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Many multivariate data sets exhibit a form of positive dependence, which can either appear globally between all variables or only locally within particular subgroups. A popular notion of positive dependence that allows for localized positivity is positive association. In this work we introduce the notion of extremal positive association for multivariate extremes from threshold exceedances. Via a sufficient condition for extremal association, we show that extremal association generalizes extremal tree models. For H\"usler--Reiss distributions the sufficient condition permits a parametric description that we call the metric property. As the parameter of a H\"usler--Reiss distribution is a Euclidean distance matrix, the metric property relates to research in electrical network theory and Euclidean geometry. We show that the metric property can be localized with respect to a graph and study surrogate likelihood inference. This gives rise to a two-step estimation procedure for locally metrical H\"usler--Reiss graphical models. The second step allows for a simple dual problem, which is implemented via a gradient descent algorithm. Finally, we demonstrate our results on simulated and real data.

The optimal branch number of MDS matrices makes them a preferred choice for designing diffusion layers in many block ciphers and hash functions. Consequently, various methods have been proposed for designing MDS matrices, including search and direct methods. While exhaustive search is suitable for small order MDS matrices, direct constructions are preferred for larger orders due to the vast search space involved. In the literature, there has been extensive research on the direct construction of MDS matrices using both recursive and nonrecursive methods. On the other hand, in lightweight cryptography, Near-MDS (NMDS) matrices with sub-optimal branch numbers offer a better balance between security and efficiency as a diffusion layer compared to MDS matrices. However, no direct construction method is available in the literature for constructing recursive NMDS matrices. This paper introduces some direct constructions of NMDS matrices in both nonrecursive and recursive settings. Additionally, it presents some direct constructions of nonrecursive MDS matrices from the generalized Vandermonde matrices. We propose a method for constructing involutory MDS and NMDS matrices using generalized Vandermonde matrices. Furthermore, we prove some folklore results that are used in the literature related to the NMDS code.

The optimal branch number of MDS matrices makes them a preferred choice for designing diffusion layers in many block ciphers and hash functions. However, in lightweight cryptography, Near-MDS (NMDS) matrices with sub-optimal branch numbers offer a better balance between security and efficiency as a diffusion layer, compared to MDS matrices. In this paper, we study NMDS matrices, exploring their construction in both recursive and nonrecursive settings. We provide several theoretical results and explore the hardware efficiency of the construction of NMDS matrices. Additionally, we make comparisons between the results of NMDS and MDS matrices whenever possible. For the recursive approach, we study the DLS matrices and provide some theoretical results on their use. Some of the results are used to restrict the search space of the DLS matrices. We also show that over a field of characteristic 2, any sparse matrix of order $n\geq 4$ with fixed XOR value of 1 cannot be an NMDS when raised to a power of $k\leq n$. Following that, we use the generalized DLS (GDLS) matrices to provide some lightweight recursive NMDS matrices of several orders that perform better than the existing matrices in terms of hardware cost or the number of iterations. For the nonrecursive construction of NMDS matrices, we study various structures, such as circulant and left-circulant matrices, and their generalizations: Toeplitz and Hankel matrices. In addition, we prove that Toeplitz matrices of order $n>4$ cannot be simultaneously NMDS and involutory over a field of characteristic 2. Finally, we use GDLS matrices to provide some lightweight NMDS matrices that can be computed in one clock cycle. The proposed nonrecursive NMDS matrices of orders 4, 5, 6, 7, and 8 can be implemented with 24, 50, 65, 96, and 108 XORs over $\mathbb{F}_{2^4}$, respectively.

A Low-rank Spectral Optimization Problem (LSOP) minimizes a linear objective subject to multiple two-sided linear matrix inequalities intersected with a low-rank and spectral constrained domain set. Although solving LSOP is, in general, NP-hard, its partial convexification (i.e., replacing the domain set by its convex hull) termed "LSOP-R," is often tractable and yields a high-quality solution. This motivates us to study the strength of LSOP-R. Specifically, we derive rank bounds for any extreme point of the feasible set of LSOP-R and prove their tightness for the domain sets with different matrix spaces. The proposed rank bounds recover two well-known results in the literature from a fresh angle and also allow us to derive sufficient conditions under which the relaxation LSOP-R is equivalent to the original LSOP. To effectively solve LSOP-R, we develop a column generation algorithm with a vector-based convex pricing oracle, coupled with a rank-reduction algorithm, which ensures the output solution satisfies the theoretical rank bound. Finally, we numerically verify the strength of the LSOP-R and the efficacy of the proposed algorithms.

We revisit the problem of computing with noisy information considered in Feige et al. 1994, which includes computing the OR function from noisy queries, and computing the MAX, SEARCH and SORT functions from noisy pairwise comparisons. For $K$ given elements, the goal is to correctly recover the desired function with probability at least $1-\delta$ when the outcome of each query is flipped with probability $p$. We consider both the adaptive sampling setting where each query can be adaptively designed based on past outcomes, and the non-adaptive sampling setting where the query cannot depend on past outcomes. The prior work provides tight bounds on the worst-case query complexity in terms of the dependence on $K$. However, the upper and lower bounds do not match in terms of the dependence on $\delta$ and $p$. We improve the lower bounds for all the four functions under both adaptive and non-adaptive query models. Most of our lower bounds match the upper bounds up to constant factors when either $p$ or $\delta$ is bounded away from $0$, while the ratio between the best prior upper and lower bounds goes to infinity when $p\rightarrow 0$ or $p\rightarrow 1/2$. On the other hand, we also provide matching upper and lower bounds for the number of queries in expectation, improving both the upper and lower bounds for the variable-length query model.

