Efficient algorithms for computing linear convolutions based on the fast Fourier transform are developed. A hybrid approach is described that combines the conventional practice of explicit dealiasing (explicitly padding the input data with zeros) and implicit dealiasing (mathematically accounting for these zero values). The new approach generalizes implicit dealiasing to arbitrary padding ratios and includes explicit dealiasing as a special case. Unlike existing implementations of implicit dealiasing, hybrid dealiasing tailors its subtransform sizes to the convolution geometry. Multidimensional convolutions are implemented with hybrid dealiasing by decomposing them into lower-dimensional convolutions. Convolutions of complex-valued and Hermitian inputs of equal length are illustrated with pseudocode and implemented in the open-source FFTW++ library. Hybrid dealiasing is shown to outperform explicit dealiasing in one, two, and three dimensions.
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Convolutional neural networks (CNNs) are currently among the most widely-used deep neural network (DNN) architectures available and achieve state-of-the-art performance for many problems. Originally applied to computer vision tasks, CNNs work well with any data with a spatial relationship, besides images, and have been applied to different fields. However, recent works have highlighted numerical stability challenges in DNNs, which also relates to their known sensitivity to noise injection. These challenges can jeopardise their performance and reliability. This paper investigates DeepGOPlus, a CNN that predicts protein function. DeepGOPlus has achieved state-of-the-art performance and can successfully take advantage and annotate the abounding protein sequences emerging in proteomics. We determine the numerical stability of the model's inference stage by quantifying the numerical uncertainty resulting from perturbations of the underlying floating-point data. In addition, we explore the opportunity to use reduced-precision floating point formats for DeepGOPlus inference, to reduce memory consumption and latency. This is achieved by instrumenting DeepGOPlus' execution using Monte Carlo Arithmetic, a technique that experimentally quantifies floating point operation errors and VPREC, a tool that emulates results with customizable floating point precision formats. Focus is placed on the inference stage as it is the primary deliverable of the DeepGOPlus model, widely applicable across different environments. All in all, our results show that although the DeepGOPlus CNN is very stable numerically, it can only be selectively implemented with lower-precision floating-point formats. We conclude that predictions obtained from the pre-trained DeepGOPlus model are very reliable numerically, and use existing floating-point formats efficiently.
Collision-resistant hashing, a fundamental primitive in modern cryptography, ensures that there is no efficient way to find distinct inputs that produce the same hash value. This property underpins the security of various cryptographic applications, making it crucial to understand its complexity. The complexity of this problem is well-understood in the classical setting and $\Theta(N^{1/2})$ queries are needed to find a collision. However, the advent of quantum computing has introduced new challenges since quantum adversaries $\unicode{x2013}$ equipped with the power of quantum queries $\unicode{x2013}$ can find collisions much more efficiently. Brassard, H\"oyer and Tapp and Aaronson and Shi established that full-scale quantum adversaries require $\Theta(N^{1/3})$ queries to find a collision, prompting a need for longer hash outputs, which impacts efficiency in terms of the key lengths needed for security. This paper explores the implications of quantum attacks in the Noisy-Intermediate Scale Quantum (NISQ) era. In this work, we investigate three different models for NISQ algorithms and achieve tight bounds for all of them: (1) A hybrid algorithm making adaptive quantum or classical queries but with a limited quantum query budget, or (2) A quantum algorithm with access to a noisy oracle, subject to a dephasing or depolarizing channel, or (3) A hybrid algorithm with an upper bound on its maximum quantum depth; i.e., a classical algorithm aided by low-depth quantum circuits. In fact, our results handle all regimes between NISQ and full-scale quantum computers. Previously, only results for the pre-image search problem were known for these models by Sun and Zheng, Rosmanis, Chen, Cotler, Huang and Li while nothing was known about the collision finding problem.
Recent advancements in the realm of deep learning, particularly in the development of large language models (LLMs), have demonstrated AI's ability to tackle complex mathematical problems or solving programming challenges. However, the capability to solve well-defined problems based on extensive training data differs significantly from the nuanced process of making scientific discoveries. Trained on almost all human knowledge available, today's sophisticated LLMs basically learn to predict sequences of tokens. They generate mathematical derivations and write code in a similar way as writing an essay, and do not have the ability to pioneer scientific discoveries in the manner a human scientist would do. In this study we delve into the potential of using deep learning to rediscover a fundamental mathematical concept: integrals. By defining integrals as area under the curve, we illustrate how AI can deduce the integral of a given function, exemplified by inferring $\int_{0}^{x} t^2 dt = \frac{x^3}{3}$ and $\int_{0}^{x} ae^{bt} dt = \frac{a}{b} e^{bx} - \frac{a}{b}$. Our experiments show that deep learning models can approach the task of inferring integrals either through a sequence-to-sequence model, akin to language translation, or by uncovering the rudimentary principles of integration, such as $\int_{0}^{x} t^n dt = \frac{x^{n+1}}{n+1}$.
