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We show that there exists no algorithm that decides for any bilinear system $(B,v)$ if the growth rate of $(B,v)$ is $1$. This answers a question of Bui who showed that if the coefficients are positive the growth rate is computable (i.e., there is an algorithm that outputs the sequence of digits of the growth rate of $(B,v)$). Our proof is based on a reduction of the computation of the joint spectral radius of a set of matrices to the computation of the growth rate of a bilinear system. We also use our reduction to deduce that there exists no algorithm that approximates the growth rate of a bilinear system with relative accuracy $\varepsilon$ in time polynomial in the size of the system and of $\varepsilon$. Our two results hold even if all the coefficients are nonnegative rationals.

We determine the exact minimax rate of a Gaussian sequence model under bounded convex constraints, purely in terms of the local geometry of the given constraint set $K$. Our main result shows that the minimax risk (up to constant factors) under the squared $L_2$ loss is given by $\epsilon^{*2} \wedge \operatorname{diam}(K)^2$ with \begin{align*} \epsilon^* = \sup \bigg\{\epsilon : \frac{\epsilon^2}{\sigma^2} \leq \log M^{\operatorname{loc}}(\epsilon)\bigg\}, \end{align*} where $\log M^{\operatorname{loc}}(\epsilon)$ denotes the local entropy of the set $K$, and $\sigma^2$ is the variance of the noise. We utilize our abstract result to re-derive known minimax rates for some special sets $K$ such as hyperrectangles, ellipses, and more generally quadratically convex orthosymmetric sets. Finally, we extend our results to the unbounded case with known $\sigma^2$ to show that the minimax rate in that case is $\epsilon^{*2}$.

Most of the popular dependence measures for two random variables $X$ and $Y$ (such as Pearson's and Spearman's correlation, Kendall's $\tau$ and Gini's $\gamma$) vanish whenever $X$ and $Y$ are independent. However, neither does a vanishing dependence measure necessarily imply independence, nor does a measure equal to $1$ imply that one variable is a measurable function of the other. Yet, both properties are natural desiderata for a convincing dependence measure. In this paper, we present a general approach to transforming a given dependence measure into a new one which exactly characterizes independence as well as functional dependence. Our approach uses the concept of monotone rearrangements as introduced by Hardy and Littlewood and is applicable to a broad class of measures. In particular, we are able to define a rearranged Spearman's $\rho$ and a rearranged Kendall's $\tau$ which do attain the value $1$ if, and only if, one variable is a measurable function of the other. We also present simple estimators for the rearranged dependence measures, prove their consistency and illustrate their finite sample properties by means of a simulation study.

Deciding whether a given function is quasiconvex is generally a difficult task. Here, we discuss a number of numerical approaches that can be used in the search for a counterexample to the quasiconvexity of a given function $W$. We will demonstrate these methods using the planar isotropic rank-one convex function \[ W_{\rm magic}^+(F)=\frac{\lambda_{\rm max}}{\lambda_{\rm min}}-\log\frac{\lambda_{\rm max}}{\lambda_{\rm min}}+\log\det F=\frac{\lambda_{\rm max}}{\lambda_{\rm min}}+2\log\lambda_{\rm min}\,, \] where $\lambda_{\rm max}\geq\lambda_{\rm min}$ are the singular values of $F$, as our main example. In a previous contribution, we have shown that quasiconvexity of this function would imply quasiconvexity for all rank-one convex isotropic planar energies $W:\operatorname{GL}^+(2)\rightarrow\mathbb{R}$ with an additive volumetric-isochoric split of the form \[ W(F)=W_{\rm iso}(F)+W_{\rm vol}(\det F)=\widetilde W_{\rm iso}\bigg(\frac{F}{\sqrt{\det F}}\bigg)+W_{\rm vol}(\det F) \] with a concave volumetric part. This example is therefore of particular interest with regard to Morrey's open question whether or not rank-one convexity implies quasiconvexity in the planar case.

The problem of $d$-Path Vertex Cover, $d$-PVC lies in determining a subset $F$ of vertices of a given graph $G=(V,E)$ such that $G \setminus F$ does not contain a path on $d$ vertices. The paths we aim to cover need not to be induced. It is known that the $d$-PVC problem is NP-complete for any $d \ge 2$. When parameterized by the size of the solution $k$, 5-PVC has direct trivial algorithm with $\mathcal{O}(5^kn^{\mathcal{O}(1)})$ running time and, since $d$-PVC is a special case of $d$-Hitting Set, an algorithm running in $\mathcal{O}(4.0755^kn^{\mathcal{O}(1)})$ time is known. In this paper we present an iterative compression algorithm that solves the 5-PVC problem in $\mathcal{O}(4^kn^{\mathcal{O}(1)})$ time.

The commonly quoted error rates for QMC integration with an infinite low discrepancy sequence is $O(n^{-1}\log(n)^r)$ with $r=d$ for extensible sequences and $r=d-1$ otherwise. Such rates hold uniformly over all $d$ dimensional integrands of Hardy-Krause variation one when using $n$ evaluation points. Implicit in those bounds is that for any sequence of QMC points, the integrand can be chosen to depend on $n$. In this paper we show that rates with any $r<(d-1)/2$ can hold when $f$ is held fixed as $n\to\infty$. This is accomplished following a suggestion of Erich Novak to use some unpublished results of Trojan from the 1980s as given in the information based complexity monograph of Traub, Wasilkowski and Wo\'zniakowski. The proof is made by applying a technique of Roth with the theorem of Trojan. The proof is non constructive and we do not know of any integrand of bounded variation in the sense of Hardy and Krause for which the QMC error exceeds $(\log n)^{1+\epsilon}/n$ for infinitely many $n$ when using a digital sequence such as one of Sobol's. An empirical search when $d=2$ for integrands designed to exploit known weaknesses in certain point sets showed no evidence that $r>1$ is needed. An example with $d=3$ and $n$ up to $2^{100}$ might possibly require $r>1$.

