Given a set of squares and a strip of bounded width and infinite height, we consider a square strip packaging problem, which we call the square independent packing problem (SIPP), to minimize the strip height so that all the squares are packed into independent cells separated by horizontal and vertical partitions. For the SIPP, we first investigate efficient solution representations and propose a compact representation that reduces the search space from $\Omega(n!)$ to $O(2^n)$, with $n$ the number of given squares, while guaranteeing that there exists a solution representation that corresponds to an optimal solution. Based on the solution representation, we show that the problem is NP-hard, and then we propose a fully polynomial-time approximation scheme (FPTAS) to solve it. We also propose three mathematical programming formulations based on different solution representations and confirm the performance of these algorithms through computational experiments. Finally, we discuss several extensions that are relevant to practical applications.
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In this work, we develop first-order (Hessian-free) and zero-order (derivative-free) implementations of the Cubically regularized Newton method for solving general non-convex optimization problems. For that, we employ finite difference approximations of the derivatives. We use a special adaptive search procedure in our algorithms, which simultaneously fits both the regularization constant and the parameters of the finite difference approximations. It makes our schemes free from the need to know the actual Lipschitz constants. Additionally, we equip our algorithms with the lazy Hessian update that reuse a previously computed Hessian approximation matrix for several iterations. Specifically, we prove the global complexity bound of $\mathcal{O}( n^{1/2} \epsilon^{-3/2})$ function and gradient evaluations for our new Hessian-free method, and a bound of $\mathcal{O}( n^{3/2} \epsilon^{-3/2} )$ function evaluations for the derivative-free method, where $n$ is the dimension of the problem and $\epsilon$ is the desired accuracy for the gradient norm. These complexity bounds significantly improve the previously known ones in terms of the joint dependence on $n$ and $\epsilon$, for the first-order and zeroth-order non-convex optimization.
In this paper, we develop a non-asymptotic local normal approximation for multinomial probabilities. First, we use it to find non-asymptotic total variation bounds between the measures induced by uniformly jittered multinomials and the multivariate normals with the same means and covariances. From the total variation bounds, we also derive a comparison of the cumulative distribution functions and quantile coupling inequalities between Pearson's chi-square statistic (written as the normalized quadratic form of a multinomial vector) and its multivariate normal analogue. We apply our results to find confidence intervals for the negative entropy of discrete distributions. Our method can be applied more generally to find confidence intervals for strictly convex functions of the weights of discrete distributions.
We study matrix sensing, which is the problem of reconstructing a low-rank matrix from a few linear measurements. It can be formulated as an overparameterized regression problem, which can be solved by factorized gradient descent when starting from a small random initialization. Linear neural networks, and in particular matrix sensing by factorized gradient descent, serve as prototypical models of non-convex problems in modern machine learning, where complex phenomena can be disentangled and studied in detail. Much research has been devoted to studying special cases of asymmetric matrix sensing, such as asymmetric matrix factorization and symmetric positive semi-definite matrix sensing. Our key contribution is introducing a continuous differential equation that we call the $\textit{perturbed gradient flow}$. We prove that the perturbed gradient flow converges quickly to the true target matrix whenever the perturbation is sufficiently bounded. The dynamics of gradient descent for matrix sensing can be reduced to this formulation, yielding a novel proof of asymmetric matrix sensing with factorized gradient descent. Compared to directly analyzing the dynamics of gradient descent, the continuous formulation allows bounding key quantities by considering their derivatives, often simplifying the proofs. We believe the general proof technique may prove useful in other settings as well.
The smoothing distribution of dynamic probit models with Gaussian state dynamics was recently proved to belong to the unified skew-normal family. Although this is computationally tractable in small-to-moderate settings, it may become computationally impractical in higher dimensions. In this work, adapting a recent more general class of expectation propagation (EP) algorithms, we derive an efficient EP routine to perform inference for such a distribution. We show that the proposed approximation leads to accuracy gains over available approximate algorithms in a financial illustration.
The Multilevel Monte Carlo (MLMC) approach usually works well when estimating the expected value of a quantity which is a Lipschitz function of intermediate quantities, but if it is a discontinuous function it can lead to a much slower decay in the variance of the MLMC correction. This article reviews the literature on techniques which can be used to overcome this challenge in a variety of different contexts, and discusses recent developments using either a branching diffusion or adaptive sampling.
