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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.

In this paper, we consider feature screening for ultrahigh dimensional clustering analyses. Based on the observation that the marginal distribution of any given feature is a mixture of its conditional distributions in different clusters, we propose to screen clustering features by independently evaluating the homogeneity of each feature's mixture distribution. Important cluster-relevant features have heterogeneous components in their mixture distributions and unimportant features have homogeneous components. The well-known EM-test statistic is used to evaluate the homogeneity. Under general parametric settings, we establish the tail probability bounds of the EM-test statistic for the homogeneous and heterogeneous features, and further show that the proposed screening procedure can achieve the sure independent screening and even the consistency in selection properties. Limiting distribution of the EM-test statistic is also obtained for general parametric distributions. The proposed method is computationally efficient, can accurately screen for important cluster-relevant features and help to significantly improve clustering, as demonstrated in our extensive simulation and real data analyses.

The algebraic degree is an important parameter of Boolean functions used in cryptography. When a function in a large number of variables is not given explicitly in algebraic normal form, it might not be feasible to compute its degree. Instead, one can try to estimate the degree using probabilistic tests. We propose a probabilistic test for deciding whether the algebraic degree of a Boolean function $f$ is below a certain value $k$. The test involves picking an affine space of dimension $k$ and testing whether the values on $f$ on that space sum up to zero. If $deg(f)<k$, then $f$ will always pass the test, otherwise it will sometimes pass and sometimes fail the test, depending on which affine space was chosen. The probability of failing the proposed test is closely related to the number of monomials of degree $k$ in a polynomial $g$, averaged over all the polynomials $g$ which are affine equivalent to $f$. We initiate the study of the probability of failing the proposed ``$deg(f)<k$'' test. We show that in the particular case when the degree of $f$ is actually equal to $k$, the probability will be in the interval $(0.288788, 0.5]$, and therefore a small number of runs of the test is sufficient to give, with very high probability, the correct answer. Exact values of this probability for all the polynomials in 8 variables were computed using the representatives listed by Hou and by Langevin and Leander.

A growing line of work shows how learned predictions can be used to break through worst-case barriers to improve the running time of an algorithm. However, incorporating predictions into data structures with strong theoretical guarantees remains underdeveloped. This paper takes a step in this direction by showing that predictions can be leveraged in the fundamental online list labeling problem. In the problem, n items arrive over time and must be stored in sorted order in an array of size Theta(n). The array slot of an element is its label and the goal is to maintain sorted order while minimizing the total number of elements moved (i.e., relabeled). We design a new list labeling data structure and bound its performance in two models. In the worst-case learning-augmented model, we give guarantees in terms of the error in the predictions. Our data structure provides strong guarantees: it is optimal for any prediction error and guarantees the best-known worst-case bound even when the predictions are entirely erroneous. We also consider a stochastic error model and bound the performance in terms of the expectation and variance of the error. Finally, the theoretical results are demonstrated empirically. In particular, we show that our data structure has strong performance on real temporal data sets where predictions are constructed from elements that arrived in the past, as is typically done in a practical use case.

Consider a two-person zero-sum search game between a Hider and a Searcher. The Hider chooses to hide in one of $n$ discrete locations (or "boxes") and the Searcher chooses a search sequence specifying which order to look in these boxes until finding the Hider. A search at box $i$ takes $t_i$ time units and finds the Hider - if hidden there - independently with probability $q_i$, for $i=1,\ldots,n$. The Searcher wants to minimize the expected total time needed to find the Hider, while the Hider wants to maximize it. It is shown in the literature that the Searcher has an optimal search strategy that mixes up to $n$ distinct search sequences with appropriate probabilities. This paper investigates the existence of optimal pure strategies for the Searcher - a single deterministic search sequence that achieves the optimal expected total search time regardless of where the Hider hides. We identify several cases in which the Searcher has an optimal pure strategy, and several cases in which such optimal pure strategy does not exist. An optimal pure search strategy has significant practical value because the Searcher does not need to randomize their actions and will avoid second guessing themselves if the chosen search sequence from an optimal mixed strategy does not turn out well.

