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Polynomials are common algebraic structures, which are often used to approximate functions including probability distributions. This paper proposes to directly define polynomial distributions in order to describe stochastic properties of systems rather than to assume polynomials for only approximating known or empirically estimated distributions. Polynomial distributions offer a great modeling flexibility, and often, also mathematical tractability. However, unlike canonical distributions, polynomial functions may have non-negative values in the interval of support for some parameter values, the number of their parameters is usually much larger than for canonical distributions, and the interval of support must be finite. In particular, polynomial distributions are defined here assuming three forms of polynomial function. The transformation of polynomial distributions and fitting a histogram to a polynomial distribution are considered. The key properties of polynomial distributions are derived in closed-form. A piecewise polynomial distribution construction is devised to ensure that it is non-negative over the support interval. Finally, the problems of estimating parameters of polynomial distributions and generating polynomially distributed samples are also studied.

Charge dynamics play essential role in many practical applications such as semiconductors, electrochemical devices and transmembrane ion channels. A Maxwell-Amp\`{e}re Nernst-Planck (MANP) model that describes charge dynamics via concentrations and the electric displacement is able to take effects beyond mean-field approximations into account. To obtain physically faithful numerical solutions, we develop a structure-preserving numerical method for the MANP model whose solution has several physical properties of importance. By the Slotboom transform with entropic-mean approximations, a positivity preserving scheme with Scharfetter-Gummel fluxes is derived for the generalized Nernst-Planck equations. To deal with the curl-free constraint, the dielectric displacement from the Maxwell-Amp\`{e}re equation is further updated with a local relaxation algorithm of linear computational complexity. We prove that the proposed numerical method unconditionally preserves the mass conservation and the solution positivity at the discrete level, and satisfies the discrete energy dissipation law with a time-step restriction. Numerical experiments verify that our numerical method has expected accuracy and structure-preserving properties. Applications to ion transport with large convection, arising from boundary-layer electric field and Born solvation interactions, further demonstrate that the MANP formulation with the proposed numerical scheme has attractive performance and can effectively describe charge dynamics with large convection of high numerical cell P\'{e}clet numbers.

We investigate the problem of recovering a partially observed high-rank matrix whose columns obey a nonlinear structure such as a union of subspaces, an algebraic variety or grouped in clusters. The recovery problem is formulated as the rank minimization of a nonlinear feature map applied to the original matrix, which is then further approximated by a constrained non-convex optimization problem involving the Grassmann manifold. We propose two sets of algorithms, one arising from Riemannian optimization and the other as an alternating minimization scheme, both of which include first- and second-order variants. Both sets of algorithms have theoretical guarantees. In particular, for the alternating minimization, we establish global convergence and worst-case complexity bounds. Additionally, using the Kurdyka-Lojasiewicz property, we show that the alternating minimization converges to a unique limit point. We provide extensive numerical results for the recovery of union of subspaces and clustering under entry sampling and dense Gaussian sampling. Our methods are competitive with existing approaches and, in particular, high accuracy is achieved in the recovery using Riemannian second-order methods.

Arbitrary pattern formation (\textsc{Apf}) is a well-studied problem in swarm robotics. The problem has been considered in two different settings so far; one is in a plane and another is in an infinite grid. This work deals with the problem in an infinite rectangular grid setting. The previous works in literature dealing with \textsc{Apf} problem in infinite grid had a fundamental issue. These deterministic algorithms use a lot of space in the grid to solve the problem mainly because of maintaining the asymmetry of the configuration or to avoid a collision. These solution techniques can not be useful if there is a space constraint in the application field. In this work, we consider luminous robots (with one light that can take two colors) to avoid symmetry, but we carefully designed a deterministic algorithm that solves the \textsc{Apf} problem using minimal required space in the grid. The robots are autonomous, identical, and anonymous and they operate in Look-Compute-Move cycles under a fully asynchronous scheduler. The \textsc{Apf} algorithm proposed in [WALCOM'2019] by Bose et al. can be modified using luminous robots so that it uses minimal space but that algorithm is not move-optimal. The algorithm proposed in this paper not only uses minimal space but also asymptotically move-optimal. The algorithm proposed in this work is designed for an infinite rectangular grid but it can be easily modified to work in a finite grid as well.

We show that there is no $2^{o(k^2)} n^{O(1)}$ time algorithm for Independent Set on $n$-vertex graphs with rank-width $k$, unless the Exponential Time Hypothesis (ETH) fails. Our lower bound matches the $2^{O(k^2)} n^{O(1)}$ time algorithm given by Bui-Xuan, Telle, and Vatshelle [Discret. Appl. Math., 2010] and it answers the open question of Bergougnoux and Kant\'{e} [SIAM J. Discret. Math., 2021]. We also show that the known $2^{O(k^2)} n^{O(1)}$ time algorithms for Weighted Dominating Set, Maximum Induced Matching and Feedback Vertex Set parameterized by rank-width $k$ are optimal assuming ETH. Our results are the first tight ETH lower bounds parameterized by rank-width that do not follow directly from lower bounds for $n$-vertex graphs.

