This paper is concerned with a Bayesian approach to testing hypotheses in statistical inverse problems. Based on the posterior distribution $\Pi \left(\cdot |Y = y\right)$, we want to infer whether a feature $\left\langle\varphi, u^\dagger\right\rangle$ of the unknown quantity of interest $u^\dagger$ is positive. This can be done by the so-called maximum a posteriori test. We provide a frequentistic analysis of this test's properties such as level and power, and prove that it is a regularized test in the sense of Kretschmann et al. (2024). Furthermore we provide lower bounds for its power under classical spectral source conditions in case of Gaussian priors. Numerical simulations illustrate its superior performance both in moderately and severely ill-posed situations.
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The recent paper (IEEE Trans. IT 69, 1680) introduced an analytical method for calculating the channel capacity without the need for iteration. This method has certain limitations that restrict its applicability. Furthermore, the paper does not provide an explanation as to why the channel capacity can be solved analytically in this particular case. In order to broaden the scope of this method and address its limitations, we turn our attention to the reverse em-problem, proposed by Toyota (Information Geometry, 3, 1355 (2020)). This reverse em-problem involves iteratively applying the inverse map of the em iteration to calculate the channel capacity, which represents the maximum mutual information. However, several open problems remained unresolved in Toyota's work. To overcome these challenges, we formulate the reverse em-problem based on Bregman divergence and provide solutions to these open problems. Building upon these results, we transform the reverse em-problem into em-problems and derive a non-iterative formula for the reverse em-problem. This formula can be viewed as a generalization of the aforementioned analytical calculation method. Importantly, this derivation sheds light on the information geometrical structure underlying this special case. By effectively addressing the limitations of the previous analytical method and providing a deeper understanding of the underlying information geometrical structure, our work significantly expands the applicability of the proposed method for calculating the channel capacity without iteration.
Distillation is the task of replacing a complicated machine learning model with a simpler model that approximates the original [BCNM06,HVD15]. Despite many practical applications, basic questions about the extent to which models can be distilled, and the runtime and amount of data needed to distill, remain largely open. To study these questions, we initiate a general theory of distillation, defining PAC-distillation in an analogous way to PAC-learning [Val84]. As applications of this theory: (1) we propose new algorithms to extract the knowledge stored in the trained weights of neural networks -- we show how to efficiently distill neural networks into succinct, explicit decision tree representations when possible by using the ``linear representation hypothesis''; and (2) we prove that distillation can be much cheaper than learning from scratch, and make progress on characterizing its complexity.
In this paper, a class of high-order methods to numerically solve Functional Differential Equations with Piecewise Continuous Arguments (FDEPCAs) is discussed. The framework stems from the expansion of the vector field associated with the reference differential equation along the shifted and scaled Legendre polynomial orthonormal basis, working on a suitable extension of Hamiltonian Boundary Value Methods. Within the design of the methods, a proper generalization of the perturbation results coming from the field of ordinary differential equations is considered, with the aim of handling the case of FDEPCAs. The error analysis of the devised family of methods is performed, while a few numerical tests on Hamiltonian FDEPCAs are provided, to give evidence to the theoretical findings and show the effectiveness of the obtained resolution strategy.
In logistic regression modeling, Firth's modified estimator is widely used to address the issue of data separation, which results in the nonexistence of the maximum likelihood estimate. Firth's modified estimator can be formulated as a penalized maximum likelihood estimator in which Jeffreys' prior is adopted as the penalty term. Despite its widespread use in practice, the formal verification of the corresponding estimate's existence has not been established. In this study, we establish the existence theorem of Firth's modified estimate in binomial logistic regression models, assuming only the full column rankness of the design matrix. We also discuss multinomial logistic regression models. Unlike the binomial regression case, we show through an example that the Jeffreys-prior penalty term does not necessarily diverge to negative infinity as the parameter diverges.
This paper proposes a~simple, yet powerful, method for balancing distributions of covariates for causal inference based on observational studies. The method makes it possible to balance an arbitrary number of quantiles (e.g., medians, quartiles, or deciles) together with means if necessary. The proposed approach is based on the theory of calibration estimators (Deville and S\"arndal 1992), in particular, calibration estimators for quantiles, proposed by Harms and Duchesne (2006). The method does not require numerical integration, kernel density estimation or assumptions about the distributions. Valid estimates can be obtained by drawing on existing asymptotic theory. An~illustrative example of the proposed approach is presented for the entropy balancing method and the covariate balancing propensity score method. Results of a~simulation study indicate that the method efficiently estimates average treatment effects on the treated (ATT), the average treatment effect (ATE), the quantile treatment effect on the treated (QTT) and the quantile treatment effect (QTE), especially in the presence of non-linearity and mis-specification of the models. The proposed approach can be further generalized to other designs (e.g. multi-category, continuous) or methods (e.g. synthetic control method). An open source software implementing proposed methods is available.
