Interior point methods (IPMs) that handle nonconvex constraints such as IPOPT, KNITRO and LOQO have had enormous practical success. We consider IPMs in the setting where the objective and constraints are thrice differentiable, and have Lipschitz first and second derivatives on the feasible region. We provide an IPM that, starting from a strictly feasible point, finds a $\mu$-approximate Fritz John point by solving $\mathcal{O}( \mu^{-7/4})$ trust-region subproblems. For IPMs that handle nonlinear constraints, this result represents the first iteration bound with a polynomial dependence on $1/\mu$. We also show how to use our method to find scaled-KKT points starting from an infeasible solution and improve on existing complexity bounds.
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There are several tools available to infer phylogenetic trees, which depict the evolutionary relationships among biological entities such as viral and bacterial strains in infectious outbreaks, or cancerous cells in tumor progression trees. These tools rely on several inference methods available to produce phylogenetic trees, with resulting trees not being unique. Thus, methods for comparing phylogenies that are capable of revealing where two phylogenetic trees agree or differ are required. An approach is then to compute a similarity or dissimilarity measure between trees, with the Robinson- Foulds distance being one of the most used, and which can be computed in linear time and space. Nevertheless, given the large and increasing volume of phylogenetic data, phylogenetic trees are becoming very large with hundreds of thousands of leafs. In this context, space requirements become an issue both while computing tree distances and while storing trees. We propose then an efficient implementation of the Robinson-Foulds distance over trees succinct representations. Our implementation generalizes also the Robinson-Foulds distances to labelled phylogenetic trees, i.e., trees containing labels on all nodes, instead of only on leaves. Experimental results show that we are able to still achieve linear time while requiring less space. Our implementation is available as an open-source tool at //github.com/pedroparedesbranco/TreeDiff.
Deep generative models require large amounts of training data. This often poses a problem as the collection of datasets can be expensive and difficult, in particular datasets that are representative of the appropriate underlying distribution (e.g. demographic). This introduces biases in datasets which are further propagated in the models. We present an approach to construct an unbiased generative adversarial network (GAN) from an existing biased GAN by rebalancing the model distribution. We do so by generating balanced data from an existing imbalanced deep generative model using an evolutionary algorithm and then using this data to train a balanced generative model. Additionally, we propose a bias mitigation loss function that minimizes the deviation of the learned class distribution from being equiprobable. We show results for the StyleGAN2 models while training on the Flickr Faces High Quality (FFHQ) dataset for racial fairness and see that the proposed approach improves on the fairness metric by almost 5 times, whilst maintaining image quality. We further validate our approach by applying it to an imbalanced CIFAR10 dataset where we show that we can obtain comparable fairness and image quality as when training on a balanced CIFAR10 dataset which is also twice as large. Lastly, we argue that the traditionally used image quality metrics such as Frechet inception distance (FID) are unsuitable for scenarios where the class distributions are imbalanced and a balanced reference set is not available.
Stabbing Planes (also known as Branch and Cut) is a proof system introduced very recently which, informally speaking, extends the DPLL method by branching on integer linear inequalities instead of single variables. The techniques known so far to prove size and depth lower bounds for Stabbing Planes are generalizations of those used for the Cutting Planes proof system. For size lower bounds these are established by monotone circuit arguments, while for depth these are found via communication complexity and protection. As such these bounds apply for lifted versions of combinatorial statements. Rank lower bounds for Cutting Planes are also obtained by geometric arguments called protection lemmas. In this work we introduce two new geometric approaches to prove size/depth lower bounds in Stabbing Planes working for any formula: (1) the antichain method, relying on Sperner's Theorem and (2) the covering method which uses results on essential coverings of the boolean cube by linear polynomials, which in turn relies on Alon's combinatorial Nullenstellensatz. We demonstrate their use on classes of combinatorial principles such as the Pigeonhole principle, the Tseitin contradictions and the Linear Ordering Principle. By the first method we prove almost linear size lower bounds and optimal logarithmic depth lower bounds for the Pigeonhole principle and analogous lower bounds for the Tseitin contradictions over the complete graph and for the Linear Ordering Principle. By the covering method we obtain a superlinear size lower bound and a logarithmic depth lower bound for Stabbing Planes proof of Tseitin contradictions over a grid graph.
Rational best approximations (in a Chebyshev sense) to real functions are characterized by an equioscillating approximation error. Similar results do not hold true for rational best approximations to complex functions in general. In the present work, we consider unitary rational approximations to the exponential function on the imaginary axis, which map the imaginary axis to the unit circle. In the class of unitary rational functions, best approximations are shown to exist, to be uniquely characterized by equioscillation of a phase error, and to possess a super-linear convergence rate. Furthermore, the best approximations have full degree (i.e., non-degenerate), achieve their maximum approximation error at points of equioscillation, and interpolate at intermediate points. Asymptotic properties of poles, interpolation nodes, and equioscillation points of these approximants are studied. Three algorithms, which are found very effective to compute unitary rational approximations including candidates for best approximations, are sketched briefly. Some consequences to numerical time-integration are discussed. In particular, time propagators based on unitary best approximants are unitary, symmetric and A-stable.