I consider the natural infinitary variations of the games Wordle and Mastermind, as well as their game-theoretic variations Absurdle and Madstermind, considering these games with infinitely long words and infinite color sequences and allowing transfinite game play. For each game, a secret codeword is hidden, which the codebreaker attempts to discover by making a series of guesses and receiving feedback as to their accuracy. In Wordle with words of any size from a finite alphabet of $n$ letters, including infinite words or even uncountable words, the codebreaker can nevertheless always win in $n$ steps. Meanwhile, the mastermind number, defined as the smallest winning set of guesses in infinite Mastermind for sequences of length $\omega$ over a countable set of colors without duplication, is uncountable, but the exact value turns out to be independent of ZFC, for it is provably equal to the eventually different number $\frak{d}({\neq^*})$, which is the same as the covering number of the meager ideal $\text{cov}(\mathcal{M})$. I thus place all the various mastermind numbers, defined for the natural variations of the game, into the hierarchy of cardinal characteristics of the continuum.
In this paper, efficient alternating direction implicit (ADI) schemes are proposed to solve three-dimensional heat equations with irregular boundaries and interfaces. Starting from the well-known Douglas-Gunn ADI scheme, a modified ADI scheme is constructed to mitigate the issue of accuracy loss in solving problems with time-dependent boundary conditions. The unconditional stability of the new ADI scheme is also rigorously proven with the Fourier analysis. Then, by combining the ADI schemes with a 1D kernel-free boundary integral (KFBI) method, KFBI-ADI schemes are developed to solve the heat equation with irregular boundaries. In 1D sub-problems of the KFBI-ADI schemes, the KFBI discretization takes advantage of the Cartesian grid and preserves the structure of the coefficient matrix so that the fast Thomas algorithm can be applied to solve the linear system efficiently. Second-order accuracy and unconditional stability of the KFBI-ADI schemes are verified through several numerical tests for both the heat equation and a reaction-diffusion equation. For the Stefan problem, which is a free boundary problem of the heat equation, a level set method is incorporated into the ADI method to capture the time-dependent interface. Numerical examples for simulating 3D dendritic solidification phenomenons are also presented.
Determining the number of factors in high-dimensional factor modeling is essential but challenging, especially when the data are heavy-tailed. In this paper, we introduce a new estimator based on the spectral properties of Spearman sample correlation matrix under the high-dimensional setting, where both dimension and sample size tend to infinity proportionally. Our estimator is robust against heavy tails in either the common factors or idiosyncratic errors. The consistency of our estimator is established under mild conditions. Numerical experiments demonstrate the superiority of our estimator compared to existing methods.
Deep neural networks have shown remarkable performance when trained on independent and identically distributed data from a fixed set of classes. However, in real-world scenarios, it can be desirable to train models on a continuous stream of data where multiple classification tasks are presented sequentially. This scenario, known as Continual Learning (CL) poses challenges to standard learning algorithms which struggle to maintain knowledge of old tasks while learning new ones. This stability-plasticity dilemma remains central to CL and multiple metrics have been proposed to adequately measure stability and plasticity separately. However, none considers the increasing difficulty of the classification task, which inherently results in performance loss for any model. In that sense, we analyze some limitations of current metrics and identify the presence of setup-induced forgetting. Therefore, we propose new metrics that account for the task's increasing difficulty. Through experiments on benchmark datasets, we demonstrate that our proposed metrics can provide new insights into the stability-plasticity trade-off achieved by models in the continual learning environment.
A standard approach to solve ordinary differential equations, when they describe dynamical systems, is to adopt a Runge-Kutta or related scheme. Such schemes, however, are not applicable to the large class of equations which do not constitute dynamical systems. In several physical systems, we encounter integro-differential equations with memory terms where the time derivative of a state variable at a given time depends on all past states of the system. Secondly, there are equations whose solutions do not have well-defined Taylor series expansion. The Maxey-Riley-Gatignol equation, which describes the dynamics of an inertial particle in nonuniform and unsteady flow, displays both challenges. We use it as a test bed to address the questions we raise, but our method may be applied to all equations of this class. We show that the Maxey-Riley-Gatignol equation can be embedded into an extended Markovian system which is constructed by introducing a new dynamical co-evolving state variable that encodes memory of past states. We develop a Runge-Kutta algorithm for the resultant Markovian system. The form of the kernels involved in deriving the Runge-Kutta scheme necessitates the use of an expansion in powers of $t^{1/2}$. Our approach naturally inherits the benefits of standard time-integrators, namely a constant memory storage cost, a linear growth of operational effort with simulation time, and the ability to restart a simulation with the final state as the new initial condition.
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