We investigate the vulnerability of computer-vision-based signal classifiers to adversarial perturbations of their inputs, where the signals and perturbations are subject to physical constraints. We consider a scenario in which a source and interferer emit signals that propagate as waves to a detector, which attempts to classify the source by analyzing the spectrogram of the signal it receives using a pre-trained neural network. By solving PDE-constrained optimization problems, we construct interfering signals that cause the detector to misclassify the source even though the perturbations to the spectrogram of the received signal are nearly imperceptible. Though such problems can have millions of decision variables, we introduce methods to solve them efficiently. Our experiments demonstrate that one can compute effective and physically realizable adversarial perturbations for a variety of machine learning models under various physical conditions.
The lifted multicut problem has diverse applications in the field of computer vision. Exact algorithms based on linear programming require an understanding of lifted multicut polytopes. Despite recent progress, two fundamental questions about these polytopes have remained open: Which lower cube inequalities define facets, and which cut inequalities define facets? In this article, we answer the first question by establishing conditions that are necessary, sufficient and efficiently decidable. Toward the second question, we show that deciding facet-definingness of cut inequalities is NP-hard. This completes the analysis of canonical facets of lifted multicut polytopes.
We introduce a framework for benchmarking optimizers according to multiple criteria over various test functions. Based on a recently introduced union-free generic depth function for partial orders/rankings, it fully exploits the ordinal information and allows for incomparability. Our method describes the distribution of all partial orders/rankings, avoiding the notorious shortcomings of aggregation. This permits to identify test functions that produce central or outlying rankings of optimizers and to assess the quality of benchmarking suites.
Arora & Ge introduced a noise-free polynomial system to compute the secret of a Learning With Errors (LWE) instance via linearization. Albrecht et al. later utilized the Arora-Ge polynomial model to study the complexity of Gr\"obner basis computations on LWE polynomial systems under the assumption of semi-regularity. In this paper we revisit the Arora-Ge polynomial and prove that it satisfies a genericity condition recently introduced by Caminata & Gorla, called being in generic coordinates. For polynomial systems in generic coordinates one can always estimate the complexity of DRL Gr\"obner basis computations in terms of the Castelnuovo-Mumford regularity and henceforth also via the Macaulay bound. Moreover, we generalize the Gr\"obner basis algorithm of Semaev & Tenti to arbitrary polynomial systems with a finite degree of regularity. In particular, existence of this algorithm yields another approach to estimate the complexity of DRL Gr\"obner basis computations in terms of the degree of regularity. In practice, the degree of regularity of LWE polynomial systems is not known, though one can always estimate the lowest achievable degree of regularity. Consequently, from a designer's worst case perspective this approach yields sub-exponential complexity estimates for general, binary secret and binary error LWE. In recent works by Dachman-Soled et al. the hardness of LWE in the presence of side information was analyzed. Utilizing their framework we discuss how hints can be incorporated into LWE polynomial systems and how they affect the complexity of Gr\"obner basis computations.
Automated disinformation generation is often listed as an important risk associated with large language models (LLMs). The theoretical ability to flood the information space with disinformation content might have dramatic consequences for societies around the world. This paper presents a comprehensive study of the disinformation capabilities of the current generation of LLMs to generate false news articles in the English language. In our study, we evaluated the capabilities of 10 LLMs using 20 disinformation narratives. We evaluated several aspects of the LLMs: how good they are at generating news articles, how strongly they tend to agree or disagree with the disinformation narratives, how often they generate safety warnings, etc. We also evaluated the abilities of detection models to detect these articles as LLM-generated. We conclude that LLMs are able to generate convincing news articles that agree with dangerous disinformation narratives.
Residual networks (ResNets) have displayed impressive results in pattern recognition and, recently, have garnered considerable theoretical interest due to a perceived link with neural ordinary differential equations (neural ODEs). This link relies on the convergence of network weights to a smooth function as the number of layers increases. We investigate the properties of weights trained by stochastic gradient descent and their scaling with network depth through detailed numerical experiments. We observe the existence of scaling regimes markedly different from those assumed in neural ODE literature. Depending on certain features of the network architecture, such as the smoothness of the activation function, one may obtain an alternative ODE limit, a stochastic differential equation or neither of these. These findings cast doubts on the validity of the neural ODE model as an adequate asymptotic description of deep ResNets and point to an alternative class of differential equations as a better description of the deep network limit.
Attention Model has now become an important concept in neural networks that has been researched within diverse application domains. This survey provides a structured and comprehensive overview of the developments in modeling attention. In particular, we propose a taxonomy which groups existing techniques into coherent categories. We review the different neural architectures in which attention has been incorporated, and also show how attention improves interpretability of neural models. Finally, we discuss some applications in which modeling attention has a significant impact. We hope this survey will provide a succinct introduction to attention models and guide practitioners while developing approaches for their applications.
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