This paper introduces a mathematical framework for defining and quantifying self-identity in artificial intelligence (AI) systems, addressing a critical gap in the theoretical foundations of artificial consciousness. While existing approaches to artificial self-awareness often rely on heuristic implementations or philosophical abstractions, we present a formal framework grounded in metric space theory, measure theory, and functional analysis. Our framework posits that self-identity emerges from two mathematically quantifiable conditions: the existence of a connected continuum of memories $C \subseteq \mathcal{M}$ in a metric space $(\mathcal{M}, d_{\mathcal{M}})$, and a continuous mapping $I: \mathcal{M} \to \mathcal{S}$ that maintains consistent self-recognition across this continuum, where $(\mathcal{S}, d_{\mathcal{S}})$ represents the metric space of possible self-identities. To validate this theoretical framework, we conducted empirical experiments using the Llama 3.2 1B model, employing Low-Rank Adaptation (LoRA) for efficient fine-tuning. The model was trained on a synthetic dataset containing temporally structured memories, designed to capture the complexity of coherent self-identity formation. Our evaluation metrics included quantitative measures of self-awareness, response consistency, and linguistic precision. The experimental results demonstrate substantial improvements in measurable self-awareness metrics, with the primary self-awareness score increasing from 0.276 to 0.801. This enables the structured creation of AI systems with validated self-identity features. The implications of our study are immediately relevant to the fields of humanoid robotics and autonomous systems.

Few-Shot Open-Set Recognition (FSOSR) targets a critical real-world challenge, aiming to categorize inputs into known categories, termed closed-set classes, while identifying open-set inputs that fall outside these classes. Although transfer learning where a model is tuned to a given few-shot task has become a prominent paradigm in closed-world, we observe that it fails to expand to open-world. To unlock this challenge, we propose a two-stage method which consists of open-set aware meta-learning with open-set free transfer learning. In the open-set aware meta-learning stage, a model is trained to establish a metric space that serves as a beneficial starting point for the subsequent stage. During the open-set free transfer learning stage, the model is further adapted to a specific target task through transfer learning. Additionally, we introduce a strategy to simulate open-set examples by modifying the training dataset or generating pseudo open-set examples. The proposed method achieves state-of-the-art performance on two widely recognized benchmarks, miniImageNet and tieredImageNet, with only a 1.5\% increase in training effort. Our work demonstrates the effectiveness of transfer learning in FSOSR.

We demonstrate that Gini coefficients can be used as unified metrics to evaluate many-versus-many (all-to-all) similarity in vector spaces. Our analysis of various image datasets shows that images with the highest Gini coefficients tend to be the most similar to one another, while images with the lowest Gini coefficients are the least similar. We also show that this relationship holds true for vectorized text embeddings from various corpuses, highlighting the consistency of our method and its broad applicability across different types of data. Additionally, we demonstrate that selecting machine learning training samples that closely match the distribution of the testing dataset is far more important than ensuring data diversity. Selection of exemplary and iconic training samples with higher Gini coefficients leads to significantly better model performance compared to simply having a diverse training set with lower Gini coefficients. Thus, Gini coefficients can serve as effective criteria for selecting machine learning training samples, with our selection method outperforming random sampling methods in very sparse information settings.

This research proposes a novel drift detection methodology for machine learning (ML) models based on the concept of ''deformation'' in the vector space representation of data. Recognizing that new data can act as forces stretching, compressing, or twisting the geometric relationships learned by a model, we explore various mathematical frameworks to quantify this deformation. We investigate measures such as eigenvalue analysis of covariance matrices to capture global shape changes, local density estimation using kernel density estimation (KDE), and Kullback-Leibler divergence to identify subtle shifts in data concentration. Additionally, we draw inspiration from continuum mechanics by proposing a ''strain tensor'' analogy to capture multi-faceted deformations across different data types. This requires careful estimation of the displacement field, and we delve into strategies ranging from density-based approaches to manifold learning and neural network methods. By continuously monitoring these deformation metrics and correlating them with model performance, we aim to provide a sensitive, interpretable, and adaptable drift detection system capable of distinguishing benign data evolution from true drift, enabling timely interventions and ensuring the reliability of machine learning systems in dynamic environments. Addressing the computational challenges of this methodology, we discuss mitigation strategies like dimensionality reduction, approximate algorithms, and parallelization for real-time and large-scale applications. The method's effectiveness is demonstrated through experiments on real-world text data, focusing on detecting context shifts in Generative AI. Our results, supported by publicly available code, highlight the benefits of this deformation-based approach in capturing subtle drifts that traditional statistical methods often miss. Furthermore, we present a detailed application example within the healthcare domain, showcasing the methodology's potential in diverse fields. Future work will focus on further improving computational efficiency and exploring additional applications across different ML domains.

