Random utility maximisation (RUM) models are one of the cornerstones of discrete choice modelling. However, specifying the utility function of RUM models is not straightforward and has a considerable impact on the resulting interpretable outcomes and welfare measures. In this paper, we propose a new discrete choice model based on artificial neural networks (ANNs) named "Alternative-Specific and Shared weights Neural Network (ASS-NN)", which provides a further balance between flexible utility approximation from the data and consistency with two assumptions: RUM theory and fungibility of money (i.e., "one euro is one euro"). Therefore, the ASS-NN can derive economically-consistent outcomes, such as marginal utilities or willingness to pay, without explicitly specifying the utility functional form. Using a Monte Carlo experiment and empirical data from the Swissmetro dataset, we show that ASS-NN outperforms (in terms of goodness of fit) conventional multinomial logit (MNL) models under different utility specifications. Furthermore, we show how the ASS-NN is used to derive marginal utilities and willingness to pay measures.
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Multivariate Cryptography is one of the main candidates for Post-quantum Cryptography. Multivariate schemes are usually constructed by applying two secret affine invertible transformations $\mathcal S,\mathcal T$ to a set of multivariate polynomials $\mathcal{F}$ (often quadratic). The secret polynomials $\mathcal{F}$ posses a trapdoor that allows the legitimate user to find a solution of the corresponding system, while the public polynomials $\mathcal G=\mathcal S\circ\mathcal F\circ\mathcal T$ look like random polynomials. The polynomials $\mathcal G$ and $\mathcal F$ are said to be affine equivalent. In this article, we present a more general way of constructing a multivariate scheme by considering the CCZ equivalence, which has been introduced and studied in the context of vectorial Boolean functions.
Many state-of-the-art models trained on long-range sequences, for example S4, S5 or LRU, are made of sequential blocks combining State-Space Models (SSMs) with neural networks. In this paper we provide a PAC bound that holds for these kind of architectures with stable SSM blocks and does not depend on the length of the input sequence. Imposing stability of the SSM blocks is a standard practice in the literature, and it is known to help performance. Our results provide a theoretical justification for the use of stable SSM blocks as the proposed PAC bound decreases as the degree of stability of the SSM blocks increases.
Differential abundance analysis is a key component of microbiome studies. While dozens of methods for it exist, currently, there is no consensus on the preferred methods. Correctness of results in differential abundance analysis is an ambiguous concept that cannot be evaluated without employing simulated data, but we argue that consistency of results across datasets should be considered as an essential quality of a well-performing method. We compared the performance of 14 differential abundance analysis methods employing datasets from 54 taxonomic profiling studies based on 16S rRNA gene or shotgun sequencing. For each method, we examined how the results replicated between random partitions of each dataset and between datasets from independent studies. While certain methods showed good consistency, some widely used methods were observed to produce a substantial number of conflicting findings. Overall, the highest consistency without unnecessary reduction in sensitivity was attained by analyzing relative abundances with a non-parametric method (Wilcoxon test or ordinal regression model) or linear regression (MaAsLin2). Comparable performance was also attained by analyzing presence/absence of taxa with logistic regression.
We present LinApart, a routine designed for efficiently performing the univariate partial fraction decomposition of large symbolic expressions. Our method is based on an explicit closed formula for the decomposition of rational functions with fully factorized denominators. We provide implementations in both the Wolfram Mathematica and C languages, made available at //github.com/fekeshazy/LinApart . The routine can provide very significant performance gains over available tools such as the Apart command in Mathematica.
Multi-target linear shrinkage is an extension of the standard single-target linear shrinkage for covariance estimation. We combine several constant matrices - the targets - with the sample covariance matrix. We derive the oracle and a \textit{bona fide} multi-target linear shrinkage estimator with exact and empirical mean. In both settings, we proved its convergence towards the oracle under Kolmogorov asymptotics. Finally, we show empirically that it outperforms other standard estimators in various situations.
The problems of optimal recovering univariate functions and their derivatives are studied. To solve these problems, two variants of the truncation method are constructed, which are order-optimal both in the sense of accuracy and in terms of the amount of involved Galerkin information. For numerical summation, it has been established how the parameters characterizing the problem being solved affect its stability.
Semi-implicit spectral deferred correction (SDC) methods provide a systematic approach to construct time integration methods of arbitrarily high order for nonlinear evolution equations including conservation laws. They converge towards $A$- or even $L$-stable collocation methods, but are often not sufficiently robust themselves. In this paper, a family of SDC methods inspired by an implicit formulation of the Lax-Wendroff method is developed. Compared to fully implicit approaches, the methods have the advantage that they only require the solution of positive definite or semi-definite linear systems. Numerical evidence suggests that the proposed semi-implicit SDC methods with Radau points are $L$-stable up to order 11 and require very little diffusion for orders 13 and 15. The excellent stability and accuracy of these methods is confirmed by numerical experiments with 1D conservation problems, including the convection-diffusion, Burgers, Euler and Navier-Stokes equations.
Predictive posterior densities (PPDs) are of interest in approximate Bayesian inference. Typically, these are estimated by simple Monte Carlo (MC) averages using samples from the approximate posterior. We observe that the signal-to-noise ratio (SNR) of such estimators can be extremely low. An analysis for exact inference reveals SNR decays exponentially as there is an increase in (a) the mismatch between training and test data, (b) the dimensionality of the latent space, or (c) the size of the test data relative to the training data. Further analysis extends these results to approximate inference. To remedy the low SNR problem, we propose replacing simple MC sampling with importance sampling using a proposal distribution optimized at test time on a variational proxy for the SNR and demonstrate that this yields greatly improved estimates.
We propose a semiparametric model for dyadic link formations in directed networks. The model contains a set of degree parameters that measure different effects of popularity or outgoingness across nodes, a regression parameter vector that reflects the homophily effect resulting from the nodal attributes or pairwise covariates associated with edges, and a set of latent random noises with unknown distributions. Our interest lies in inferring the unknown degree parameters and homophily parameters. The dimension of the degree parameters increases with the number of nodes. Under the high-dimensional regime, we develop a kernel-based least squares approach to estimate the unknown parameters. The major advantage of our estimator is that it does not encounter the incidental parameter problem for the homophily parameters. We prove consistency of all the resulting estimators of the degree parameters and homophily parameters. We establish high-dimensional central limit theorems for the proposed estimators and provide several applications of our general theory, including testing the existence of degree heterogeneity, testing sparse signals and recovering the support. Simulation studies and a real data application are conducted to illustrate the finite sample performance of the proposed methods.
Given a finite set of matrices with integer entries, the matrix mortality problem asks if there exists a product of these matrices equal to the zero matrix. We consider a special case of this problem where all entries of the matrices are nonnegative. This case is equivalent to the NFA mortality problem, which, given an NFA, asks for a word $w$ such that the image of every state under $w$ is the empty set. The size of the alphabet of the NFA is then equal to the number of matrices in the set. We study the length of shortest such words depending on the size of the alphabet. We show that this length for an NFA with $n$ states can be at least $2^n - 1$, $2^{(n - 4)/2}$ and $2^{(n - 2)/3}$ if the size of the alphabet is, respectively, equal to $n$, three and two.
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