The theory of finite simple groups is a (rather unexplored) area likely to provide interesting computational problems and modelling tools useful in a cryptographic context. In this note, we review some applications of finite non-abelian simple groups to cryptography and discuss different scenarios in which this theory is clearly central, providing the relevant definitions to make the material accessible to both cryptographers and group theorists, in the hope of stimulating further interaction between these two (non-disjoint) communities. In particular, we look at constructions based on various group-theoretic factorization problems, review group theoretical hash functions, and discuss fully homomorphic encryption using simple groups. The Hidden Subgroup Problem is also briefly discussed in this context.
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Reconstruction of interaction network between random events is a critical problem arising from statistical physics and politics to sociology, biology, and psychology, and beyond. The Ising model lays the foundation for this reconstruction process, but finding the underlying Ising model from the least amount of observed samples in a computationally efficient manner has been historically challenging for half a century. By using the idea of sparsity learning, we present a approach named SIMPLE that has a dominant sample complexity from theoretical limit. Furthermore, a tuning-free algorithm is developed to give a statistically consistent solution of SIMPLE in polynomial time with high probability. On extensive benchmarked cases, the SIMPLE approach provably reconstructs underlying Ising models with global optimality. The application on the U.S. senators voting in the last six congresses reveals that both the Republicans and Democrats noticeably assemble in each congresses; interestingly, the assembling of Democrats is particularly pronounced in the latest congress.
In the context of computer models, calibration is the process of estimating unknown simulator parameters from observational data. Calibration is variously referred to as model fitting, parameter estimation/inference, an inverse problem, and model tuning. The need for calibration occurs in most areas of science and engineering, and has been used to estimate hard to measure parameters in climate, cardiology, drug therapy response, hydrology, and many other disciplines. Although the statistical method used for calibration can vary substantially, the underlying approach is essentially the same and can be considered abstractly. In this survey, we review the decisions that need to be taken when calibrating a model, and discuss a range of computational methods that can be used to compute Bayesian posterior distributions.
We introduce a novel modeling approach for time series imputation and forecasting, tailored to address the challenges often encountered in real-world data, such as irregular samples, missing data, or unaligned measurements from multiple sensors. Our method relies on a continuous-time-dependent model of the series' evolution dynamics. It leverages adaptations of conditional, implicit neural representations for sequential data. A modulation mechanism, driven by a meta-learning algorithm, allows adaptation to unseen samples and extrapolation beyond observed time-windows for long-term predictions. The model provides a highly flexible and unified framework for imputation and forecasting tasks across a wide range of challenging scenarios. It achieves state-of-the-art performance on classical benchmarks and outperforms alternative time-continuous models.
We establish an efficient approximation algorithm for the partition functions of a class of quantum spin systems at low temperature, which can be viewed as stable quantum perturbations of classical spin systems. Our algorithm is based on combining the contour representation of quantum spin systems of this type due to Borgs, Koteck\'y, and Ueltschi with the algorithmic framework developed by Helmuth, Perkins, and Regts, and Borgs et al.
Diffusion models have demonstrated compelling generation quality by optimizing the variational lower bound through a simple denoising score matching loss. In this paper, we provide theoretical evidence that the prevailing practice of using a constant loss weight strategy in diffusion models leads to biased estimation during the training phase. Simply optimizing the denoising network to predict Gaussian noise with constant weighting may hinder precise estimations of original images. To address the issue, we propose an elegant and effective weighting strategy grounded in the theoretically unbiased principle. Moreover, we conduct a comprehensive and systematic exploration to dissect the inherent bias problem deriving from constant weighting loss from the perspectives of its existence, impact and reasons. These analyses are expected to advance our understanding and demystify the inner workings of diffusion models. Through empirical evaluation, we demonstrate that our proposed debiased estimation method significantly enhances sample quality without the reliance on complex techniques, and exhibits improved efficiency compared to the baseline method both in training and sampling processes.
The numerical solution of the Stokes equations on an evolving domain with a moving boundary is studied based on the arbitrary Lagrangian-Eulerian finite element method and a second-order projection method along the trajectories of the evolving mesh for decoupling the unknown solutions of velocity and pressure. The error of the semidiscrete arbitrary Lagrangian-Eulerian method is shown to be $O(h^{r+1})$ for the Taylor--Hood finite elements of degree $r\ge 2$, using Nitsche's duality argument adapted to an evolving mesh, by proving that the material derivative and the Stokes--Ritz projection commute up to terms which have optimal-order convergence in the $L^2$ norm. Additionally, the error of the fully discrete finite element method, with a second-order projection method along the trajectories of the evolving mesh, is shown to be $O(\ln(1/\tau+1)\tau^{2}+\ln(1/h+1)h^{r+1})$ in the discrete $L^\infty(0,T; L^2)$ norm using newly developed energy techniques and backward parabolic duality arguments that are applicable to the Stokes equations with an evolving mesh. To maintain consistency between the notations of the numerical scheme in a moving domain and those in a fixed domain, we introduce the equivalence class of finite element spaces across time levels. Numerical examples are provided to support the theoretical analysis and to illustrate the performance of the method in simulating Navier--Stokes flow in a domain with a rotating propeller.
