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In this paper, we employ general results on the value distributions of perfect nonlinear functions from $\mathbb{F}_{p^m}$ to $\mathbb{F}_p$ together with a specific group action to give a unified approach to determining the weight distributions of two classes of linear codes over $\mathbb{F}_p$ constructed from perfect nonlinear functions, where $p$ is an odd prime number and $m\in\mathbb{N}_+$.

The circuit class $\mathsf{QAC}^0$ was introduced by Moore (1999) as a model for constant depth quantum circuits where the gate set includes many-qubit Toffoli gates. Proving lower bounds against such circuits is a longstanding challenge in quantum circuit complexity; in particular, showing that polynomial-size $\mathsf{QAC}^0$ cannot compute the parity function has remained an open question for over 20 years. In this work, we identify a notion of the \emph{Pauli spectrum} of $\mathsf{QAC}^0$ circuits, which can be viewed as the quantum analogue of the Fourier spectrum of classical $\mathsf{AC}^0$ circuits. We conjecture that the Pauli spectrum of $\mathsf{QAC}^0$ circuits satisfies \emph{low-degree concentration}, in analogy to the famous Linial, Nisan, Mansour theorem on the low-degree Fourier concentration of $\mathsf{AC}^0$ circuits. If true, this conjecture immediately implies that polynomial-size $\mathsf{QAC}^0$ circuits cannot compute parity. We prove this conjecture for the class of depth-$d$, polynomial-size $\mathsf{QAC}^0$ circuits with at most $n^{O(1/d)}$ auxiliary qubits. We obtain new circuit lower bounds and learning results as applications: this class of circuits cannot correctly compute -- the $n$-bit parity function on more than $(\frac{1}{2} + 2^{-\Omega(n^{1/d})})$-fraction of inputs, and -- the $n$-bit majority function on more than $(1 - 1/\mathrm{poly}(n))$-fraction of inputs. \end{itemize} Additionally we show that this class of $\mathsf{QAC}^0$ circuits with limited auxiliary qubits can be learned with quasipolynomial sample complexity, giving the first learning result for $\mathsf{QAC}^0$ circuits. More broadly, our results add evidence that ``Pauli-analytic'' techniques can be a powerful tool in studying quantum circuits.

This work focuses on developing methods for approximating the solution operators of a class of parametric partial differential equations via neural operators. Neural operators have several challenges, including the issue of generating appropriate training data, cost-accuracy trade-offs, and nontrivial hyperparameter tuning. The unpredictability of the accuracy of neural operators impacts their applications in downstream problems of inference, optimization, and control. A framework based on the linear variational problem that gives the correction to the prediction furnished by neural operators is considered based on earlier work in JCP 486 (2023) 112104. The operator, called Residual-based Error Corrector Operator or simply Corrector Operator, associated with the corrector problem is analyzed further. Numerical results involving a nonlinear reaction-diffusion model in two dimensions with PCANet-type neural operators show almost two orders of increase in the accuracy of approximations when neural operators are corrected using the correction scheme. Further, topology optimization involving a nonlinear reaction-diffusion model is considered to highlight the limitations of neural operators and the efficacy of the correction scheme. Optimizers with neural operator surrogates are seen to make significant errors (as high as 80 percent). However, the errors are much lower (below 7 percent) when neural operators are corrected.

We show that the problem of whether a query is equivalent to a query of tree-width $k$ is decidable, for the class of Unions of Conjunctive Regular Path Queries with two-way navigation (UC2RPQs). A previous result by Barcel\'o, Romero, and Vardi [SIAM Journal on Computing, 2016] has shown decidability for the case $k=1$, and here we extend this result showing that decidability in fact holds for any arbitrary $k\geq 1$. The algorithm is in 2ExpSpace, but for the restricted but practically relevant case where all regular expressions of the query are of the form $a^*$ or $(a_1 + \dotsb + a_n)$ we show that the complexity of the problem drops to $\Pi^P_2$. We also investigate the related problem of approximating a UC2RPQ by queries of small tree-width. We exhibit an algorithm which, for any fixed number $k$, builds the maximal under-approximation of tree-width $k$ of a UC2RPQ. The maximal under-approximation of tree-width $k$ of a query $q$ is a query $q'$ of tree-width $k$ which is contained in $q$ in a maximal and unique way, that is, such that for every query $q''$ of tree-width $k$, if $q''$ is contained in $q$ then $q''$ is also contained in $q'$. Our approach is shown to be robust, in the sense that it allows also to test equivalence with queries of a given path-width, it also covers the previously known result for $k=1$, and it allows to test for equivalence of whether a (one-way) UCRPQ is equivalent to a UCRPQ of a given tree-width (or path-width).

Let $\Omega = [0,1]^d$ be the unit cube in $\mathbb{R}^d$. We study the problem of how efficiently, in terms of the number of parameters, deep neural networks with the ReLU activation function can approximate functions in the Sobolev spaces $W^s(L_q(\Omega))$ and Besov spaces $B^s_r(L_q(\Omega))$, with error measured in the $L_p(\Omega)$ norm. This problem is important when studying the application of neural networks in a variety of fields, including scientific computing and signal processing, and has previously been solved only when $p=q=\infty$. Our contribution is to provide a complete solution for all $1\leq p,q\leq \infty$ and $s > 0$ for which the corresponding Sobolev or Besov space compactly embeds into $L_p$. The key technical tool is a novel bit-extraction technique which gives an optimal encoding of sparse vectors. This enables us to obtain sharp upper bounds in the non-linear regime where $p > q$. We also provide a novel method for deriving $L_p$-approximation lower bounds based upon VC-dimension when $p < \infty$. Our results show that very deep ReLU networks significantly outperform classical methods of approximation in terms of the number of parameters, but that this comes at the cost of parameters which are not encodable.