We bring in here a novel algebraic approach for attacking the McEliece cryptosystem. It consists in introducing a subspace of matrices representing quadratic forms. Those are associated with quadratic relationships for the component-wise product in the dual of the code used in the cryptosystem. Depending on the characteristic of the code field, this space of matrices consists only of symmetric matrices or skew-symmetric matrices. This matrix space is shown to contain unusually low-rank matrices (rank $2$ or $3$ depending on the characteristic) which reveal the secret polynomial structure of the code. Finding such matrices can then be used to recover the secret key of the scheme. We devise a dedicated approach in characteristic $2$ consisting in using a Gr\"obner basis modeling that a skew-symmetric matrix is of rank $2$. This allows to analyze the complexity of solving the corresponding algebraic system with Gr\"obner bases techniques. This computation behaves differently when applied to the skew-symmetric matrix space associated with a random code rather than with a Goppa or an alternant code. This gives a distinguisher of the latter code family. We give a bound on its complexity which turns out to interpolate nicely between polynomial and exponential depending on the code parameters. A distinguisher for alternant/Goppa codes was already known [FGO+11]. It is of polynomial complexity but works only in a narrow parameter regime. This new distinguisher is also polynomial for the parameter regime necessary for [FGO+11] but contrarily to the previous one is able to operate for virtually all code parameters relevant to cryptography. Moreover, we use this matrix space to find a polynomial time attack of the McEliece cryptosystem provided that the Goppa code is distinguishable by the method of [FGO+11] and its degree is less than $q-1$, where $q$ is the alphabet size of the code.

We propose a family of recursive cutting-plane algorithms to solve feasibility problems with constrained memory, which can also be used for first-order convex optimization. Precisely, in order to find a point within a ball of radius $\epsilon$ with a separation oracle in dimension $d$ -- or to minimize $1$-Lipschitz convex functions to accuracy $\epsilon$ over the unit ball -- our algorithms use $\mathcal O(\frac{d^2}{p}\ln \frac{1}{\epsilon})$ bits of memory, and make $\mathcal O((C\frac{d}{p}\ln \frac{1}{\epsilon})^p)$ oracle calls, for some universal constant $C \geq 1$. The family is parametrized by $p\in[d]$ and provides an oracle-complexity/memory trade-off in the sub-polynomial regime $\ln\frac{1}{\epsilon}\gg\ln d$. While several works gave lower-bound trade-offs (impossibility results) -- we explicit here their dependence with $\ln\frac{1}{\epsilon}$, showing that these also hold in any sub-polynomial regime -- to the best of our knowledge this is the first class of algorithms that provides a positive trade-off between gradient descent and cutting-plane methods in any regime with $\epsilon\leq 1/\sqrt d$. The algorithms divide the $d$ variables into $p$ blocks and optimize over blocks sequentially, with approximate separation vectors constructed using a variant of Vaidya's method. In the regime $\epsilon \leq d^{-\Omega(d)}$, our algorithm with $p=d$ achieves the information-theoretic optimal memory usage and improves the oracle-complexity of gradient descent.

This manuscript portrays optimization as a process. In many practical applications the environment is so complex that it is infeasible to lay out a comprehensive theoretical model and use classical algorithmic theory and mathematical optimization. It is necessary as well as beneficial to take a robust approach, by applying an optimization method that learns as one goes along, learning from experience as more aspects of the problem are observed. This view of optimization as a process has become prominent in varied fields and has led to some spectacular success in modeling and systems that are now part of our daily lives.

Knowledge is a formal way of understanding the world, providing a human-level cognition and intelligence for the next-generation artificial intelligence (AI). One of the representations of knowledge is the structural relations between entities. An effective way to automatically acquire this important knowledge, called Relation Extraction (RE), a sub-task of information extraction, plays a vital role in Natural Language Processing (NLP). Its purpose is to identify semantic relations between entities from natural language text. To date, there are several studies for RE in previous works, which have documented these techniques based on Deep Neural Networks (DNNs) become a prevailing technique in this research. Especially, the supervised and distant supervision methods based on DNNs are the most popular and reliable solutions for RE. This article 1)introduces some general concepts, and further 2)gives a comprehensive overview of DNNs in RE from two points of view: supervised RE, which attempts to improve the standard RE systems, and distant supervision RE, which adopts DNNs to design the sentence encoder and the de-noise method. We further 3)cover some novel methods and describe some recent trends and discuss possible future research directions for this task.

Dynamic programming (DP) solves a variety of structured combinatorial problems by iteratively breaking them down into smaller subproblems. In spite of their versatility, DP algorithms are usually non-differentiable, which hampers their use as a layer in neural networks trained by backpropagation. To address this issue, we propose to smooth the max operator in the dynamic programming recursion, using a strongly convex regularizer. This allows to relax both the optimal value and solution of the original combinatorial problem, and turns a broad class of DP algorithms into differentiable operators. Theoretically, we provide a new probabilistic perspective on backpropagating through these DP operators, and relate them to inference in graphical models. We derive two particular instantiations of our framework, a smoothed Viterbi algorithm for sequence prediction and a smoothed DTW algorithm for time-series alignment. We showcase these instantiations on two structured prediction tasks and on structured and sparse attention for neural machine translation.