Broad Class Of Functions Research Articles

Multi-Server Multi-Function Distributed Computation.

The work here studies the communication cost for a multi-server multi-task distributed computation framework, as well as for a broad class of functions and data statistics. Considering the framework where a user seeks the computation of multiple complex (conceivably non-linear) tasks from a set of distributed servers, we establish the communication cost upper bounds for a variety of data statistics, function classes, and data placements across the servers. To do so, we proceed to apply, for the first time here, Körner's characteristic graph approach-which is known to capture the structural properties of data and functions-to the promising framework of multi-server multi-task distributed computing. Going beyond the general expressions, and in order to offer clearer insight, we also consider the well-known scenario of cyclic dataset placement and linearly separable functions over the binary field, in which case, our approach exhibits considerable gains over the state of the art. Similar gains are identified for the case of multi-linear functions.

On Constrained Mixed-Integer DR-Submodular Minimization

Diminishing returns (DR)–submodular functions encompass a broad class of functions that are generally nonconvex and nonconcave. We study the problem of minimizing any DR-submodular function with continuous and general integer variables under box constraints and, possibly, additional monotonicity constraints. We propose valid linear inequalities for the epigraph of any DR-submodular function under the constraints. We further provide the complete convex hull of such an epigraph, which, surprisingly, turns out to be polyhedral. We propose a polynomial-time exact separation algorithm for our proposed valid inequalities with which we first establish the polynomial-time solvability of this class of mixed-integer nonlinear optimization problems. Funding: This work was supported by the Office of Naval Research Global [Grant N00014-22-1-2602].

The Name Is Barrel, β-Barrel.

β-barrels are a class of membrane proteins made up of a cylindrical, anti-parallel β-sheet with a hydrophobic exterior and a hydrophilic interior. The majority of proteins found in the outer membranes (OMs) of Gram-negative bacteria, mitochondria, and chloroplasts are β-barrel outer membrane proteins (OMPs). β-barrel OMPs have a diverse repertoire of functions, including nutrient transport, secretion, bacterial virulence, and enzymatic activity. Here, we discuss the broad functional classes of β-barrel OMPs, how they are folded into the membrane, and the future of β-barrel OMP research and its applications.

Language for Description of Worlds. Part 1: Theoretical Foundation

We will reduce the task of creating AI to the task of finding an appropriate language for description of the world. This will not be a programing language because programing languages describe only computable functions, while our language will describe a somewhat broader class of functions. Another specificity of this language will be that the description will consist of separate modules. This will enable us look for the description of the world automatically such that we discover it module after module. Our approach to the creation of this new language will be to start with a particular world and write the description of that particular world. The point is that the language which can describe this particular world will be appropriate for describing any world. This is the first part of the paper. In this part we will define the basic theoretical concepts which will we will need when creating the language we are looking for.

A possible genomic footprint of polygenic adaptation on population divergence in seed beetles?

Efforts to unravel the genomic basis of incipient speciation are hampered by a mismatch between our toolkit and our understanding of the ecology and genetics of adaptation. While the former is focused on detecting selective sweeps involving few independently acting or linked speciation genes, the latter states that divergence typically occurs in polygenic traits under stabilizing selection. Here, we ask whether a role of stabilizing selection on polygenic traits in population divergence may be unveiled by using a phenotypically informed integrative approach, based on genome‐wide variation segregating in divergent populations. We compare three divergent populations of seed beetles (Callosobruchus maculatus) where previous work has demonstrated a prominent role for stabilizing selection on, and population divergence in, key life history traits that reflect rate‐dependent metabolic processes. We derive and assess predictions regarding the expected pattern of covariation between genetic variation segregating within populations and genetic differentiation between populations. Population differentiation was considerable (mean F ST = 0.23–0.26) and was primarily built by genes showing high selective constraints and an imbalance in inferred selection in different populations (positive Tajima's D NS in one and negative in one), and this set of genes was enriched with genes with a metabolic function. Repeatability of relative population differentiation was low at the level of individual genes but higher at the level of broad functional classes, again spotlighting metabolic genes. Absolute differentiation (d XY) showed a very different general pattern at this scale of divergence, more consistent with an important role for genetic drift. Although our exploration is consistent with stabilizing selection on polygenic metabolic phenotypes as an important engine of genome‐wide relative population divergence and incipient speciation in our study system, we note that it is exceedingly difficult to firmly exclude other scenarios.