In this note we prove almost sure unisolvence of RBF interpolation on randomly distributed sequences by a wide class of polyharmonic splines (including Thin-Plate Splines), without polynomial addition.
Any experiment with climate models relies on a potentially large set of spatio-temporal boundary conditions. These can represent both the initial state of the system and/or forcings driving the model output throughout the experiment. Whilst these boundary conditions are typically fixed using available reconstructions in climate modelling studies, they are highly uncertain, that uncertainty is unquantified, and the effect on the output of the experiment can be considerable. We develop efficient quantification of these uncertainties that combines relevant data from multiple models and observations. Starting from the coexchangeability model, we develop a coexchangable process model to capture multiple correlated spatio-temporal fields of variables. We demonstrate that further exchangeability judgements over the parameters within this representation lead to a Bayes linear analogy of a hierarchical model. We use the framework to provide a joint reconstruction of sea-surface temperature and sea-ice concentration boundary conditions at the last glacial maximum (19-23 ka) and use it to force an ensemble of ice-sheet simulations using the FAMOUS-Ice coupled atmosphere and ice-sheet model. We demonstrate that existing boundary conditions typically used in these experiments are implausible given our uncertainties and demonstrate the impact of using more plausible boundary conditions on ice-sheet simulation.
Fusing measurements from multiple, heterogeneous, partial sources, observing a common object or process, poses challenges due to the increasing availability of numbers and types of sensors. In this work we propose, implement and validate an end-to-end computational pipeline in the form of a multiple-auto-encoder neural network architecture for this task. The inputs to the pipeline are several sets of partial observations, and the result is a globally consistent latent space, harmonizing (rigidifying, fusing) all measurements. The key enabler is the availability of multiple slightly perturbed measurements of each instance:, local measurement, "bursts", that allows us to estimate the local distortion induced by each instrument. We demonstrate the approach in a sequence of examples, starting with simple two-dimensional data sets and proceeding to a Wi-Fi localization problem and to the solution of a "dynamical puzzle" arising in spatio-temporal observations of the solutions of Partial Differential Equations.
While generalized linear mixed models (GLMMs) are a fundamental tool in applied statistics, many specifications -- such as those involving categorical factors with many levels or interaction terms -- can be computationally challenging to estimate due to the need to compute or approximate high-dimensional integrals. Variational inference (VI) methods are a popular way to perform such computations, especially in the Bayesian context. However, naive VI methods can provide unreliable uncertainty quantification. We show that this is indeed the case in the GLMM context, proving that standard VI (i.e. mean-field) dramatically underestimates posterior uncertainty in high-dimensions. We then show how appropriately relaxing the mean-field assumption leads to VI methods whose uncertainty quantification does not deteriorate in high-dimensions, and whose total computational cost scales linearly with the number of parameters and observations. Our theoretical and numerical results focus on GLMMs with Gaussian or binomial likelihoods, and rely on connections to random graph theory to obtain sharp high-dimensional asymptotic analysis. We also provide generic results, which are of independent interest, relating the accuracy of variational inference to the convergence rate of the corresponding coordinate ascent variational inference (CAVI) algorithm for Gaussian targets. Our proposed partially-factorized VI (PF-VI) methodology for GLMMs is implemented in the R package vglmer, see //github.com/mgoplerud/vglmer . Numerical results with simulated and real data examples illustrate the favourable computation cost versus accuracy trade-off of PF-VI.
In this contribution, we derive a consistent variational formulation for computational homogenization methods and show that traditional FE2 and IGA2 approaches are special discretization and solution techniques of this most general framework. This allows us to enhance dramatically the numerical analysis as well as the solution of the arising algebraic system. In particular, we expand the dimension of the continuous system, discretize the higher dimensional problem consistently and apply afterwards a discrete null-space matrix to remove the additional dimensions. A benchmark problem, for which we can obtain an analytical solution, demonstrates the superiority of the chosen approach aiming to reduce the immense computational costs of traditional FE2 and IGA2 formulations to a fraction of the original requirements. Finally, we demonstrate a further reduction of the computational costs for the solution of general non-linear problems.
Coordinate exchange (CEXCH) is a popular algorithm for generating exact optimal experimental designs. The authors of CEXCH advocated for a highly greedy implementation - one that exchanges and optimizes single element coordinates of the design matrix. We revisit the effect of greediness on CEXCHs efficacy for generating highly efficient designs. We implement the single-element CEXCH (most greedy), a design-row (medium greedy) optimization exchange, and particle swarm optimization (PSO; least greedy) on 21 exact response surface design scenarios, under the $D$- and $I-$criterion, which have well-known optimal designs that have been reproduced by several researchers. We found essentially no difference in performance of the most greedy CEXCH and the medium greedy CEXCH. PSO did exhibit better efficacy for generating $D$-optimal designs, and for most $I$-optimal designs than CEXCH, but not to a strong degree under our parametrization. This work suggests that further investigation of the greediness dimension and its effect on CEXCH efficacy on a wider suite of models and criterion is warranted.
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