We demonstrate that Assembly Theory, pathway complexity, the assembly index, and the assembly number are subsumed and constitute a weak version of algorithmic (Kolmogorov-Solomonoff-Chaitin) complexity reliant on an approximation method based upon statistical compression, their results obtained due to the use of methods strictly equivalent to the LZ family of compression algorithms used in compressing algorithms such as ZIP, GZIP, or JPEG. Such popular algorithms have been shown to empirically reproduce the results of AT's assembly index and their use had already been reported in successful application to separating organic from non-organic molecules, and the study of selection and evolution. Here we exhibit and prove the connections and full equivalence of Assembly Theory to Shannon Entropy and statistical compression, and AT's disconnection as a statistical approach from causality. We demonstrate that formulating a traditional statistically compressed description of molecules, or the theory underlying it, does not imply an explanation or quantification of biases in generative (physical or biological) processes, including those brought about by selection and evolution, when lacking in logical consistency and empirical evidence. We argue that in their basic arguments, the authors of AT conflate how objects may assemble with causal directionality, and conclude that Assembly Theory does nothing to explain selection or evolution beyond known and previously established connections, some of which are reviewed.
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Splitting methods are a widely used numerical scheme for solving convection-diffusion problems. However, they may lose stability in some situations, particularly when applied to convection-diffusion problems in the presence of an unbounded convective term. In this paper, we propose a new splitting method, called the "Adapted Lie splitting method", which successfully overcomes the observed instability in certain cases. Assuming that the unbounded coefficient belongs to a suitable Lorentz space, we show that the adapted Lie splitting converges to first-order under the analytic semigroup framework. Furthermore, we provide numerical experiments to illustrate our newly proposed splitting approach.
Many analyses of multivariate data focus on evaluating the dependence between two sets of variables, rather than the dependence among individual variables within each set. Canonical correlation analysis (CCA) is a classical data analysis technique that estimates parameters describing the dependence between such sets. However, inference procedures based on traditional CCA rely on the assumption that all variables are jointly normally distributed. We present a semiparametric approach to CCA in which the multivariate margins of each variable set may be arbitrary, but the dependence between variable sets is described by a parametric model that provides low-dimensional summaries of dependence. While maximum likelihood estimation in the proposed model is intractable, we propose two estimation strategies: one using a pseudolikelihood for the model and one using a Markov chain Monte Carlo (MCMC) algorithm that provides Bayesian estimates and confidence regions for the between-set dependence parameters. The MCMC algorithm is derived from a multirank likelihood function, which uses only part of the information in the observed data in exchange for being free of assumptions about the multivariate margins. We apply the proposed Bayesian inference procedure to Brazilian climate data and monthly stock returns from the materials and communications market sectors.
In this work is considered a spectral problem, involving a second order term on the domain boundary: the Laplace-Beltrami operator. A variational formulation is presented, leading to a finite element discretization. For the Laplace-Beltrami operator to make sense on the boundary, the domain is smooth: consequently the computational domain (classically a polygonal domain) will not match the physical one. Thus, the physical domain is discretized using high order curved meshes so as to reduce the \textit{geometric error}. The \textit{lift operator}, which is aimed to transform a function defined on the mesh domain into a function defined on the physical one, is recalled. This \textit{lift} is a key ingredient in estimating errors on eigenvalues and eigenfunctions. A bootstrap method is used to prove the error estimates, which are expressed both in terms of \textit{finite element approximation error} and of \textit{geometric error}, respectively associated to the finite element degree $k\ge 1$ and to the mesh order~$r\ge 1$. Numerical experiments are led on various smooth domains in 2D and 3D, which allow us to validate the presented theoretical results.
Generalized cross-validation (GCV) is a widely-used method for estimating the squared out-of-sample prediction risk that employs a scalar degrees of freedom adjustment (in a multiplicative sense) to the squared training error. In this paper, we examine the consistency of GCV for estimating the prediction risk of arbitrary ensembles of penalized least-squares estimators. We show that GCV is inconsistent for any finite ensemble of size greater than one. Towards repairing this shortcoming, we identify a correction that involves an additional scalar correction (in an additive sense) based on degrees of freedom adjusted training errors from each ensemble component. The proposed estimator (termed CGCV) maintains the computational advantages of GCV and requires neither sample splitting, model refitting, or out-of-bag risk estimation. The estimator stems from a finer inspection of the ensemble risk decomposition and two intermediate risk estimators for the components in this decomposition. We provide a non-asymptotic analysis of the CGCV and the two intermediate risk estimators for ensembles of convex penalized estimators under Gaussian features and a linear response model. Furthermore, in the special case of ridge regression, we extend the analysis to general feature and response distributions using random matrix theory, which establishes model-free uniform consistency of CGCV.
