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The recent successes in analyzing images with deep neural networks are almost exclusively achieved with Convolutional Neural Networks (CNNs). The training of these CNNs, and in fact of all deep neural network architectures, uses the backpropagation algorithm where the output of the network is compared with the desired result and the difference is then used to tune the weights of the network towards the desired outcome. In a 2022 preprint, Geoffrey Hinton suggested an alternative way of training which passes the desired results together with the images at the input of the network. This so called Forward Forward (FF) algorithm has up to now only been used in fully connected networks. In this paper, we show how the FF paradigm can be extended to CNNs. Our FF-trained CNN, featuring a novel spatially-extended labeling technique, achieves a classification accuracy of 99.0% on the MNIST hand-written digits dataset. We show how different hyperparameters affect the performance of the proposed algorithm and compare the results with CNN trained with the standard backpropagation approach. Furthermore, we use Class Activation Maps to investigate which type of features are learnt by the FF algorithm.

Error bounds are derived for sampling and estimation using a discretization of an intrinsically defined Langevin diffusion with invariant measure $d\mu_\phi \propto e^{-\phi} \mathrm{dvol}_g $ on a compact Riemannian manifold. Two estimators of linear functionals of $\mu_\phi $ based on the discretized Markov process are considered: a time-averaging estimator based on a single trajectory and an ensemble-averaging estimator based on multiple independent trajectories. Imposing no restrictions beyond a nominal level of smoothness on $\phi$, first-order error bounds, in discretization step size, on the bias and variances of both estimators are derived. The order of error matches the optimal rate in Euclidean and flat spaces, and leads to a first-order bound on distance between the invariant measure $\mu_\phi$ and a stationary measure of the discretized Markov process. Generality of the proof techniques, which exploit links between two partial differential equations and the semigroup of operators corresponding to the Langevin diffusion, renders them amenable for the study of a more general class of sampling algorithms related to the Langevin diffusion. Conditions for extending analysis to the case of non-compact manifolds are discussed. Numerical illustrations with distributions, log-concave and otherwise, on the manifolds of positive and negative curvature elucidate on the derived bounds and demonstrate practical utility of the sampling algorithm.

Bias correction can often improve the finite sample performance of estimators. We show that the choice of bias correction method has no effect on the higher-order variance of semiparametrically efficient parametric estimators, so long as the estimate of the bias is asymptotically linear. It is also shown that bootstrap, jackknife, and analytical bias estimates are asymptotically linear for estimators with higher-order expansions of a standard form. In particular, we find that for a variety of estimators the straightforward bootstrap bias correction gives the same higher-order variance as more complicated analytical or jackknife bias corrections. In contrast, bias corrections that do not estimate the bias at the parametric rate, such as the split-sample jackknife, result in larger higher-order variances in the i.i.d. setting we focus on. For both a cross-sectional MLE and a panel model with individual fixed effects, we show that the split-sample jackknife has a higher-order variance term that is twice as large as that of the `leave-one-out' jackknife.

EODECA (Engineered Ordinary Differential Equations as Classification Algorithm) is a novel approach at the intersection of machine learning and dynamical systems theory, presenting a unique framework for classification tasks [1]. This method stands out with its dynamical system structure, utilizing ordinary differential equations (ODEs) to efficiently handle complex classification challenges. The paper delves into EODECA's dynamical properties, emphasizing its resilience against random perturbations and robust performance across various classification scenarios. Notably, EODECA's design incorporates the ability to embed stable attractors in the phase space, enhancing reliability and allowing for reversible dynamics. In this paper, we carry out a comprehensive analysis by expanding on the work [1], and employing a Euler discretization scheme. In particular, we evaluate EODECA's performance across five distinct classification problems, examining its adaptability and efficiency. Significantly, we demonstrate EODECA's effectiveness on the MNIST and Fashion MNIST datasets, achieving impressive accuracies of $98.06\%$ and $88.21\%$, respectively. These results are comparable to those of a multi-layer perceptron (MLP), underscoring EODECA's potential in complex data processing tasks. We further explore the model's learning journey, assessing its evolution in both pre and post training environments and highlighting its ability to navigate towards stable attractors. The study also investigates the invertibility of EODECA, shedding light on its decision-making processes and internal workings. This paper presents a significant step towards a more transparent and robust machine learning paradigm, bridging the gap between machine learning algorithms and dynamical systems methodologies.

