Given a polyline on $n$ vertices, the polyline simplification problem asks for a minimum size subsequence of these vertices defining a new polyline whose distance to the original polyline is at most a given threshold under some distance measure. In this paper, we improve the long-standing running time bound for the simplification of polylines under the local Fr\'echet distance. The best algorithm known so far is by Imai and Iri and has a cubic running time in $n$. We present an algorithm with a running time of $O(n^2)$ under the $L_1$ and $L_\infty$ norm, and $O(n^2 \log n)$ under $L_{p \in (1,\infty)}$ (including the Euclidean norm $L_2$). Our approach is based on the ideas of Chan and Chin, who showed that under the local Hausdorff distance, the Imai-Iri algorithm can be improved to run in quadratic time for $L_1$, $L_2$, and $L_\infty$. However, the Hausdorff distance does not take the order of the points along the polyline into account. The Fr\'echet distance, which is sensitive to the course of the polylines, is hence often deemed the superior distance measure for polyline similarity but it also more intricate to compute. So far, the significantly faster simplification algorithms for the Hausdorff distance made them preferable for practical application. But our new algorithm for simplification under the Fr\'echet distance matches the running time bounds for the Hausdorff distance up to logarithmic factors and thus allows the usage of this more suitable distance measure.
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This paper outlines the ethical implications of text simplification within the framework of assistive systems. We argue that a distinction should be made between the technologies that perform text simplification and the realisation of these in assistive technologies. When using the latter as a motivation for research, it is important that the subsequent ethical implications be carefully considered. We provide guidelines for the framing of text simplification independently of assistive systems, as well as suggesting directions for future research and discussion based on the concerns raised.
Isogeometric analysis with the boundary element method (IGABEM) has recently gained interest. In this paper, the approximability of IGABEM on 3D acoustic scattering problems will be investigated and a new improved BeTSSi submarine will be presented as a benchmark example. Both Galerkin and collocation are considered in combination with several boundary integral equations (BIE). In addition to the conventional BIE, regularized versions of this BIE will be considered. Moreover, the hyper-singular BIE and the Burton--Miller formulation are also considered. A new adaptive integration routine is presented, and the numerical examples show the importance of the integration procedure in the boundary element method. The numerical examples also include comparison between standard BEM and IGABEM, which again verifies the higher accuracy obtained from the increased inter-element continuity of the spline basis functions. One of the main objectives in this paper is benchmarking acoustic scattering problems, and the method of manufactured solution will be used frequently in this regard.
Escaping from saddle points and finding local minimum is a central problem in nonconvex optimization. Perturbed gradient methods are perhaps the simplest approach for this problem. However, to find $(\epsilon, \sqrt{\epsilon})$-approximate local minima, the existing best stochastic gradient complexity for this type of algorithms is $\tilde O(\epsilon^{-3.5})$, which is not optimal. In this paper, we propose LENA (Last stEp shriNkAge), a faster perturbed stochastic gradient framework for finding local minima. We show that LENA with stochastic gradient estimators such as SARAH/SPIDER and STORM can find $(\epsilon, \epsilon_{H})$-approximate local minima within $\tilde O(\epsilon^{-3} + \epsilon_{H}^{-6})$ stochastic gradient evaluations (or $\tilde O(\epsilon^{-3})$ when $\epsilon_H = \sqrt{\epsilon}$). The core idea of our framework is a step-size shrinkage scheme to control the average movement of the iterates, which leads to faster convergence to the local minima.
Transformers have achieved state-of-the-art results across multiple NLP tasks. However, the self-attention mechanism complexity scales quadratically with the sequence length, creating an obstacle for tasks involving long sequences, like in the speech domain. In this paper, we discuss the usefulness of self-attention for Direct Speech Translation. First, we analyze the layer-wise token contributions in the self-attention of the encoder, unveiling local diagonal patterns. To prove that some attention weights are avoidable, we propose to substitute the standard self-attention with a local efficient one, setting the amount of context used based on the results of the analysis. With this approach, our model matches the baseline performance, and improves the efficiency by skipping the computation of those weights that standard attention discards.
We formulate the quadratic eigenvalue problem underlying the mathematical model of a linear vibrational system as an eigenvalue problem of a diagonal-plus-low-rank matrix $A$. The eigenvector matrix of $A$ has a Cauchy-like structure. Optimal viscosities are those for which $trace(X)$ is minimal, where $X$ is the solution of the Lyapunov equation $AX+XA^{*}=GG^{*}$. Here $G$ is a low-rank matrix which depends on the eigenfrequencies that need to be damped. After initial eigenvalue decomposition of linearized problem which requires $O(n^3)$ operations, our algorithm computes optimal viscosities for each choice of external dampers in $O(n^2)$ operations, provided that the number of dampers is small. Hence, the subsequent optimization is order of magnitude faster than in the standard approach which solves Lyapunov equation in each step, thus requiring $O(n^3)$ operations. Our algorithm is based on $O(n^2)$ eigensolver for complex symmetric diagonal-plus-rank-one matrices and fast $O(n^2)$ multiplication of linked Cauchy-like matrices.