A δ

Over the last decade, automatic differentiation (AD) has profoundly impacted graphics and vision applications --- both broadly via deep learning and specifically for inverse rendering. Traditional AD methods ignore gradients at discontinuities, instead treating functions as continuous. Rendering algorithms intrinsically rely on discontinuities, crucial at object silhouettes and in general for any branching operation. Researchers have proposed fully- automatic differentiation approaches for handling discontinuities by restricting to affine functions, or semi- automatic processes restricted either to invertible functions or to specialized applications like vector graphics. This paper describes a compiler-based approach to extend reverse mode AD so as to accept arbitrary programs involving discontinuities. Our novel gradient rules generalize differentiation to work correctly, assuming there is a single discontinuity in a local neighborhood, by approximating the prefiltered gradient over a box kernel oriented along a 1D sampling axis. We describe when such approximation rules are first-order correct, and show that this correctness criterion applies to a relatively broad class of functions. Moreover, we show that the method is effective in practice for arbitrary programs, including features for which we cannot prove correctness. We evaluate this approach on procedural shader programs, where the task is to optimize unknown parameters in order to match a target image, and our method outperforms baselines in terms of both convergence and efficiency. Our compiler outputs gradient programs in TensorFlow, PyTorch (for quick prototypes) and Halide with an optional auto-scheduler (for efficiency). The compiler also outputs GLSL that renders the target image, allowing users to interactively modify and animate the shader, which would otherwise be cumbersome in other representations such as triangle meshes or vector art.

Global Linear Convergence of Evolution Strategies on More than Smooth Strongly Convex Functions

Evolution strategies (ESs) are zeroth-order stochastic black-box optimization heuristics invariant to monotonic transformations of the objective function. They evolve a multivariate normal distribution, from which candidate solutions are generated. Among different variants, CMA-ES is nowadays recognized as one of the state-of-the-art zeroth-order optimizers for difficult problems. Despite ample empirical evidence that ESs with a step-size control mechanism converge linearly, theoretical guarantees of linear convergence of ESs have been established only on limited classes of functions. In particular, theoretical results on convex functions are missing, where zeroth-order and also first-order optimization methods are often analyzed. In this paper, we establish almost sure linear convergence and a bound on the expected hitting time of an ES family, namely, the $(1+1)_\kappa$-ES, which includes the (1+1)-ES with (generalized) one-fifth success rule and an abstract covariance matrix adaptation with bounded condition number, on a broad class of functions. The analysis holds for monotonic transformations of positively homogeneous functions and of quadratically bounded functions, the latter of which particularly includes monotonic transformation of strongly convex functions with Lipschitz continuous gradient. As far as the authors know, this is the first work that proves linear convergence of ES on such a broad class of functions.

Stabilization of spline bases by extension

We present a method to stabilize bases with local supports by means of extension. It generalizes the known approach for tensor product B-splines to a much broader class of functions, which includes hierarchical and weighted variants of polynomial, trigonometric, and exponential splines, but also box splines, T-splines, and other function spaces of interest with a local basis. Extension removes elements that cause instabilities from a given basis by linking them with the remaining ones by means of a specific linear combination. The two guiding principles for this process are locality and persistence. Locality aims at coupling basis functions whose supports are close together, while persistence guarantees that a given set of globally supported functions, like certain monomials in the case of polynomial splines, remain in the span of the basis after extension. Furthermore, we study how extension influences the approximation power and the condition of Gramian matrices associated with the basis, and present a series of examples illustrating the potential of the method.

ТОЧНЫЕ ОЦЕНКИ И РАДИУСЫ ВЫПУКЛОСТИ НЕКОТОРЫХ КЛАССОВ АНАЛИТИЧЕСКИХ ФУНКЦИЙ

The study of the geometric properties of analytic functions is one of the classical problems of the theory of functions of a complex variable and has been of steady interest to many mathematicians for more than half a century now. At the same time, a separate area is the building of sufficient conditions of one-leaf analytic functions, including finding the conditions for simple geometric properties of analytic functions (convex or star-shaped, almost starshaped, etc.). The solution of these problems in many cases is associated with finding estimates in different classes of analytical functions, which in itself is also a relevant problem. This article is devoted to finding exact estimates of analytic functions and their derivatives in fairly broad classes of functions, which are distinguished in the form of some restrictions on the domains obtained from the domains of values of these functions by circular symmetrization or symmetrization with respect to a straight line. Based on these results, the exact radii of convexity in some classes of functions are found.

PANNZER-A practical tool for protein function prediction.