We propose an approach utilizing gamma-distributed random variables, coupled with log-Gaussian modeling, to generate synthetic datasets suitable for training neural networks. This addresses the challenge of limited real observations in various applications. We apply this methodology to both Raman and coherent anti-Stokes Raman scattering (CARS) spectra, using experimental spectra to estimate gamma process parameters. Parameter estimation is performed using Markov chain Monte Carlo methods, yielding a full Bayesian posterior distribution for the model which can be sampled for synthetic data generation. Additionally, we model the additive and multiplicative background functions for Raman and CARS with Gaussian processes. We train two Bayesian neural networks to estimate parameters of the gamma process which can then be used to estimate the underlying Raman spectrum and simultaneously provide uncertainty through the estimation of parameters of a probability distribution. We apply the trained Bayesian neural networks to experimental Raman spectra of phthalocyanine blue, aniline black, naphthol red, and red 264 pigments and also to experimental CARS spectra of adenosine phosphate, fructose, glucose, and sucrose. The results agree with deterministic point estimates for the underlying Raman and CARS spectral signatures.
The field of visual computing is rapidly advancing due to the emergence of generative artificial intelligence (AI), which unlocks unprecedented capabilities for the generation, editing, and reconstruction of images, videos, and 3D scenes. In these domains, diffusion models are the generative AI architecture of choice. Within the last year alone, the literature on diffusion-based tools and applications has seen exponential growth and relevant papers are published across the computer graphics, computer vision, and AI communities with new works appearing daily on arXiv. This rapid growth of the field makes it difficult to keep up with all recent developments. The goal of this state-of-the-art report (STAR) is to introduce the basic mathematical concepts of diffusion models, implementation details and design choices of the popular Stable Diffusion model, as well as overview important aspects of these generative AI tools, including personalization, conditioning, inversion, among others. Moreover, we give a comprehensive overview of the rapidly growing literature on diffusion-based generation and editing, categorized by the type of generated medium, including 2D images, videos, 3D objects, locomotion, and 4D scenes. Finally, we discuss available datasets, metrics, open challenges, and social implications. This STAR provides an intuitive starting point to explore this exciting topic for researchers, artists, and practitioners alike.
We introduce a generic framework that reduces the computational cost of object detection while retaining accuracy for scenarios where objects with varied sizes appear in high resolution images. Detection progresses in a coarse-to-fine manner, first on a down-sampled version of the image and then on a sequence of higher resolution regions identified as likely to improve the detection accuracy. Built upon reinforcement learning, our approach consists of a model (R-net) that uses coarse detection results to predict the potential accuracy gain for analyzing a region at a higher resolution and another model (Q-net) that sequentially selects regions to zoom in. Experiments on the Caltech Pedestrians dataset show that our approach reduces the number of processed pixels by over 50% without a drop in detection accuracy. The merits of our approach become more significant on a high resolution test set collected from YFCC100M dataset, where our approach maintains high detection performance while reducing the number of processed pixels by about 70% and the detection time by over 50%.
While it is nearly effortless for humans to quickly assess the perceptual similarity between two images, the underlying processes are thought to be quite complex. Despite this, the most widely used perceptual metrics today, such as PSNR and SSIM, are simple, shallow functions, and fail to account for many nuances of human perception. Recently, the deep learning community has found that features of the VGG network trained on the ImageNet classification task has been remarkably useful as a training loss for image synthesis. But how perceptual are these so-called "perceptual losses"? What elements are critical for their success? To answer these questions, we introduce a new Full Reference Image Quality Assessment (FR-IQA) dataset of perceptual human judgments, orders of magnitude larger than previous datasets. We systematically evaluate deep features across different architectures and tasks and compare them with classic metrics. We find that deep features outperform all previous metrics by huge margins. More surprisingly, this result is not restricted to ImageNet-trained VGG features, but holds across different deep architectures and levels of supervision (supervised, self-supervised, or even unsupervised). Our results suggest that perceptual similarity is an emergent property shared across deep visual representations.
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