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We present simple conditions for Bayesian consistency in the supremum metric. The key to the technique is a triangle inequality which allows us to explicitly use weak convergence, a consequence of the standard Kullback--Leibler support condition for the prior. A further condition is to ensure that smoothed versions of densities are not too far from the original density, thus dealing with densities which could track the data too closely. A key result of the paper is that we demonstrate supremum consistency using weaker conditions compared to those currently used to secure $\mathbb{L}_1$ consistency.

There has been a surge of interest in developing robust estimators for models with heavy-tailed data in statistics and machine learning. This paper proposes a log-truncated M-estimator for a large family of statistical regressions and establishes its excess risk bound under the condition that the data have $(1+\varepsilon)$-th moment with $\varepsilon \in (0,1]$. With an additional assumption on the associated risk function, we obtain an $\ell_2$-error bound for the estimation. Our theorems are applied to establish robust M-estimators for concrete regressions. Besides convex regressions such as quantile regression and generalized linear models, many non-convex regressions can also be fit into our theorems, we focus on robust deep neural network regressions, which can be solved by the stochastic gradient descent algorithms. Simulations and real data analysis demonstrate the superiority of log-truncated estimations over standard estimations.

The best subset selection (or "best subsets") estimator is a classic tool for sparse regression, and developments in mathematical optimization over the past decade have made it more computationally tractable than ever. Notwithstanding its desirable statistical properties, the best subsets estimator is susceptible to outliers and can break down in the presence of a single contaminated data point. To address this issue, a robust adaption of best subsets is proposed that is highly resistant to contamination in both the response and the predictors. The adapted estimator generalizes the notion of subset selection to both predictors and observations, thereby achieving robustness in addition to sparsity. This procedure, referred to as "robust subset selection" (or "robust subsets"), is defined by a combinatorial optimization problem for which modern discrete optimization methods are applied. The robustness of the estimator in terms of the finite-sample breakdown point of its objective value is formally established. In support of this result, experiments on synthetic and real data are reported that demonstrate the superiority of robust subsets over best subsets in the presence of contamination. Importantly, robust subsets fares competitively across several metrics compared with popular robust adaptions of continuous shrinkage estimators.

Recently, minimal linear codes have been extensively studied due to their applications in secret sharing schemes, secure two-party computations, and so on. Constructing minimal linear codes violating the Ashikhmin-Barg condition and then determining their weight distributions have been interesting in coding theory and cryptography. In this paper, a generic construction for binary linear codes with dimension $m+2$ is presented, then a necessary and sufficient condition for this binary linear code to be minimal is derived. Based on this condition and exponential sums, a new class of minimal binary linear codes violating the Ashikhmin-Barg condition is obtained, and then their weight enumerators are determined.

We provide an algorithm that implements the indicator function of NURBS-shaped planar domains, tailored to the fast computation on huge point clouds, together with the corresponding Matlab code.

Consider a random graph process with $n$ vertices corresponding to points $v_{i} \sim {Unif}[0,1]$ embedded randomly in the interval, and where edges are inserted between $v_{i}, v_{j}$ independently with probability given by the graphon $w(v_{i},v_{j}) \in [0,1]$. Following Chuangpishit et al. (2015), we call a graphon $w$ diagonally increasing if, for each $x$, $w(x,y)$ decreases as $y$ moves away from $x$. We call a permutation $\sigma \in S_{n}$ an ordering of these vertices if $v_{\sigma(i)} < v_{\sigma(j)}$ for all $i < j$, and ask: how can we accurately estimate $\sigma$ from an observed graph? We present a randomized algorithm with output $\hat{\sigma}$ that, for a large class of graphons, achieves error $\max_{1 \leq i \leq n} | \sigma(i) - \hat{\sigma}(i)| = O^{*}(\sqrt{n})$ with high probability; we also show that this is the best-possible convergence rate for a large class of algorithms and proof strategies. Under an additional assumption that is satisfied by some popular graphon models, we break this "barrier" at $\sqrt{n}$ and obtain the vastly better rate $O^{*}(n^{\epsilon})$ for any $\epsilon > 0$. These improved seriation bounds can be combined with previous work to give more efficient and accurate algorithms for related tasks, including: estimating diagonally increasing graphons, and testing whether a graphon is diagonally increasing.

The problem of selecting optimal backdoor adjustment sets to estimate causal effects in graphical models with hidden and conditioned variables is addressed. Previous work has defined optimality as achieving the smallest asymptotic estimation variance and derived an optimal set for the case without hidden variables. For the case with hidden variables there can be settings where no optimal set exists and currently only a sufficient graphical optimality criterion of limited applicability has been derived. In the present work optimality is characterized as maximizing a certain adjustment information which allows to derive a necessary and sufficient graphical criterion for the existence of an optimal adjustment set and a definition and algorithm to construct it. Further, the optimal set is valid if and only if a valid adjustment set exists and has higher (or equal) adjustment information than the Adjust-set proposed in Perkovi{\'c} et al. [Journal of Machine Learning Research, 18: 1--62, 2018] for any graph. The results translate to minimal asymptotic estimation variance for a class of estimators whose asymptotic variance follows a certain information-theoretic relation. Numerical experiments indicate that the asymptotic results also hold for relatively small sample sizes and that the optimal adjustment set or minimized variants thereof often yield better variance also beyond that estimator class. Surprisingly, among the randomly created setups more than 90\% fulfill the optimality conditions indicating that also in many real-world scenarios graphical optimality may hold. Code is available as part of the python package \url{//github.com/jakobrunge/tigramite}.

We present the Olsson.wl Mathematica package which aims to find linear transformations for some classes of multivariable hypergeometric functions. It is based on a well-known method developed by P. O. M. Olsson in J. Math. Phys. 5, 420 (1964) in order to derive the analytic continuations of the Appell $F_1$ double hypergeometric series from the linear transformations of the Gauss $_2F_1$ hypergeometric function. We provide a brief description of Olsson's method and demonstrate the commands of the package, along with examples. We also provide a companion package, called ROC2.wl and dedicated to the derivation of the regions of convergence of double hypergeometric series. This package can be used independently of Olsson.wl.

For neural networks (NNs) with rectified linear unit (ReLU) or binary activation functions, we show that their training can be accomplished in a reduced parameter space. Specifically, the weights in each neuron can be trained on the unit sphere, as opposed to the entire space, and the threshold can be trained in a bounded interval, as opposed to the real line. We show that the NNs in the reduced parameter space are mathematically equivalent to the standard NNs with parameters in the whole space. The reduced parameter space shall facilitate the optimization procedure for the network training, as the search space becomes (much) smaller. We demonstrate the improved training performance using numerical examples.

In this paper, we study the optimal convergence rate for distributed convex optimization problems in networks. We model the communication restrictions imposed by the network as a set of affine constraints and provide optimal complexity bounds for four different setups, namely: the function $F(\xb) \triangleq \sum_{i=1}^{m}f_i(\xb)$ is strongly convex and smooth, either strongly convex or smooth or just convex. Our results show that Nesterov's accelerated gradient descent on the dual problem can be executed in a distributed manner and obtains the same optimal rates as in the centralized version of the problem (up to constant or logarithmic factors) with an additional cost related to the spectral gap of the interaction matrix. Finally, we discuss some extensions to the proposed setup such as proximal friendly functions, time-varying graphs, improvement of the condition numbers.