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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.

The machine learning of lattice operators has three possible bottlenecks. From a statistical standpoint, it is necessary to design a constrained class of operators based on prior information with low bias, and low complexity relative to the sample size. From a computational perspective, there should be an efficient algorithm to minimize an empirical error over the class. From an understanding point of view, the properties of the learned operator need to be derived, so its behavior can be theoretically understood. The statistical bottleneck can be overcome due to the rich literature about the representation of lattice operators, but there is no general learning algorithm for them. In this paper, we discuss a learning paradigm in which, by overparametrizing a class via elements in a lattice, an algorithm for minimizing functions in a lattice is applied to learn. We present the stochastic lattice gradient descent algorithm as a general algorithm to learn on constrained classes of operators as long as a lattice overparametrization of it is fixed, and we discuss previous works which are proves of concept. Moreover, if there are algorithms to compute the basis of an operator from its overparametrization, then its properties can be deduced and the understanding bottleneck is also overcome. This learning paradigm has three properties that modern methods based on neural networks lack: control, transparency and interpretability. Nowadays, there is an increasing demand for methods with these characteristics, and we believe that mathematical morphology is in a unique position to supply them. The lattice overparametrization paradigm could be a missing piece for it to achieve its full potential within modern machine learning.

Pomset logic and BV are both logics that extend multiplicative linear logic (with Mix) with a third connective that is self-dual and non-commutative. Whereas pomset logic originates from the study of coherence spaces and proof nets, BV originates from the study of series-parallel orders, cographs, and proof systems. Both logics enjoy a cut-admissibility result, but for neither logic can this be done in the sequent calculus. Provability in pomset logic can be checked via a proof net correctness criterion and in BV via a deep inference proof system. It has long been conjectured that these two logics are the same. In this paper we show that this conjecture is false. We also investigate the complexity of the two logics, exhibiting a huge gap between the two. Whereas provability in BV is NP-complete, provability in pomset logic is $\Sigma_2^p$-complete. We also make some observations with respect to possible sequent systems for the two logics.

Large language models (LLMs) fine-tuned with reinforcement learning from human feedback (RLHF) have been used in some of the most widely deployed AI models to date, such as OpenAI's ChatGPT, Anthropic's Claude, or Meta's LLaMA-2. While there has been significant work developing these methods, our understanding of the benefits and downsides of each stage in RLHF is still limited. To fill this gap, we present an extensive analysis of how each stage of the process (i.e. supervised fine-tuning (SFT), reward modelling, and RLHF) affects two key properties: out-of-distribution (OOD) generalisation and output diversity. OOD generalisation is crucial given the wide range of real-world scenarios in which these models are being used, while output diversity refers to the model's ability to generate varied outputs and is important for a variety of use cases. We perform our analysis across two base models on both summarisation and instruction following tasks, the latter being highly relevant for current LLM use cases. We find that RLHF generalises better than SFT to new inputs, particularly as the distribution shift between train and test becomes larger. However, RLHF significantly reduces output diversity compared to SFT across a variety of measures, implying a tradeoff in current LLM fine-tuning methods between generalisation and diversity. Our results provide guidance on which fine-tuning method should be used depending on the application, and show that more research is needed to improve the trade-off between generalisation and diversity.

Recent works have demonstrated a double descent phenomenon in over-parameterized learning. Although this phenomenon has been investigated by recent works, it has not been fully understood in theory. In this paper, we investigate the multiple descent phenomenon in a class of multi-component prediction models. We first consider a ''double random feature model'' (DRFM) concatenating two types of random features, and study the excess risk achieved by the DRFM in ridge regression. We calculate the precise limit of the excess risk under the high dimensional framework where the training sample size, the dimension of data, and the dimension of random features tend to infinity proportionally. Based on the calculation, we further theoretically demonstrate that the risk curves of DRFMs can exhibit triple descent. We then provide a thorough experimental study to verify our theory. At last, we extend our study to the ''multiple random feature model'' (MRFM), and show that MRFMs ensembling $K$ types of random features may exhibit $(K+1)$-fold descent. Our analysis points out that risk curves with a specific number of descent generally exist in learning multi-component prediction models.