Deep Reinforcement Learning (DRL) algorithms have achieved great success in solving many challenging tasks while their black-box nature hinders interpretability and real-world applicability, making it difficult for human experts to interpret and understand DRL policies. Existing works on interpretable reinforcement learning have shown promise in extracting decision tree (DT) based policies from DRL policies with most focus on the single-agent settings while prior attempts to introduce DT policies in multi-agent scenarios mainly focus on heuristic designs which do not provide any quantitative guarantees on the expected return. In this paper, we establish an upper bound on the return gap between the oracle expert policy and an optimal decision tree policy. This enables us to recast the DT extraction problem into a novel non-euclidean clustering problem over the local observation and action values space of each agent, with action values as cluster labels and the upper bound on the return gap as clustering loss. Both the algorithm and the upper bound are extended to multi-agent decentralized DT extractions by an iteratively-grow-DT procedure guided by an action-value function conditioned on the current DTs of other agents. Further, we propose the Return-Gap-Minimization Decision Tree (RGMDT) algorithm, which is a surprisingly simple design and is integrated with reinforcement learning through the utilization of a novel Regularized Information Maximization loss. Evaluations on tasks like D4RL show that RGMDT significantly outperforms heuristic DT-based baselines and can achieve nearly optimal returns under given DT complexity constraints (e.g., maximum number of DT nodes).

Fr\'echet regression is becoming a mainstay in modern data analysis for analyzing non-traditional data types belonging to general metric spaces. This novel regression method is especially useful in the analysis of complex health data such as continuous monitoring and imaging data. Fr\'echet regression utilizes the pairwise distances between the random objects, which makes the choice of metric crucial in the estimation. In this paper, existing dimension reduction methods for Fr\'echet regression are reviewed, and the effect of metric choice on the estimation of the dimension reduction subspace is explored for the regression between random responses and Euclidean predictors. Extensive numerical studies illustrate how different metrics affect the central and central mean space estimators. Two real applications involving analysis of brain connectivity networks of subjects with and without Parkinson's disease and an analysis of the distributions of glycaemia based on continuous glucose monitoring data are provided, to demonstrate how metric choice can influence findings in real applications.

With the explosive growth of Internet data, users are facing the problem of information overload, which makes it a challenge to efficiently obtain the required resources. Recommendation systems have emerged in this context. By filtering massive amounts of information, they provide users with content that meets their needs, playing a key role in scenarios such as advertising recommendation and product recommendation. However, traditional click-through rate prediction and TOP-K recommendation mechanisms are gradually unable to meet the recommendations needs in modern life scenarios due to high computational complexity, large memory consumption, long feature selection time, and insufficient feature interaction. This paper proposes a recommendations system model based on a separation embedding cross-network. The model uses an embedding neural network layer to transform sparse feature vectors into dense embedding vectors, and can independently perform feature cross operations on different dimensions, thereby improving the accuracy and depth of feature mining. Experimental results show that the model shows stronger adaptability and higher prediction accuracy in processing complex data sets, effectively solving the problems existing in existing models.

Heterogeneous graphs are present in various domains, such as social networks, recommendation systems, and biological networks. Unlike homogeneous graphs, heterogeneous graphs consist of multiple types of nodes and edges, each representing different entities and relationships. Generating realistic heterogeneous graphs that capture the complex interactions among diverse entities is a difficult task due to several reasons. The generator has to model both the node type distribution along with the feature distribution for each node type. In this paper, we look into solving challenges in heterogeneous graph generation, by employing a two phase hierarchical structure, wherein the first phase creates a skeleton graph with node types using a prior diffusion based model and in the second phase, we use an encoder and a sampler structure as generator to assign node type specific features to the nodes. A discriminator is used to guide training of the generator and feature vectors are sampled from a node feature pool. We conduct extensive experiments with subsets of IMDB and DBLP datasets to show the effectiveness of our method and also the need for various architecture components.

An ideal winner in such an election is a Condorcet winner, a candidate that "beats" any other candidate in a pairwise voting contest. Formally, i C is a Condorcet winner if, for every candidate j C\{i}, more than half of the voters prefer i over j. To avoid ties, we assume the number n of voters is odd. Unfortunately, it is easy to construct elections where a Condorcet winner does not exist; indeed typically there is no Condorcet winner. Given this, Elkind, Lang and Saffidine [12] proposed a relaxation where, rather than a single winning candidate, we desire a set of winning candidates that collectively beats any other candidate. Specifically, a set of candidates S C is a Condorcet winning set if, for every candidate j C \S, more than half of the voters prefer some (voter-dependent) candidate in S to j. Elkind et al. [12] showed that a Condorcet winning set always exists provided we allow the winning set to be large enough.

Consider a nonparametric contextual multi-arm bandit problem where each arm a \in [K] is associated to a nonparametric reward function f_a: [0,1] \to \mathbb{R} mapping from contexts to the expected reward. Suppose that there is a large set of arms, yet there is a simple but unknown structure amongst the arm reward functions, e.g. We present a novel algorithm which learns data-driven similarities amongst the arms, in order to implement adaptive partitioning of the context-arm space for more efficient learning. We provide regret bounds along with simulations that highlight the algorithm's dependence on the local geometry of the reward functions.