We consider Shor's quantum factoring algorithm in the setting of noisy quantum gates. Under a generic model of random noise for (controlled) rotation gates, we prove that the algorithm does not factor integers of the form $pq$ when the noise exceeds a vanishingly small level in terms of $n$ -- the number of bits of the integer to be factored, where $p$ and $q$ are from a well-defined set of primes of positive density. We further prove that with probability $1 - o(1)$ over random prime pairs $(p,q)$, Shor's factoring algorithm does not factor numbers of the form $pq$, with the same level of random noise present.
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Virtual try-on is a critical image synthesis task that aims to transfer clothes from one image to another while preserving the details of both humans and clothes. While many existing methods rely on Generative Adversarial Networks (GANs) to achieve this, flaws can still occur, particularly at high resolutions. Recently, the diffusion model has emerged as a promising alternative for generating high-quality images in various applications. However, simply using clothes as a condition for guiding the diffusion model to inpaint is insufficient to maintain the details of the clothes. To overcome this challenge, we propose an exemplar-based inpainting approach that leverages a warping module to guide the diffusion model's generation effectively. The warping module performs initial processing on the clothes, which helps to preserve the local details of the clothes. We then combine the warped clothes with clothes-agnostic person image and add noise as the input of diffusion model. Additionally, the warped clothes is used as local conditions for each denoising process to ensure that the resulting output retains as much detail as possible. Our approach, namely Diffusion-based Conditional Inpainting for Virtual Try-ON (DCI-VTON), effectively utilizes the power of the diffusion model, and the incorporation of the warping module helps to produce high-quality and realistic virtual try-on results. Experimental results on VITON-HD demonstrate the effectiveness and superiority of our method.
Large Language Models (LLMs) have emerged as a transformative force, revolutionizing numerous fields well beyond the conventional domain of Natural Language Processing (NLP) and garnering unprecedented attention. As LLM technology continues to progress, the telecom industry is facing the prospect of its potential impact on its landscape. To elucidate these implications, we delve into the inner workings of LLMs, providing insights into their current capabilities and limitations. We also examine the use cases that can be readily implemented in the telecom industry, streamlining numerous tasks that currently hinder operational efficiency and demand significant manpower and engineering expertise. Furthermore, we uncover essential research directions that deal with the distinctive challenges of utilizing the LLMs within the telecom domain. Addressing these challenges represents a significant stride towards fully harnessing the potential of LLMs and unlocking their capabilities to the fullest extent within the telecom domain.
Regression experts consistently recommend plotting residuals for model diagnosis, despite the availability of many numerical hypothesis test procedures designed to use residuals to assess problems with a model fit. Here we provide evidence for why this is good advice using data from a visual inference experiment. We show how conventional tests are too sensitive, which means that too often the conclusion would be that the model fit is inadequate. The experiment uses the lineup protocol which puts a residual plot in the context of null plots. This helps generate reliable and consistent reading of residual plots for better model diagnosis. It can also help in an obverse situation where a conventional test would fail to detect a problem with a model due to contaminated data. The lineup protocol also detects a range of departures from good residuals simultaneously.
We aim to identify the time-dependent source term in the diffusion equation using boundary measurements. This facilitates tracing back the origins of environmental pollutants. Based on the idea of dynamic complex geometrical optics (CGO) solutions, we analyze a variational formulation of the inverse source problem and prove the uniqueness result. We propose a two-step reconstruction algorithm. Initially, the locations of the point sources are determined, followed by the reconstruction of the Fourier components of the emission concentration functions. Numerical experiments on simulated data are conducted. The results demonstrate that our proposed two-step reconstruction algorithm can reliably reconstruct multiple point sources and accurately reconstruct the emission concentration functions. Additionally, we partition the algorithm into online and offline computations, with the bulk of the work done offline. This paves the way for real-time traceability of pollutants. Our proposed method, applicable in various fields - especially those related to water pollution, can identify the source of a contaminant in the environment, thus serving as a valuable tool in environmental protection.
Bayesian inference paradigms are regarded as powerful tools for solution of inverse problems. However, when applied to inverse problems in physical sciences, Bayesian formulations suffer from a number of inconsistencies that are often overlooked. A well known, but mostly neglected, difficulty is connected to the notion of conditional probability densities. Borel, and later Kolmogorov's (1933/1956), found that the traditional definition of conditional densities is incomplete: In different parameterizations it leads to different results. We will show an example where two apparently correct procedures applied to the same problem lead to two widely different results. Another type of inconsistency involves violation of causality. This problem is found in model selection strategies in Bayesian inversion, such as Hierarchical Bayes and Trans-Dimensional Inversion where so-called hyperparameters are included as variables to control either the number (or type) of unknowns, or the prior uncertainties on data or model parameters. For Hierarchical Bayes we demonstrate that the calculated 'prior' distributions of data or model parameters are not prior-, but posterior information. In fact, the calculated 'standard deviations' of the data are a measure of the inability of the forward function to model the data, rather than uncertainties of the data. For trans-dimensional inverse problems we show that the so-called evidence is, in fact, not a measure of the success of fitting the data for the given choice (or number) of parameters, as often claimed. We also find that the notion of Natural Parsimony is ill-defined, because of its dependence on the parameter prior. Based on this study, we find that careful rethinking of Bayesian inversion practices is required, with special emphasis on ways of avoiding the Borel-Kolmogorov inconsistency, and on the way we interpret model selection results.