Many existing algorithms for streaming geometric data analysis have been plagued by exponential dependencies in the space complexity, which are undesirable for processing high-dimensional data sets. In particular, once $d\geq\log n$, there are no known non-trivial streaming algorithms for problems such as maintaining convex hulls and L\"owner-John ellipsoids of $n$ points, despite a long line of work in streaming computational geometry since [AHV04]. We simultaneously improve these results to $\mathrm{poly}(d,\log n)$ bits of space by trading off with a $\mathrm{poly}(d,\log n)$ factor distortion. We achieve these results in a unified manner, by designing the first streaming algorithm for maintaining a coreset for $\ell_\infty$ subspace embeddings with $\mathrm{poly}(d,\log n)$ space and $\mathrm{poly}(d,\log n)$ distortion. Our algorithm also gives similar guarantees in the \emph{online coreset} model. Along the way, we sharpen results for online numerical linear algebra by replacing a log condition number dependence with a $\log n$ dependence, answering a question of [BDM+20]. Our techniques provide a novel connection between leverage scores, a fundamental object in numerical linear algebra, and computational geometry. For $\ell_p$ subspace embeddings, we give nearly optimal trade-offs between space and distortion for one-pass streaming algorithms. For instance, we give a deterministic coreset using $O(d^2\log n)$ space and $O((d\log n)^{1/2-1/p})$ distortion for $p>2$, whereas previous deterministic algorithms incurred a $\mathrm{poly}(n)$ factor in the space or the distortion [CDW18]. Our techniques have implications in the offline setting, where we give optimal trade-offs between the space complexity and distortion of subspace sketch data structures. To do this, we give an elementary proof of a "change of density" theorem of [LT80] and make it algorithmic.
The stochastic gradient Langevin Dynamics is one of the most fundamental algorithms to solve sampling problems and non-convex optimization appearing in several machine learning applications. Especially, its variance reduced versions have nowadays gained particular attention. In this paper, we study two variants of this kind, namely, the Stochastic Variance Reduced Gradient Langevin Dynamics and the Stochastic Recursive Gradient Langevin Dynamics. We prove their convergence to the objective distribution in terms of KL-divergence under the sole assumptions of smoothness and Log-Sobolev inequality which are weaker conditions than those used in prior works for these algorithms. With the batch size and the inner loop length set to $\sqrt{n}$, the gradient complexity to achieve an $\epsilon$-precision is $\tilde{O}((n+dn^{1/2}\epsilon^{-1})\gamma^2 L^2\alpha^{-2})$, which is an improvement from any previous analyses. We also show some essential applications of our result to non-convex optimization.
For any small positive real $\varepsilon$ and integer $t > \frac{1}{\varepsilon}$, we build a graph with a vertex deletion set of size $t$ to a tree, and twin-width greater than $2^{(1-\varepsilon) t}$. In particular, this shows that the twin-width is sometimes exponential in the treewidth, in the so-called oriented twin-width and grid number, and that adding an apex may multiply the twin-width by at least $2-\varepsilon$. Except for the one in oriented twin-width, these lower bounds are essentially tight.
Learning accurate classifiers for novel categories from very few examples, known as few-shot image classification, is a challenging task in statistical machine learning and computer vision. The performance in few-shot classification suffers from the bias in the estimation of classifier parameters; however, an effective underlying bias reduction technique that could alleviate this issue in training few-shot classifiers has been overlooked. In this work, we demonstrate the effectiveness of Firth bias reduction in few-shot classification. Theoretically, Firth bias reduction removes the $O(N^{-1})$ first order term from the small-sample bias of the Maximum Likelihood Estimator. Here we show that the general Firth bias reduction technique simplifies to encouraging uniform class assignment probabilities for multinomial logistic classification, and almost has the same effect in cosine classifiers. We derive an easy-to-implement optimization objective for Firth penalized multinomial logistic and cosine classifiers, which is equivalent to penalizing the cross-entropy loss with a KL-divergence between the uniform label distribution and the predictions. Then, we empirically evaluate that it is consistently effective across the board for few-shot image classification, regardless of (1) the feature representations from different backbones, (2) the number of samples per class, and (3) the number of classes. Finally, we show the robustness of Firth bias reduction, in the case of imbalanced data distribution. Our implementation is available at //github.com/ehsansaleh/firth_bias_reduction
Earables (ear wearables) is rapidly emerging as a new platform encompassing a diverse range of personal applications. The traditional authentication methods hence become less applicable and inconvenient for earables due to their limited input interface. Nevertheless, earables often feature rich around-the-head sensing capability that can be leveraged to capture new types of biometrics. In this work, we proposeToothSonic which leverages the toothprint-induced sonic effect produced by users performing teeth gestures for earable authentication. In particular, we design representative teeth gestures that can produce effective sonic waves carrying the information of the toothprint. To reliably capture the acoustic toothprint, it leverages the occlusion effect of the ear canal and the inward-facing microphone of the earables. It then extracts multi-level acoustic features to reflect the intrinsic toothprint information for authentication. The key advantages of ToothSonic are that it is suitable for earables and is resistant to various spoofing attacks as the acoustic toothprint is captured via the user's private teeth-ear channel that modulates and encrypts the sonic waves. Our experiment studies with 25 participants show that ToothSonic achieves up to 95% accuracy with only one of the users' tooth gestures.
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