We consider the fast in-place computation of the Euclidean polynomial modular remainder R(X) $\not\equiv$ A(X) mod B(X) with A and B of respective degrees n and m $\le$ n. If the multiplication of two polynomials of degree k can be performed with M(k) operations and O(k) extra space, then standard algorithms for the remainder require O(n/m M(m)) arithmetic operations and, apart from that of A and B, at least O(n -- m) extra memory. This extra space is notably usually used to store the whole quotient Q(X) such that A = BQ + R with deg R < deg B.We avoid the storage of the whole of this quotient, and propose an algorithm still using O(n/m M(m)) arithmetic operations but only O(m) extra space.When the divisor B is sparse with a constant number of non-zero terms, the arithmetic complexity bound reduces to O(n).When it is allowed to use the input space of A or B for intermediate computations, but putting A and B back to their initial states after the completion of the remainder computation, we further propose an in-place algorithm (that is with its extra required space reduced to O(1) only) using at mostO(n/m M(m) log(m) arithmetic operations.To achieve this, we develop techniques for Toeplitz matrix operations which output is also part of the input. In-place accumulated versions are obtained for the latter and for polynomial remaindering via reductions to accumulated polynomial multiplication, for which a recent fast in-place algorithm hasbeen developed.
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Online speech recognition, where the model only accesses context to the left, is an important and challenging use case for ASR systems. In this work, we investigate augmenting neural encoders for online ASR by incorporating structured state-space sequence models (S4), which are a family of models that provide a parameter-efficient way of accessing arbitrarily long left context. We perform systematic ablation studies to compare variants of S4 models and propose two novel approaches that combine them with convolutions. We find that the most effective design is to stack a small S4 using real-valued recurrent weights with a local convolution, allowing them to work complementarily. Our best model achieves WERs of 4.01%/8.53% on test sets from Librispeech, outperforming Conformers with extensively tuned convolution.
A general a posteriori error analysis applies to five lowest-order finite element methods for two fourth-order semi-linear problems with trilinear non-linearity and a general source. A quasi-optimal smoother extends the source term to the discrete trial space, and more importantly, modifies the trilinear term in the stream-function vorticity formulation of the incompressible 2D Navier-Stokes and the von K\'{a}rm\'{a}n equations. This enables the first efficient and reliable a posteriori error estimates for the 2D Navier-Stokes equations in the stream-function vorticity formulation for Morley, two discontinuous Galerkin, $C^0$ interior penalty, and WOPSIP discretizations with piecewise quadratic polynomials.
We prove that the scaled maximum steady-state waiting time and the scaled maximum steady-state queue length among $N$ $GI/GI/1$-queues in the $N$-server fork-join queue, converge to a normally distributed random variable as $N\to\infty$. The maximum steady-state waiting time in this queueing system scales around $\frac{1}{\gamma}\log N$, where $\gamma$ is determined by the cumulant generating function $\Lambda$ of the service distribution and solves the Cram\'er-Lundberg equation with stochastic service times and deterministic inter-arrival times. This value $\frac{1}{\gamma}\log N$ is reached at a certain hitting time. The number of arrivals until that hitting time satisfies the central limit theorem, with standard deviation $\frac{\sigma_A}{\sqrt{\Lambda'(\gamma)\gamma}}$. By using distributional Little's law, we can extend this result to the maximum queue length. Finally, we extend these results to a fork-join queue with different classes of servers.
Standard multiparameter eigenvalue problems (MEPs) are systems of $k\ge 2$ linear $k$-parameter square matrix pencils. Recently, a new form of multiparameter eigenvalue problems has emerged: a rectangular MEP (RMEP) with only one multivariate rectangular matrix pencil, where we are looking for combinations of the parameters for which the rank of the pencil is not full. Applications include finding the optimal least squares autoregressive moving average (ARMA) model and the optimal least squares realization of autonomous linear time-invariant (LTI) dynamical system. For linear and polynomial RMEPs, we give the number of solutions and show how these problems can be solved numerically by a transformation into a standard MEP. For the transformation we provide new linearizations for quadratic multivariate matrix polynomials with a specific structure of monomials and consider mixed systems of rectangular and square multivariate matrix polynomials. This numerical approach seems computationally considerably more attractive than the block Macaulay method, the only other currently available numerical method for polynomial RMEPs.
This work proposes the extended functional tensor train (EFTT) format for compressing and working with multivariate functions on tensor product domains. Our compression algorithm combines tensorized Chebyshev interpolation with a low-rank approximation algorithm that is entirely based on function evaluations. Compared to existing methods based on the functional tensor train format, our approach often reduces the required storage, sometimes considerably, while achieving the same accuracy. In particular, we reduce the number of function evaluations required to achieve a prescribed accuracy by up to over 96% compared to the algorithm from [Gorodetsky, Karaman and Marzouk, Comput. Methods Appl. Mech. Eng., 347 (2019)] .
In the Activation Edge-Multicover problem we are given a multigraph $G=(V,E)$ with activation costs $\{c_{e}^u,c_{e}^v\}$ for every edge $e=uv \in E$, and degree requirements $r=\{r_v:v \in V\}$. The goal is to find an edge subset $J \subseteq E$ of minimum activation cost $\sum_{v \in V}\max\{c_{uv}^v:uv \in J\}$,such that every $v \in V$ has at least $r_v$ neighbors in the graph $(V,J)$. Let $k= \max_{v \in V} r_v$ be the maximum requirement and let $\theta=\max_{e=uv \in E} \frac{\max\{c_e^u,c_e^v\}}{\min\{c_e^u,c_e^v\}}$ be the maximum quotient between the two costs of an edge. For $\theta=1$ the problem admits approximation ratio $O(\log k)$. For $k=1$ it generalizes the Set Cover problem (when $\theta=\infty$), and admits a tight approximation ratio $O(\log n)$. This implies approximation ratio $O(k \log n)$ for general $k$ and $\theta$, and no better approximation ratio was known. We obtain the first logarithmic approximation ratio $O(\log k +\log\min\{\theta,n\})$, that bridges between the two known ratios -- $O(\log k)$ for $\theta=1$ and $O(\log n)$ for $k=1$. This implies approximation ratio $O\left(\log k +\log\min\{\theta,n\}\right) +\beta \cdot (\theta+1)$ for the Activation $k$-Connected Subgraph problem, where $\beta$ is the best known approximation ratio for the ordinary min-cost version of the problem.
