation with cross-conformal prediction is, however, different: whereas for cross-conformal prediction validity is only an empirical fact (and can be broken with excessive randomization), this note draws the reader's attention to the obvious fact that cross-conformal e-prediction enjoys a guaranteed property of validity.
General first order methods (GFOMs), including various gradient descent and AMP algorithms, constitute a broad class of iterative algorithms in modern statistical learning problems. Some GFOMs also serve as constructive proof devices, iteratively characterizing the empirical distributions of statistical estimators in the large system limits for any fixed number of iterations. This paper develops a non-asymptotic, entrywise characterization for a general class of GFOMs. Our characterizations capture the precise entrywise behavior of the GFOMs, and hold universally across a broad class of heterogeneous random matrix models. As a corollary, we provide the first non-asymptotic description of the empirical distributions of the GFOMs beyond Gaussian ensembles. We demonstrate the utility of these general results in two applications. In the first application, we prove entrywise universality for regularized least squares estimators in the linear model, by controlling the entrywise error relative to a suitably constructed GFOM. This algorithmic proof method also leads to systematically improved averaged universality results for regularized regression estimators in the linear model, and resolves the universality conjecture for (regularized) MLEs in logistic regression. In the second application, we obtain entrywise Gaussian approximations for a class of gradient descent algorithms. Our approach provides non-asymptotic state evolution for the bias and variance of the algorithm along the iteration path, applicable for non-convex loss functions. The proof relies on a new recursive leave-k-out method that provides almost delocalization for the GFOMs and their derivatives. Crucially, our method ensures entrywise universality for up to poly-logarithmic many iterations, which facilitates effective $\ell_2/\ell_\infty$ control between certain GFOMs and statistical estimators in applications.
We propose a variational symplectic numerical method for the time integration of dynamical systems issued from the least action principle. We assume a quadratic internal interpolation of the state between two time steps and we approximate the action in one time step by the Simpson's quadrature formula. The resulting scheme is nonlinear and symplectic. First numerical experiments concern a nonlinear pendulum and we have observed experimentally very good convergence properties.
A non-linear complex system governed by multi-spatial and multi-temporal physics scales cannot be fully understood with a single diagnostic, as each provides only a partial view and much information is lost during data extraction. Combining multiple diagnostics also results in imperfect projections of the system's physics. By identifying hidden inter-correlations between diagnostics, we can leverage mutual support to fill in these gaps, but uncovering these inter-correlations analytically is too complex. We introduce a groundbreaking machine learning methodology to address this issue. Our multimodal approach generates super resolution data encompassing multiple physics phenomena, capturing detailed structural evolution and responses to perturbations previously unobservable. This methodology addresses a critical problem in fusion plasmas: the Edge Localized Mode (ELM), a plasma instability that can severely damage reactor walls. One method to stabilize ELM is using resonant magnetic perturbation to trigger magnetic islands. However, low spatial and temporal resolution of measurements limits the analysis of these magnetic islands due to their small size, rapid dynamics, and complex interactions within the plasma. With super-resolution diagnostics, we can experimentally verify theoretical models of magnetic islands for the first time, providing unprecedented insights into their role in ELM stabilization. This advancement aids in developing effective ELM suppression strategies for future fusion reactors like ITER and has broader applications, potentially revolutionizing diagnostics in fields such as astronomy, astrophysics, and medical imaging.
JAX is widely used in machine learning and scientific computing, the latter of which often relies on existing high-performance code that we would ideally like to incorporate into JAX. Reimplementing the existing code in JAX is often impractical and the existing interface in JAX for binding custom code either limits the user to a single Jacobian product or requires deep knowledge of JAX and its C++ backend for general Jacobian products. With JAXbind we drastically reduce the effort required to bind custom functions implemented in other programming languages with full support for Jacobian-vector products and vector-Jacobian products to JAX. Specifically, JAXbind provides an easy-to-use Python interface for defining custom, so-called JAX primitives. Via JAXbind, any function callable from Python can be exposed as a JAX primitive. JAXbind allows a user to interface the JAX function transformation engine with custom derivatives and batching rules, enabling all JAX transformations for the custom primitive.