Generative AIs, especially Large Language Models (LLMs) such as ChatGPT or Llama, have advanced significantly, positioning them as valuable tools for digital forensics. While initial studies have explored the potential of ChatGPT in the context of investigations, the question of to what extent LLMs can assist the forensic report writing process remains unresolved. To answer the question, this article first examines forensic reports with the goal of generalization (e.g., finding the `average structure' of a report). We then evaluate the strengths and limitations of LLMs for generating the different parts of the forensic report using a case study. This work thus provides valuable insights into the automation of report writing, a critical facet of digital forensics investigations. We conclude that combined with thorough proofreading and corrections, LLMs may assist practitioners during the report writing process but at this point cannot replace them.

Neural Radiance Fields (NeRFs) have demonstrated the remarkable potential of neural networks to capture the intricacies of 3D objects. By encoding the shape and color information within neural network weights, NeRFs excel at producing strikingly sharp novel views of 3D objects. Recently, numerous generalizations of NeRFs utilizing generative models have emerged, expanding its versatility. In contrast, Gaussian Splatting (GS) offers a similar renders quality with faster training and inference as it does not need neural networks to work. We encode information about the 3D objects in the set of Gaussian distributions that can be rendered in 3D similarly to classical meshes. Unfortunately, GS are difficult to condition since they usually require circa hundred thousand Gaussian components. To mitigate the caveats of both models, we propose a hybrid model that uses GS representation of the 3D object's shape and NeRF-based encoding of color and opacity. Our model uses Gaussian distributions with trainable positions (i.e. means of Gaussian), shape (i.e. covariance of Gaussian), color and opacity, and neural network, which takes parameters of Gaussian and viewing direction to produce changes in color and opacity. Consequently, our model better describes shadows, light reflections, and transparency of 3D objects.

We study and introduce new gradient operators in the complex and bicomplex settings, inspired from the well-known Least Mean Square (LMS) algorithm invented in 1960 by Widrow and Hoff for Adaptive Linear Neuron (ADALINE). These gradient operators will be used to formulate new learning rules for the Bicomplex Least Mean Square (BLMS) algorithms and we will also formulate these learning rules will for the case of multicomplex LMS algorithms (MLMS). This approach extends both the classical real and complex LMS algorithms.

In the ever-evolving landscape of Artificial Intelligence (AI), the synergy between generative AI and Software Engineering emerges as a transformative frontier. This whitepaper delves into the unexplored realm, elucidating how generative AI techniques can revolutionize software development. Spanning from project management to support and updates, we meticulously map the demands of each development stage and unveil the potential of generative AI in addressing them. Techniques such as zero-shot prompting, self-consistency, and multimodal chain-of-thought are explored, showcasing their unique capabilities in enhancing generative AI models. The significance of vector embeddings, context, plugins, tools, and code assistants is underscored, emphasizing their role in capturing semantic information and amplifying generative AI capabilities. Looking ahead, this intersection promises to elevate productivity, improve code quality, and streamline the software development process. This whitepaper serves as a guide for stakeholders, urging discussions and experiments in the application of generative AI in Software Engineering, fostering innovation and collaboration for a qualitative leap in the efficiency and effectiveness of software development.

May's Theorem [K. O. May, Econometrica 20 (1952) 680-684] characterizes majority voting on two alternatives as the unique preferential voting method satisfying several simple axioms. Here we show that by adding some desirable axioms to May's axioms, we can uniquely determine how to vote on three alternatives. In particular, we add two axioms stating that the voting method should mitigate spoiler effects and avoid the so-called strong no show paradox. We prove a theorem stating that any preferential voting method satisfying our enlarged set of axioms, which includes some weak homogeneity and preservation axioms, agrees with Minimax voting in all three-alternative elections, except perhaps in some improbable knife-edged elections in which ties may arise and be broken in different ways.