In this paper, we provide bounds for the genus of the pancake graph $\mathbb{P}_n$, burnt pancake graph $\mathbb{BP}_n$, and undirected generalized pancake graph $\mathbb{P}_m(n)$. Our upper bound for $\mathbb{P}_n$ is sharper than the previously-known bound, and the other bounds presented are the first of their kind. Our proofs are constructive and rely on finding an appropriate rotation system (also referred to in the literature as Edmonds' permutation technique) where certain cycles in the graphs we consider become boundaries of regions of a 2-cell embedding. A key ingredient in the proof of our bounds for the genus $\mathbb{P}_n$ and $\mathbb{BP}_n$ is a labeling algorithm of their vertices that allows us to implement rotation systems to bound the number of regions of a 2-cell embedding of said graphs.

We examine the problem of variance components testing in general mixed effects models using the likelihood ratio test. We account for the presence of nuisance parameters, i.e. the fact that some untested variances might also be equal to zero. Two main issues arise in this context leading to a non regular setting. First, under the null hypothesis the true parameter value lies on the boundary of the parameter space. Moreover, due to the presence of nuisance parameters the exact location of these boundary points is not known, which prevents from using classical asymptotic theory of maximum likelihood estimation. Then, in the specific context of nonlinear mixed-effects models, the Fisher information matrix is singular at the true parameter value. We address these two points by proposing a shrinked parametric bootstrap procedure, which is straightforward to apply even for nonlinear models. We show that the procedure is consistent, solving both the boundary and the singularity issues, and we provide a verifiable criterion for the applicability of our theoretical results. We show through a simulation study that, compared to the asymptotic approach, our procedure has a better small sample performance and is more robust to the presence of nuisance parameters. A real data application is also provided.

An intensive line of research on fixed parameter tractability of integer programming is focused on exploiting the relation between the sparsity of a constraint matrix $A$ and the norm of the elements of its Graver basis. In particular, integer programming is fixed parameter tractable when parameterized by the primal tree-depth and the entry complexity of $A$, and when parameterized by the dual tree-depth and the entry complexity of $A$; both these parameterization imply that $A$ is sparse, in particular, the number of its non-zero entries is linear in the number of columns or rows, respectively. We study preconditioners transforming a given matrix to a row-equivalent sparse matrix if it exists and provide structural results characterizing the existence of a sparse row-equivalent matrix in terms of the structural properties of the associated column matroid. In particular, our results imply that the $\ell_1$-norm of the Graver basis is bounded by a function of the maximum $\ell_1$-norm of a circuit of $A$. We use our results to design a parameterized algorithm that constructs a matrix row-equivalent to an input matrix $A$ that has small primal/dual tree-depth and entry complexity if such a row-equivalent matrix exists. Our results yield parameterized algorithms for integer programming when parameterized by the $\ell_1$-norm of the Graver basis of the constraint matrix, when parameterized by the $\ell_1$-norm of the circuits of the constraint matrix, when parameterized by the smallest primal tree-depth and entry complexity of a matrix row-equivalent to the constraint matrix, and when parameterized by the smallest dual tree-depth and entry complexity of a matrix row-equivalent to the constraint matrix.

Prediction, in regression and classification, is one of the main aims in modern data science. When the number of predictors is large, a common first step is to reduce the dimension of the data. Sufficient dimension reduction (SDR) is a well established paradigm of reduction that keeps all the relevant information in the covariates X that is necessary for the prediction of Y . In practice, SDR has been successfully used as an exploratory tool for modelling after estimation of the sufficient reduction. Nevertheless, even if the estimated reduction is a consistent estimator of the population, there is no theory that supports this step when non-parametric regression is used in the imputed estimator. In this paper, we show that the asymptotic distribution of the non-parametric regression estimator is the same regardless if the true SDR or its estimator is used. This result allows making inferences, for example, computing confidence intervals for the regression function avoiding the curse of dimensionality.

In order to understand structural relationships among sets of variables at extreme levels, we develop an extremes analogue to partial correlation. We begin by developing an inner product space constructed from transformed-linear combinations of independent regularly varying random variables. We define partial tail correlation via the projection theorem for the inner product space. We show that the partial tail correlation can be understood as the inner product of the prediction errors from transformed-linear prediction. We connect partial tail correlation to the inverse of the inner product matrix and show that a zero in this inverse implies a partial tail correlation of zero. We then show that under a modeling assumption that the random variables belong to a sensible subset of the inner product space, the matrix of inner products corresponds to the previously-studied tail pairwise dependence matrix. We develop a hypothesis test for partial tail correlation of zero. We demonstrate the performance in two applications: high nitrogen dioxide levels in Washington DC and extreme river discharges in the upper Danube basin.