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We present SCULPT, a novel 3D generative model for clothed and textured 3D meshes of humans. Specifically, we devise a deep neural network that learns to represent the geometry and appearance distribution of clothed human bodies. Training such a model is challenging, as datasets of textured 3D meshes for humans are limited in size and accessibility. Our key observation is that there exist medium-sized 3D scan datasets like CAPE, as well as large-scale 2D image datasets of clothed humans and multiple appearances can be mapped to a single geometry. To effectively learn from the two data modalities, we propose an unpaired learning procedure for pose-dependent clothed and textured human meshes. Specifically, we learn a pose-dependent geometry space from 3D scan data. We represent this as per vertex displacements w.r.t. the SMPL model. Next, we train a geometry conditioned texture generator in an unsupervised way using the 2D image data. We use intermediate activations of the learned geometry model to condition our texture generator. To alleviate entanglement between pose and clothing type, and pose and clothing appearance, we condition both the texture and geometry generators with attribute labels such as clothing types for the geometry, and clothing colors for the texture generator. We automatically generated these conditioning labels for the 2D images based on the visual question answering model BLIP and CLIP. We validate our method on the SCULPT dataset, and compare to state-of-the-art 3D generative models for clothed human bodies. We will release the codebase for research purposes.

This paper introduces the "GPT-in-the-loop" approach, a novel method combining the advanced reasoning capabilities of Large Language Models (LLMs) like Generative Pre-trained Transformers (GPT) with multiagent (MAS) systems. Venturing beyond traditional adaptive approaches that generally require long training processes, our framework employs GPT-4 for enhanced problem-solving and explanation skills. Our experimental backdrop is the smart streetlight Internet of Things (IoT) application. Here, agents use sensors, actuators, and neural networks to create an energy-efficient lighting system. By integrating GPT-4, these agents achieve superior decision-making and adaptability without the need for extensive training. We compare this approach with both traditional neuroevolutionary methods and solutions provided by software engineers, underlining the potential of GPT-driven multiagent systems in IoT. Structurally, the paper outlines the incorporation of GPT into the agent-driven Framework for the Internet of Things (FIoT), introduces our proposed GPT-in-the-loop approach, presents comparative results in the IoT context, and concludes with insights and future directions.

In this paper we develop a numerical method for efficiently approximating solutions of certain Zakai equations in high dimensions. The key idea is to transform a given Zakai SPDE into a PDE with random coefficients. We show that under suitable regularity assumptions on the coefficients of the Zakai equation, the corresponding random PDE admits a solution random field which, for almost all realizations of the random coefficients, can be written as a classical solution of a linear parabolic PDE. This makes it possible to apply the Feynman--Kac formula to obtain an efficient Monte Carlo scheme for computing approximate solutions of Zakai equations. The approach achieves good results in up to 25 dimensions with fast run times.

In this paper, we introduce a novel Artificial Intelligence (AI) system inspired by the philosophical and psychoanalytical concept of imagination as a ``Re-construction of Experiences". Our AI system is equipped with an imagination-inspired module that bridges the gap between textual inputs and other modalities, enriching the derived information based on previously learned experiences. A unique feature of our system is its ability to formulate independent perceptions of inputs. This leads to unique interpretations of a concept that may differ from human interpretations but are equally valid, a phenomenon we term as ``Interpretable Misunderstanding". We employ large-scale models, specifically a Multimodal Large Language Model (MLLM), enabling our proposed system to extract meaningful information across modalities while primarily remaining unimodal. We evaluated our system against other large language models across multiple tasks, including emotion recognition and question-answering, using a zero-shot methodology to ensure an unbiased scenario that may happen by fine-tuning. Significantly, our system outperformed the best Large Language Models (LLM) on the MELD, IEMOCAP, and CoQA datasets, achieving Weighted F1 (WF1) scores of 46.74%, 25.23%, and Overall F1 (OF1) score of 17%, respectively, compared to 22.89%, 12.28%, and 7% from the well-performing LLM. The goal is to go beyond the statistical view of language processing and tie it to human concepts such as philosophy and psychoanalysis. This work represents a significant advancement in the development of imagination-inspired AI systems, opening new possibilities for AI to generate deep and interpretable information across modalities, thereby enhancing human-AI interaction.