This paper studies the influence of probabilism and non-determinism on some quantitative aspect X of the execution of a system modeled as a Markov decision process (MDP). To this end, the novel notion of demonic variance is introduced: For a random variable X in an MDP M, it is defined as 1/2 times the maximal expected squared distance of the values of X in two independent execution of M in which also the non-deterministic choices are resolved independently by two distinct schedulers. It is shown that the demonic variance is between 1 and 2 times as large as the maximal variance of X in M that can be achieved by a single scheduler. This allows defining a non-determinism score for M and X measuring how strongly the difference of X in two executions of M can be influenced by the non-deterministic choices. Properties of MDPs M with extremal values of the non-determinism score are established. Further, the algorithmic problems of computing the maximal variance and the demonic variance are investigated for two random variables, namely weighted reachability and accumulated rewards. In the process, also the structure of schedulers maximizing the variance and of scheduler pairs realizing the demonic variance is analyzed.
We present and analyze a structure-preserving method for the approximation of solutions to nonlinear cross-diffusion systems, which combines a Local Discontinuous Galerkin spatial discretization with the backward Euler time stepping scheme. The proposed method makes use of the underlying entropy structure of the system, expressing the main unknown in terms of the entropy variable by means of a nonlinear transformation. Such a transformation allows for imposing the physical positivity or boundedness constraints on the approximate solution in a strong sense. Moreover, nonlinearities do not appear explicitly within differential operators or interface terms in the scheme, which significantly improves its efficiency and ease its implementation. We prove the existence of discrete solutions and their asymptotic convergence to continuous weak solutions. Numerical results for some one- and two-dimensional problems illustrate the accuracy and entropy stability of the proposed method.
Classical statistics deals with determined and precise data analysis. But in reality, there are many cases where the information is not accurate and a degree of impreciseness, uncertainty, incompleteness, and vagueness is observed. In these situations, uncertainties can make classical statistics less accurate. That is where neutrosophic statistics steps in to improve accuracy in data analysis. In this article, we consider the Birnbaum-Saunders distribution (BSD) which is very flexible and practical for real world data modeling. By integrating the neutrosophic concept, we improve the BSD's ability to manage uncertainty effectively. In addition, we provide maximum likelihood parameter estimates. Subsequently, we illustrate the practical advantages of the neutrosophic model using two cases from the industrial and environmental fields. This paper emphasizes the significance of the neutrosophic BSD as a robust solution for modeling and analysing imprecise data, filling a crucial gap left by classical statistical methods.
In relational verification, judicious alignment of computational steps facilitates proof of relations between programs using simple relational assertions. Relational Hoare logics (RHL) provide compositional rules that embody various alignments of executions. Seemingly more flexible alignments can be expressed in terms of product automata based on program transition relations. A single degenerate alignment rule (self-composition), atop a complete Hoare logic, comprises a RHL for $\forall\forall$ properties that is complete in the ordinary logical sense (Cook'78). The notion of alignment completeness was previously proposed as a more satisfactory measure, and some rules were shown to be alignment complete with respect to a few ad hoc forms of alignment automata. This paper proves alignment completeness with respect to a general class of $\forall\forall$ alignment automata, for a RHL comprised of standard rules together with a rule of semantics-preserving rewrites based on Kleene algebra with tests. A new logic for $\forall\exists$ properties is introduced and shown to be alignment complete. The $\forall\forall$ and $\forall\exists$ automata are shown to be semantically complete. Thus the logics are both complete in the ordinary sense. Recent work by D'Osualdo et al highlights the importance of completeness relative to assumptions (which we term entailment completeness), and presents $\forall\forall$ examples seemingly beyond the scope of RHLs. Additional rules enable these examples to be proved in our RHL, shedding light on the open problem of entailment completeness.
We derive information-theoretic generalization bounds for supervised learning algorithms based on the information contained in predictions rather than in the output of the training algorithm. These bounds improve over the existing information-theoretic bounds, are applicable to a wider range of algorithms, and solve two key challenges: (a) they give meaningful results for deterministic algorithms and (b) they are significantly easier to estimate. We show experimentally that the proposed bounds closely follow the generalization gap in practical scenarios for deep learning.
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