This paper develops a clustering method that takes advantage of the sturdiness of model-based clustering, while attempting to mitigate some of its pitfalls. First, we note that standard model-based clustering likely leads to the same number of clusters per margin, which seems a rather artificial assumption for a variety of datasets. We tackle this issue by specifying a finite mixture model per margin that allows each margin to have a different number of clusters, and then cluster the multivariate data using a strategy game-inspired algorithm to which we call Reign-and-Conquer. Second, since the proposed clustering approach only specifies a model for the margins -- but leaves the joint unspecified -- it has the advantage of being partially parallelizable; hence, the proposed approach is computationally appealing as well as more tractable for moderate to high dimensions than a `full' (joint) model-based clustering approach. A battery of numerical experiments on artificial data indicate an overall good performance of the proposed methods in a variety of scenarios, and real datasets are used to showcase their application in practice.

Gradient-based first-order convex optimization algorithms find widespread applicability in a variety of domains, including machine learning tasks. Motivated by the recent advances in fixed-time stability theory of continuous-time dynamical systems, we introduce a generalized framework for designing accelerated optimization algorithms with strongest convergence guarantees that further extend to a subclass of non-convex functions. In particular, we introduce the \emph{GenFlow} algorithm and its momentum variant that provably converge to the optimal solution of objective functions satisfying the Polyak-{\L}ojasiewicz (PL) inequality, in a fixed-time. Moreover for functions that admit non-degenerate saddle-points, we show that for the proposed GenFlow algorithm, the time required to evade these saddle-points is bounded uniformly for all initial conditions. Finally, for strongly convex-strongly concave minimax problems whose optimal solution is a saddle point, a similar scheme is shown to arrive at the optimal solution again in a fixed-time. The superior convergence properties of our algorithm are validated experimentally on a variety of benchmark datasets.

Binarized Neural Networks (BNNs) are receiving increasing attention due to their lightweight architecture and ability to run on low-power devices. The state-of-the-art for training classification BNNs restricted to few-shot learning is based on a Mixed Integer Programming (MIP) approach. This paper proposes the BeMi ensemble, a structured architecture of BNNs based on training a single BNN for each possible pair of classes and applying a majority voting scheme to predict the final output. The training of a single BNN discriminating between two classes is achieved by a MIP model that optimizes a lexicographic multi-objective function according to robustness and simplicity principles. This approach results in training networks whose output is not affected by small perturbations on the input and whose number of active weights is as small as possible, while good accuracy is preserved. We computationally validate our model using the MNIST and Fashion-MNIST datasets using up to 40 training images per class. Our structured ensemble outperforms both BNNs trained by stochastic gradient descent and state-of-the-art MIP-based approaches. While the previous approaches achieve an average accuracy of 51.1% on the MNIST dataset, the BeMi ensemble achieves an average accuracy of 61.7% when trained with 10 images per class and 76.4% when trained with 40 images per class.

The optimal error estimate that depending only on the polynomial degree of $ \varepsilon^{-1}$ is established for the temporal semi-discrete scheme of the Cahn-Hilliard equation, which is based on the scalar auxiliary variable (SAV) formulation. The key to our analysis is to convert the structure of the SAV time-stepping scheme back to a form compatible with the original format of the Cahn-Hilliard equation, which makes it feasible to use spectral estimates to handle the nonlinear term. Based on the transformation of the SAV numerical scheme, the optimal error estimate for the temporal semi-discrete scheme which depends only on the low polynomial order of $\varepsilon^{-1}$ instead of the exponential order, is derived by using mathematical induction, spectral arguments, and the superconvergence properties of some nonlinear terms. Numerical examples are provided to illustrate the discrete energy decay property and validate our theoretical convergence analysis.

The last decade has witnessed an experimental revolution in data science and machine learning, epitomised by deep learning methods. Indeed, many high-dimensional learning tasks previously thought to be beyond reach -- such as computer vision, playing Go, or protein folding -- are in fact feasible with appropriate computational scale. Remarkably, the essence of deep learning is built from two simple algorithmic principles: first, the notion of representation or feature learning, whereby adapted, often hierarchical, features capture the appropriate notion of regularity for each task, and second, learning by local gradient-descent type methods, typically implemented as backpropagation. While learning generic functions in high dimensions is a cursed estimation problem, most tasks of interest are not generic, and come with essential pre-defined regularities arising from the underlying low-dimensionality and structure of the physical world. This text is concerned with exposing these regularities through unified geometric principles that can be applied throughout a wide spectrum of applications. Such a 'geometric unification' endeavour, in the spirit of Felix Klein's Erlangen Program, serves a dual purpose: on one hand, it provides a common mathematical framework to study the most successful neural network architectures, such as CNNs, RNNs, GNNs, and Transformers. On the other hand, it gives a constructive procedure to incorporate prior physical knowledge into neural architectures and provide principled way to build future architectures yet to be invented.