The paper presents a numerical method for simulating flow and mechanics in fractured rock. The governing equations that couple the effects in the rock mass and in the fractures are obtained using the discrete fracture-matrix approach. The fracture flow is driven by the cubic law, and the contact conditions prevent fractures from self-penetration. A stable finite element discretization is proposed for the displacement-pressure-flux formulation. The resulting nonlinear algebraic system of equations and inequalities is decoupled using a robust iterative splitting into the linearized flow subproblem, and the quadratic programming problem for the mechanical part. The non-penetration conditions are solved by means of dualization and an optimal quadratic programming algorithm. The capability of the numerical scheme is demonstrated on a benchmark problem for tunnel excavation with hundreds of fractures in 3D. The paper's novelty consists in a combination of three crucial ingredients: (i) application of discrete fracture-matrix approach to poroelasticity, (ii) robust iterative splitting of resulting nonlinear algebraic system working for real-world 3D problems, and (iii) efficient solution of its mechanical quadratic programming part with a large number of fractures in mutual contact by means of own solvers implemented into an in-house software library.
This paper is devoted to the analysis of a numerical scheme based on the Finite Element Method for approximating the solution of Koiter's model for a linearly elastic elliptic membrane shell subjected to remaining confined in a prescribed half-space. First, we show that the solution of the obstacle problem under consideration is uniquely determined and satisfies a set of variational inequalities which are governed by a fourth order elliptic operator, and which are posed over a non-empty, closed, and convex subset of a suitable space. Second, we show that the solution of the obstacle problem under consideration can be approximated by means of the penalty method. Third, we show that the solution of the corresponding penalised problem is more regular up to the boundary. Fourth, we write down the mixed variational formulation corresponding to the penalised problem under consideration, and we show that the solution of the mixed variational formulation is more regular up to the boundary as well. In view of this result concerning the augmentation of the regularity of the solution of the mixed penalised problem, we are able to approximate the solution of the one such problem by means of a Finite Element scheme. Finally, we present numerical experiments corroborating the validity of the mathematical results we obtained.
Maximum entropy (Maxent) models are a class of statistical models that use the maximum entropy principle to estimate probability distributions from data. Due to the size of modern data sets, Maxent models need efficient optimization algorithms to scale well for big data applications. State-of-the-art algorithms for Maxent models, however, were not originally designed to handle big data sets; these algorithms either rely on technical devices that may yield unreliable numerical results, scale poorly, or require smoothness assumptions that many practical Maxent models lack. In this paper, we present novel optimization algorithms that overcome the shortcomings of state-of-the-art algorithms for training large-scale, non-smooth Maxent models. Our proposed first-order algorithms leverage the Kullback-Leibler divergence to train large-scale and non-smooth Maxent models efficiently. For Maxent models with discrete probability distribution of $n$ elements built from samples, each containing $m$ features, the stepsize parameters estimation and iterations in our algorithms scale on the order of $O(mn)$ operations and can be trivially parallelized. Moreover, the strong $\ell_{1}$ convexity of the Kullback--Leibler divergence allows for larger stepsize parameters, thereby speeding up the convergence rate of our algorithms. To illustrate the efficiency of our novel algorithms, we consider the problem of estimating probabilities of fire occurrences as a function of ecological features in the Western US MTBS-Interagency wildfire data set. Our numerical results show that our algorithms outperform the state of the arts by one order of magnitude and yield results that agree with physical models of wildfire occurrence and previous statistical analyses of wildfire drivers.
Inverse problem theory is often studied in the ideal infinite-dimensional setting. Through the lens of the PDE-constrained optimization, the well-posedness PDE theory suggests unique reconstruction of the parameter function that attain the zero-loss property of the mismatch function, when infinite amount of data is provided. Unfortunately, this is not the case in practice, when we are limited to finite amount of measurements due to experimental or economical reasons. Consequently, one must compromise the inference goal to a discrete approximation of the unknown smooth function. What is the reconstruction power of a fixed number of data observations? How many parameters can one reconstruct? Here we describe a probabilistic approach, and spell out the interplay of the observation size $(r)$ and the number of parameters to be uniquely identified $(m)$. The technical pillar is the random sketching strategy, in which the matrix concentration inequality and sampling theory are largely employed. By analyzing randomly sub-sampled Hessian matrix, we attain well-conditioned reconstruction problem with high probability. Our main theory is finally validated in numerical experiments. We set tests on both synthetic and the data from an elliptic inverse problem. The empirical performance shows that given suitable sampling quality, the well-conditioning of the sketched Hessian is certified with high probability.
In this paper we consider the generalized inverse iteration for computing ground states of the Gross-Pitaevskii eigenvector problem (GPE). For that we prove explicit linear convergence rates that depend on the maximum eigenvalue in magnitude of a weighted linear eigenvalue problem. Furthermore, we show that this eigenvalue can be bounded by the first spectral gap of a linearized Gross-Pitaevskii operator, recovering the same rates as for linear eigenvector problems. With this we establish the first local convergence result for the basic inverse iteration for the GPE without damping. We also show how our findings directly generalize to extended inverse iterations, such as the Gradient Flow Discrete Normalized (GFDN) proposed in [W. Bao, Q. Du, SIAM J. Sci. Comput., 25 (2004)] or the damped inverse iteration suggested in [P. Henning, D. Peterseim, SIAM J. Numer. Anal., 53 (2020)]. Our analysis also reveals why the inverse iteration for the GPE does not react favourably to spectral shifts. This empirical observation can now be explained with a blow-up of a weighting function that crucially contributes to the convergence rates. Our findings are illustrated by numerical experiments.
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