Numerous approaches have attempted to interpret deep neural networks (DNNs) by attributing the prediction of DNN to its input features. One of the well-studied attribution methods is Integrated Gradients (IG). Specifically, the choice of baselines for IG is a critical consideration for generating meaningful and unbiased explanations for model predictions in different scenarios. However, current practice of exploiting a single baseline fails to fulfill this ambition, thus demanding multiple baselines. Fortunately, the inherent connection between IG and Aumann-Shapley Value forms a unique perspective to rethink the design of baselines. Under certain hypothesis, we theoretically analyse that a set of baseline aligns with the coalitions in Shapley Value. Thus, we propose a novel baseline construction method called Shapley Integrated Gradients (SIG) that searches for a set of baselines by proportional sampling to partly simulate the computation path of Shapley Value. Simulations on GridWorld show that SIG approximates the proportion of Shapley Values. Furthermore, experiments conducted on various image tasks demonstrate that compared to IG using other baseline methods, SIG exhibits an improved estimation of feature's contribution, offers more consistent explanations across diverse applications, and is generic to distinct data types or instances with insignificant computational overhead.

The multiple scattering theory (MST) is a Green's function method that has been widely used in electronic structure calculations for crystalline disordered systems. The key property of the MST method is the scattering path matrix (SPM) that characterizes the Green's function within a local solution representation. This paper studies various approximations of the SPM, under the condition that an appropriate reference is used for perturbation. In particular, we justify the convergence of the SPM approximations with respect to the size of scattering region and scattering length of reference, which are the central numerical parameters to achieve a linear scaling method with MST. We also present some numerical experiments on several typical systems to support the theory.

We analyze the number of queries that a whitebox adversary needs to make to a private learner in order to reconstruct its training data. For $(\epsilon, \delta)$ DP learners with training data drawn from any arbitrary compact metric space, we provide the \emph{first known lower bounds on the adversary's query complexity} as a function of the learner's privacy parameters. \emph{Our results are minimax optimal for every $\epsilon \geq 0, \delta \in [0, 1]$, covering both $\epsilon$-DP and $(0, \delta)$ DP as corollaries}. Beyond this, we obtain query complexity lower bounds for $(\alpha, \epsilon)$ R\'enyi DP learners that are valid for any $\alpha > 1, \epsilon \geq 0$. Finally, we analyze data reconstruction attacks on locally compact metric spaces via the framework of Metric DP, a generalization of DP that accounts for the underlying metric structure of the data. In this setting, we provide the first known analysis of data reconstruction in unbounded, high dimensional spaces and obtain query complexity lower bounds that are nearly tight modulo logarithmic factors.

We propose a new distributed-computing model, inspired by permissionless distributed systems such as Bitcoin and Ethereum, that allows studying permissionless consensus in a mathematically regular setting. Like in the sleepy model of Pass and Shi, we consider a synchronous, round-by-round message-passing system in which the set of online processors changes each round. Unlike the sleepy model, the set of processors may be infinite. Moreover, processors never fail; instead, an adversary can temporarily or permanently impersonate some processors. Finally, processors have access to a strong form of message-authentication that authenticates not only the sender of a message but also the round in which the message was sent. Assuming that, each round, the adversary impersonates less than 1/2 of the online processors, we present two consensus algorithms. The first ensures deterministic safety and constant latency in expectation, assuming a probabilistic leader-election oracle. The second ensures deterministic safety and deterministic liveness assuming irrevocable impersonation and eventually-stabilizing participation. The model is unrealistic in full generality. However, if we assume finitely many processes and that the set of faulty processes remains constant, the model coincides with a practically-motivated model: the static version of the sleepy model.

Discrete particle simulations have become the standard in science and industrial applications exploring the properties of particulate systems. Most of such simulations rely on the concept of interacting spherical particles to describe the properties of particulates, although, the correct representation of the nonspherical particle shape is crucial for a number of applications. In this work we describe the implementation of clumps, i.e. assemblies of rigidly connected spherical particles, which can approximate given nonspherical shapes, within the \textit{MercuryDPM} particle dynamics code. \textit{MercuryDPM} contact detection algorithm is particularly efficient for polydisperse particle systems, which is essential for multilevel clumps approximating complex surfaces. We employ the existing open-source \texttt{CLUMP} library to generate clump particles. We detail the pre-processing tools providing necessary initial data, as well as the necessary adjustments of the algorithms of contact detection, collision/migration and numerical time integration. The capabilities of our implementation are illustrated for a variety of examples.