Auditory spatial attention detection (ASAD) aims to decode the attended spatial location with EEG in a multiple-speaker setting. ASAD methods are inspired by the brain lateralization of cortical neural responses during the processing of auditory spatial attention, and show promising performance for the task of auditory attention decoding (AAD) with neural recordings. In the previous ASAD methods, the spatial distribution of EEG electrodes is not fully exploited, which may limit the performance of these methods. In the present work, by transforming the original EEG channels into a two-dimensional (2D) spatial topological map, the EEG data is transformed into a three-dimensional (3D) arrangement containing spatial-temporal information. And then a 3D deep convolutional neural network (DenseNet-3D) is used to extract temporal and spatial features of the neural representation for the attended locations. The results show that the proposed method achieves higher decoding accuracy than the state-of-the-art (SOTA) method (94.4% compared to XANet's 90.6%) with 1-second decision window for the widely used KULeuven (KUL) dataset, and the code to implement our work is available on Github: //github.com/xuxiran/ASAD_DenseNet
We say that a Hamilton cycle $C=(x_1,\ldots,x_n)$ in a graph $G$ is $k$-symmetric, if the mapping $x_i\mapsto x_{i+n/k}$ for all $i=1,\ldots,n$, where indices are considered modulo $n$, is an automorphism of $G$. In other words, if we lay out the vertices $x_1,\ldots,x_n$ equidistantly on a circle and draw the edges of $G$ as straight lines, then the drawing of $G$ has $k$-fold rotational symmetry, i.e., all information about the graph is compressed into a $360^\circ/k$ wedge of the drawing. The maximum $k$ for which there exists a $k$-symmetric Hamilton cycle in $G$ is referred to as the Hamilton compression of $G$. We investigate the Hamilton compression of four different families of vertex-transitive graphs, namely hypercubes, Johnson graphs, permutahedra and Cayley graphs of abelian groups. In several cases we determine their Hamilton compression exactly, and in other cases we provide close lower and upper bounds. The constructed cycles have a much higher compression than several classical Gray codes known from the literature. Our constructions also yield Gray codes for bitstrings, combinations and permutations that have few tracks and/or that are balanced.
We study the classical problem of approximating a non-decreasing function $f: \mathcal{X} \to \mathcal{Y}$ in $L^p(\mu)$ norm by sequentially querying its values, for known compact real intervals $\mathcal{X}$, $\mathcal{Y}$ and a known probability measure $\mu$ on $\cX$. For any function~$f$ we characterize the minimum number of evaluations of $f$ that algorithms need to guarantee an approximation $\hat{f}$ with an $L^p(\mu)$ error below $\epsilon$ after stopping. Unlike worst-case results that hold uniformly over all $f$, our complexity measure is dependent on each specific function $f$. To address this problem, we introduce GreedyBox, a generalization of an algorithm originally proposed by Novak (1992) for numerical integration. We prove that GreedyBox achieves an optimal sample complexity for any function $f$, up to logarithmic factors. Additionally, we uncover results regarding piecewise-smooth functions. Perhaps as expected, the $L^p(\mu)$ error of GreedyBox decreases much faster for piecewise-$C^2$ functions than predicted by the algorithm (without any knowledge on the smoothness of $f$). A simple modification even achieves optimal minimax approximation rates for such functions, which we compute explicitly. In particular, our findings highlight multiple performance gaps between adaptive and non-adaptive algorithms, smooth and piecewise-smooth functions, as well as monotone or non-monotone functions. Finally, we provide numerical experiments to support our theoretical results.
We consider the performance of Glauber dynamics for the random cluster model with real parameter $q>1$ and temperature $\beta>0$. Recent work by Helmuth, Jenssen and Perkins detailed the ordered/disordered transition of the model on random $\Delta$-regular graphs for all sufficiently large $q$ and obtained an efficient sampling algorithm for all temperatures $\beta$ using cluster expansion methods. Despite this major progress, the performance of natural Markov chains, including Glauber dynamics, is not yet well understood on the random regular graph, partly because of the non-local nature of the model (especially at low temperatures) and partly because of severe bottleneck phenomena that emerge in a window around the ordered/disordered transition. Nevertheless, it is widely conjectured that the bottleneck phenomena that impede mixing from worst-case starting configurations can be avoided by initialising the chain more judiciously. Our main result establishes this conjecture for all sufficiently large $q$ (with respect to $\Delta$). Specifically, we consider the mixing time of Glauber dynamics initialised from the two extreme configurations, the all-in and all-out, and obtain a pair of fast mixing bounds which cover all temperatures $\beta$, including in particular the bottleneck window. Our result is inspired by the recent approach of Gheissari and Sinclair for the Ising model who obtained a similar-flavoured mixing-time bound on the random regular graph for sufficiently low temperatures. To cover all temperatures in the RC model, we refine appropriately the structural results of Helmuth, Jenssen and Perkins about the ordered/disordered transition and show spatial mixing properties "within the phase", which are then related to the evolution of the chain.
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