We consider the following general model of a sorting procedure: we fix a hereditary permutation class $\mathcal{C}$, which corresponds to the operations that the procedure is allowed to perform in a single step. The input of sorting is a permutation $\pi$ of the set $[n]=\{1,2,\dotsc,n\}$, i.e., a sequence where each element of $[n]$ appears once. In every step, the sorting procedure picks a permutation $\sigma$ of length $n$ from $\mathcal{C}$, and rearranges the current permutation of numbers by composing it with $\sigma$. The goal is to transform the input $\pi$ into the sorted sequence $1,2,\dotsc,n$ in as few steps as possible. This model of sorting captures not only classical sorting algorithms, like insertion sort or bubble sort, but also sorting by series of devices, like stacks or parallel queues, as well as sorting by block operations commonly considered, e.g., in the context of genome rearrangement. Our goal is to describe the possible asymptotic behavior of the worst-case number of steps needed when sorting with a hereditary permutation class. As the main result, we show that any hereditary permutation class $\mathcal{C}$ falls into one of five distinct categories. Disregarding the classes that cannot sort all permutations, the number of steps needed to sort any permutation of $[n]$ with $\mathcal{C}$ is either $\Theta(n^2)$, a function between $O(n)$ and $\Omega(\sqrt{n})$, a function betwee $O(\log^2 n)$ and $\Omega(\log n), or $1$, and for each of these cases we provide a structural characterization of the corresponding hereditary classes.

We study the approximability of general convex sets in $\mathbb{R}^n$ by intersections of halfspaces, where the approximation quality is measured with respect to the standard Gaussian distribution $N(0,I_n)$ and the complexity of an approximation is the number of halfspaces used. While a large body of research has considered the approximation of convex sets by intersections of halfspaces under distance metrics such as the Lebesgue measure and Hausdorff distance, prior to our work there has not been a systematic study of convex approximation under the Gaussian distribution. We establish a range of upper and lower bounds, both for general convex sets and for specific natural convex sets that are of particular interest. Our results demonstrate that the landscape of approximation is intriguingly different under the Gaussian distribution versus previously studied distance measures. For example, we show that $2^{\Theta(\sqrt{n})}$ halfspaces are both necessary and sufficient to approximate the origin-centered $\ell_2$ ball of Gaussian volume 1/2 to any constant accuracy, and that for $1 \leq p < 2$, the origin-centered $\ell_p$ ball of Gaussian volume 1/2 can be approximated to any constant accuracy as an intersection of $2^{\widetilde{O}(n^{3/4})}$ many halfspaces. These bounds are quite different from known approximation results under more commonly studied distance measures. Our results are proved using techniques from many different areas. These include classical results on convex polyhedral approximation, Cram\'er-type bounds on large deviations from probability theory, and -- perhaps surprisingly -- a range of topics from computational complexity, including computational learning theory, unconditional pseudorandomness, and the study of influences and noise sensitivity in the analysis of Boolean functions.

Over the last decades, the family of $\alpha$-stale distributions has proven to be useful for modelling in telecommunication systems. Particularly, in the case of radar applications, finding a fast and accurate estimation for the amplitude density function parameters appears to be very important. In this work, the maximum likelihood estimator (MLE) is proposed for parameters of the amplitude distribution. To do this, the amplitude data are \emph{projected} on the horizontal and vertical axes using two simple transformations. It is proved that the \emph{projected} data follow a zero-location symmetric $\alpha$-stale distribution for which the MLE can be computed quite fast. The average of computed MLEs based on two \emph{projections} is considered as estimator for parameters of the amplitude distribution. Performance of the proposed \emph{projection} method is demonstrated through simulation study and analysis of two sets of real radar data.

In this paper, we propose a randomized $\tilde{O}(\mu(G))$-round algorithm for the maximum cardinality matching problem in the CONGEST model, where $\mu(G)$ means the maximum size of a matching of the input graph $G$. The proposed algorithm substantially improves the current best worst-case running time. The key technical ingredient is a new randomized algorithm of finding an augmenting path of length $\ell$ with high probability within $\tilde{O}(\ell)$ rounds, which positively settles an open problem left in the prior work by Ahmadi and Kuhn [DISC'20]. The idea of our augmenting path algorithm is based on a recent result by Kitamura and Izumi [IEICE Trans.'22], which efficiently identifies a sparse substructure of the input graph containing an augmenting path, following a new concept called \emph{alternating base trees}. Their algorithm, however, resorts to a centralized approach of collecting the entire information of the substructure into a single vertex for constructing an augmenting path. The technical highlight of this paper is to provide a fully-decentralized counterpart of such a centralized method. To develop the algorithm, we prove several new structural properties of alternating base trees, which are of independent interest.

Efficiently approximating the probability of system failure has gained increasing importance as expensive simulations begin to play a larger role in reliability quantification tasks in areas such as structural design, power grid design, and safety certification among others. This work derives credible intervals on the probability of failure for a simulation which we assume is a realizations of a Gaussian process. We connect these intervals to binary classification error and comment on their applicability to a broad class of iterative schemes proposed throughout the literature. A novel iterative sampling scheme is proposed which can suggest multiple samples per batch for simulations with parallel implementations. We empirically test our scalable, open-source implementation on a variety simulations including a Tsunami model where failure is quantified in terms of maximum wave hight.