In this paper, we plan to show an eigenvalue algorithm for block Hessenberg matrices by using the idea of non-commutative integrable systems and matrix-valued orthogonal polynomials. We introduce adjacent families of matrix-valued $\theta$-deformed bi-orthogonal polynomials, and derive corresponding discrete non-commutative hungry Toda lattice from discrete spectral transformations for polynomials. It is shown that this discrete system can be used as a pre-precessing algorithm for block Hessenberg matrices. Besides, some convergence analysis and numerical examples of this algorithm are presented.
A CUR factorization is often utilized as a substitute for the singular value decomposition (SVD), especially when a concrete interpretation of the singular vectors is challenging. Moreover, if the original data matrix possesses properties like nonnegativity and sparsity, a CUR decomposition can better preserve them compared to the SVD. An essential aspect of this approach is the methodology used for selecting a subset of columns and rows from the original matrix. This study investigates the effectiveness of \emph{one-round sampling} and iterative subselection techniques and introduces new iterative subselection strategies based on iterative SVDs. One provably appropriate technique for index selection in constructing a CUR factorization is the discrete empirical interpolation method (DEIM). Our contribution aims to improve the approximation quality of the DEIM scheme by iteratively invoking it in several rounds, in the sense that we select subsequent columns and rows based on the previously selected ones. Thus, we modify $A$ after each iteration by removing the information that has been captured by the previously selected columns and rows. We also discuss how iterative procedures for computing a few singular vectors of large data matrices can be integrated with the new iterative subselection strategies. We present the results of numerical experiments, providing a comparison of one-round sampling and iterative subselection techniques, and demonstrating the improved approximation quality associated with using the latter.
A dataset with two labels is linearly separable if it can be split into its two classes with a hyperplane. This inflicts a curse on some statistical tools (such as logistic regression) but forms a blessing for others (e.g. support vector machines). Recently, the following question has regained interest: What is the probability that the data are linearly separable? We provide a formula for the probability of linear separability for Gaussian features and labels depending only on one marginal of the features (as in generalized linear models). In this setting, we derive an upper bound that complements the recent result by Hayakawa, Lyons, and Oberhauser [2023], and a sharp upper bound for sign-flip noise. To prove our results, we exploit that this probability can be expressed as a sum of the intrinsic volumes of a polyhedral cone of the form $\text{span}\{v\}\oplus[0,\infty)^n$, as shown in Cand\`es and Sur [2020]. After providing the inequality description for this cone, and an algorithm to project onto it, we calculate its intrinsic volumes. In doing so, we encounter Youden's demon problem, for which we provide a formula following Kabluchko and Zaporozhets [2020]. The key insight of this work is the following: The number of correctly labeled observations in the data affects the structure of this polyhedral cone, allowing the translation of insights from geometry into statistics.
This paper studies the convergence of a spatial semidiscretization of a three-dimensional stochastic Allen-Cahn equation with multiplicative noise. For non-smooth initial values, the regularity of the mild solution is investigated, and an error estimate is derived with the spatial $ L^2 $-norm. For smooth initial values, two error estimates with the general spatial $ L^q $-norms are established.
We propose new linear combinations of compositions of a basic second-order scheme with appropriately chosen coefficients to construct higher order numerical integrators for differential equations. They can be considered as a generalization of extrapolation methods and multi-product expansions. A general analysis is provided and new methods up to order 8 are built and tested. The new approach is shown to reduce the latency problem when implemented in a parallel environment and leads to schemes that are significantly more efficient than standard extrapolation when the linear combination is delayed by a number of steps.
When writing high-performance code for numerical computation in a scripting language like MATLAB, it is crucial to have the operations in a large for-loop vectorized. If not, the code becomes too slow to use, even for a moderately large problem. However, in the process of vectorizing, the code often loses its original structure and becomes less readable. This is particularly true in the case of a finite element implementation, even though finite element methods are inherently structured. A basic remedy to this is the separation of the vectorization part from the mathematics part of the code, which is easily achieved through building the code on top of the basic linear algebra subprograms that are already vectorized codes, an idea that has been used in a series of papers over the last fifteen years, developing codes that are fast and still structured and readable. We discuss the vectorized basic linear algebra package and introduce a formalism using multi-linear algebra to explain and define formally the functions in the package, as well as MATLAB pagetime functions. We provide examples from computations of varying complexity, including the computation of normal vectors, volumes, and finite element methods. Benchmarking shows that we also get fast computations. Using the library, we can write codes that closely follow our mathematical thinking, making writing, following, reusing, and extending the code easier.
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