Relative perturbation theory for eigenvalues of Hermitian positive definite matrices has been well-studied, and the major results were later derived analogously for Hermitian non-singular matrices. In this dissertation we extend several relative perturbation results to Hermitian matrices that are potentially singular, and also develop a general class of relative bounds for Hermitian matrices. As a result, corresponding relative bounds for singular values of rank-deficient $m\times n$ matrices are also obtained using related Jordan-Wielandt matrices. We also discuss a comparison between the main relative bound derived and the Weyl's absolute perturbation bound in terms of their sharpness and derivation in practice.

This paper presents a new accelerated proximal Markov chain Monte Carlo methodology to perform Bayesian inference in imaging inverse problems with an underlying convex geometry. The proposed strategy takes the form of a stochastic relaxed proximal-point iteration that admits two complementary interpretations. For models that are smooth or regularised by Moreau-Yosida smoothing, the algorithm is equivalent to an implicit midpoint discretisation of an overdamped Langevin diffusion targeting the posterior distribution of interest. This discretisation is asymptotically unbiased for Gaussian targets and shown to converge in an accelerated manner for any target that is $\kappa$-strongly log-concave (i.e., requiring in the order of $\sqrt{\kappa}$ iterations to converge, similarly to accelerated optimisation schemes), comparing favorably to [M. Pereyra, L. Vargas Mieles, K.C. Zygalakis, SIAM J. Imaging Sciences, 13, 2 (2020), pp. 905-935] which is only provably accelerated for Gaussian targets and has bias. For models that are not smooth, the algorithm is equivalent to a Leimkuhler-Matthews discretisation of a Langevin diffusion targeting a Moreau-Yosida approximation of the posterior distribution of interest, and hence achieves a significantly lower bias than conventional unadjusted Langevin strategies based on the Euler-Maruyama discretisation. For targets that are $\kappa$-strongly log-concave, the provided non-asymptotic convergence analysis also identifies the optimal time step which maximizes the convergence speed. The proposed methodology is demonstrated through a range of experiments related to image deconvolution with Gaussian and Poisson noise, with assumption-driven and data-driven convex priors.

In this paper, we present two non-overlapping Schwarz algorithms for the hybridizable discontinuous Galerkin (HDG) method. The first algorithm is based on the Neumann-Neumann method. The second one is an iterative algorithm uses both trace and flux interface unknowns on interfaces between subdomains. Numerical results are provided to verify the validity of our algorithms.

We consider two classes of natural stochastic processes on finite unlabeled graphs. These are Euclidean stochastic optimization algorithms on the adjacency matrix of weighted graphs and a modified version of the Metropolis MCMC algorithm on stochastic block models over unweighted graphs. In both cases we show that, as the size of the graph goes to infinity, the random trajectories of the stochastic processes converge to deterministic limits. These deterministic limits are curves on the space of measure-valued graphons. Measure-valued graphons, introduced by Lov\'{a}sz and Szegedy, are a refinement of the concept of graphons that can distinguish between two infinite exchangeable arrays that give rise to the same graphon limit. We introduce new metrics on this space which provide us with a natural notion of convergence for our limit theorems. This notion is equivalent to the convergence of infinite-exchangeable arrays. Under a suitable time-scaling, the Metropolis chain admits a diffusion limit as the number of vertices go to infinity. We then demonstrate that, in an appropriately formulated zero-noise limit, the stochastic process of adjacency matrices of this diffusion converge to a deterministic gradient flow curve on the space of graphons introduced in arXiv:2111.09459 [math.PR]. Under suitable assumptions, this allows us to estimate an exponential convergence rate for the Metropolis chain in a certain limiting regime. To the best of our knowledge, both the actual rate and the connection between a natural Metropolis chain commonly used in exponential random graph models and gradient flows on graphons are new in the literature.

This paper is dedicated to the mathematical analysis of finite difference schemes for the angular diffusion operator present in the azimuth-independent Fokker-Planck equation. The study elucidates the reasons behind the lack of convergence in half range mode for certain widely recognized discrete ordinates methods, and establishes sets of sufficient conditions to ensure that the schemes achieve convergence of order $2$. In the process, interesting properties regarding Gaussian nodes and weights, which until now have remained unnoticed by mathematicians, naturally emerge.

A new generalization of Reed-Solomon codes is given. This new generalization has similar information rate bound and similar distance rate bound as BCH codes. It also approaches to the Gilbert bound as Goppa codes. Nevertheless, decoding these new codes is much faster than decoding BCH codes.