The facility of next-generation sequencing has led to an explosion of gene catalogs for novel genomes, transcriptomes and metagenomes, which are functionally uncharacterized. Computational inference has emerged as a necessary substitute for first-hand experimental evidence. PANNZER (Protein ANNotation with Z-scoRE) is a high-throughput functional annotation web server that stands out among similar publically accessible web servers in supporting submission of up to 100,000 protein sequences at once and providing both Gene Ontology (GO) annotations and free text description predictions. Here, we demonstrate the use of PANNZER and discuss future plans and challenges. We present two case studies to illustrate problems related to data quality and method evaluation. Some commonly used evaluation metrics and evaluation datasets promote methods that favor unspecific and broad functional classes over more informative and specific classes. We argue that this can bias the development of automated function prediction methods. The PANNZER web server and source code are available at http://ekhidna2.biocenter.helsinki.fi/sanspanz/.

MIXED FRACTIONAL INTEGRATION AND DIFFERENTIATION AS RECIPROCAL OPERATIONS

We study the question of the composition of the mixed fractional integral and the mixed fractional derivative in sufficiently broad class of functions. The treatment formula for mixed fractional derivative is obtained.

The extension of the linear inequality method for generalized rational Chebyshev approximation to approximation by general quasilinear functions

In this paper, we demonstrate that a well-known linear inequality method developed for rational Chebyshev approximation is equivalent to the application of the bisection method used in quasiconvex optimization. Although this correspondence is not surprising, it naturally connects rational and generalized rational Chebyshev approximation problems with modern developments in the area of quasiconvex functions and, therefore, offers more theoretical and computational tools for solving this problem. The second important contribution of this paper is the extension of the linear inequality method to a broader class of Chebyshev approximation problems, where the corresponding objective functions remain quasiconvex. In this broader class of functions, the inequalities are no longer required to be linear: it is enough for each inequality to define a convex set and the computational challenge is in solving the corresponding convex feasibility problems. Therefore, we propose a more systematic and general approach for treating Chebyshev approximation problems. In particular, we are looking at the problems where the approximations are quasilinear functions with respect to their parameters that are also the decision variables in the corresponding optimization problems.

Geometric Programming for Optimal Positive Linear Systems

This paper studies the parameter tuning problem of positive linear systems for optimizing their stability properties. We specifically show that, under certain regularity assumptions on the parametrization, the problem of finding the minimum-cost parameters that achieve a given requirement on a system norm reduces to a \emph{geometric program}, which in turn can be exactly and efficiently solved by convex optimization. The flexibility of geometric programming allows the state, input, and output matrices of the system to simultaneously depend on the parameters to be tuned. The class of system norms under consideration includes the $H^2$ norm, $H^\infty$ norm, Hankel norm, and Schatten $p$-norm. Also, the parameter tuning problem for ensuring the robust stability of the system under structural uncertainties is shown to be solved by geometric programming. The proposed optimization framework is further extended to delayed positive linear systems, where it is shown that the parameter tunning problem jointly constrained by the exponential decay rate, the $\mathcal L^1$-gain, and the $\mathcal L^\infty$-gain can be solved by convex optimization. The assumption on the system parametrization is stated in terms of posynomial functions, which form a broad class of functions and thus allow us to deal with various interesting positive linear systems arising from, for example, dynamical buffer networks and epidemic spreading processes. We present numerical examples to illustrate the effectiveness of the proposed optimization framework.

A distributed control approach for heterogeneous linear multiagent systems

This paper considers an internal model-based distributed control approach to the cooperative output regulation problem of heterogeneous linear time-invariant multiagent systems over fixed directed communication graph topologies. First, a new definition of the linear cooperative output regulation problem is introduced in order to allow a broad class of functions to be tracked and rejected by a network of agents. Second, the solvability of this problem with three distributed control laws, namely dynamic state feedback, dynamic output feedback with local measurement and dynamic output feedback, is investigated by first considering a global condition and then providing an agent-wise local sufficient condition under standard assumptions. Finally, two numerical examples are provided to illustrate the selected contributions of this paper.

Greedy Maximization of Functions with Bounded Curvature under Partition Matroid Constraints

We investigate the performance of a deterministic GREEDY algorithm for the problem of maximizing functions under a partition matroid constraint. We consider non-monotone submodular functions and monotone subadditive functions. Even though constrained maximization problems of monotone submodular functions have been extensively studied, little is known about greedy maximization of non-monotone submodular functions or monotone subadditive functions.
 We give approximation guarantees for GREEDY on these problems, in terms of the curvature. We find that this simple heuristic yields a strong approximation guarantee on a broad class of functions.
 We discuss the applicability of our results to three real-world problems: Maximizing the determinant function of a positive semidefinite matrix, and related problems such as the maximum entropy sampling problem, the constrained maximum cut problem on directed graphs, and combinatorial auction games.
 We conclude that GREEDY is well-suited to approach these problems. Overall, we present evidence to support the idea that, when dealing with constrained maximization problems with bounded curvature, one needs not search for (approximate) monotonicity to get good approximate solutions.