Critical points mark locations in the domain where the level-set topology of a scalar function undergoes fundamental changes and thus indicate potentially interesting features in the data. Established methods exist to locate and relate such points in a deterministic setting, but it is less well understood how the concept of critical points can be extended to the analysis of uncertain data. Most methods for this task aim at finding likely locations of critical points or estimate the probability of their occurrence locally but do not indicate if critical points at potentially different locations in different realizations of a stochastic process are manifestations of the same feature, which is required to characterize the spatial uncertainty of critical points. Previous work on relating critical points across different realizations reported challenges for interpreting the resulting spatial distribution of critical points but did not investigate the causes. In this work, we provide a mathematical formulation of the problem of finding critical points with spatial uncertainty and computing their spatial distribution, which leads us to the notion of uncertain critical points. We analyze the theoretical properties of these structures and highlight connections to existing works for special classes of uncertain fields. We derive conditions under which well-interpretable results can be obtained and discuss the implications of those restrictions for the field of visualization. We demonstrate that the discussed limitations are not purely academic but also arise in real-world data.
Physics Informed Neural Networks is a numerical method which uses neural networks to approximate solutions of partial differential equations. It has received a lot of attention and is currently used in numerous physical and engineering problems. The mathematical understanding of these methods is limited, and in particular, it seems that, a consistent notion of stability is missing. Towards addressing this issue we consider model problems of partial differential equations, namely linear elliptic and parabolic PDEs. We consider problems with different stability properties, and problems with time discrete training. Motivated by tools of nonlinear calculus of variations we systematically show that coercivity of the energies and associated compactness provide the right framework for stability. For time discrete training we show that if these properties fail to hold then methods may become unstable. Furthermore, using tools of $\Gamma-$convergence we provide new convergence results for weak solutions by only requiring that the neural network spaces are chosen to have suitable approximation properties.
Game comonads provide a categorical syntax-free approach to finite model theory, and their Eilenberg-Moore coalgebras typically encode important combinatorial parameters of structures. In this paper, we develop a framework whereby the essential properties of these categories of coalgebras are captured in a purely axiomatic fashion. To this end, we introduce arboreal categories, which have an intrinsic process structure, allowing dynamic notions such as bisimulation and back-and-forth games, and resource notions such as number of rounds of a game, to be defined. These are related to extensional or "static" structures via arboreal covers, which are resource-indexed comonadic adjunctions. These ideas are developed in a general, axiomatic setting, and applied to relational structures, where the comonadic constructions for pebbling, Ehrenfeucht-Fra\"iss\'e and modal bisimulation games recently introduced by Abramsky et al. are recovered, showing that many of the fundamental notions of finite model theory and descriptive complexity arise from instances of arboreal covers.
Even though the analysis of unsteady 2D flow fields is challenging, fluid mechanics experts generally have an intuition on where in the simulation domain specific features are expected. Using this intuition, showing similar regions enables the user to discover flow patterns within the simulation data. When focusing on similarity, a solid mathematical framework for a specific flow pattern is not required. We propose a technique that visualizes similar and dissimilar regions with respect to a region selected by the user. Using infinitesimal strain theory, we capture the strain and rotation progression and therefore the dynamics of fluid parcels along pathlines, which we encode as distributions. We then apply the Jensen-Shannon divergence to compute the (dis)similarity between pathline dynamics originating in a user-defined flow region and the pathline dynamics of the flow field. We validate our method by applying it to two simulation datasets of two-dimensional unsteady flows. Our results show that our approach is suitable for analyzing the similarity of time-dependent flow fields.
In multi-turn dialog, utterances do not always take the full form of sentences \cite{Carbonell1983DiscoursePA}, which naturally makes understanding the dialog context more difficult. However, it is essential to fully grasp the dialog context to generate a reasonable response. Hence, in this paper, we propose to improve the response generation performance by examining the model's ability to answer a reading comprehension question, where the question is focused on the omitted information in the dialog. Enlightened by the multi-task learning scheme, we propose a joint framework that unifies these two tasks, sharing the same encoder to extract the common and task-invariant features with different decoders to learn task-specific features. To better fusing information from the question and the dialog history in the encoding part, we propose to augment the Transformer architecture with a memory updater, which is designed to selectively store and update the history dialog information so as to support downstream tasks. For the experiment, we employ human annotators to write and examine a large-scale dialog reading comprehension dataset. Extensive experiments are conducted on this dataset, and the results show that the proposed model brings substantial improvements over several strong baselines on both tasks. In this way, we demonstrate that reasoning can indeed help better response generation and vice versa. We release our large-scale dataset for further research.
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