A strict bramble of a graph $G$ is a collection of pairwise-intersecting connected subgraphs of $G.$ The order of a strict bramble ${\cal B}$ is the minimum size of a set of vertices intersecting all sets of ${\cal B}.$ The strict bramble number of $G,$ denoted by ${\sf sbn}(G),$ is the maximum order of a strict bramble in $G.$ The strict bramble number of $G$ can be seen as a way to extend the notion of acyclicity, departing from the fact that (non-empty) acyclic graphs are exactly the graphs where every strict bramble has order one. We initiate the study of this graph parameter by providing three alternative definitions, each revealing different structural characteristics. The first is a min-max theorem asserting that ${\sf sbn}(G)$ is equal to the minimum $k$ for which $G$ is a minor of the lexicographic product of a tree and a clique on $k$ vertices (also known as the lexicographic tree product number). The second characterization is in terms of a new variant of a tree decomposition called lenient tree decomposition. We prove that ${\sf sbn}(G)$ is equal to the minimum $k$ for which there exists a lenient tree decomposition of $G$ of width at most $k.$ The third characterization is in terms of extremal graphs. For this, we define, for each $k,$ the concept of a $k$-domino-tree and we prove that every edge-maximal graph of strict bramble number at most $k$ is a $k$-domino-tree. We also identify three graphs that constitute the minor-obstruction set of the class of graphs with strict bramble number at most two. We complete our results by proving that, given some $G$ and $k,$ deciding whether ${\sf sbn}(G) \leq k$ is an ${\sf NP}$-complete problem.

Given polynomials $f_0,\dots, f_k$ the Ideal Membership Problem, IMP for short, asks if $f_0$ belongs to the ideal generated by $f_1,\dots, f_k$. In the search version of this problem the task is to find a proof of this fact. The IMP is a well-known fundamental problem with numerous applications, for instance, it underlies many proof systems based on polynomials such as Nullstellensatz, Polynomial Calculus, and Sum-of-Squares. Although the IMP is in general intractable, in many important cases it can be efficiently solved. Mastrolilli [SODA'19] initiated a systematic study of IMPs for ideals arising from Constraint Satisfaction Problems (CSPs), parameterized by constraint languages, denoted IMP($\Gamma$). The ultimate goal of this line of research is to classify all such IMPs accordingly to their complexity. Mastrolilli achieved this goal for IMPs arising from CSP($\Gamma$) where $\Gamma$ is a Boolean constraint language, while Bulatov and Rafiey [ArXiv'21] advanced these results to several cases of CSPs over finite domains. In this paper we consider IMPs arising from CSPs over `affine' constraint languages, in which constraints are subgroups (or their cosets) of direct products of Abelian groups. This kind of CSPs include systems of linear equations and are considered one of the most important types of tractable CSPs. Some special cases of the problem have been considered before by Bharathi and Mastrolilli [MFCS'21] for linear equation modulo 2, and by Bulatov and Rafiey [ArXiv'21] to systems of linear equations over $GF(p)$, $p$ prime. Here we prove that if $\Gamma$ is an affine constraint language then IMP($\Gamma$) is solvable in polynomial time assuming the input polynomial has bounded degree.

While existing work in robust deep learning has focused on small pixel-level $\ell_p$ norm-based perturbations, this may not account for perturbations encountered in several real world settings. In many such cases although test data might not be available, broad specifications about the types of perturbations (such as an unknown degree of rotation) may be known. We consider a setup where robustness is expected over an unseen test domain that is not i.i.d. but deviates from the training domain. While this deviation may not be exactly known, its broad characterization is specified a priori, in terms of attributes. We propose an adversarial training approach which learns to generate new samples so as to maximize exposure of the classifier to the attributes-space, without having access to the data from the test domain. Our adversarial training solves a min-max optimization problem, with the inner maximization generating adversarial perturbations, and the outer minimization finding model parameters by optimizing the loss on adversarial perturbations generated from the inner maximization. We demonstrate the applicability of our approach on three types of naturally occurring perturbations -- object-related shifts, geometric transformations, and common image corruptions. Our approach enables deep neural networks to be robust against a wide range of naturally occurring perturbations. We demonstrate the usefulness of the proposed approach by showing the robustness gains of deep neural networks trained using our adversarial training on MNIST, CIFAR-10, and a new variant of the CLEVR dataset.

Both generative adversarial network models and variational autoencoders have been widely used to approximate probability distributions of datasets. Although they both use parametrized distributions to approximate the underlying data distribution, whose exact inference is intractable, their behaviors are very different. In this report, we summarize our experiment results that compare these two categories of models in terms of fidelity and mode collapse. We provide a hypothesis to explain their different behaviors and propose a new model based on this hypothesis. We further tested our proposed model on MNIST dataset and CelebA dataset.