The V-line transform with some generalizations and cone differentiation

The paper studies various properties of the V-line transform (VLT) in the plane and the conical Radon transform (CRT) in . The VLT maps a function to a family of its integrals along trajectories made of two rays emanating from a common point. The CRT considered in this paper maps a function to a set of its integrals over surfaces of polyhedral cones. These types of operators appear in mathematical models of single scattering optical tomography, Compton camera imaging and other applications. We derive new explicit inversion formulae for the VLT and the CRT, as well as proving some previously known results using more intuitive geometric ideas. Using our inversion formula for the VLT, we describe the range of that transformation when applied to a fairly broad class of functions and prove some support theorems. The efficiency of our method is demonstrated on several numerical examples. As an auxiliary result that plays a big role in this article, we derive a generalization of the fundamental theorem of calculus, which we call the cone differentiation theorem.

Scalable Proximal Jacobian Iteration Method With Global Convergence Analysis for Nonconvex Unconstrained Composite Optimizations.

The recent studies have found that the nonconvex relaxation functions usually perform better than the convex counterparts in the l0 -norm and rank function minimization problems. However, due to the absence of convexity in these nonconvex problems, developing efficient algorithms with convergence guarantee becomes very challenging. Inspired by the basic ideas of both the Jacobian alternating direction method of multipliers (JADMMs) for solving linearly constrained problems with separable objectives and the proximal gradient methods (PGMs) for optimizing the unconstrained problems with one variable, this paper focuses on extending the PGMs to the proximal Jacobian iteration methods (PJIMs) for handling with a family of nonconvex composite optimization problems with two splitting variables. To reduce the total computational complexity by decreasing the number of iterations, we devise the accelerated version of PJIMs through the well-known Nesterov's acceleration strategy and further extend both to solve the multivariable cases. Most importantly, we provide a rigorous convergence analysis, in theory, to show that the generated variable sequence globally converges to a critical point by exploiting the Kurdyka-Łojasiewica (KŁ) property for a broad class of functions. Furthermore, we also establish the linear and sublinear convergence rates of the obtained variable sequence in the objective function. As the specific application to the nonconvex sparse and low-rank recovery problems, several numerical experiments can verify that the newly proposed algorithms not only keep fast convergence speed but also have high precision.

Bounded projectors and duality in the spaces of functions holomorphic in the unit ball

The paper studies the Banach spaces h ∞(φ), h 0(φ), and h 1(η) of harmonic functions over the unit ball in R n . These spaces depend on a weight function φ and a weight measure η. For a given function φ from a sufficiently broad class of functions, we solve the duality problem. that is, we construct measures η such that h 1(η)* ∼ h ∞(φ) and h 0(φ)* ∼ h 1(η).

On the positive definiteness of some functions related to the Schoenberg problem

For a broad class of functions f: [0,+∞) → ℝ, we prove that the function f(ρ λ(x)) is positive definite on a nontrivial real linear space E if and only if 0 ≤ λ ≤ α(E, ρ). Here ρ is a nonnegative homogeneous function on E such that ρ(x) ≢ 0 and α(E, ρ) is the Schoenberg constant.

Bone and ivory manufacturing at Ebla (Syria) during the Early and Middle Bronze Age (c. 2500–1600 BC)

The corpus of bone tools and other objects from Tell Mardikh-Ebla constitutes a basis for studying the role and development of bone working at this urban centre between 2500 BC and 1600 BC. A preliminary assessment of bone artefact production that takes into account typology, distribution, and relation with other broad functional classes of items is presented, underlining trends and changes over time. During the Early Bronze Age III–IV a specific local production is attested through decorated bone sticks/pins. Decorated tubes are documented by three exemplars, showing that this kind of bone artefact — widespread in the Levant and Anatolia — was also present at Ebla. Bone tools were also widely used. Craft workers producing figurative and patterned inlays and beads preferred other materials, such as semi-precious stones and shell. A trend towards a dichotomy between the highly-specialized manufacture of valuable objects, such as inlays and incrustations linked to the ivory industry, and the production of tools that seem to have been focused upon spinning and weaving activities is evident during the Middle Bronze Age, when the number of bone tools decreased sharply.